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BRAKING PERFORMANCE
For the majority of airplane configurations
and runways conditions, the airplane brakes
furnish the most powerful means of deceler-
ation. While specific techniques of braking
are required for specific situations, there
are various fundamentals which are common
to all conditions.
Solid ftictim is the resistance to relative
motion of two surfaces in contact. When
relative motion exists between the surfaces,
the resistance to relative motion is termed
“kinetic” or “sliding” friction; when no
relative motion exists between the surfaces,
the resistance to the impending relative mo-
tion is termed “static” friction. The minute
discontinuities of the surfaces in contact are
able to mate quite closely when relative
motion impends rather than exists, so static
friction will generally exceed kinetic friction.
The magnitude of the friction force between
two surfaces will depend in great part on the
types of surfaces in contact and the magnitude
of force pressing the surfaces together. A
convenient method of relating the friction
charactersitics of surfaces in contact is a
proportion of the friction force to the normal
(or perpendicular) force pressing the surfaces
together. This proportion defines the coeffi-
cient of friction, ~1.
,L= F/N
where
p =coefFicient of friction (mu)
F =friction force, Ibs.
N = normal force, lbs.
The coefficient of friction of tires on a runway
surface is a function of many factors. Runway
surface condition, rubber composition, tread,
inflation pressure, surface friction shearing
stress, relative slip speed, etc., all are factors
which affect the coefficient of friction. When
the tire is rolling along the runway without
the use of brakes, the friction force resulting is
simple rolling resistance. The coefficient of
NAVWEPS O&ROT-R0
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
rolling friction is of an approximate magnitude
of 0.015 to 0.030 for dry, hard runway surface.
The application of brakes supplies a torque
to the wheel which tends co retard wheel rota-
tion. However, the initial application of
brakes creates a braking torque but the initial
retarding torque is balanced by the increase in
friction force which produces a driving or
rolling torque. Of course, when the braking
torque is equal to the rolling torque, the wheel
experiences no acceleration in rotation and the
equilibrium of a constant rotational speed is
maintained. Thus, the application of brake
develops a retarding torque and causes an
increase in friction force between the tire and
runway surface. A common problem of brak-
ing technique is application of excessive brake
pressure which creates a braking torque greater
than the maximum possible rolling torque.
In this case, the wheel loses rotational speed
and decelerates until the wheel is stationary
and the result is a locked wheel with the tire
surface subject to a full slip condition.
The relationship of friction force, normal
force, braking torque, and rolling torque is
illustrated in figure 6.11.
The effect of slip velocity on the coefficient
of friction is illustrated by the graph of figure
6.11. The conditions of zero slip corresponds
to the rolling wheel without brake application
while the condition of full, 100 percent slip
corresponds to the locked wheel where the
relative velocity between the tire surface and
the runway equals the actual velocity. With
the application of brakes, the coefficient of
friction increases but incurs a small but meas-
urable apparent slip. Continued increase in
friction coefficient is obtained until some max-
imum is achieved then decreases as the slip
increases and approaching the 100 percent slip
condition. Actually, the peak value of co-
efficient of friction occurs at an incipient skid
condition and the relative slip apparent at this
point consists primarily of elastic shearing
deflection of the tire structure.
387 | 404 | 404 | 00-80T-80.pdf |
NAVWEPS DD-BOl-80
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
When the runway surface is dry, brush-
finished concrete, the maximum value for the
coefficient of friction for most aircraft tires is
on the order of 0.6 to 0.8. Many factors can
determine small differences in this peak value
of friction coefhcient for dry surface conditions.
For example, a soft gum rubber composition
can develop a very high value of coefficient of
friction but only for low values of surface
shearing stress. At high values of surface
shearing stress, the soft gum rubber will shear
or scrub off before high values of friction co-
efficient are developed. The higher strength
compounds used in the production of aircraft
tires produce greater resistance to surface shear
and scrubbing but the harder rubber has lower
intrinsic friction coeflicient. Since the high
performance airplane cannot afford the luxury
of excessive tire weight or size, the majority
of airplane tires will be of relatively hard
rubber and will operate at or near the rated
load capacities. As a result, there will be
little difference between the peak values of
friction coefficient for the dry, hard surface
runway for the majority of aircraft tires.
If high traction on dry surfaces were the
only consideration in the design of tires, the
result would be a soft rubber tire of extreme
width to create a large footprint and reduce
surface shearing stresses, e.g., driving tires on
a drag racer. However, such a tire has many
other characteristics which are undesirable
such as high rolling friction, large size, poor
side force characteristics, etc.
When the runway has water or ice on the
surface, the maximum value for the coefficient
of friction is reduced greatly below the value
obtained for the dry runway condition. When
water is on the surface, the tread design be-
comes of greater importance to maintain con-
tact between the rubber and the runway and
prevent a film of water from lubricating the
surfaces. When the rainfall is light, the peak
value for friction coefficient is on the order of
0.5. With heavy rainfall it is more likely
that sufficient water will stand to form a liquid
film between the tire and the runway. In this
case, the peak coefficient of friction rarely
exceeds 0.3. In some extreme conditions, the
tire may simply plane along the water without
contact of the runway and the coefficient of
friction is much lower than 0.3. Smooth,
clear ice on the runway will cause extremely
low values for the coefficient of friction. In
such a condition, the peak value for the co-
efhcient of friction may be on the order of 0.2
or 0.15.
Note that immediately past the incipient
skidding condition the coefficient of friction
decreases with increased slip speed, especially
for the wet or icy runway conditions. Thus,
once skid begins, a reduction in friction force
and rolling torque must be met with a reduc-
tion in braking torque, otherwise the wheel
will decelerate and lock. This is an important
factor to consider in braking technique because
the skidding tire surface on the locked wheel
produces considerably less retarding force than
when at the incipient skid condition which
causes the peak coeflicient of friction. If the
wheel locks from excessive braking, the sliding
tire surface produces less than the maximum
retarding force and the tires become relatively
incapable of developing any significant side
force. Stop distance will increase and it may
be difficult-if not impossible-to control the
airplane when full slip is developed. In addi-
tion, at high rolling vel,ocities on the dry sur-
face runway, the immediate problem of a skid-
ding tire is not necessarily the loss of retard-
ing force but the imminence of tire failure. The
pilot must insure that the application of brakes
does not produce some excessive braking torque
which is greater than the maximum rolling
torque and particular care must be taken when
the runway conditions produce low values of
friction coefficient and when the normal force
on the braking surfaces is small. When it is
difficult to perceive or distinguish a skidding
condition, the value of an antiskid or auto-
matic braking system will be appreciated. | 405 | 405 | 00-80T-80.pdf |
NAVWEPS OD4OT-80
APPLICATION OF AERODYNAMICS
I
TO SPECIFIC PRO&EMS OF FLYING
NORMAL FORCE
COEFFICIENT
OF
FRICTION 0.6
F/N 0.4
0.2 -
ROLLING
WHEEL \
DRY CONCRETE
--CONCRETE
\-
\ \ LIGHT RAIN
\
\
HEAVY RAIN
I i- LOCKED
WHEEL
IO 20 30 40 50 60 70 80 90 100
PER CENT SLIP
I
Figure 6.7 1. Braking Perhnance
389 | 406 | 406 | 00-80T-80.pdf |
NAVWEPS DD-BOT-BO
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
BRAKING TECHNIQUE. It must be
clearly distinguished that the techniques for
minimum stop distance may differ greatly from
the techniques required to minimize wear and
tear on the tires and brakes. For the majority
of airplane configurations, brakes will provide
the most important source of deceleration for
all but the most severe of icy runway condi-
tions. Of course, aerodynamic drag is very
durable and should be utilized to decelerate the
airplane if the runway is long enough and the
drag high enough. Aerodynamic drag will be
of importance only for the initial 20 to 30 per-
cent of speed reduction from the point of touch-
down. At speeds less than 60 to 70 percent of
the landing speed, aerodynamic drag is of little
consequence and brakes will be the principal
source of deceleration regardless of the runway
surface. For the conditions of minimum land-
ing distance, aerodynamic drag will be a prin-
cipal source of deceleration only for the initial
portion of landing roll for very high drag con-
figurations on very poor runway conditions.
These cases are quite limited so considerable
importance must be assigned to proper use of
the brakes to produce maximum effectiveness.
In order to provide the maximum possible
retarding force, effort must be directed to pro-
duce the maximum normal force on the braking
surfaces. (See figure 6.11.) The pilot will be
able to influence the normal force on the brak-
ing surfaces during the initial part of the land-
ing roll when dynamic pressure is large and
aerodynamic forces and moments are of conse-
quence. During this portion of the landing
roll the pilot can control the airplane lift and
the distribution of normal force to the landing
gears.
First to consider is that any positive lift will
support a part of the airplane weight and reduce
the normal force on the landing gear. Of
course, for the purposes of braking friction, it
would be to advantage to create negative lift
but this is not the usual capability of the air-
plane with the tricycle landing gear. Since
the airplane lift may be considerable immedi-
ately after landing, retraction of flaps or ex-
tension of spoilers immediately after touch-
down will reduce the wing lift and increase
the normal force on the landing gear. With
the retraction of flaps, the reduced drag is more
than compensated for by the increased braking
friction force afforded by the increased normal
force on the braking surfaces.
A second possible factor to control braking
effectiveness is the distribution of normal force
to the landing gear surfaces. The nose wheel
of the tricycle landing gear configuration usu-
ally has no brakes and any normal force dis-
tributed to this wheel is useful only for pro-
ducing side force for control of the airplane.
Under conditions of deceleration, the nose-
down pitching moment created by the friction
force and the inertia force tends to transfer
a significant amount of normal force to nose
wheel where it is unavailable to assist in
creating friction force. For the instant after
landing touchdown, the pilot may control
this condition to some extent and regain or
increase the normal force on the main wheels.
After touchdown, the nose is lowered until the
nose wheel contacts the runway then brakes
are applied while the stick is eased back with-
out lifting the nose wheel back off the runway.
The effect is to minimize the normal force on
the nose wheel and increase the normal force
on braking surfaces. While the principal
effect is to transfer normal force to the main
wheels, there may be a significant increase in
normal force due to a reduction in net lift, i.e.,
tail download is noticeable. This reduction
in net lift tends to be particular to tailless or
short coupled airplane configurations.
The combined effect of flap retraction and
aft stickis a significant increase in braking
friction force. Of course, the flaps should not
be retracted while still airborne and aft stick
should be used just enough without lifting
the nosewheel off the runway. These tech-
niques are to no avail if proper use of the
390 | 407 | 407 | 00-80T-80.pdf |
brakes does not produce the maximum coefi-
cient of friction. The incipient skid condi-
tion will produce the maximum coefficient of
friction but this peak is difficult to recognize
and maintain without an antiskid system.
Judicious use of the brakes is necessary to
obtain the peak coefficient of friction but not
develop a skid or locked wheel which could
cause tire failure, loss of control, or consider-
able reduction in the friction coefficient.
The capacity of the brakes must be sufficient
to create adequate braking torque and produce
the high coefficient of friction. In addition,
the brakes must be capable of withstanding
the heat generated without fading or losing
effectiveness. The most critical requirements
of the brakes occur during landing at the
maximum allowable landing weight.
TYPICAL ERRORS OF BRAKING TECH-
NIQUE. Errors in braking technique are usu-
ally coincident with errors of other sorts. For
example, if the pilot lands an airplane with
excessive airspeed, poor braking technique
could accompany the original error to produce
an unsafe situation. One common error of
of braking technique is the application of
braking torque in excess of the maximum
possible rolling torque. The result will be
that the wheel decelerates and locks and the
skid reduces the coefficient of friction, lowers
the capability for side force, and enhances the
possibility of tire failure. If maximum brak-
ing is necessary, caution must be used to
modulate the braking torque to prevent lock-
ing the wheel and causing a skid. On the
other hand, maximum coefficient of friction is
obtained at the incipient skidding condition
so sufficient brake torque must be applied to
produce maximum friction force. Intermittent
braking serves no useful purpose when the
objective is maximum deceleration because
the periods between brake application produce
only slight or negligible cooling. Brake
should be applied smoothly and braking
torque modulated at or near the peak value
to insure that skid does not develop.
NAVWEPS OD-BOT-80
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
One of the important factors affecting the
landing roll distance is landing touchdown
speed. Any excess velocity at landing causes
a large increase in the minimum stop distance
and it is necessary that the pilot control the
landing precisely so to land at the appropriate
speed. When landing on the dry, hard surface
runway of adequate length, a tendency is to
take advantage of any excess runway and
allow the airplane to touchdown with excess
speed. Of course, such- errors in technique
cannot be tolerated and the pilot must strive
for precision in all landings. Immediately
after touchdown, the airp1ar.e lift may be
considerable and the normal force on the
braking surfaces quite low. Thus, if excessive
braking torque is applied, the wheel may lock
easily at high speeds and tire failure may take
place suddenly.
Landing on a wet or icy runway requires
judicious use of the brakes because of the re-
duction in the maximum coefficient of friction.
Because of reduction in the maximum attain-
able value of the coefficient of friction, the
pilot must anticipate an increase in the mini-
mum landing distance above that applicable
for the dry runway conditions. When there
is considerable water or ice on rhe runway, an
increase in landing distance on the order of 40
to 100 percent must be expected for similar
conditions of gross weight, density altitude,
wind, etc. Unfortunately, the conditions
likely to produce poor braking action also
will cause high idle thrust of the turbojet
engine and the extreme case (smooth, glazed
ice or heavy rain) may dictate shutting down
the engine to effect a reasonable stopping
distance.
REFUSAL SPEEDS, LINE SPEEDS, AND
CRITICAL FIELD LENGTH
During takeoff, it is necessary to monitor
the performance of the airplane and evaluate
the acceleration to insure that the airplane will
391 | 408 | 408 | 00-80T-80.pdf |
NAVWEPS 00401-80
APPLICATION OF AERODYNAlvllCS
TO SPECIFIC PROBLEMS OF FLYING
achieve the takeoff speed in the specified dis-
tance. If it is apparent that the airplane is
not accelerating normally or that the airplane
or powerplant is not functioning properly,
a decision must be made to refuse or continue
takeoff. If the decision to refuse takeoff is
made early in the takeoff roll, no problem
exists because the airplane has not gained
much speed and a large portion of runway
distance is unused. However, at speeds near
the takeoff speed, the airplane has used a large
portion of the takeoff distance and the distance
required to stop is appreciable. The problem
which exists is to define the highest speed
attained during takeoff acceleration from which
the airplane may be decelerated to a stop on
the runway length remaining, i.e., the “refusal
speed. ’ ’
The refusal speed will be a function of take-
off performance, stopping performance, and
the length of available runway. The ideal
situation would be to have a runway length
which exceeds the total distance required to
accelerate to the takeoff speed then decelerate
from the takeoff speed. In this case, the
refusal speed would exceed the takeoff speed
and there would be little concern for the case
of refused takeoff. While this may be the
case for some instances, the usual case is that
the runway length is less than the “accelerate-
stop” distance and the refusal speed is less than
the takeoff speed. A graphical representation
of the refused takeoff condition is illustrated
in figure 6.12 by a plot of velocity versus dis-
tance. At the beginning of the runway, the
airplane starts accelerating and the variation
of velocity and distance is defined by the
takeoff acceleration profile. The deceleration
profile describes the variation of velocity with
distance where the airplane is brought to a
stop at the end of the runway. The inter-
section of the acceleration and deceleration
profiles then defines the refusal speed and the
refusal distance along the runway. Of course,
an allowance must be made for the time spent
at the refusal speed as the power is reduced
and braking action is initiated.
During takeoff, the airplane could be accel-
erated to any speed up to the refusal speed,
then decelerated to a stop on- the runway
remaining. Once past the refusal speed, the
airplane cannot be brought to a stop on the
runway remaining and the airplane is com-
mitted to an unsafe stop. If takeoff is refused
when above the refusal speed, the only hope
is for assistance from the arresting gear, run-
way barrier, or an extensive overrun at the
end of the runway. This fact points to the
need for planning of the takeoff and the require-
ment to monitor the takeoff acceleration.
If the refusal speed data are not available,
the following equations may be used to ap-
proximate the refusal speed and distance:
where
V,= refusal speed
S,= refusal distance
and for the appropriate takeoff configuration,
V,*= takeoff speed
S,,= takeoff distance
V,= landing speed
J,=landing distance
R. = runway length available
These approximate relationships do not ac-
count for the time spent at the refusal point and
must not be used in lieu of accurate handbook
data.
In the case of the single-engine airplane, the
pilot must monitor the takeoff performance to
recognize malfunctions or lack of adequate ac-
celeration prior to reaching the refusal speed.
Obviously, it is to advantage to recognize rhe
3’92 | 409 | 409 | 00-80T-80.pdf |
NAVWEPS OO-EOT-80
APP,LlCATlON OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
TAKEOFF SPEED TAKEOFF SPEED
REF”z
/
ACCELERATION
PROFILE
F REFUSAL DISTANCE I P P c c
TAKEOFF DISTANCE TAKEOFF DISTANCE
.DECELERATloN
PROFlLE
RUNWAY AVAILABLE I
VARIATION OF SPEED WITH DISTANCE FOR
UNIFORMLY ACCELERATED MOTION DURING
TAKEOFF ROLL
O~mi-rrmLurauLurauupuLnUFFFPI.vm
0 10 20 30 40 50 60 i-o 60 90 100
PER CENT OF TAKEOFF DISTANCE
Figure 6.12. Refused Take& and Takeof\ Velocity Variafion
393 | 410 | 410 | 00-80T-80.pdf |
NAVWEPS DD4OT-80
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYI’NG
possibility of a refused takeoff before exceed-
ing the refusal speed. To this end, the pilot
must carefully evaluate airplane and power-
plant performance and judge the acceleration
of the airplane by the use of “line speeds.”
The accelerated motion of the airplane during
takeoff roll will define certain relationships be-
tween velocity and distance when the acceler-
ation of the airplane is normal. By comparison
of predicted and actual speeds at various points
along the runway, the pilot can evaluate the
acceleration and assess the takeoff perfotm-
axe.
An example of an acceleration profile is
shown by the second illustration of figure 6.12,
where the variation of velocity and distance is
defined for the case of uniformly accelerated
motion, i.e., constant acceleration. While the
case of uniformly accelerated motion doesnot
correspond exactly to the takeoff performance
of all airplanes, it is sufficiently applicable to
illustrate the principle of line speeds and ac-
celeration checks. If the takeoff acceleration
of the airplane were constant, the airplane
would develop specific percentages of the take-
off speed at specific percentages of the takeoff
distance. Representative values from figure
6.12 are as follows:
Pmmr 0, ‘&Ofl diJh?%Z Pmmr of r‘iko~
"docit~
PInmr 0, &off
rim
0 0 0
25 so. 0 50.0
SO 70.7 70.7
75 86. 5 86.5
100 100 100
As an example of this uniformly accelerated
motion, the airplane upon reaching the half-
way point of takeoff roll would have spent
70.7 percent of the total takeoff time and ac-
celerated to 70.7 percent of the takeoff speed.
If the airplane has not reached a specific speed
at a specific distance, it is obvious that the ac-
celeration is below the predicted value and the
airplane surely will not achieve the takeoff
speed in the specified takeoff distance. There-
fore, properly computed line speeds at various
points along the runway will allow the pilot
to monitor the takeoff performance and recog-
nize a deficiency of acceleration. Of course,
a deficiency of acceleration must be recognized
prior to reaching some point along the runway
where takeoff cannot be safely achieved or
refused
The fundamental principles of refusal speeds
and line speeds are applicable equally well to
single-engine and multiengine airplanes. How-
ever, in the case of the multiengine airplane
additional consideration must be given to the
decision to continue or refuse takeoff when
engine failure occurs during the takeoff roll.
IF failure of one engine occurs prior to reaching
the-refusal speed, takeoff should be discon-
tinued and the airplane brought to a stop on
the remaining runway. If failure of one engine
c~curs after exceeding the refusal speed, the
airplane is committed to continue takeoff with
the remaining engines operative or an unsafe
refused take&. Sn some cases, the remaining
runway may not be sufficient to allow acceler-
ation to the takeoff speed and the airplane can
neither takeoff or stop on the runway rcmain-
ibg. To facilitate consideration of this prob-
lem, several specific defmitions are necessary.
(1) Takeoff and initial climb speed: A speed,
usually a fixed percentage above the stall speed,
at which the airplane will become airborne and
best clear obstacles immediately after takeoff.
For a particular airplane in the takeoff con-
figuration, this speed (in EAS or CM) is a
function of gross weight but in no circumstances
should it he less than the minimum directional
control speed for the critical asymmetrical
power condition. Generally, the takeoff and
initial climb speed is referred to as the “V,”
speed.
(2) Critical engine jaih~~ speed: A speed
achieved during the takeoff roll at which fail-
ure of one engine will require the same distance
to continue accelerating with the operative en-
gines to accomplish safe takeoff or refuse
takeoff and decelerate to a stop utilizing the
airplane brakes. At critical engine failure
394 | 411 | 411 | 00-80T-80.pdf |
NAVWEPS OD-BOT-80
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
RUNWAY LENGTH EQUAL TO CRITICAL
FIELD LENGTH
TAKEOFF d:;‘,
CLIMB SP
CRITICAL ENGINE LACCELERATION WITH 1
FAILURE SPEED q ENGINE INOPERATIVE
ACCELERATION
WITH ALL ENGINES
OPERATIVE
\
PROFILE
\
RUNWAY LENGTH
RUNWAY LENGTH LESS THAN CRITICAL
FIELD LENGTH
MINIMUM SPEED NECESSAR
TO CONTINUE TAKEOFF WITH
ONE ENGINE IN
REFUSAL SPEE LERATION WITH
ONE ENGINE INOPERATIVE
ACCELERATION DECELERATION
TH ALL ENGINES
OPERATIVE
Figure 6.13. Critical Field Len&
395 | 412 | 412 | 00-80T-80.pdf |
NAVWEPS OD-ROT-80
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
speed, the distance necessary to continue take-
off with one engine inoperative is equal to the
stopping distance. The critical engine failure
speed is generally referred to as the “I’,”
speed and it is a function of the same factors
which determine the takeoff performance, e.g.,
density altitude, gross weight, temperature,
humidity, etc.
(3) &t&l jield kngrh: The runway length
necessary to accelerate with all engines opera-
tive to the critical engine failure speed (VJ
then continue accelerating to the takeoff and
initial climb speed (VJ with one engine
inoperative and achieve safe takeoff or refuse
takeoff. By this definition, critical field
length describes the minimum length of run-
way necessary for safe operation of the multi-
engine airplane. Obviously, the critical field
length is a function of the same factors affect-
ing the takeoff distance of the airplane.
The conditions of Vi, V,, and critical field
length are illustrated by figure 6.13. The
first illustration of figure 6.13 depicts the
case where the runway length is equal to the
rritical jield kngth. In this case, the airplane
could accelerate to Vi with all engines opera-
tive then either continue takeoff safely with
one engine inoperative or refuse takeoff and
decelerate to a stop on the remaining runway.
For this condition, an engine failure occurring
at lefs than I’, speed dictates that takeoff must
be refused because inadequate distance remains
to effect a safe takeoff at V, speed. However,
at or below V, speed, adequate distance re-
mains to bring the airplane to a stop. If
engine failure occurs at some speed greater
than Vi speed, takeoff should be continued
because adequate distance remains to accelerate
to V, speed and effect a safe takeoff with one
engine inoperative. If engine failure occurs
beyond Vi speed, inadequate distance remains
to brake the airplane to a stop on the runway,
The second illustration of figure 6.13 depicts
the case where the ranway length is 1~s than
the titical field length. In this case, the term
of “V,” speed is not applicable because of
inadequate distance and the refusal speed is
less than the minimum speed necessary to
continue a safe takeoff with one engine inoper-
ative. If engine failure occurs below refusal
speed, the takeoff must be refused and adequate
distance remains to effect a stop on the runway.
If engine failure occurs above refusal speed
but below the minimum speed necessary to
continue takeoff with one engine inoperative,
an accident is inevitable. Within this range
of speeds, the airplane cannot effect a safe
takeoff at L’s with one engine inoperative or
a safe stop on the remaining runway. For this
reason, the pilot must properly plan the
takeoff and insure that the runway available
is equal to or greater than the critical field
length. If the runway available is less than
the critical field length, there must be sufficient
justification for the particular operation be-
cause of the hazardous consequences of engine
failure between the refusal speed and the
minimum speed necessary to continue takeoff
with one engine inoperative. Otherwise, the
gross weight of the airplane should be reduced
in attempt to decrease the critical field length
to equal the available runway.
SONIC “BOOMS’
From the standpoint of public relations
and the maintaining of friendly public support
for Naval Aviation, great care must be taken
to prevent sonic booms in populated areas.
While the ordinary sonic boom does not carry
any potential of physical damage, the disturb-
ance must be avoided because of the undesirable
annoyance and apprehension. As supersonic
flight becomes more commonplace and an
ordinary consequence of flying operations, the
prevention of sonic booms in populated areas
becomes a difficult and perplexing job.
When the airplane is in supersonic flight,
the local pressure and velocity changes on the
airplane surfaces are coincident with the
formation of shock waves. The pressure jump
through the shock waves in the immediate
vicinity of the airplane surfaces is determined | 413 | 413 | 00-80T-80.pdf |
by the local flow changes at these surfaces.
Of course, the strength of the shock waves
and the pressure jump through the wave
decreases rapidly with distance away from the
airplane. While the pressure jump through
the shock wave decreases with distance away
from the surface, it does not disappear com-
pletely and a measurable-but very small-
pressure will exist at a considerable distance
from the airplane.
Sound is transmitted through the air as a
series of very weak pressure waves. In the
ordinary range of audible frequencies, the
threshold of audibility for intensity of sound
is for pressure waves with an approximate
R.M.S. value of pressure as low as 0.0000002
psf. Within this same range of frequencies,
the threshold of feeling for intensity of sound
is for pressure values with an approximate
R.M.S. value of pressure of 0.2 to 0.5 psf.
Continuous sound at the threshold of feeling
is of the intensity to cause painful hearing.
Thus, the shock waves generated by an air-
plane in supersonic flight are capable of creat-
ing audible sound and, in the extreme case, can
be of a magnitude to cause considerable dis-
turbance. Pressure jumps of 0.02 to 0.3 psf
have been recorded during the passage of an
airplane in supersonic flight. As a result,
the sonic “booms” are the pressure waves
generated by the shock waves formed on the
airplane in supersonic flight.
The source of sonic booms is illustrated by
figure 6.14. When the airplane is in level
supersonic flight, a pattern of shock waves is
developed which is much dependent on the
configuration and flight Mach number of the
airplane. At a considerable distance from the
airplane, these shock waves tend to combine
along two common fronts and extend away
from the airplane in a sort of conical surface.
The waves decrease in strength with distance
away from the airplane but the pressure jump
remains of an audible intensity for a consider-
able distance from the airplane. If the wave
extends to the ground or water surface, it will
NAVWEPS OD-8OT-RO
APP,LICATlON OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
be reflected and attenuated to some extent
depending on the character of the reflecting
surface. Of course, if this attached wave form
is carried,across a populated area at the surface,
th.- population will experience the pressure
waves as a sonic boom.
The intensity of the boom will depend on
many different factors. The characteristics of
the airplane generating the shock waves will
be of some importance since a large, high drag,
high gross weight airplane in flight at high
Mach number will be transferring a greater
energy to the air mass. Flight altitude will
have an important bearing on boom intensity
since at high altitude the pressure jump across
a given wave form is much less. In addition,
at high altitude a greater distance exists be-
tween the generating source of the pressure
disturbance and the ground level and the
strength of the wave will have a greater dis-
tance in which to decay. The ordinary vari-
ation of temperature and density plus the
natural turbulence of atmosphere will tend to
reflect or dissipate the shock wave generated
at high altitude. However, in a stable, quies-
cent atmosphere, the pressure wave from the
airplane in high supersonic flight at high alti-
tude may be of an audible magnitude at lateral
distances as great as 10 to 30 miles. Thus,
supersonic flight over or adjacent to populated
areas will produce a sonic boom.
Actually, it is not necessary for any airplane
to fly supersonic over or adjacent to a popu-
lated area to create a sonic boom. This
possibility is shown by the second illustration
of figure 6.14 where an airplane decelerates to
subsonic from a supersonic dive. As the air-
plane slows to subsonic from supersonic speed,
the airplane will release the leading bow and
tail waves which formed as the airplane accel-
erated from subsonic to supersonic speed. The
release of these shock waves is analogous to
the case where a surface ship slows to below
the wave propagation speed and releases the
bow wave which then travels out ahead of | 414 | 414 | 00-80T-80.pdf |
NAVWEPS oo-8OT-80
APPUCATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYllNG
AIRPLANE IN SUPERSONIC FLIGHT
‘AIL WAVE
FLIGHT
Revised JonuaV 1965
Figure 6.14. Sonic Booms
398 | 415 | 415 | 00-80T-80.pdf |
the ship. When the airplane slows to sub-
sonic, the shock wave travels out ahead of the
airplane in a form which is somewhat spheri-
cal. Because there are density variations
through the shock wave, the shock wave
moving ahead of the airplane can cause aber-
rations in light waves and it may appear.to
the pilot as if a large sheet of clear cellulose
or plastic were in front of the airplane. In
addition, the density variation and initial
shape of the wave leaving the airplane may
cause reflection of sunlight which would
appear as a sudden, brilliant “flash” to the
pilot.
Of course, the wave released by decelerating
to subsonic speed can travel out ahead of the
airplane and traverse a populated area to cause
a sonic boom. The initial direction of the
released wave will be the flight path of the
airplane at the instant it decelerates to sub-
sonic speed. To be sure, the released wave
should not be aimed in the direction of a popu-
lated area, even if a considerable distance
away. There are instances where a released
wave has been of an audible magnitude as far
as 30 to 40 miles ahead of the point of release.
The released pressure wave will be of greatest
intensity when created by a large, high drag
configuration at low altitude. Since the wave
intensity decreases rapidly with distance away
from the source, the boom will be of strongest
audibility near the point of release.
It should become apparent that sonic booms
are a byproduct of supersonic aviation and,
with supersonic flight becoming more common-
place, the problem is more perplexing. The
potential of sonic booms is mostly of the
audible nature and nuisance of the disturbance.
The damage potential of the ordinary sonic
boom is quite small and the principal effects
are confined to structures which are extremely
brittle, low strength, and have characteristic
high residual stresses. In other words, only
the extremes of pressure waves generated by
airplanes in flight could possibly cause cracked
plaster and window glass. Such materials
NAVWEPS OD-ROT-80
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
are quite prone to sharp dynamic stresses and,
when .superimposed on the high residual
stresses common to the products and building
construction, slight but insignificant damage
may result. Actually, the most objectionable
feature of the sonic boom is the audibility and
the anxiety or apprehension caused by the
sharp, loud noise which resembles a blast.
The pressure jump through the shock waves
in the immediate vicinity of the airplane is
much greater than those common to the
audible “booms” at ground level. Thus,
airplanes in close formation at supersonic
speeds may encounter considerable interference
between airplanes. In addition, to eliminate
even the most remote possibility of structural
damage, a high speed airplane should not
make a supersonic pass close to a large air-
plane which may have low limit load factor
and be prone to be easily disturbed or damaged
by a strong pressure wave.
HELICOPTER PROBLEMS
The main difference between helicopter and
an airplane is the main source of lift. The
airplane derives its lift from a fixed airfoil
surface while the helicopter derives lift from
a rotating airfoil called the rotor. Hence, the
aircraft will be classified as either “fixed-
wing” or “rotating wing.” The word “heli-
copter” is derived from the Greek words
meaning “helical wing” or “rotating wing. ”
Lift generation, by a “rotating wing” enables
the helicopter to accomplish its unique mission
of hovering motionless in the air, taking off
and landing in a confined or restricted area,
and autorotating to a safe landing following
a power failure. Lift generation by “rotating
wing’ ’ is also responsible for some of the
unusual problems the helicopter can encounter.
Since the helicopter problems are due to par-
ticular nature of .the rotor aerodynamics, the
basic flow conditions within the rotor must be
considered in detail. For simplicity, the
initial discussion will consider only the hover-
ing rotor. Although the term hovering | 416 | 416 | 00-80T-80.pdf |
NAVWEPS DO-ROT-80
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
usually means remaining over a particular
spot on the ground, it shall be considered
here as flight at zero airspeed. This is necessary
because the aerodynamic characteristics of the
rotor depend on its motion with respect to
the air and not the ground. Hovering in a 20
knot wind is aerodynamically equivalent to
flying at an airspeed of 20 knots in a no-wind
condition, and the characteristics will be
identical in the two conditions.
The first point to realize is that the rotor is
subject to the same physical laws of aero-
dynamics and motion that govern flight of
the fixed-wing airplane. The manner in which
the rotor is subject to these laws is much more
complicated due to the complex flow con-
ditions.
Rotor lift can be explained by either of two
methods. The first method, utilizing simple
momentum theory based on Newton’s Laws,
merely states that lift results from the rotor
accelerating a mass of air downward in the
same way that the jet engine develops thrust
by accelerating a mass of air out the tailpipe.
The second method of viewing rotor lift
concerns the pressure forces acting on the
various sections of the blade from root to tip.
The simple momentum theory is useful in
determining only lift characteristics while
the “blade element” theory gives drag as
well as lift characteristics and is useful in
giving a picture of the forces at work on the
rotor. In the “blade element” theory, the
blade is divided up into “blade elements” as
shown in figure 6.15. The forces acting on
each blade element are analyzed. Then the
forces on all elements are summed up to give
the characteristics of the whole rotor. The
relative wind acting on each segment is the
resultant of two velocity components: (1) the
velocity due CO the rotation of the blades
about the hub and (2) the induced velocity,
or downwash velocity caused by the rotor.
the velocity due to rotation at a particular
element is proportional to the rotor speed and
the distance of the element from the rotor hub.
Thus, the velocity due to rotation varies line-
arly from zero at the hub to a maximum at
the tip. A typical blade section with the
forces acting on it is shown in figure 6.15.
A summation of the forces acting perpen-
dicular to the plane of rotation (tip path plane)
will determine the rotor thrust (or lift) char-
acteristics while summation of the moments
resulting from forces acting in the plane of
rotation will determine the rotor torque char-
acteristics. As a result of this analysis, the
rotor thrust (or lift) is found to be propoc-
tional to the air density, a nondimensional
thrust coefficient, and the square of the tip
speed, or linear speed of the tip of the blade.
The thrust coefficient is a function of the aver-
age blade section lift coefficient and the rotor
solidity, which is the proportion of blade area
to disc area. The lift coefficient is identical to
that used in airplane aerodynamics while the
solidity is analagous to the aspect ratio in air-
plane aerodynamics. The rotor torque is
found to be proportional to a nondimensional
torque coefficient, the air density, the disc
area, the square of the tip speed, and the blade
radius. The torque coefficient is dependent
upon the average profile drag coefficient of the
blades, the blade pitch angle, and the average
lift coefficient of the blades. The torque can
be thought to result from components of profile
and induced drag forces acting on the blades,
similar to those on an airplane.
As in the airplane, there is one angle of
attack or blade pitch condition that will result
in the most efficient operation. Unfortu-
nately, the typical helicopter rotor operates
at a near constant RPM and thus a constant
true airspeed and cannot operate at this most
eficient condition over a wide range of altitude
and gross weight as the fixed-wing airplane.
The airplane is able to maintain an efficient
angle of attack at various altitudes, and gross
weights by flying at various airspeeds but the
helicopter will operate with a near constant
rotor velocity and vary blade angle to contend
with variations in altitude and gross weight. | 417 | 417 | 00-80T-80.pdf |
MAVWEPS 00-BOT-80
APPLICATION OF AEBODYNAMICS
TO SPECIFIC PBOBLEMS OF FLYING
ROTOR BLADE ELEMENT
RELATIVE -
ROTATION WIND COMPONENT
DUE TO ROTATION
LOCAL FLOW AT BLADE ELEMENT
VELOCITY DUE
RESULTANT
RELATIVE VELOCITY
--_
TIP PATH PLANE
Figure 6.75. Rotor Blade Flemenf Aerodynamics
401 | 418 | 418 | 00-80T-80.pdf |
NAVWEPS oo-80~-80
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYl,NG
If the rotor could operate within a wide
range of rotor speed, the efficiency and per-
formance could be improved.
With the previous relationships established
for the rotor in hovering flight, the effect of
forward flight br rotor translation can be con-
sidered. With forward flight, a third velocity
component, that tif the forward velocity of the
helicopter, must be considered in determining
the relative wind acting on each rotor blade
element. Since the entire rotor moves with the
helicopter, the velocity of air passing over
each of the elements on the advancfng blade is
increased by the forward speed of the heli-
copter and the velocity of the air passing over
each element of the retreating blade is de-
creased by the same amount. This is shown in
figure 6.16.
If the blade angles of attack on both advanc-
ing and retreating blades remained the same
as in hovering flight, the higher velocity on
the advancing blade would cause a dissym-
metry of lift and the helicopter would tend to
roll to the left. It was this effect that created
great difficu!ty during many early helicopter
and autogiro projects. Juan De La Cierva was
the first to realize what caused this effect and
he solved the problem by mounting his auto-
giro blades individually on flapping hinges,
thus allowing a flapping action to automati-
cally correct the dissymmetry of lift that re-
sulted from forward flight. This is the method
still used in an articulated rotor system today.
The see-saw, or semi-rigid, rotor corrects the
lift dissymmetry by rocking the entire hub and
blades about a gimbal joint, By rocking the
entire rotor system forward, the angle of at-
tack on the advancing blade is reduced and
the angle of attack on the retreating blade is
increased. The rigid rotor must produce cyclic
variation of the blade pitch mechanically as
the blade rotates to eliminate the lift dissym-
metry. Irrespective of the method used to
correct the dissymmetry of lift, identical aero-
dynamic characteristics result. Thus, what is
said about rotor aerodynamics is equally valid
for all types of rotor systems.
By analyzing the velocity components acting
on the rotor blade sections from the blade
root to the tip on both advancing and retreat-
ing blades, a large variation of blade section
angle of attack is found. Figure 6.16 illus-
trates a typical variation of the local blade
angle of attack for various spanwise positions
along the advancing and retreating blades of
a rotor at high forward speed. There is a
region of positive angles of attack resulting
in positive lift over the entire advancing blade.
Immediately next to the hub of the retreating
blade there is an area of reversed flow where
the velocity due to the forward motion of the
helicopter is greater than the rearward velocity
due to the blade rotation. The next area is a
negative stall region where, although the
flow is in the proper direction relative the
blade, the angle of attack exceeds that for
negative stall. Progressing out the retreating
blade, the blade angle of attack becomes less
negative, resulting in an area of negative lift.
Then the blade angle becomes positive again,
resulting in a positive lift region. The blade
angle continues to increase, until near the tip
of the retreating blade the positive stall angle
of attack is exceeded, resulting in stalling of
the tip section. This wide variation in blade
section angles of attack results in a large
variation in blade section lift and drag coefK-
cients. The overall lift force on the left and
right sides of the rotor disc are equalized by
cyclically varying the blade pitch as explained
previously, but the drag variation is not
eliminated. This drag variation causes a
shaking force on the rotor system and con-
tributes to the vibration of the helicopter.
RETREATING BLADE STALL. Retreat-
ing blade stall results whenever the angle of
attack of the blade exceeds the stall angle of
attack of the blade section. This condition
occurs in high speed flight at the tip of the
retreating blade since, in order to develop the
same lift as the advancing blade, the retreating | 419 | 419 | 00-80T-80.pdf |
NAVWEPS OO-ROT-RO
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
VELOCITY DUE
TO FORWARD
FLIGHT
RETREATING GLADE
VELOCITY DUE
TO ROTATION
REVERSED FLOW
NEGATIVE STALL
NEGATIVE LIFT
VELOCITY DUE
I - POSITIVE STALL ANGLE
LOCAL I/
BLADE
ANGLE OD- nrc7 I
OF HUB
ATTACK
ETRiATING BLADE
-lop I _- - - NEGATIVE STALL ANGLE
,
-ADVANCING
BLADE
Figure 6.16. Rotor Flow Conditions in Forward f/i&
403 | 420 | 420 | 00-80T-80.pdf |
NAVWEPS OD-80140
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
blade must operate at a greater angle of attack.
If the blade pitch is increased or the forward
speed increased the stalled portion of the rotor
disc becomes larger with the stall progressing
in toward the hub from the tip of the retreating
blade. When approximately 15 percent of
the rotor disc is stalled, control of the heli-
copter will be impossible. Flight tests have
determined that control becomes marginal and
the stall is considered severe when the outer
one-quarter of the retreating blade is stalled.
Retreating blade stall can be recognized by
rotor toughness, erratic stick forces, a vibration
and stick shake with a frequency determined
by the number of blades and the rotor speed.
Each of the blades of a three-bladed rotor will
stall as it passes through the stall region and
create a vibration with three beats pet rotor
revolution. Other evidence of retreating blade
stall is partial or complete loss of control or a
pitch-up tendency which can be uncontrollable
if the stall is severe.
Conditions favorable for the occurrence of
retreating blade stall are those conditions that
result in high retreating blade angles of attack.
Each of the following conditions results in a
higher angle of attack on the retreating blade
and may contribute to retreating blade stall:
1. High airspeed
2. Low totor RPM-operation at low
rotor RPM necessitates the use of
higher blade pitch to get a given
thrust from the rotor, thus a higher
angle of attack
3. High gross weight
4. High density altitude
5. Accelerated flight, high load factor
6. Flight through turbulent air or gusts-
sharp updrafts result in temporary
increase in blade angle of attack
7. Excessive ot abrupt control deflections
during maneuvers
Recovery from a stalled condition can be
effected only by decreasing the blade angle of
attack below the stall angle. This can be
accomplished by one or a combination of the
following items depending on severity of the
stall:
1. Decrease collective pitch
2. Decrease airspeed
3. Increase rotor RPM
4. Decrease severity of accelerated ma-
newer or control deflection
If the stall is severe enough to result in pitch-up,
forward cyclic to attempt to control pitch-up
is ineffective and may aggravate the stall since
forward cyclic results in an increase in blade
angle of attack on the retreating blade. The
helicopter will automatically recover from
a severe stall since the airspeed is decreased in
the nose high attitude but recovery can be
assisted by gradual reduction in collective
pitch, increasing RPM, and leveling the heli-
copter with pedal and cyclic stick.
From the previous discussion, it is apparent
that there is some degree of retreating blade
stall even at moderate airspeeds. However,
the helicopter is able to perform satisfactorily
until a sufficiently large area of the rotor disc
is stalled. Adequate warning of the impend-
ing stall is present when the stall condition
is approached slowly. There is inadequate
warni,ng of the stall only when the blade pitch
or blade angle of attack is increased rapidly.
Therefore, unintentional severe stall is most
likely to occur during abrupt control motions
or rapid accelerated maneuvers.
COMPRESSIBILITY EFFECTS. The highest
relative velocities occur at the tip of the ad-
vancing blade since the speed of the helicopter
is added to the speed due to rotation at this
point. When the Mach number of the tip
section of the advancing blade exceeds the
critical Mach number for the rotor blade sec-
tion, compressibility effects result. The criti-
cal Mach number is reduced by thick, highly
cambered airfoils and critical Mach number
decreases with increased lift coefficient. Most
helicopter blades have symmetrical sections
and therefore have relatively high critical
Mach numbers at low lift coefhcients. Since
the principal effects of compressibility are the | 421 | 421 | 00-80T-80.pdf |
large increase in drag and rearward shift of the
airfoil aerodynamic center, compressibility ef-
fects on the helicopter increase the power te-
quired to maintain rotor RPM and cause rotor
roughness, vibration, stick shake, and an un-
desirable structural twisting of the blade.
Since compressibility effects become more
severe at higher lift coefficients (higher blade
angles of attack) and higher Mach numbers,
the following operating conditions represent
the most adverse conditions from the stand-
point of compressibility:
1. High airspeed
2. High rotor RPM
3. High gross weight
4. High density altitude
5. Low temperature-the speed of sound
is proportional to the square root of
the absolute temperature. Therefore,
sonic velocity will be more easily
obtained at low temperatures when
the sonic speed is lower.
6. Turbulent air-sharp gusts momen-
tarily increase the blade angle of
attack and thus lower the critical
Mach number to the point where
compressibility effects may be en-
countered on the blade.
Compressibility effects will vanish by de-
creasing the blade pitch. The similarities in
the critical conditions for retreating blade
stall and compressibility should be noticed
but one basic difference must be appreciated-
compressibility occurs at HIGH RPM while
retreating blade stall occurs at LOW RPM.
Recovery technique is identical for both with
the exception of RPM control.
AUTOROTATION CHARACTERISTICS.
One of the unique characteristics of helicopters
is their ability to take part of the energy of
the airstream to keep the rotor turning and
glide down to a landing with no power.
Consideration of the rotor during a vertical
autorotation will provide an understanding of
why the rotor continues to rotate without
power. During autorotation, the flow of air
NAVWEPS CID-ROT-R0
APP,LlCATlON OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYmING
is upward through the rotor disc and there is
a vertical velocity component equal to the
rate of descent of the helicopter. In addition,
there is a velocity component due to rotation
of the rotor. The vector sum of these two
velocities is the relative wind for the blade
element. The forces resulting from the relative
wind on each particular blade section will
provide the reason why the rotor will continue
to operate without power. First, consider, a
blade element near the tip of the blade as illus-
trated in figure 6.17. At this point there is
a lift force acting perpendicular to the relative
wind and a drag force acting parallel to the
relative wind through the aerodynamic center.
Since the rotation of the rotor is affected only
by forces acting in the plane of rotation, the
important forces are components of the lift
and drag force in the plane of rotation. In
this low angle of attack high speed tip section,
the net in-plane force is a drag force which
would tend to retard the rotor. Next, con-
sider a blade section at about the half-span
position as illustrated in figure 6.17. In this
case, the same forces are present, but the iti-
plane component of lift force is greater than
the drag force and this results in a net thrust
or forward force in the plane of rotation which
tends to drive the rotor.
During a steady autorotation, there is a
balance of torque from the forces along the
blade so that the RPM is maintained in equi-
librium at some particular value. The region
of the rotor disc where there is a net drag force
on the blade is called the “propeller region”
and the region of the rotor disc where there
is a net in-plane thrust force is called the
“autorotation region.” These regions are
shown for vertical autorotation and forward
speed (or normal) autorotation in figure 6.17.
Forces acting on the rotor blades in forward
flight autorotation are similar to those in
vertical autorotation but the difference will
consist mainly of shifts of the autorotation
region to the left and the addition of reverse
405 | 422 | 422 | 00-80T-80.pdf |
NAVWEPS 004oT-80
APPLICATION 0~ AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
LIFT FORCES IN THE
AUTOROTATION REGION
FORCES IN THE
PROPELLER REGION I
t
LIFT
I
’ \ DRAG
RESULTANT
VELOCITi DUE TO
ROTe;TION
FLIGHT 1
I DIRECTION
UTOROTATION
VERTICAL AUTOROTATION FORWARD FLIGHT
AUTQRQTa-i\QN
Figure 6.17. Rotor Autorotation Flaw Conditions
406 | 423 | 423 | 00-80T-80.pdf |
flow and negative stall regions similar to the
powered flight condition.
Autorotation is essentially a stable flight
condition. If external disturbances cause the
rotor to slow down, the autorotation region of
the disc automatically expands to restore the
rotor speed to the original equilibrium condi-
tion. On the other hand, if an external
disturbance causes the rotor to speed up, the
propeller region automatically expands and
tends to accelerate the rotor to the original
equilibrium condition. Actually the stable
autorotation condition will exist only when
the autorotational speed is within certain
limits. If the rotor speed is allowed to slow
some excessive amount, then the rotor becomes
unstable and the RPM will decrease even
further unless the pilot immediately corrects
the condition by proper control action.
In case of engine failure, the fixed-wing
airplane will be glided at maximum lift-drag
ratio to produce maximum glide distance. If
minimum rate of descent is desired in power-off
flight rather than maximum glide distance, the
fixed-wing airplane will be flown at some
lower airspeed. Actually, the minimum rate
of descent will occur at minimum power
required. The helicopter exhibits similar char-
acteristics but ordinarily the best autorota~tion
speed may be considered that speed that results
in the minimum rate of descent rather than
maximum glide distance. The aerodynamic
condition of the rotor which produces mini-
mum rate of descent is:
Maximum ratio of
(Mean blade lift coetIicient)3”
Mean blade drag coefficient
It is this ratio which determines the auto-
rotation rate of descent. Figure 6.18 illus-
trates the variation of autorotation rate of
descent with equivalent airspeed for a typical
helicopter. Point A on this curve defines the
point which produces autorotation with mini-
mum rate of descent. Maximum glide distance
NAVWEPS OO-BOT-BO
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYl,NG
during autorotation descent would be obtained
at the flight condition which produces the
greatest proportion between airspeed and rate
of descent. Thus, a straight line from the
origin tangent to the curve will define the
point for maximum autorotative glide dis-
tance. This corresponds to Point B of figure
6.18. If the helicopter is being glided at the
speed for maximum glide distance, a decrease
in airspeed would reduce the rate of descent
but the glide distance would decrease. If the
helicopter is being glided at the speed for
minimum rate of descent, the rate of descent
(steady state) can not be reduced but the glide
distance can be increased by increasing the
glide speed to that for maximum distance.
Weight and wind affect the glide character-
istics of a helicopter the same way an airplane
is affected. Ideally, the helicopter autorotates
at a higher equivalent airspeed at higher gross
weight or when autorotating into a headwind.
In addition to aerodynamic forces which act
on the rotor during autorotation, inertia forces
are also important. These effects are usually
associated with the pilot’s response time be-
cause the rate a pilot reacts to a power failure
is quite critical. The time necessary to reduce
collective pitch and enter autorotation be-
comes critical if the rotor inertia characteristics
are such as to allow the rotor to slow down to
a dangerous level before the pilot can react.
With power on, the blade pitch is relatively
high and the engine supplies enough torque to
overcome the drag of the blades. At the
instant of power failure the blades are at a high
pitch with high drag. If there is no engine
torque to maintain the RPM, the rotor will
decelerate depending on the rotor torque and
rotor inertia. If the rotor has high rotational
energy the rotor will lose RPM less rapidly,
giving the pilot more time to reduce collective
pitch and enter autorotation. If the rotor has
low rotational energy, the rotor will lose RPM
rapidly and the pilot may not be able to react
quickly enough to prevent a serious loss of
rotor RPM. Once the collective pitch is at | 424 | 424 | 00-80T-80.pdf |
PdAVWEPS OD-BOT-80
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
the low pitch limit, the rotor RPM can be in-
creased only by a sacrifice in altitude or air-
speed. If insufficient altitude is available to
exchange for rotor speed, a hard landing is
inevitable. SufIicient rotor rotational energy
must be available to permit adding collective
pitch to reduce the helicopter’s rate of descent
before final ground contact.
In the case of most small helicopters, at
least 300 feet of altitude is necessary for an
average pilot to set up a steady autorotation
and land the helicopter safely without damage.
This minimum becomes 500 to 600 feet for the
larger helicopters, and will be even greater for
helicopters with increased disc loading. These
characteristics are usually presented in the
flight handbook in the form of a “dead man’s
curve” which shows the combinations of air-
speed and altitude above the terrain where a
successful autorotative landing would be dish-
cult, if not impossible.
A typical “dead man’s curve” is shown in
figure 6.18. The most critical combinations
are due to low altitude and low airspeed illus-
trated by area A of figure 6.18. Less critical
conditions exist at higher airspeeds because of
the greater energy available to set up a steady
autorotation. The lower limit of area A is a
finite altitude because the helicopter can be
landed successfully if collective pitch is held
rather than reduced. In this specific case there
is not sufficient energy to reach a steady state
autorotation. The maximum altitude at which
this is possible is approximately ten feet on
most helicopters.
Area B on the “dead man’s curve” of figure
6.18 is critical because of ground contact flight
speed or rate of descent, which is based on the
strength of the landing gear. The average
pilot may have difhculty in successfully flaring
the helicopter from a high speed flight con-
dition without allowing the tail rotor to strike
the ground or contacting the ground at an ex-
cessive airspeed. A less critical zone is some-
times shown on this curve to indicate that
higher ground contact speeds can be permitted
when the landing surface is smooth. In ad-
dition, various stability and control character-
istics of a helicopter may produce critical con-
ditions in this area. The critical areas of the
“dead man’s curve” should be avoided unless
such operation is a specific mission require-
ment.
POWER SETTLING. The term “power
settling” has been used to describe a variety of
flight conditions of the helicopter. True
“power settling” occurs only when the heli-
copter rotor is operating in a rotary flow
condition called the “vortex ring state.”
The flow through the rotor in the “vortex
ring state” is upward near the center of the
disc and downward in the outer portion,
resulting in a condition of zero net thrust on
the rotor. If the rotor thrust is zero, the
helicopter is effectively free-falling and ex-
tremely high rates of descent can result.
The downwash distribution within the
rotor is shown in figure 6.19 for the conditions
of normal hovering and power settling. Part
A of figure 6.19 illustrates the typical down-
wash distribution for hovering flight. If
sufficient power were not available to hover
at this condition, the helicopter would begin
to settle at some rate of descent depending on
the deficiency of power. This rate of descent
would effectively decrease the downwash
throughout the rotor and result in a redistri-
bution of downwash similar to Part B of
figure 6.19. At the outer portion of the
rotor disc, the local induced downwash veloc-
ity is greater than the rate of descent and
downflow exists. At the center of the rotor
disc, the rate of descent is greater than the
local induced downwash velocity and the
resultant flow is upward. This flow condition
results in the rotary “vortex ring” state. By
reference to the basic momentum theory it is
apparent that the rotor will produce no thrust
in this condition if the net mass flow of air
through the rotor is zero. It is important to
note that the main lifting part of the rotor is
not stalled. The rotor roughness and loss of | 425 | 425 | 00-80T-80.pdf |
NAVWEPS OO-BOT-80
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
RATE
OF
DESCENT,
FPM
POWER
OFF
MINIMUM
RATE OF
/ DESCENT
I VELOCITY, KNOTS
DEAD MAN’S CURVE
ALTITUDE ~
ABOVE
TERRAIN
SAFE
F T.
VELOCITY, KNOTS
Figure 6.1%. Autorotation Characteristics
409 | 426 | 426 | 00-80T-80.pdf |
NAVWEPS DO-BOT-BO
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
VARIATION OF INDUCED VELOCITY
ALONG THE BLADE SPAN DURING
HOVERING FLIGHT
VARIATION OF INDUCED VELOCITY
ALONG THE BLADE SPAN DURING
VORTEX RING STATE
VORTEX RING STATE
Figure 6.79. Rotor Downwash Distribution
410 | 427 | 427 | 00-80T-80.pdf |
control experienced during “power settling”
results from the turbulent rotational flow on
the blades and the unsteady shifting of the
flow in and out spanwise along the blade.
There is an area of positive thrust in the outer
portion of the rotor as a result of the mass of
air accelerated downward and an area of
negative thrust at the center of the rotor as a
result of the mass of air flowing upward. The
rotor is stalled only near the hub but no
important effect is contributed because of the
low local velocities.
Operation in the “vortex ring” state is a
transient condition and the helicopter will
seek equilibrium by descending. As the heli-
copter descends, a greater upflow through the
disc results until eventually the flow is entirely
up through the rotor and the rotor enters auto-
rotation where lower rates of descent can be
achieved. Unfortunately, considerable alti-
tude will be lost before the autorotative type
of flow is achieved and a positive recovery
technique must be applied to minimize the loss
of altitude. “Power settling” can be recog-
nized by rotor roughness, loss of control due
to the turbulent rotational flow, and a very
high rate of descent (as high as 3,000 fpm).
It is most likely to be encountered inadvert-
ently when attempting to hover when suf-
ficient power is not available because of high
gross weight or high density altitude.
Recovery from “power settling” can be ac-
complished by getting the rotor out of the
“vortex ring state.” If the condition is en-
countered with low power, rapid application
of full power may increase the downwash suf-
ficiently to get .the rotor out of the condition.
If the condition is encountered at high or
maximum power or, if maximum power does
not effect a recovery, increasing airspeed by
diving will result in recovery with minimum
loss of altitude. This type of recovery is most
effective but adequate cyclic control must be
available. If cyclic control has been lost, re-
covery must be effected by reducing power and
collective pitch and entering autorotation.
NAVWEPS OD-EOT-80
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYING
When normal autorotation has been estab-
lished, a normal power recovery from the auto-
rotation can be made. While such a recovery
technique is effective, considerable altitude
may be lost. Hence, diving out of the power
settling condition provides the most favorable
means of recovery.
Actually, real instances of true “power
settling” are quite rare. A condition often
described incorrectly as “power settling” is
merely a high sink rate as a result of insufficient
power to terminate an approach to landing.
This situation frequently occurs during high
gross weight or high density altitude operation.
The flow conditions within the rotor are quite
normal and there is merely insufficient power
to reduce rate of descent and terminate an
approach. Such a situation becomes more
critical with a steep approach since the more
rapid descent will require more power to
terminate the approach.
THE FLIGHT HANDBOOK
For the professional aviator, there are few
documents which are as important as the air-
plane flight handbook. The information and
data contained in the various sections of the
flight handbook provide the basis for safe and
effective operation of the airplane.
Various sections of the flight handbook are
devoted to the following subjects:
(1) Equipment and Systems. With the me-
chanical complexity of the modern airplane,
it is imperative that the pilot be familiar with
every item of the aircraft. Only through exact
knowledge of the equipment can the pilot
properly operate the airplane and contend
with malfunctions.
(2) Operating Procedures. Good procedures
are mandatory to effect safe operation of the
airplane and its equipment. The complexity
of modern equipment dictates the use of special
and exact procedures of operation and any
haphazard or non-standard procedure is an
411 | 428 | 428 | 00-80T-80.pdf |
NAVWEPS O&ROT-SO
APPLICATION OF AERODYNAMICS
TO SPECIFIC PROBLEMS OF FLYl.NG
invitation for trouble of many sorts. The
normal and emergency procedures applicable
to each specific airplane will insure the
proper operation of the equipment.
(3) Operating Limitatiom. The operation of
the airplane and powerplant must be conducted
within the established limitations. Failure
to do so will invite failure or malfunction of
the equipment and increase the operating cost
or possibly cause an accident.
(4) Flight Characteristics. While all aircraft
will have certain minimum requirements for
flying qualities, the actual peculiarities and
special features of specific airplanes will differ.
These particular flight characteristics must be
well known and understood by the pilot.
(5) Operating Data. The performance of
each specific airplane defines its application to
various uses and missions. The handbook
operating data must be available at all times
to properly plan and elnccate the flight of an
aircraft. Constant reference to the operating
data will insure safe and effective operation
of the airplane.
Great time and effort are expended in the 1
preparation of the flight handbook to provide
the most exact information, data, and pro-
cedures. Diligent study and continuous UC
of the flight handbook will ensure that the
greatest effectiveness is achieved from the
airplane while still operating within the
inherent capabilities of the design.
412 | 429 | 429 | 00-80T-80.pdf |
NAVWEPS 00-802-30
SELECTED REFERENCES
SELECTED REFERENCES
1. Dommasch, Sherby, and Connolly
“Airplane Aerodynamics”
Pitman Publishing Co.
2d Edition, 1957
2. Perkins and Hage
“Airplane Performance, Stability, and Control”
John Wiley and Sons
1949
3. E. A. Bonney
“Engineering Supersonic Aerodynamics”
McGraw-Hill Book Co.
1950
4. Hurt, Vernon, and Martin
“Aeronautical Engineering, Section I, Manual of Instruction, Avi-
ation Safety Officer Course”
University of Southern California
1958
5. Fairchild, Magill, and Brye
“Principles of Helicopter Engineering”
University of Southern California
1959
413
Revised January 1965 | 430 | 430 | 00-80T-80.pdf |
NAVWEPS 00-8OT-80
INDEX
INDEX
P.W
accelerated motion. .......................... 182
adverseyaw ................................. 291
aerodynamic center. .......................... 47
Aerodynamics, Basic, Chapter I. ............... 1
Aerodynamics, High Speed, Chapter III. ........ 201
aeroelastic effects. ............................ 330
afterburner. ................................. 129
aileron reversal. .............................. 339
airfoil
drag characteristics. ........................ 33
lift characteristics. ......................... 27
pitching moments. ......................... 47
terminology. .............................. 20
airspeed
calibrated. ................................ 10
equivalent ................................. 11
indicated .................................. 10
mcasurcmcnt ............................... 9
primary control of airspeed. ................ 27,350
fT”C ....................................... 14
altitude
density altitude. ........................... 4
pressure altitude. ........................... 4
primary control of altitude. ............... 154,352
angle of attack. .............................. 22
angle of attack indicator. ..................... 357
angleofbank ............................... 37,176
angle of climb. .............................. 152
Application of Aerodynamics to Specific Prob-
lems of Flying, Chapter VI. ............... 349
approach .................................... 360
aspect ratio. ................................. 61
asymmetrical Power, multi-engine airplane, 294
atmosphere
properties ................................. 1
standard ................................... 4
autorotation characteristics. ................... 405
autorotative rolling. ....................... 309,317
available thrust and power. ................... 104
axis system, airplane reference. ................ 249
Bernoulli equation. ............................ 6
bobweight .................................. 273
boundarylayer ............................... 52
boundary layer control. ....................... 43
brake mean effective pressure, BMEP. .......... 137
*IL*e
brake horsepower, BHP. ...................... 137
braking performance. .................. ..... 387
braking technique. ......... .... ..... ..... 390
calibrated airspeed, CAS ....................... 10
center of gravity limitations. ........... 259,275
center of pressure. ............... ..... ..... 47
circulation. .................................. 16
climb angle. ................................. 152
climb performance. ...... ..... ...... .... 150
climb rate. .. ........................ ...... 154
components of the gas turbine. ................ 109
compressible flow. .......................... 204
compressibility ............................. 201
compressor stall or surge. .. .... ...... .... 125
control force stability. ........................ 264
control of airspeed and altitude. ................ 349
control, Stability and Control, Chapter IV. ...... 243
control systems, longitudinal. ................. 281
creep considerations. ......................... 330
critical altitude. .......... .................. 143
critical field length. ............ ............. 396
critical Mach number. ........................ 215
cumulative damage. ... ..... ...... ..... 328
cycle of operation
gas turbineengine .......................... 106
reciprocating engine. ....................... 135
damping ..................................... 247
density, density ratio. ........................ 2
density altitude. .. .......................... 4
detonation .. .:. ....................... 140, 194
dihedral ..................................... 295
directional control .. .......... ............. 290
directional stability. ......................... 284
divergence ................................ 245, 342
downspring, ................................. 270
downwash ................................... 66
drag
coefficient. ........ ................. 29
equation ................................... 29
induced ................................... 66
parasite. .. .. ..... ... ............. 87
total ...................................... 92
dynamic pressure. .... ... .... ............ : 9
dynamic stability., ... .... .... ... ....... 245
efficiency factor. ..... ... ...... .. ....... 89
414 | 431 | 431 | 00-80T-80.pdf |
endurance Pwe
off-optimum. .............................. 172
performance. ............... .... ......... 170
specific ................................. 158, 170
engine failure
effect on multiengine airplane. ............ 294, 376
power off glide performance. ................ 369
equilibrium conditions. ....................... 150
equivalent airspeed, EAT. .................... 11
equivalent parasite area. ...................... 89
equivalent shaft horsepower, ESHP. ........... 133
expansionwave .............................. 211
factorofsafcty ............................... 326
fatigue considerations. ........................ 328
feathering and governing of propellers. ......... 148
flap
aerodynamic effects. ....................... 37, 43
typCS ...................................... 41
flutter ....................................... 342
force divergence. ............................. 218
friction
braking, ............................. 388
cocfficicnt. ................................ 388
skin friction. .............................. 54
frost ........................................ 373
fuel qualities. ................................ 141
glide performance. ........................... 369
governing apparatus, turbine engines. .......... 121
governing and feathering of propellers. ......... 147
gusts and wind shear. .................... ... 367
gust load factor. ........................... 332
groundeffect ................................. 379
heating, aerodynamic. ........................ 242
helicopter, problems. ......................... 399
helicopter stability and control. ............. 319
high lift
devices .. ...... ............ ... ., ., 39
flight at high lift conditions ................. 35
High Speed Aerodynamics, Chapter III. ........ 201
humidity, effect on power. .................... 144
ice .......................................... 373
indicated airspeed, IAS. ......... ............. 10
induced
angle of attack. ........................... 66
drag. .................................... 66
drag coefficient. ........................... 68
flow. ..................................... 63
inertia coupling. ................... ....... 315
inlets, supersonic powerplant. ................. 238
interference between airplanes in flight. ........ 383
items of airplane performance. ................. 150
landing and ground loads. .................... 343
landing flare and touchdown, ............. 362
NAVWEPS 00-8OT-80
INDEX
Psge
!a”ding gear configuration stability. ........... 305
landing performance. ...... ... .... ... 192
factors affecting performance. .... ........... 196
lateral co”trol, ............................... 300
lateral stability. ....................... .... 294
lift
characteristics. ........................... 24
coefficient .................................. 23
equation ................................... 23
generation .. ............ ............ ... 16, 63
lift-drag ratio. ............................... 32
limit load. ................................. 326
linespeeds ................................... 394
load factor ................................ 37, 331
load spectrum. ............. _ ................. 328
longitudinal
control .................................... 275
dynamic stability. .......................... 279
static stability. ........................... 250
Mach number
definition. .................................. 202
critical Mach “umber. ...................... 2!5
maneuvering load factor ..................... 331
maneuvering performance. ........ ........... 176
maneuvering stability. ...................... 268
mean aerodynamic chord, MAC. ............. 63
mirror landing system ......................... 358
normal shock wave. .......................... 207
obliqueshockwave .......................... 207
operating limitations
propellers, ..................... .... 148
reciproczring c ... .................. 144
turbojet ................................... 124
.. turboprop. ............................... 133
Operating Strength Limitations, Chapter V. .... 325
overstress, effect on service life. ................ 344
parasite area, equivalent. .... .............. 89
parasitedrag ................................. 87
performance, Airplane Performance, Chapter II. 95
pilot induced oscillarion. ..................... 314
pitching moment
airfoil. .................................... 47
longitudinal. .................. ........ 249, 251
pitch-up ..................................... 313
pitot-static system. ............... ........... 9
planform effects. ............................. 61
power effects on stability. ..................... 259
power off stability ... ........................ 259
power required. ................... .......... 96
power settling. ................... ........ 4c3
preignition, ........... .... .... 140
pressure altitude. ......... .... ... ... .. 4
pressure distribution. .. ......... ..... 14
415 | 432 | 432 | 00-80T-80.pdf |
NAVWEPS OO-EOT-80
INDEX
proprllcrs lmd
charactcrlstics. ........... : ............... 14s
efficiency .................................. 145
opcrarmg limitations. ............ : ......... 148
propulsion
etlicicncy .................................. 106
principles. ............. : _ ................. 104
ram tempcraturc rise. ......................... 242
range performance. $ .................... ..... 158
off-optimum conditions. .................... 172
propeller airplanes. ......................... 160
turbojet airplanes ........................... 164
rate of climb. ................................ 154
reciprocating engines
operating characteristics ..................... 13s
operating limitations. ....................... 144
refusal speed, ....... .+: .................... :, .... 392
tetreating blade stall. ......................... 402
reversed command region. ..................... 353
Reynolds number. ............. :. ............. 54
scale effect. .................................. 59
separation, airflow. ............................ 56
service life. ................................ 328
shock induced separation. ..................... 218
shock wave formation. ....................... 218
sideslip angle. ............................... 284
slipstream rotation. .......................... 294
sonic booms. ................................. 396
spanwise lift distribution. ..................... 74
specificendurance ............................. 170
specific fuel consumption
reciprocating engine. ....................... 141
turbojet cngi.ne. ......................... 117
specificrange ................................. 158
speed, maximum and minimum. ............... 150
spin,spinrecovery..........................291, 307
Srability and Control, Chapter IV. ............. 243
stability
directional. ................................ 284
dynamic ................................... 245
helicopter .................................. 319
lateral ..................................... 294
longitudinal. ... : .......................... 250
miscellaneous problems. ..................... 305
static ............................ ......... 243
stallspeeds ................................. 35
Page
seal pattern. ................................ 77
stall rec”very. ............................... 39
standard atmosphere. ......................... 4
static strength. ............................... 326
streamline pattern. ............. ............. 14
supercharging. ............................... 141
supersonic airfoil sections. .................... 223
sweepback ................................... 63
advantages .................................. 226
disadvantages. ............................. 231
takeoff ...................................... 365
takeoff performance. .......................... 184
factors affecting performance. ................ 187
taper, taper ratio. ............................ 63
thrust augmentation. ......................... 129
thrust required. .............................. 96
time limitations, powcrplants. ............... 128, 144
tip stall. .................................... 77
tip vortex. .................................. 63
torque. ............................... ,;, ........ 137
transition of boundary layer. .................. 52 .. transon1c aIrfoIl scctlo”s. ...................... 220
true airspeed, TAS. ........................... 14
turbojetengines .............................. 107
operating characteristics. ................... 116
operating limitations. ...................... 124
turboprop, gas turbine-propeller combination. ... 132
turbulence.................................332, 339
turning performance. ......................... 178
turn rate, turn radius. ........................ 176
unsymmetrical power, see asymmetrical power.
viscosity. ................................... 4
V-n or V-g diagram. ......................... 334
vortex system. ............................... 63
line or bound vortex. ....................... 64
tip or trailing vortex. ...................... 64
w&r injection
reciprocating engine. ....................... 144
turbojetengine ............................. 131
wavedrag................................. , 215
wind, effect on range. ........................ 168
windshear ................................. 367
yaw,adverse ................................ 291
yawangle ................................... 284
YPWm”me”t ................................ 284
t?evised Jcmuarv 1965 | 433 | 433 | 00-80T-80.pdf |
arXiv:1001.0266v2 [cond-mat.str-el] 4 May 2010
Realization of the Exactly Solvable Kitaev Honeycomb Lattice Model in a Spin
Rotation Invariant System
Fa Wang 1
1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
The exactly solvable Kitaev honeycomb lattice model is real ized as the low energy effect Hamil-
tonian of a spin-1/2 model with spin rotation and time-rever sal symmetry. The mapping to low
energy effective Hamiltonian is exact, without truncation e rrors in traditional perturbation series
expansions. This model consists of a honeycomb lattice of cl usters of four spin-1/2 moments, and
contains short-range interactions up to six-spin(or eight -spin) terms. The spin in the Kitaev model
is represented not as these spin-1/2 moments, but as pseudo-spin of the two-dimensional spin singlet
sector of the four antiferromagnetically coupled spin-1/2 moments within each cluster. Spin corre-
lations in the Kitaev model are mapped to dimer correlations or spin-chirality correlations in this
model. This exact construction is quite general and can be us ed to make other interesting spin-1/2
models from spin rotation invariant Hamiltonians. We discu ss two possible routes to generate the
high order spin interactions from more natural couplings, w hich involves perturbative expansions
thus breaks the exact mapping, although in a controlled mann er.
PACS numbers: 75.10.Jm, 75.10.Kt
Contents
I. Introduction. 1
II. Formulation of the Pseudo-spin-1/2 from
Four-spin Cluster. 2
III. Realization of the Kitaev Model. 3
IV. Generate the High Order Physical Spin
Interactions by Perturbative Expansion. 5
A. Generate the High Order Terms by Coupling
to Optical Phonon. 5
B. Generate the High Order Terms by Magnetic
Interactions between Clusters. 7
V. Conclusions. 8
Acknowledgments 8
A. Coupling between Distortions of a
Tetrahedron and the Pseudo-spins 8
B. Derivation of the Terms Generated by
Second Order Perturbation of Inter-cluster
Magnetic Interactions9
References 10
I. INTRODUCTION.
Kitaev’s exactly solvable spin-1/2 honeycomb lattice
model1 (noted as the Kitaev model hereafter) has in-
spired great interest since its debut, due to its exact
solvability, fractionalized excitations, and the potential
to realize non-Abelian anyons. The model simply reads
HKitaev = −
∑
x−links <jk>
Jxτx
j τx
k −
∑
y−links <jk>
Jyτy
j τy
k
−
∑
z−links <jk>
Jzτz
j τz
k
(1)
where τx,y,z are Pauli matrices, and x, y, z -links are de-
fined in FIG. 1. It was shown by Kitaev 1 that this spin-
1/2 model can be mapped to a model with one Majo-
rana fermion per site coupled to Ising gauge fields on the
links. And as the Ising gauge flux has no fluctuation, the
model can be regarded as, under each gauge flux config-
uration, a free Majorana fermion problem. The ground
state is achieved in the sector of zero gauge flux through
each hexagon. The Majorana fermions in this sector have
Dirac-like gapless dispersion resembling that of graphene,
as long as|Jx|, |Jy|, and |Jz| satisfy the triangular rela-
tion, sum of any two of them is greater than the third
one1. It was further proposed by Kitaev 1 that opening of
fermion gap by magnetic field can give the Ising vortices
non-Abelian anyonic statistics, because the Ising vortex
will carry a zero-energy Majorana mode, although mag-
netic field destroys the exact solvability.
Great efforts have been invested to better understand
the properties of the Kitaev model. For example, sev-
eral groups have pointed out that the fractionalized Ma-
jorana fermion excitations may be understood from the
more familiar Jordan-Wigner transformation of 1D spin
systems2,3. The analogy between the non-Abelian Ising
vortices and vortices in p + ip superconductors has been
raised in serveral works 4–7. Exact diagonalization has
been used to study the Kitaev model on small lattices 8.
And perturbative expansion methods have been devel-
oped to study the gapped phases of the Kitaev-type
models9.
Many generalizations of the Kitaev model have been | 0 | 0 | 1001.0266.pdf |
2
y
x
zz z
z zzz
z z
y y
y
x
xx
x x x
y yy
y
x
y
x
z
z
y y
x x x
y
z
x
x
FIG. 1: The honeycomb lattice for the Kitaev model. Filled
and open circles indicate two sublattices.x, y, z label the links
along three different directions used in (1).
derived as well. There have been several proposals to
open the fermion gap for the non-Abelian phase without
spoiling exact solvability4,6. And many generalizations
to other(even 3D) lattices have been developed in the
last few years10–16. All these efforts have significantly
enriched our knowledge of exactly solvable models and
quantum phases of matter.
However, in the original Kitaev model and its later
generalizations in the form of spin models, spin rotation
symmetry is explicitly broken. This makes them harder
to realize in solid state systems. There are many pro-
posals to realized the Kitaev model in more controllable
situations, e.g. in cold atom optical lattices17,18, or in
superconducting circuits 19. But it is still desirable for
theoretical curiosity and practical purposes to realize the
Kitaev-type models in spin rotation invariant systems.
In this paper we realize the Kitaev honeycomb lattice
model as the low energy Hamiltonian for a spin rotation
invariant system. The trick isnot to use the physical spin
as the spin in the Kitaev model, instead the spin-1/2 in
Kitaev model is from some emergent two-fold degener-
ate low energy states in the elementary unit of physical
system. This type of idea has been explored recently by
Jackeli and Khaliullin20, in which the spin-1/2 in the Ki-
taev model is the low energy Kramers doublet created by
strong spin-orbit coupling oft2g orbitals. In the model
presented below, the Hilbert space of spin-1/2 in the Ki-
taev model is actually the two dimensional spin singlet
sector of four antiferromagnetically coupled spin-1/2 mo-
ments, and the role of spin-1/2 operators(Pauli matrices)
in the Kitaev model is replaced by certain combinations
ofSj · Sk [or the spin-chirality Sj · (Sk × Sℓ)] between the
four spins.
One major drawback of the model to be presented is
that it contains high order spin interactions(involves up
to six or eight spins), thus is still unnatural. However it
opens the possibility to realize exotic (exactly solvable)
models from spin-1/2 Hamiltonian with spin rotation in-
variant interactions. We will discuss two possible routes
to reduce this artificialness through controlled perturba-
tive expansions, by coupling to optical phonons or by
magnetic couplings between the elementary units.
The outline of this paper is as follows. In Section II
we will lay out the pseudo-spin-1/2 construction. In Sec-
4
2 3
1
z z
x
x
y
y
23
4
1
FIG. 2: Left: the physical spin lattice for the model (8). The
dash circles are honeycomb lattice sites, each of which is ac-
tually a cluster of four physical spins. The dash straight li nes
are honeycomb lattice bonds, with their type x, y, z labeled.
The interaction between clusters connected by x, y, z bonds
are the Jx,y,z terms in (8) or (9) respectively. Note this is not
the 3-12 lattice used in Ref. 9,10. Right: enlarged picture of
the clusters with the four physical spins labeled as 1 , . . . , 4.
Thick solid bonds within one cluster have large antiferromag-
netic Heisenberg coupling Jcluster.
tion III the Kitaev model will be explicitly constructed
using this formalism, and some properties of this con-
struction will be discussed. In Section IV we will discuss
two possible ways to generate the high order spin in-
teractions involved in the construction of Section III by
perturbative expansions. Conclusions and outlook will
be summarized in Section V.
II. FORMULATION OF THE PSEUDO-SPIN-1/2
FROM FOUR-SPIN CLUSTER.
In this Section we will construct the pseudo-spin-1/2
from a cluster of four physical spins, and map the phys-
ical spin operators to pseudo-spin operators. The map-
ping constructed here will be used in later Sections to
construct the effective Kitaev model. In this Section we
will work entirely within the four-spin cluster, all unspec-
ified physical spin subscripts take values 1, . . . , 4.
Consider a cluster of four spin-1/2 moments(called
physical spins hereafter), labeled by S1,...,4, antiferro-
magnetically coupled to each other (see the right bot-
tom part of FIG. 2). The Hamiltonian within the clus-
ter(up to a constant) is simply the Heisenberg antiferro-
magnetic(AFM) interactions,
Hcluster = ( Jcluster/2) (S1 + S2 + S3 + S4)2 (2)
The energy levels should be apparent from this form:
one group of spin-2 quintets with energy 3Jcluster, three
groups of spin-1 triplets with energy Jcluster, and two spin
singlets with energy zero. We will consider large positive | 1 | 1 | 1001.0266.pdf |
3
Jcluster limit. So only the singlet sector remains in low
energy.
The singlet sector is then treated as a pseudo-spin-1/2
Hilbert space. From now on we denote the pseudo-spin-
1/2 operators asT = (1 /2)⃗ τ, with ⃗ τthe Pauli matri-
ces. It is convenient to choose the following basis of the
pseudo-spin
|τz = ±1⟩ = 1
√
6
(
| ↓↓↑↑⟩ + ω−τ z
| ↓↑↓↑⟩ + ωτ z
| ↓↑↑↓⟩
+ | ↑↑↓↓⟩ + ω−τ z
| ↑↓↑↓⟩ + ωτ z
| ↑↓↓↑⟩
)
(3)
where ω = e2πi/3 is the complex cubic root of unity,
| ↓↓↑↑⟩ and other states on the right-hand-side(RHS) are
basis states of the four-spin system, in terms of Sz quan-
tum numbers of physical spins 1 , . . . , 4 in sequential or-
der. This pseudo-spin representation has been used by
Harriset al. to study magnetic ordering in pyrochlore
antiferromagnets21.
We now consider the effect of Heisenberg-type inter-
actions Sj · Sk inside the physical singlet sector. Note
that since any Sj · Sk within the cluster commutes with
the cluster Hamiltonian Hcluster (2), their action do not
mix physical spin singlet states with states of other total
physical spin. This property is also true for the spin-
chirality operator used later. So the pseudo-spin Hamil-
tonian constructed below will beexact low energy Hamil-
tonian, without truncation errors in typical perturbation
series expansions.
It is simpler to consider the permutation operators
Pjk ≡ 2Sj · Sk + 1 /2, which just exchange the states
of the two physical spin-1/2 moments j and k (j ̸= k).
As an example we consider the action of P34,
P34|τz = −1⟩ = 1
√
6
(
| ↓↓↑↑⟩ + ω| ↓↑↑↓⟩ + ω2| ↓↑↓↑⟩
+ | ↑↑↓↓⟩ + ω| ↑↓↓↑⟩ + ω2| ↑↓↑↓⟩
)
= |τz = +1 ⟩
and similarly P34|τz = −1⟩ = |τz = +1 ⟩. Therefore P34
is just τx in the physical singlet sector. A complete list
of all permutation operators is given in TABLE I. We
can choose the following representation ofτx and τy ,
τx = P12 = 2 S1 · S2 + 1/2
τy = ( P13 − P14)/
√
3 = (2 /
√
3)S1 · (S3 − S4)
(4)
Many other representations are possible as well, because
several physical spin interactions may correspond to the
same pseudo-spin interaction in the physical singlet sec-
tor, and we will take advantage of this later.
For τz we can use τz = −iτxτy , where i is the imagi-
nary unit,
τz = −i(2/
√
3)(2S1 · S2 + 1/2)S1 · (S3 − S4) (5)
physical spin pseudo-spin
P12, and P34 τx
P13, and P24 − (1/ 2)τx + (
√
3/ 2)τy
P14, and P23 − (1/ 2)τx − (
√
3/ 2)τy
− χ 234, χ 341, − χ 412, and χ 123 (
√
3/ 4)τz
TABLE I: Correspondence between physical spin operators
and pseudo-spin operators in the physical spin singlet sector of
the four antiferromagnetically coupled physical spins. Pjk =
2Sj ·Sk +1/ 2 are permutation operators, χ jkℓ = Sj ·(Sk × Sℓ)
are spin-chirality operators. Note that several physical s pin
operators may correspond to the same pseudo-spin operator.
However there is another simpler representation ofτz ,
by the spin-chirality operator χjkℓ = Sj · (Sk × Sℓ). Ex-
plicit calculation shows that the effect of S2 · (S3 × S4) is
−(
√
3/4)τz in the physical singlet sector. This can also
be proved by using the commutation relation [ S2 ·S3, S2 ·
S4] = iS2 · (S3 × S4). A complete list of all chirality
operators is given in TABLE I. Therefore we can choose
another representation ofτz ,
τz = −χ234/(
√
3/4) = −(4/
√
3)S2 · (S3 × S4) (6)
The above representations of τx,y,z are all invariant under
global spin rotation of the physical spins.
With the machinery of equations (4), (5), and (6), it
will be straightforward to construct various pseudo-spin-
1/2 Hamiltonians on various lattices, of the Kitaev vari-
ety and beyond, as the exact low energy effective Hamil-
tonian of certain spin-1/2 models with spin-rotation sym-
metry. In these constructions a pseudo-spin lattice site
actually represents a cluster of four spin-1/2 moments.
III. REALIZATION OF THE KITAEV MODEL.
In this Section we will use directly the results of the
previous Section to write down a Hamiltonian whose low
energy sector is described by the Kitaev model. The
Hamiltonian will be constructed on the physical spin lat-
tice illustrated in FIG. 2. In this Section we will use
j, kto label four-spin clusters (pseudo-spin-1/2 sites), the
physical spins in cluster j are labeled as Sj1, . . . , Sj4.
Apply the mappings developed in Section II, we have
the desired Hamiltonian in short notation,
H =
∑
cluster
Hcluster −
∑
x−links <jk>
Jxτx
j τx
k
−
∑
y−links <jk>
Jyτy
j τy
k −
∑
z−links <jk>
Jzτz
j τz
k
(7)
where j, k label the honeycomb lattice sites thus the four-
spin clusters, Hcluster is given by (2), τx,y,z should be
replaced by the corresponding physical spin operators in
(4) and (5) or (6), or some other equivalent representa-
tions of personal preference. | 2 | 2 | 1001.0266.pdf |
4
Plug in the expressions (4) and (6) into (7), the Hamil- tonian reads e xplicitly as
H =
∑
j
(Jcluster/2)(Sj1 + Sj2 + Sj3 + Sj4)2 −
∑
z−links <jk>
Jz (16/9)[Sj2 · (Sj3 × Sj4)][Sk2 · (Sk3 × Sk4)]
−
∑
x−links <jk>
Jx (2Sj1 · Sj2 + 1/2)(2Sk1 · Sk2 + 1/2) −
∑
y−links <jk>
Jy (4/3)[Sj1 · (Sj3 − Sj4)][Sk1 · (Sk3 − Sk4)]
(8)
While by the represenation (4) and (5), the Hamilto- nian becomes
H =
∑
j
(Jcluster/2)(Sj1 + Sj2 + Sj3 + Sj4)2
−
∑
x−links <jk>
Jx (2Sj1 · Sj2 + 1/2)(2Sk1 · Sk2 + 1/2) −
∑
y−links <jk>
Jy (4/3)[Sj1 · (Sj3 − Sj4)][Sk1 · (Sk3 − Sk4)]
−
∑
z−links <jk>
Jz (−4/3)(2Sj3 · Sj4 + 1/2)[Sj1 · (Sj3 − Sj4)](2Sk3 · Sk4 + 1/2)[Sk1 · (Sk3 − Sk4)]
(9)
This model, in terms of physical spins S, has full
spin rotation symmetry and time-reversal symmetry. A
pseudo-magnetic field term∑
j
⃗h · ⃗ τj term can also be
included under this mapping, however the resulting Ki-
taev model with magnetic field is not exactly solvable.
It is quite curious that such a formidably looking Hamil-
tonian (8), with biquadratic and six-spin(or eight-spin)
terms, has an exactly solvable low energy sector.
We emphasize that because the first intra-cluster term∑
cluster Hcluster commutes with the latter Kitaev terms
independent of the representation used, the Kitaev model
is realized as theexact low energy Hamiltonian of this
model without truncation errors of perturbation theories,
namely no (|Jx,y,z |/Jcluster)2 or higher order terms will
be generated under the projection to low energy clus-
ter singlet space. This is unlike, for example, thet/U
expansion of the half-filled Hubbard model 22,23, where
at lowest t2/U order the effective Hamiltonian is the
Heisenberg model, but higher order terms ( t4/U3 etc.)
should in principle still be included in the low energy ef-
fective Hamiltonian for any finitet/U. Similar compari-
son can be made to the perturbative expansion studies of
the Kitaev-type models by Vidalet al.9, where the low
energy effective Hamiltonians were obtained in certian
anisotropic (strong bond/triangle) limits. Although the
spirit of this work, namely projection to low energy sec-
tor, is the same as all previous perturbative approaches
to effective Hamiltonians.
Note that the original Kitaev model (1) has three-
fold rotation symmetry around a honeycomb lattice site,
combined with a three-fold rotation in pseudo-spin space
(cyclic permutation ofτx, τy, τz ). This is not apparent
in our model (8) in terms of physical spins, under the
current representation ofτx,y,z . We can remedy this by
using a different set of pseudo-spin Pauli matrices τ′x,y,z
in (7),
τ′x =
√
1/3τz +
√
2/3τx,
τ′y =
√
1/3τz −
√
1/6τx +
√
1/2τy,
τ′z =
√
1/3τz −
√
1/6τx −
√
1/2τy
With proper representation choice, they have a symmet-
ric form in terms of physical spins,
τ′x = −(4/3)S2 · (S3 × S4) +
√
2/3(2S1 · S2 + 1/2)
τ′y = −(4/3)S3 · (S4 × S2) +
√
2/3(2S1 · S3 + 1/2)
τ′z = −(4/3)S4 · (S2 × S3) +
√
2/3(2S1 · S4 + 1/2)
(10)
So the symmetry mentioned above can be realized by a
three-fold rotation of the honeycomb lattice, with a cyclic
permutation ofS2, S3 and S4 in each cluster. This is in
fact the three-fold rotation symmetry of the physical spin
lattice illustrated in FIG. 2. However this more symmet-
ric representation will not be used in later part of this
paper. | 3 | 3 | 1001.0266.pdf |
5
Another note to take is that it is not necessary to have
such a highly symmetric cluster Hamiltonian (2). The
mappings to pseudo-spin-1/2 should work as long as the
ground states of the cluster Hamiltonian are the two-fold
degenerate singlets. One generalization, which conforms
the symmetry of the lattice in FIG. 2, is to have
Hcluster = ( Jcluster/2)(r · S1 + S2 + S3 + S4)2 (11)
with Jcluster > 0 and 0 < r < 3. However this is not
convenient for later discussions and will not be used.
We briefly describe some of the properties of (8). Its
low energy states are entirely in the space that each of the
clusters is a physical spin singlet (called cluster singlet
subspace hereafter). Therefore physical spin correlations
are strictly confined within each cluster. The excitations
carrying physical spin are gapped, and their dynamics
are ‘trivial’ in the sense that they do not move from one
cluster to another. But there are non-trivial low energy
physical spin singlet excitations, described by the pseudo-
spins defined above. The correlations of the pseudo-spins
can be mapped to correlations of their corresponding
physical spin observables (the inverse mappings are not
unique, c.f. TABLE I). For exampleτx,y correlations
become certain dimer-dimer correlations, τz correlation
becomes chirality-chirality correlation, or four-dimer cor-
relation. It will be interesting to see the corresponding
picture of the exotic excitations in the Kitaev model, e.g.
the Majorana fermion and the Ising vortex. However this
will be deferred to future studies.
It is tempting to call this as an exactly solved spin liq-
uid with spin gap ( ∼ Jcluster), an extremely short-range
resonating valence bond(RVB) state, from a model with
spin rotation and time reversal symmetry. However it
should be noted that the unit cell of this model contains
an even number of spin-1/2 moments (so does the orig-
inal Kitaev model) which does not satisfy the stringent
definition of spin liquid requiring odd number of elec-
trons per unit cell. Several parent Hamiltonians of spin
liquids have already been constructed. See for example,
Ref.24–27 .
IV. GENERATE THE HIGH ORDER PHYSICAL
SPIN INTERACTIONS BY PERTURBATIVE
EXPANSION.
One major drawback of the present construction is that
it involves high order interactions of physical spins[see
(8) and (9)], thus is ‘unnatural’. In this Section we will
make compromises between exact solvability and natu-
ralness. We consider two clustersj and k and try to
generate the Jx,y,z interactions in (7) from perturbation
series expansion of more natural(lower order) physical
spin interactions. Two different approaches for this pur-
pose will be laid out in the following two Subsections. In
Subsection IV A we will consider the two clusters as two
tetrahedra, and couple the spin system to certain opti-
cal phonons, further coupling between the phonon modes
(a) (b) (c) (d)
(b) (c) (d)
Q E
2
Q E
1
(a)
1
1 1 1
1 1 1
2
2
2
2
2
2 2
2
3
3
3 3 3
3 3 4 4
4 4
3
4
4
4
4
1
FIG. 3: Illustration of the tetragonal to orthorhombic
QE
1(top) and QE
2(bottom) distortion modes. (a) Perspective
view of the tetrahedron. 1 , . . . , 4 label the spins. Arrows in-
dicate the motion of each spin under the distortion mode. (b)
Top view of (a). (c)(d) Side view of (a).
of the two clusters can generate at lowest order the de-
sired high order spin interactions. In Subsection IV B we
will introduce certain magnetic, e.g. Heisenberg-type, in-
teractions between physical spins of different clusters, at
lowest order(second order) of perturbation theory the de-
sired high order spin interactions can be achieved. These
approaches involve truncation errors in the perturbation
series, thus the mapping to low energy effect Hamilto-
nian will no longer be exact. However the error intro-
duced may be controlled by small expansion parameters.
In this Section we denote the physical spins on cluster
j(k) as j1, . . . , j 4 ( k1, . . . , k 4), and denote pseudo-spins
on cluster j(k) as ⃗ τj (⃗ τk).
A. Generate the High Order Terms by Coupling to
Optical Phonon.
In this Subsection we regard each four-spin cluster
as a tetrahedron, and consider possible optical phonon
modes(distortions) and their couplings to the spin sys-
tem. The basic idea is that the intra-cluster Heisen-
berg couplingJcluster can linearly depend on the dis-
tance between physical spins. Therefore certain distor-
tions of the tetrahedron couple to certain linear combi-
nations ofSℓ · Sm. Integrating out phonon modes will
then generate high order spin interactions. This idea has
been extensively studied and applied to several magnetic
materials28–34. More details can be found in a recent
review by Tchernyshyov and Chern 35. And we will fre-
quently use their notations. In this Subsection we will
use the representation (5) forτz .
Consider first a single tetrahedron with four spins
1, . . . , 4. The general distortions of this tetrahedron can
be classified by their symmetry (see for example Ref. 35).
Only two tetragonal to orthorhombic distortion modes,
QE
1and QE
2(illustrated in FIG. 3), couple to the pseudo-
spins defined in Section II. A complete analysis of all
modes is given in Appendix A. The coupling is of the | 4 | 4 | 1001.0266.pdf |
6
form
J′(QE
1fE
1 + QE
2fE
2 )
where J′ is the derivative of Heisenberg coupling Jcluster
between two spins ℓ and m with respect to their distance
rℓm, J′ = d Jcluster/drℓm; QE
1,2 are the generalized coor-
dinates of these two modes; and the functions fE
1,2 are
fE
2 = (1 /2)(S2 · S4 + S1 · S3 − S1 · S4 − S2 · S3),
fE
1 =
√
1/12(S1 · S4 + S2 · S3 + S2 · S4 + S1 · S3
− 2S1 · S2 − 2S3 · S4).
According to TABLE I we have fE
1 = −(
√
3/2)τx and
fE
2 = (
√
3/2)τy. Then the coupling becomes
(
√
3/2)J′(−QE
1τx + QE
2τy ) (12)
The spin-lattice(SL) Hamiltonian on a single cluster j
is [equation (1.8) in Ref. 35],
Hcluster j, SL =Hcluster j + k
2 (QE
1j )2 + k
2 (QE
2j )2
−
√
3
2 J′(QE
1j τx
j − QE
2jτy
j ),
(13)
where k > 0 is the elastic constant for these phonon
modes, J′ is the spin-lattice coupling constant, QE
1j and
QE
2j are the generalized coordinates of the QE
1 and QE
2
distortion modes of cluster j, Hcluster j is (2). As al-
ready noted in Ref. 35, this model does not really break
the pseudo-spin rotation symmetry of a single cluster.
Now we put two clusters j and k together, and in-
clude a perturbation λ Hperturbation to the optical phonon
Hamiltonian,
Hjk,SL =Hcluster j, SL + Hcluster k, SL
+ λ Hperturbation[QE
1j , QE
2j , QE
1k, QE
2k]
where λ (in fact λ/k) is the expansion parameter.
Consider the perturbation Hperturbation = QE
1j · QE
1k,
which means a coupling between the QE
1 distortion
modes of the two tetrahedra. Integrate out the optical
phonons, at lowest non-trivial order, it produces a term
(3J′2 λ)/(4 k2) τx
j · τx
k . This can be seen by minimizing
separately the two cluster Hamiltonians with respect to
QE
1, which gives QE
1 = (
√
3 J′)/(2 k)τx, then plug this
into the perturbation term. Thus we have produced the
Jx term in the Kitaev model with Jx = −(3 J′2 λ)/(4 k2).
Similarly the perturbation Hperturbation = QE
2j · QE
2k
will generate (3 J′2 λ)/(4 k2) τy
j · τy
k at lowest non-trivial
order. So we can make Jy = −(3 J′2 λ)/(4 k2).
The τz
j · τz
k coupling is more difficult to get. We
treat it as −τx
j τy
j · τx
k τy
k . By the above reasoning, we
need an anharmonic coupling Hperturbation = QE
1jQE
2j ·
QE
1kQE
2k. It will produce at lowest non-trivial or-
der (9 J′4 λ)/(16 k4) τx
j τy
j · τx
k τy
k . Thus we have Jz =
(9 J′4 λ)/(16 k4).
Finally we have made up a spin-lattice model HSL,
which involves only Sℓ ·Sm interaction for physical spins,
HSL =
∑
cluster
Hcluster, SL +
∑
x−links <jk>
λx QE
1j · QE
1k
+
∑
y−links <jk>
λy QE
2j · QE
2k
+
∑
z−links <jk>
λz QE
1j QE
2j · QE
1kQE
2k
where QE
1j is the generalized coordinate for the QE
1mode
on cluster j, and QE
1k, QE
2j, QE
2k are similarly defined;
λx,y = −(4Jx,yk2)/(3J′2) and λz = (16 Jzk4)/(9J′4); the
single cluster spin-lattice Hamiltonian Hcluster, SL is (13).
Collect the results above we have the spin-lattice
Hamiltonian HSL explicitly written as,
HSL =
∑
cluster j
[
(Jcluster/2)(Sj1 + Sj2 + Sj3 + Sj4)2 + k
2 (QE
1j )2 + k
2 (QE
2j )2
+ J′
(
QE
1j
Sj1 · Sj4 + Sj2 · Sj3 + Sj2 · Sj4 + Sj1 · Sj3 − 2Sj1 · Sj2 − 2Sj3 · Sj4
√
12
+ QE
2j
Sj2 · Sj4 + Sj1 · Sj3 − Sj1 · Sj4 − Sj2 · Sj3
2
)]
−
∑
x−links <jk>
4Jxk2
3J′2 QE
1j · QE
1k −
∑
y−links <jk>
4Jyk2
3J′2 QE
2j · QE
2k +
∑
z−links <jk>
16Jzk4
9J′4 QE
1j QE
2j · QE
1kQE
2k
(14)
The single cluster spin-lattice Hamiltonian [first three
lines in (14)] is quite natural. However we need some
harmonic(on x- and y-links of honeycomb lattice) and an-
harmonic coupling (on z-links) between optical phonon | 5 | 5 | 1001.0266.pdf |
7
modes of neighboring tetrahedra. And these coupling
constantsλx,y,z need to be tuned to produce Jx,y,z of
the Kitaev model. This is still not easy to implement in
solid state systems. At lowest non-trivial order of pertur-
bative expansion, we do get our model (9). Higher order
terms in expansion destroy the exact solvability, but may
be controlled by the small parametersλx,y,z /k.
B. Generate the High Order Terms by Magnetic
Interactions between Clusters.
In this Subsection we consider more conventional per-
turbations, magnetic interactions between the clusters,
e.g. the Heisenberg couplingSj · Sk with j and k belong
to different tetrahedra. This has the advantage over the
previous phonon approach for not introducing additional
degrees of freedom. But it also has a significant disad-
vantage: the perturbation does not commute with the
cluster Heisenberg Hamiltonian (2), so the cluster sin-
glet subspace will be mixed with other total spin states.
In this Subsection we will use the spin-chirality represen-
tation (6) forτz .
Again consider two clusters j and k. For simplicity
of notations define a projection operator Pjk = Pj Pk,
where Pj,k is projection into the singlet subspace of clus-
ter j and k, respectively, Pj,k = ∑
s=±1 |τz
j,k = s⟩⟨τz
j,k =
s|. For a given perturbation λ Hperturbation with small
parameter λ (in factor λ/Jcluster is the expansion param-
eter), lowest two orders of the perturbation series are
λPjk HperturbationPjk + λ2 Pjk Hperturbation(1 − Pjk )
× [0 − Hcluster j − Hcluster k]−1(1 − Pjk )HperturbationPjk
(15)
With proper choice of λ and Hperturbation we can generate
the desired Jx,y,z terms in (8) from the first and second
order of perturbations.
The calculation can be dramatically simplified by the
following fact that any physical spin-1/2 operator Sx,y,z
ℓ
converts the cluster spin singlet states |τz = ±1⟩ into
spin-1 states of the cluster. This can be checked by
explicit calculations and will not be proved here. For
all the perturbations to be considered later, the above
mentioned fact can be exploited to replace the factor
[0− Hcluster j − Hcluster k]−1 in the second order pertur-
bation to a c-number ( −2Jcluster)−1.
The detailed calculations are given in Appendix B. We
will only list the results here.
The perturbation on x-links is given by
λx Hperturbation, x = λx[Sj1 · Sk1 + sgn(Jx) · (Sj2 · Sk2)]
− Jx(Sj1 · Sj2 + Sk1 · Sk2).
where λx =
√
12|Jx| · Jcluster, sgn( Jx) = ±1 is the sign
of Jx.
The perturbation on y-links is
λy Hperturbation, y
=λy[Sj1 · Sk1 + sgn(Jy) · (Sj3 − Sj4) · (Sk3 − Sk4)]
− |Jy|(Sj3 · Sj4 + Sk3 · Sk4)
with λy =
√
4|Jy| · Jcluster.
The perturbation on z-links is
λz Hperturbation, z
= λz [Sj2 · (Sk3 × Sk4) + sgn( Jz) · Sk2 · (Sj3 × Sj4)]
− |Jz|(Sj3 · Sj4 + Sk3 · Sk4).
with λz = 4
√
|Jz| · Jcluster.
The entire Hamiltonian Hmagnetic reads explicitly as,
Hmagnetic =
∑
cluster j
(Jcluster/2)(Sj1 + Sj2 + Sj3 + Sj4)2
+
∑
x−links <jk>
{√
12|Jx| · Jcluster
[
Sj1 · Sk1 + sgn(Jx) · (Sj2 · Sk2)
]
− Jx(Sj1 · Sj2 + Sk1 · Sk2)
}
+
∑
y−links <jk>
{√
4|Jy| · Jcluster
[
Sj1 · (Sk3 − Sk4) + sgn( Jy)Sk1 · (Sj3 − Sj4)
]
− |Jy|(Sj3 · Sj4 + Sk3 · Sk4)
}
+
∑
z−links <jk>
{
4
√
|Jz| · Jcluster
[
Sj2 · (Sk3 × Sk4) + sgn( Jz )Sk2 · (Sj3 × Sj4)
]
− |Jz|(Sj3 · Sj4 + Sk3 · Sk4)
}
.
(16)
In (16), we have been able to reduce the four spin in-
teractions in (8) to inter-cluster Heisenberg interactions,
and the six-spin interactions in (8) to inter-cluster spin-
chirality interactions. The inter-cluster Heisenberg cou-
plings inHperturbation x,y may be easier to arrange. The
inter-cluster spin-chirality coupling in Hperturbation z ex-
plicitly breaks time reversal symmetry and is probably
harder to implement in solid state systems. However
spin-chirality order may have important consequences
in frustrated magnets36,37, and a realization of spin- | 6 | 6 | 1001.0266.pdf |
8
chirality interactions in cold atom optical lattices has
been proposed38.
Our model (8) is achieved at second order of the per-
turbation series. Higher order terms become trunca-
tion errors but may be controlled by small parameters
λx,y,z /Jcluster ∼
√
|Jx,y,z |/Jcluster.
V. CONCLUSIONS.
We constructed the exactly solvable Kitaev honeycomb
model1 as the exact low energy effective Hamiltonian of
a spin-1/2 model [equations (8) or (9)] with spin-rotation
and time reversal symmetry. The spin in Kitaev model is
represented as the pseudo-spin in the two-fold degenerate
spin singlet subspace of a cluster of four antiferromag-
netically coupled spin-1/2 moments. The physical spin
model is a honeycomb lattice of such four-spin clusters,
with certain inter-cluster interactions. The machinery
for the exact mapping to pseudo-spin Hamiltonian was
developed (see e.g. TABLE I), which is quite general
and can be used to construct other interesting (exactly
solvable) spin-1/2 models from spin rotation invariant
systems.
In this construction the pseudo-spin correlations in the
Kitaev model will be mapped to dimer or spin-chirality
correlations in the physical spin system. The correspond-
ing picture of the fractionalized Majorana fermion exci-
tations and Ising vortices still remain to be clarified.
This exact construction contains high order physical
spin interactions, which is undesirable for practical im-
plementation. We described two possible approaches to
reduce this problem: generating the high order spin in-
teractions by perturbative expansion of the coupling to
optical phonon, or the magnetic coupling between clus-
ters. This perturbative construction will introduce trun-
cation error of perturbation series, which may be con-
trolled by small expansion parameters. Whether these
constructions can be experimentally engineered is how-
ever beyond the scope of this study. It is conceivable that
other perturbative expansion can also generate these high
order spin interactions, but this possibility will be left for
future works.
Acknowledgments
The author thanks Ashvin Vishwanath, Yong-Baek
Kim and Arun Paramekanti for inspiring discussions, and
Todadri Senthil for critical comments. The author is sup-
ported by the MIT Pappalardo Fellowship in Physics.
Appendix A: Coupling between Distortions of a
Tetrahedron and the Pseudo-spins
In this Appendix we reproduce from Ref. 35 the cou-
plings of all tetrahedron distortion modes to the spin
system. And convert them to pseudo-spin notation in
the physical spin singlet sector.
Consider a general small distortion of the tetrahedron,
the spin Hamiltonian becomes
Hcluster, SL = ( Jcluster/2)(
∑
ℓ
Sℓ)2 + J′ ∑
ℓ<m
δrℓm(Sℓ · Sm)
(A1)
where δrℓm is the change of bond length between spins
ℓ and m, J′ is the derivative of Jcluster with respect to
bond length.
There are six orthogonal distortion modes of the tetra-
hedron [TABLE 1.1 in Ref. 35]. One of the modes A is the
trivial representation of the tetrahedral group Td; two E
modes form the two dimensional irreducible representa-
tion ofTd; and three T2 modes form the three dimen-
sional irreducible representation. The E modes are also
illustrated in FIG. 3.
The generic couplings in (A1) [second term] can be
converted to couplings to these orthogonal modes,
J′(QAfA + QE
1fE
1 + QE
2fE
2 + QT2
1 fT2
1 + QT2
2 fT2
2 + QT2
3 fT2
3 )
where Q are generalized coordinates of the corresponding
modes, functions f can be read off from TABLE 1.2 of
Ref.35. For the A mode, δrℓm =
√
2/3QA, so fA is
fA =
√
2/3 (S1 · S2 + S3 · S4 + S1 · S3
+ S2 · S4 + S1 · S4 + S2 · S3).
The functions fE
1,2 for the E modes have been given before
but are reproduced here,
fE
2 = (1 /2)(S2 · S4 + S1 · S3 − S1 · S4 − S2 · S3),
fE
1 =
√
1/12(S1 · S4 + S2 · S3 + S2 · S4 + S1 · S3
− 2S1 · S2 − 2S3 · S4).
The functions fT2
1,2,3 for the T2 modes are
fT2
1 = ( S2 · S3 − S1 · S4),
fT2
2 = ( S1 · S3 − S2 · S4),
fT2
3 = ( S1 · S2 − S3 · S4)
Now we can use TABLE I to convert the above cou-
plings into pseudo-spin. It is easy to see that fA and
fT2
1,2,3 are all zero when converted to pseudo-spins, namely
projected to the physical spin singlet sector. But fE
1 =
(P14 +P23 +P24 +P13 −2P12 −2P34)/(4
√
3) = −(
√
3/2)τx
and fE
2 = ( P24 + P13 − P14 − P23)/4 = (
√
3/2)τy. This
has already been noted by Tchernyshyov et al.28, only
the E modes can lift the degeneracy of the physical spin
singlet ground states of the tetrahedron. Therefore the
general spin lattice coupling is the form of (12) given in
the main text. | 7 | 7 | 1001.0266.pdf |
9
Appendix B: Derivation of the Terms Generated by
Second Order Perturbation of Inter-cluster
Magnetic Interactions
In this Appendix we derive the second order pertur-
bations of inter-cluster Heisenberg and spin-chirality in-
teractions. The results can then be used to construct
(16).
First consider the perturbation λ Hperturbation = λ[Sj1 ·
Sk1 + r(Sj2 · Sk2)], where r is a real number to be tuned
later. Due to the fact mentioned in Subsection IV B,
the action ofHperturbation on any cluster singlet state
will produce a state with total spin-1 for both cluster j
and k. Thus the first order perturbation in (15) van-
ishes. And the second order perturbation term can be
greatly simplified: operator (1− P jk )[0 − Hcluster j −
Hcluster k]−1(1 − P jk ) can be replaced by a c-number
(−2Jcluster)−1. Therefore the perturbation up to second
order is
− λ2
2Jcluster
Pjk (Hperturbation)2Pjk
This is true for other perturbations considered later in
this Appendix. The clusterj and cluster k parts can be
separated, this term then becomes ( a, b = x, y, z ),
− λ2
2Jcluster
∑
a,b
[
Pj Sa
j1Sb
j1Pj · PkSa
k1Sb
k1Pk
+ 2r PjSa
j1Sb
j2Pj · PkSa
k1Sb
k2Pk
+ r2 Pj Sa
j2Sb
j2Pj · PkSa
k2Sb
k2Pk
]
Then use the fact that Pj Sa
jℓSb
jm Pj = δab(1/3)Pj(Sjℓ ·
Sjm )Pj by spin rotation symmetry, the perturbation be-
comes
− λ2
6Jcluster
[ 9 + 9 r2
16 + 2r Pjk (Sj1 · Sj2)(Sk1 · Sk2)Pjk
]
= − λ2
6Jcluster
[ 9 + 9 r2
16 + (r/2)τx
j τx
k − r/2
− r Pjk (Sj1 · Sj2 + Sk1 · Sk2)Pjk
]
.
So we can choose −(r λ2)/(12Jcluster) = −Jx, and include
the last intra-cluster Sj1 · Sj2 + Sk1 · Sk2 term in the first
order perturbation.
The perturbation on x-links is then (not unique),
λx Hperturbation, x =λx[Sj1 · Sk1 + sgn(Jx) · (Sj2 · Sk2)]
− Jx(Sj1 · Sj2 + Sk1 · Sk2)
with λx =
√
12|Jx| · Jcluster, and r = sgn( Jx) is the sign
of Jx. The non-trivial terms produced by up to second
order perturbation will be the τx
j τx
k term. Note that the
last term in the above equation commutes with cluster
Hamiltonians so it does not produce second or higher
order perturbations.
Similarly considering the following perturbation on y-
links, λ Hperturbation = λ[Sj1 · (Sk3 − Sk4) + r Sk1 · (Sj3 −
Sj4)]. Following similar procedures we get the second
order perturbation from this term
− λ2
6Jcluster
[ 9 + 9 r2
8
+ 2r Pjk [Sj1 · (Sj3 − Sj4)][Sk1 · (Sk3 − Sk4)]Pjk
− (3/2) Pjk (Sk3 · Sk4 + r2 Sj3 · Sj4)Pjk
]
= − λ2
6Jcluster
[ 9 + 9 r2
8 + 2r (3/4)τy
j τy
k
− (3/2) Pjk (Sk3 · Sk4 + r2 Sj3 · Sj4)Pjk
]
So we can choose −(r λ2)/(4Jcluster) = −Jy, and include
the last intra-cluster Sk3 · Sk4 + r2 Sj3 · Sj4 term in the
first order perturbation.
Therefore we can choose the following perturbation on
y-links (not unique),
λy Hperturbation, y
=λy[Sj1 · Sk1 + sgn(Jy) · (Sj3 − Sj4) · (Sk3 − Sk4)]
− |Jy|(Sj3 · Sj4 + Sk3 · Sk4)
with λy =
√
4|Jy| · Jcluster, r = sgn( Jy ) is the sign of Jy.
The τz
j τz
k term is again more difficult to get. We use
the representation of τz by spin-chirality (6). And con-
sider the following perturbation
Hperturbation = Sj2 · (Sj3 × Sj4) + r Sk2 · (Sj3 × Sj4)
The first order term in (15) vanishes due to the same
reason as before. There are four terms in the second
order perturbation. The first one is
λ2 Pjk Sj2 · (Sk3 × Sk4)(1 − Pjk )
× [0 − Hcluster j − Hcluster k]−1
× (1 − Pjk )Sj2 · (Sk3 × Sk4)Pjk
For the cluster j part we can use the same arguments
as before, the Hcluster j can be replaced by a c-number
Jcluster. For the cluster k part, consider the fact that
Sk3 × Sk4 equals to the commutator −i[Sk4, Sk3 · Sk4],
the action of Sk3 × Sk4 on physical singlet states of k will
also only produce spin-1 state. So we can replace the
Hcluster k in the denominator by a c-number Jcluster as
well. Use spin rotation symmetry to separate the j and
k parts, this term simplifies to
− λ2
6Jcluster
Pj Sj2 · Sj2Pj · Pk(Sk3 × Sk4) · (Sk3 × Sk4)Pk.
Use ( S)2 = 3 /4 and
(Sk3 × Sk4) · (Sk3 × Sk4)
=
∑
a,b
(Sa
k3Sb
k4Sa
k3Sb
k4 − Sa
k3Sb
k4Sb
k3Sa
k4)
= ( Sk3 · Sk3)(Sk4 · Sk4) −
∑
a,b
Sa
k3Sb
k3[δab/2 − Sa
k4Sb
k4]
= 9 /16 + ( Sk3 · Sk4)(Sk3 · Sk4) − (3/8) | 8 | 8 | 1001.0266.pdf |
10
this term becomes
− λ2
6Jcluster
· (3/4)[3/16 + ( τx/2 − 1/4)2]
= − (λ2)/(32Jcluster) · (2 − τx
k ).
Another second order perturbation term r2λ2 Pjk Sk2 ·
(Sj3 × Sj4)(1 − P jk )[0 − Hcluster j − Hcluster k]−1(1 −
Pjk )Sk2 · (Sj3 × Sj4)Pjk can be computed in the similar
way and gives the result −(r2 λ2)/(32Jcluster) · (2 − τx
j ).
For one of the cross term
r λ2 Pjk Sj2 · (Sk3 × Sk4)(1 − Pjk )
× [0 − Hcluster j − Hcluster k]−1
× (1 − Pjk )Sk2 · (Sj3 × Sj4)Pjk
We can use the previous argument for both cluster j and
k, so (1 −PAB)[0−Hcluster j −Hcluster k]−1(1−Pjk ) can be
replace by c-number ( −2Jcluster)−1. This term becomes
− r λ2
2Jcluster
Pjk [Sj2 · (Sk3 × Sk4)][Sk2 · (Sj3 × Sj3)]Pjk .
Spin rotation symmetry again helps to separate the terms
for clusterj and k, and we get −(r λ2)/(32Jcluster)·τz
j τz
k .
The other cross term r λ2 Pjk Sk2 · (Sj3 × Sj4)(1 −
Pjk )[0 − Hcluster j − Hcluster k]−1(1 − P jk )Sj2 · (Sk3 ×
Sk4)Pjk gives the same result.
In summary the second order perturbation from λ[Sj2 ·
(Sj3 × Sj4) + r Sk2 · (Sj3 × Sj4)] is
− r λ2
16Jcluster
· τz
j τz
k + λ2
32Jcluster
(τx
k + r2 τx
j − 2r2 − 2).
Using this result we can choose the following pertur-
bation on z-links,
λz Hperturbation, z
=λz [Sj2 · (Sk3 × Sk4) + sgn( Jz ) · Sk2 · (Sj3 × Sj4)]
− |Jz|(Sj3 · Sj4 + Sk3 · Sk4)
with λz = 4
√
|Jz|Jcluster, r = sgn( Jz ) is the sign of Jz.
The last term on the right-hand-side is to cancel the non-
trivial terms (r2 τx
j + τx
k )λ2
z/(32Jcluster) from the second
order perturbation of the first term. Up to second order
perturbation this will produce−Jzτz
j τz
k interactions.
Finally we have been able to reduce the high order
interactions to at most three spin terms, the Hamiltonian
Hmagnetic is
Hmagnetic =
∑
j
Hcluster j +
∑
x−links <jk>
λxHperturbation x
+
∑
y−links <jk>
λyHperturbation y
+
∑
z−links <jk>
λz Hperturbation z
where Hcluster j are given by (2), λx,y,z Hperturbation x,y,z
are given above. Plug in relevant equations we get (16)
in Subsection IV B.
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arXiv:1001.0510v1 [cond-mat.stat-mech] 4 Jan 2010
Interplay among helical order, surface effects and range of i nteracting layers in
ultrathin films.
F. Cinti (1, 2, 3), A. Rettori (2, 3), and A. Cuccoli (2)
(1) Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1
(2)CNISM and Department of Physics, University of Florence, 50019 Sesto Fiorentino (FI), Italy. and
(3)CNR-INFM S3 National Research Center, I-41100 Modena, Italy
(Dated: June 8, 2022)
The properties of helical thin films have been thoroughly inv estigated by classical Monte Carlo
simulations. The employed model assumes classical planar s pins in a body-centered tetragonal
lattice, where the helical arrangement along the film growth direction has been modeled by nearest
neighbor and next-nearest neighbor competing interaction s, the minimal requirement to get helical
order. We obtain that, while the in-plane transition temper atures remain essentially unchanged with
respect to the bulk ones, the helical/fan arrangement is sta bilized at more and more low temperature
when the film thickness, n, decreases; in the ordered phase, increasing the temperatu re, a softening
of the helix pitch wave-vector is also observed. Moreover, w e show also that the simulation data
around both transition temperatures lead us to exclude the p resence of a first order transition for all
analyzed sizes. Finally, by comparing the results of the pre sent work with those obtained for other
models previously adopted in literature, we can get a deeper insight about the entwined role played
by the number (range) of interlayer interactions and surfac e effects in non-collinear thin films.
PACS numbers: 64.60.an,64.60.De,75.10.Hk,75.40.Cx,75.70.Ak.
I. INTRODUCTION
The study of low dimensional frustrated magnetic
systems1 still raises great interest, both in consequence
of theoretical aspects, related to their peculiar criti-
cal properties2, and in view of possible technological
applications3. Indeed, beside conventional ferromagnetic
or antiferromagnetic phase transitions, in many new ma-
terials other nontrivial and unconventional forms of or-
dering have been observed4,5. A quantity of particular
interest in this context is the spin chirality, an order pa-
rameter which turned out to be extremely relevant in,
e.g., magnetoelectric materials6, itinerant MnSi 7, binary
compounds as FeGe 8, glass transition of spins 9, and XY
helimagnets, as Holmium, Terbium or Dysprosium 10. In
the latter case, a new universality class was predicted be-
cause aZ2 × SO(2) symmetry is spontaneously broken
in the ordered phase 2: In fact, when dealing with such
systems, in addition to the SO(2) symmetry of the spin
degrees of freedom ⃗Si, one has to consider also the Z2
symmetry of the spin chirality κij ∝
[
⃗Si × ⃗Sj
] z
.
For these rare-earth elements, the development of new
and sophisticated experimental methods 11 has allowed to
obtain ultra-thin films where the non-collinear modula-
tion is comparable with the film thickness. Under such
conditions the lack of translational invariance due to the
presence of surfaces results decisive in order to observe
a drastic change of the magnetic structures12. Recent
experimental data on ultra-thin Holmium films 13 have
been lately interpreted and discussed 14,15 on the basis
of detailed classical Monte Carlo (MC) simulations of a
spin Hamiltonian, which is believed to give a realistic
modeling of bulk Holmium. Such Hamiltonian, proposed
by Bohr et al.16, allows for competitive middle-range in-
teractions by including six different exchange constants
along thec crystallographic axis, and gives a helix pitch
wave-vector Qz such that Qzc′ ≃ 30◦, where c′ = c/2 is
the distance between nearest neighboring spin layers par-
allel to theab crystallographic planes, henceforth denoted
also as x − y planes, while z will be taken parallel to c.
For n > 16, n being the number of spin layers in the film,
a correct bulk limit is reached, while for lower n the film
properties are clearly affected by the strong competition
among the helical pitch and the surface effects, which in-
volve the majority of the spin layers. In the thickness
rangen = 9 − 16, i.e. right for thickness values com-
parable with the helical pitch, three different magnetic
phases emerged, with the high-temperature, disordered,
paramagnetic phase and the low-temperature, long-range
ordered one separated by an intriguing, intermediate-
temperature block phase, where outer ordered layers co-
exist with some inner disordered ones, the phase tran-
sition of the latter eventually displaying the signatures
of a Kosterlitz-Thouless one. Finally, forn ≤ 7 the film
collapses once and for all to a quasi-collinear order.
The complex phase diagram unveiled by such MC sim-
ulations awaken however a further intriguing question:
to what extent the observed behavior may be considered
a simple consequence of the competition between helical
order and surface effects? I.e., is it just a matter of hav-
ing such a competition or does the range of interactions
also play a relevant role? Indeed, when the range of the
interactions is large enough we have a greater number of
planes which can be thought of as ”surface planes”, i.e.
for which the number of interacting neighbors are sig-
nificantly reduced with respect to the bulk layers; there-
fore, we expect that the larger the interaction range, the
stronger should be the surface effects. But, at the same
time, the same modulation of the magnetic order can | 0 | 0 | 1001.0510.pdf |
2
x
z
y
J J
J
1
0
2
FIG. 1: (colors online) (a): body-centered tetragonal (BCT)
lattice with J0 in-plane coupling constant, and out-of-plane
J1, and J2 competing interactions.
be achieved with different number of interacting layers:
notably, nearest and next-nearest layers competitive in-
teractions are enough to get a helical structure with a
whatever pitch wavevector. Such observation gives us a
possible way to solve the conundrum previously emerged,
as we have the possibility of varying the range of inter-
actions without modifying the helical pitch, thus decou-
pling the two relevant length scales along the film growth
direction, and making accessible a range ofn of the or-
der of, or smaller than, the helical pitch, but still large
enough that a substantial number of layers can behave
as “bulk” layers. Therefore, while in the previous papers
we have studied the properties of ultrathin magnetic films
of Ho assuming a model with six interlayer exchange in-
teractions, here we investigate by MC simulations the
properties of the same system by making use of the sim-
plest model Hamiltonian able to describe the onset of a
helical magnetic order in Holmium, i.e. we consider only
two inter-layer coupling constants, as previously done in
Ref. 11.
The paper is organized as follows: In Sec. II the model
Hamiltonian will be defined, and the MC techniques, and
all the thermodynamic quantities relevant for this study,
will be introduced. In Sec. III the results obtained for
different thicknesses will be presented, both in the matter
of the critical properties of the model and of the magnetic
ordered structures observed. Finally, in Sec. IV we shall
discuss such results, drawing also some conclusions.
II. MODEL HAMILTONIAN AND MONTE
CARLO OBSER V ABLES
The model Hamiltonian we use in our simulations is the
minimal one able to describe helimagnetic structures:
H = −
J0
∑
⟨ij⟩
⃗Si · ⃗Sj + J1
∑
⟨ik⟩
⃗Si · ⃗Sk + J2
∑
⟨il⟩
⃗Si · ⃗Sl
.
(1)
⃗Si are classical planar unit vectors representing the di-
rection of the total angular momentum of the magnetic
ions, whose magnitude
√
j(j + 1) ( j = 8 for Holmium
ions) is already encompassed within the definition of the
interaction constantsJ0, 1, 2. As sketched in Fig. 1, the
magnetic ions are located on the sites of a body-centered
tetragonal (BCT) lattice; the first sum appearing in the
Hamiltonian describes the in-plane (xy) nearest neigh-
bor (NN) interaction, which is taken ferromagnetic (FM),
with exchange strengthJ0 > 0; the second sum rep-
resents the coupling, of exchange strength J1, between
spins belonging to nearest neighbor (NN) planes along
thez-direction (which we will assume to coincide with
the film growth direction); finally, the third sum takes
into account the interaction, of exchange strengthJ2, be-
tween spins lying on next-nearest neighbor (NNN) planes
alongz. In order to have frustration, giving rise to non-
collinear order along z in the bulk, NN interaction J1
can be taken both ferro- or antiferromagnetic, but NNN
couplingJ2 has necessarily to be antiferromagnetic, and
the condition |J2| > |J1|/4 must be fulfilled. Such simpli-
fied Hamiltonian was already employed to simulate he-
lical ordering in bulk systems by Diep1,17 and Loison 18.
In the bulk limit, the state of minimal energy of a sys-
tem described by Eq.(1) corresponds to a helical arrange-
ment of spins. The ground state energy per spin is equal
toeg(Qz) = [ −4J0 − 2J1 (4 cos (Qzc′) + δ cos (2Qzc′))]
where c′ is the distance between NN layers, δ = J2
J1
,
and Qzc′ = arccos
(
− 1
δ
)
is the angle between spins ly-
ing on adjacent planes along the z-direction. The ob-
served helical arrangement in bulk holmium corresponds
toQzc′ ≃ 30.5◦10: such value can be obtained from
the formula above with the set of coupling constants
J0=67.2 K, J1=20.9 K, and J2 = −24.2 K, that we have
employed in our simulations. The given values for the ex-
change constants are the same already used by Weschke
et al.in Ref. 13 to interpret experimental data on
Holmium films on the basis of a J1 − J2 model, after
a proper scaling by the numbers of NN and NNN on
neighboring layers of a BCT lattice.
In the following we will denote with n the film thick-
ness, i.e. the number of spin layers along the z direction,
and with L×L the number of spins in each layer (i.e., L
is the lattice size along both the x and y directions). In
our simulations thickness values from 1 to 24 were con-
sidered, while the range of lateral sizeL was from 8 to
64. Periodic boundary conditions were applied along x
and y, while free boundaries were obviously taken along
the film growth direction z.
Thermal equilibrium was attained by the usual
Metropolis algorithm 19, supplemented by the over-
relaxed technique 20 in order to speed-up the sampling
of the spin configuration space: a typical “Monte Carlo
step” was composed by four Metropolis and four-five
over-relaxed moves per particle. Such judicious mix of
moves is able both to get faster the thermal equilibrium
and to minimize the correlation “time” between succes-
sive samples, i.e. the undesired effects due to lack of in- | 1 | 1 | 1001.0510.pdf |
3
dependence of different samples during the measurement
stage. For each temperature we have usually performed
three independent simulations, each one containing at
least 2×105 measurements, taken after discarding up to
5×104 Monte Carlo steps in order to assure thermal equi-
libration.
In the proximity of the critical region the multiple his-
togram (MH) technique was also employed 21, as it allows
us to estimate the physical observables of interest over a
whole temperature range in a substantially continuous
way by interpolating results obtained from sets of simu-
lations performed at some different temperatures.
For all the quantities of interest, the average value and
the error estimate were obtained by the bootstrap re-
sampling method22 given that, as pointed out in Ref. 23,
for a large enough number of measurements, this method
turns out to be more accurate than the usual blocking
technique. In our implementation, we pick out randomly
a sizable number of measurements (typically, between 1
and 1×103 for the single simulation, and between 1 and
5×104 for the MH technique), and iterate the re-sampling
at least one hundred times.
The thermodynamic observables we have investigated
include the FM order parameter for each plane l:
ml =
√
(mx
l)2 + (my
l)2 , (2)
which is related to the SO(2) symmetry breaking. At the
same time, it turns out to be significant also the average
order parameter of the film, defined as
M = 1
n
n∑
l=1
ml . (3)
Turning to the helical order, which is the relevant
quantity for the Z2 × SO(2) symmetry, we can explore
it along two different directions. The first one is by the
introduction of the chirality order parameter1,2
κ = 1
4(n − 1)L2 sin Qz
∑
⟨ij⟩
[
Sx
i Sy
j − Sy
i Sx
j
]
, (4)
where the sum refers to spins belonging to NN layers
iand j, respectively, while Qz is the bulk helical pitch
vector along the z direction. The second possibility is
that of looking at the integral of the structure factor:
MHM = 1
K
∫ π
0
dqzS(⃗ q) (5)
where S(⃗ q), with ⃗ q= (0 , 0, qz), is the structure factor 24
(i.e. the Fourier transform of the spin correlation func-
tion) along the z-direction of the film, while the normal-
ization factorK is the structure factor integral at T = 0.
Although the use of the last observable can be seen as a
suitable and elegant way to overcome the intrinsic diffi-
culties met in defining a correct helical order parameter,
free of any undue external bias (as the wave-vectorQz
0 20 40 60 80 100 120 140
T (K)
0
0.5
1
1.5
2
2.5
c v / k B
L = 24
L = 32
L = 48
L = 64
20 30 40 50 60 70
2.1
2.2
2.3
2.4
2.5
2.6
c v, max
L
FIG. 2: (color online) Specific heat cv per spin vs. temper-
ature for thickness n = 16 (for lateral dimension, see the
legend inside the figure). Inset: Maximum of cv vs. L ob-
tained through MH technique. The continuum red line is a
power law fit.
entering the definition ofκ in Eq. (4)), we remind that
such quantity has generally to be managed with particu-
lar care, as discussed in details in Refs.14,15, where it was
shown that the presence of block structures prevents us
to unambiguously relate the evolution ofS(⃗ q) with the
onset of helical order. However, for the specific case of
the model under investigation such integrated quantity
can still be considered a fairly significant order parame-
ter, as no block structures emerge from the simulations
(see below).
In order to get a clear picture of the critical region and
to give an accurate estimate of the critical temperature,
we look also at the following quantities
cv = nL2β2 (
⟨e2⟩ − ⟨e⟩2)
, (6)
χo = nL2β
(
⟨o2⟩ − ⟨o⟩2)
, (7)
∂β o = nL2 (⟨oe⟩ − ⟨o⟩⟨e⟩) , (8)
u4(o) = 1 − ⟨o4⟩
3⟨o2⟩2 , (9)
where β = 1 /kBT , and o is one of the relevant observ-
ables, i.e. ml, M, κ, M HM . In this paper, we shall mainly
locate the critical temperature by looking at the intersec-
tion of the graphs of the Binder cumulant25, Eq. (9), as a
function of T obtained at different L. For clarity reasons,
we introduce also the following symbols: by TN (n) we
will denote the helical/fan phase transition temperature
for thicknessn, TC (n) will instead indicate the order-
ing temperature of the sample as deduced by looking at
the behaviour of the average order parameter (3), while
Tl
C (n) will be the l-th plane transition temperature re-
lated to the order parameter defined in Eq. (2). | 2 | 2 | 1001.0510.pdf |
4
u 4 (M)
0.62
0.64
0.66
130 131 132 133 134 135 136 137 138
T (K)
0.5
0.55
0.6
0.65
u 4 (M HM )
0 0.2 0.4 0.6 0.8 q z
S(q z ) (a.u.)
(a)
(b)
0.66
0.64
0.62
FIG. 3: (color online) Binder cumulants at thickness n =
16, colors as in Fig. 2. (a): Binder cumulant for the order
parameter defined in Eq. (3). (b): Binder cumulant extracted
from the integral of the structure factor (see Sec. II). Inse t:
structure factor for L = 64 between T = 131 K (upper curve)
and T = 140 K (lower), with 1 K temperature step.
III. RESULTS
The results obtained by MC simulations of the model
introduced in Sec. II will be presented starting from
n= 16, i.e. the highest investigated film thickness which
still displays a bulk-like behaviour. In Fig. 2 the spe-
cific heat for samples withn = 16 and lateral dimension
L = 24 , 32, 48, 64 is shown. The location of the specific
heat maximum shows a quite definite evolution toward
the bulk transition temperature,T Ho
N ≃ 132 K10 (it is
worthwhile to note that for this XY model the mean field
theory predicts a critical temperature T Ho
N,MF ≃ 198 K).
The intensity of the maximum of cv has been analyzed
by the MH technique for the same lateral dimensions (see
inset of Fig. 2): it clearly appears as it increases withL
in a smooth way.
The Binder cumulant for the average order parameter
defined in Eq. (3) was obtained close to the cv peak and is
reported in Fig. 3a; its analysis leads to an estimate of the
critical temperature of the sample (given by the location
of the common crossing point of the different curves re-
ported in the figure) ofTC (16) = 133 .2(5) This value can
be considered in a rather good agreement with the exper-
imental ordering temperature of HolmiumT Ho
N , the rel-
ative difference being about 1%. Even such a mismatch
betweenT Ho
N and TC (16) could be completely eliminated
by slightly adjusting the in-plane coupling constant J0,
but, as discussed in Sec. II, we shall preserve the value
reported in Refs. 13, and 12 in order to allow for a correct
comparison with the results reported in those papers.
The development of the helical arrangement of magne-
tization along the film growth direction was investigated
by looking at the integral of the structure factorS(⃗ q)
along the z-direction, i.e. by taking ⃗ q= (0 , 0, qz), and
making again use of the cumulant analysis in order to
locate the helical transition temperature atTN (16) =
20 40 60 80 100 120 140
T (K)
χ κ (a.u.)
0
0.2
0.4
0.6
κ
0
0.5
1
1.5
2
c v / k B
20 40 60 80 100 120 140
T (K)
0.1
0.2
0.3
0.4
0.5
0.6
u 4 (κ)
132 134 136
T (K)
∂ β κ (a.u.)
(a)
(c)
(b)
(d)
FIG. 4: (color online) Thermodynamic quantities obtained f or
thickness n = 8 in the temperature range 0-150 K. Colors and
symbols as in Fig. 2. (a): specific heat; (b): chirality order
parameter. (c): susceptibility χκ . (d): Binder cumulant for
κ.
133.1(3) K (see Fig. 3b). The crossing points of the
Binder’s cumulants of the helical order parameter imme-
diately appear to be located, within the error bars, at the
same temperature of those for the average magnetization
previously discussed. In addition, it is worthwhile to ob-
serve that the peak evolution ofS(0, 0, qz), in particular
close to TN (16) (inset of Fig. 3b), displays the typical
behaviour expected for an helical structure. We can thus
conclude that forn = 16, as it is commonly observed
in bulk samples, the establishment of the in-plane order
coincides with onset of the perpendicular helical arrange-
ment atTN (16). However, due to helix distortion in the
surface regions, the maximum of S(0, 0, qz) stabilizes at
values of qz sensibly smaller (e.g. Qz(TN (16)) ≈ 16◦,
and Qz(T = 10 K) ≈ 28◦) with respect to the bulk one
(QHo
z= 30 .5◦).
The MC simulations outcomes for n = 16 we just pre-
sented appear quite different with respect to those ob-
tained at the same thickness for the model with six cou-
pling constants along thez direction14,15. Indeed, for
the J1-J2 model here investigated, we observe that all
layers order at the same temperature, and we do not find
any hint of the block-phase, with inner disordered planes
intercalated to antiparallelquasi-FM four-layer blocks,
previously observed; sample MC runs we made using the
samehcp lattice employed in Refs. 14,15 shows that the
presence or absence of the block phase is not related to
the lattice geometry, but it is a consequence of the inter-
action range only.
We now move to describe and discuss MC simulation
data for thinner samples. A graphical synthesis of the
results obtained forn = 8 in reported in Fig. 4a-d. The
specific heat cv, shown in Figs. 4a, reveals very small
finite-size effects, which, however, cannot be unambigu-
ously detected for the largest lattice size (L = 64), as
they fall comfortably within the error range. Surpris-
ingly, the specific heat maximum is located close to the
bulk transition temperature as found forn = 16, and | 3 | 3 | 1001.0510.pdf |
5
0 2 4 6 8 10 12 14 16 18 20
n
0
20
40
60
80
100
120
140
T N (n) , T C (n) (K)
T N (n)
T C (n)
T N
bulk
FIG. 5: Transition temperatures TN (n) and TC (n) vs. film
thickness n.
the same is true for the crossing point of the Binder cu-
mulant of the average magnetizationM (not reported in
figure), which is located at TC (8) = 133 .3(3) K. These
data give a first rough indication that also for n = 8 all
the planes of the sample are still ordering almost at the
same temperature; such property has been observed for
all the investigated thicknessesn below 16, so that TC (n)
results quite n-independent (see also Fig. 5) .
Although the layer subtraction does not seem to mod-
ify TC (n), the onset of helical arrangement is observed to
shift at lower temperatures as n decreases. The chirality
κ defined in Eq. (4) is reported in Fig 4b for n = 8. As the
temperature decreases, around T ∼ 80 K we can identify
a finite-size behaviour of κ which, at variance with the
previous one, can be easily recognized as typical of an
effective phase transition. Such conclusion is confirmed
by the analysis of the chiral susceptibilityχκ (Fig. 4c),
which for the largest L has a maximum at T = 85 K. As-
suming that the order parameter (4) is the relevant one
to single out the onset of the fan arrangement, we can
get a more accurate estimate ofTN (8) by looking at the
Binder cumulant u4(κ), reported in Fig. 4d. By making
use of the MH technique, we locate the crossing point at
TN (8) = 92(2) K. Finally, it is worthwhile to observe as
the specific heat does not show any anomaly at TN (8),
being the entropy substantially removed at TC (8).
The scenario just outlined for n = 8 results to be cor-
rect in the thickness range 6 ≤ n ≲ 15, where a clear
separation between TN (n) and TC (n) can be easily fig-
ured out. In such temperature window, the strong sur-
face effects produce aquasi-FM set-up of the magnetic
film structure along the z-direction. While leaving to the
next Section a more detailed discussion of this regime, we
report in Fig. 5 a plot ofTN (n) and TC (n) vs. n for all
the simulated thicknesses. The separation between the
two critical temperatures is maximum forn = 6, where
TN (6) = 38(4), that is TN (6) ∼ 1
3 TC (6). For films with
less than six layers no fan order is observed, i.e. for n = 5
and below the chirality does not display any typical fea-
ture of fan ordering at any temperature belowTC (n). As
a representative quantity we finally look at the rotation
0 1 2 3 4 5 6
0
5
10
15
20
∆ϕ l (deg.)
T =10K
T =20K
T =30K
T =40K
T= 50K
0 1 2 3 4 5
l
0
1
2
3
4
5
(a) n = 6
(b) n = 5
FIG. 6: Rotation angle ∆ ϕl between magnetic moments on
NN layers ( l + 1 , l) at some low temperatures, for thickness
n = 5 and n = 6, and lateral dimension L = 64.
angle of the magnetization between nearest planes:
∆ ϕl = ϕl+1 − ϕl = arccos
[
Mx
l Mx
l+1 + My
l My
l+1
]
(10)
where ( Mx
l , M y
l ) is the magnetic vector profile for each
plane l. ∆ ϕl is displayed in Fig. 6a and Fig. 6b, for
n = 6 and n = 5, respectively. In Fig. 6a, a quite clear
fan stabilization is observed when the temperature de-
creases, while in Fig. 6b, i.e. forn = 5, ∆ ϕl keeps an
almost temperature independent very small value; what’s
more, ∆ϕl seems to loose any temperature dependence
as T = 0 is approached. We attribute the absence of fan
arrangement for n ≤ 5 as simply due to the lack of “bulk
planes” inside the film, so that we are left with only a 2d
trend atTC (n), i.e. at the temperature where the order
parameters defined in Eqs. (2) and (3) show a critical
behaviour.
IV. DISCUSSION AND CONCLUSION
A possible framework to analyze the results presented
in the previous Section is suggested by Fig. 5, where we
can easily distinguish three significant regions:i) high
thickness, n ⩾ 16, where the films substantially display a
bulk behaviour, with the single planes ordering tempera-
ture coinciding with the helical phase transition one;ii)
intermediate thickness, 6 ≤ n ≲ 15, where the tempera-
ture corresponding to the onset of in-plane order, TC (n),
is still ≃ T Ho
N , but where the helical/fan arrangement sta-
bilizes only below a finite temperature TN (n) < T C (n);
iii) low thickness,1 ≤ n ≤ 5, where TC (n) ≲ T Ho
N but no
fan phase is present at any temperature.
The observed behaviour in region iii) can be reason-
ably attributed to the decreasing relevance of the con-
tribution to the total energy of the system coming from
the competitive interactions among NNN planes as the
film thickness decreases; moreover, the thinness of the | 4 | 4 | 1001.0510.pdf |
6
0 20 40 60 80 100 120 140
T (K)
0
10
20
30
∆ϕ l,l+1 ( T ) (deg.)
T N (16)T N (8)
FIG. 7: (color online) ∆ ϕl(T ) vs. temperature for the surface
planes, l = 1 (triangles), l = 2 (squares), l = 3 (diamonds),
l = 4 (circles). Straight lines and full symbols: n = 8. Dashed
lines and open symbols: n = 16.
film leads to an effective 2d-like trend. Region ii) looks
however more intriguing, and requires a more accurate
discussion, which can benefit from a careful comparison
of the behaviour of a given quantity in regionsi) and ii).
For this purpose, we look at the temperature depen-
dence of the rotation angle of the magnetization between
NN planes. In Fig. 7, ∆ϕl(T ) for n = 8 and n = 16
(continuous and dashed lines, respectively), is plotted for
the outermost planes,l = 1 . . . 4. For both thicknesses, a
monotonic trend is observed for all l, but at variance with
what happens for the highest thickness, for n = 8 we see,
starting from a temperature T ≲ TN (8), an abrupt drop
of ∆ ϕ3 and ∆ ϕ4, which rapidly reach an almost con-
stant value, only slightly larger than ∆ ϕ1. In the tem-
perature range TN (8) ≲ T < T C (8) we thus substantially
observe the same small magnetic phase shifts between all
NN layers, testifying an energetically stablequasi-FM
configuration giving no contribution to the helical order
parameters. The latter point can be made clearer by
looking at the the peak positionQz,max of the structure
factor S(0, 0, qz). In Fig. 8 the average of Qz,max vs T is
reported, again for n = 8 and for different lateral dimen-
sions L26. As expected from the previous argument, we
see that Qz,max = 0 for TN (8) < T < T C (8), while it be-
gins to shift to higher values as soon as the temperature
decreases belowTN (8), making apparent a progressive
fan stabilization with Qz,max ̸= 0 and reaching a value
of about 21 ◦ for T = 10 K.
In a previous study, where the magnetic properties of
Ho thin films were investigated by MC simulations of a
Heisenberg model with easy-plane single-ion anisotropy
and six out-of-plane coupling constants (as obtained by
experimental neutron scattering measurements16) on a
HCP lattice14,15, it was found that for thicknesses compa-
rable with the helical pitch the phase diagram landscape
is quite different from what we find here. Indeed, for
n= 9 − 16, three different magnetic phases could be sin-
0 20 40 60 80 100
T (K)
0
5
10
15
20
25
30
Q z, max (deg.)
FIG. 8: (color online) Qz, position of the maximum of S(⃗ q),
vs. temperature for thickness n = 8. Inset: magnetic vector
(mx
l, my
l) profile for some temperatures for L = 64. Colors
and symbols as in Fig. 2.
0 1 2 3 4 5 6 7 8 9 10 11 12
l
0
20
40
60
80
∆ϕ l (deg)
T =100K
130K
135K
140K
145K
FIG. 9: ∆ ϕl for a BCT lattice and n = 12, when the six
coupling constants set employed in Ref. 14,15 (see text) is
used. The temperature range has been chosen aroundTC (n)
(error bars lye within point size).
gled out, with the high-temperature, paramagnetic phase
separated from the low-temperature, long-range ordered
one, by an intermediate-temperature block phase where
outer ordered 4-layers blocks coexist with some inner dis-
ordered ones. Moreover, it was observed that the phase
transition of such inner layers turns out to have the sig-
natures of a Kosterlitz-Thouless one.
The absence of the block phase in the J1 − J2 model
here investigated has to be attributed to the different
range of interactions, rather than to the different lattice
structure. We came to this conclusion by doing some
simulations using the same set of interaction constants
employed in Refs. 14,15, but using a BCT lattice: the
results we obtained for ∆ϕl with n = 12 are reported in
Fig. 9. The latter is absolutely similar to Fig.7 of Ref. 15
and clearly displays the footmarks of the block phase (see
down-triangle), with two external blocks of ordered layers
(l =1. . . 5 and 8. . . 12 ), where ∆ ϕl is roughly 10 ◦, sep-
arated by a block of disordered layers, and with almost | 5 | 5 | 1001.0510.pdf |
7
-140 -139 -138 -137 -136 -135 -134 -133 -132 -131
e
0
0.5
1
P e
90K
91K
92K
93K
94K
95K
-94 -92 -90 -88 -86 -84 -82 -80 -78
e
0
0.2
0.4
P e
129K
130K
131K
132K
133K
134K
T C (8) = 133.3(3)K
T N (8) = 92(2)K
(a)
(b)
FIG. 10: (colors online) Equilibrium probability distribu tion
of the energy for the thickness n = 8 for some temperatures
around TN (8), (a), and TC (8), (b), respectively.
opposite magnetization. We can thus confidently assert
that, regardless of the underlying lattice structure, by
decreasing the number of the out-of-plane interactions,
for thicknesses close to the helical bulk pitch, the block
phase is replaced by a quasi-FM configuration in the in-
termediate temperature range TN (n) < T < T C (n) .
As a final issue we address the problem of the order
of the transitions observed at TN (n) and TC (n), respec-
tively. In particular, we focus our attention to the thick-
ness ranges where the chiral order parameter is relevant,
i.e. regionsi) and ii) as defined at the beginning of
this Section. In Fig. 10 the equilibrium probability dis-
tribution of the energy for temperatures aroundTN (8)
(Fig. 10a) and TC (8) (Fig. 10b) is plotted: for both
temperatures, no double peak structure is observed, so
that we have no direct indication for a first order tran-
sition even if, according to precedent studies of Loison
and Diep17,18, the presence of a first-order transition at
TN (n), cannot be completely excluded, as it could reveal
itself only when the lateral dimension L are much larger
than the largest correlation length. The same conclusion
about the order of transition is reached for any other in-
vestigated film thickness, as the energy probability distri-
bution shape does not qualitatively change. This findings
agree with the results we got in previous MC simulations
discussed in Ref. 15, so that we may conclude that the
order of the observed transitions is not affected by the
range of interactions.
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arXiv:1001.0764v2 [cond-mat.str-el] 13 Jan 2010
Optical Integral and Sum Rule Violation
Saurabh Maiti, Andrey V. Chubukov
Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA
(Dated: November 9, 2018)
The purpose of this work is to investigate the role of the latt ice in the optical Kubo sum rule in
the cuprates. We compute conductivities, optical integral s W , and ∆ W between superconducting
and normal states for 2-D systems with lattice dispersion ty pical of the cuprates for four different
models – a dirty BCS model, a single Einstein boson model, a ma rginal Fermi liquid model, and a
collective boson model with a feedback from superconductiv ity on a collective boson. The goal of
the paper is two-fold. First, we analyze the dependence of W on the upper cut-off ( ωc) placed on
the optical integral because in experiments W is measured up to frequencies of order bandwidth.
For a BCS model, the Kubo sum rule is almost fully reproduced a t ωc equal to the bandwidth. But
for other models only 70%-80% of Kubo sum rule is obtained up t o this scale and even less so for
∆ W , implying that the Kubo sum rule has to be applied with cautio n. Second, we analyze the sign
of ∆ W . In all models we studied ∆ W is positive at small ωc, then crosses zero and approaches a
negative value at large ωc, i.e. the optical integral in a superconductor is smaller th an in a normal
state. The point of zero crossing, however, increases with t he interaction strength and in a collective
boson model becomes comparable to the bandwidth at strong co upling. We argue that this model
exhibits the behavior consistent with that in the cuprates.
I. INTRODUCTION
The analysis of sum rules for optical conductivity has a
long history. Kubo, in an extensive paper 1 in 1957, used
a general formalism of a statistical theory of irreversible
processes to investigate the behavior of the conductivity
in electronic systems. For a system of interacting elec-
trons, he derived the expression for the integral of the real
part of a (complex) electric conductivityσ(Ω) and found
that it is independent on the nature of the interactions
and reduces to
∫ ∞
0
Re σ(Ω) dΩ = π
2
ne2
m (1)
Here n is the density of the electrons in the system and
m is the bare mass of the electron. This expression is
exact provided that the integration extends truly up to
infinity, and its derivation uses the obvious fact that at
energies higher than the total bandwidth of a solid, elec-
trons behave as free particles.
The independence of the r.h.s. of Eq. (1) on temper-
ature and the state of a solid (e.g., a normal or a super-
conducting state – henceforth referred to as NS and SCS
respectively) implies that, while the functional form of
σ(Ω) changes with, e.g., temperature, the total spectral
weight is conserved and only gets redistributed between
different frequencies as temperature changes. This con-
servation of the total weight ofσ(Ω) is generally called a
sum rule.
One particular case, studied in detail for conventional
superconductors, is the redistribution of the spectral
weight between normal and superconducting states. This
is known as Ferrel-Glover-Tinkham (FGT) sum rule:2,3
∫ ∞
0+
Re σNS (Ω) =
∫ ∞
0+
Re σsc(Ω) + πnse2
2m (2)
where ns is the superfluid density, and πnse2/(2m) is
the spectral weight under the δ-functional piece of the
conductivity in the superconducting state.
In practice, the integration up to an infinite frequency
is hardly possible, and more relevant issue for practical
applications is whether a sum rule is satisfied, at least ap-
proximately, for a situation when there is a single electron
band which crosses the Fermi level and is well separated
from other bands. Kubo considered this case in the same
paper of 1957 and derived the expression for the “band”,
or Kubo sum rule
∫ ‘∞′
0
Re σ(Ω) dΩ = WK = πe2
2N
∑
⃗k
∇2
⃗kx
ε⃗k n⃗k (3)
where n⃗k is the electronic distribution function and ε⃗k is
the band dispersion. Prime in the upper limit of the inte-
gration has the practical implication that the upper limit
is much larger than the bandwidth of a given band which
crosses the Fermi level, but smaller than the frequencies
of interband transitions. Interactions with external ob-
jects, e.g., phonons or impurities, and interactions be-
tween fermions are indirectly present in the distribution
function which is expressed via the full fermionic Green’s
function asn⃗k = T ∑
m G(⃗k, ωm). For ǫk = k2/2m,
∇2
⃗kx
ε⃗k = 1 /m, WK = πne2/(2m), and Kubo sum rule
reduces to Eq. (1). In general, however, ε⃗k is a lattice
dispersion, and Eqs. (1) and (3) are different. Most im-
portant,WK in Eq. (3) generally depends on T and on
the state of the system because of n⃗k. In this situation,
the temperature evolution of the optical integral does not
reduce to a simple redistribution of the spectral weight
– the whole spectral weight inside the conduction band
changes withT . This issue was first studied in detail by
Hirsch 4 who introduced the now-frequently-used nota-
tion “violation of the conductivity sum rule”.
In reality, as already pointed out by Hirsch, there is no
true violation as the change of the total spectral weight | 0 | 0 | 1001.0764.pdf |
2
in a given band is compensated by an appropriate change
of the spectral weight in other bands such that the total
spectral weight, integrated over all bands, is conserved,
as in Eq. (1). Still, non-conservation of the spectral
weight within a given band is an interesting phenomenon
as the degree of non-conservation is an indicator of rele-
vant energy scales in the problem. Indeed, when relevant
energy scales are much smaller than the Fermi energy,
i.e., changes in the conductivity are confined to a near
vicinity of a Fermi surface (FS), one can expandεk near
kF as εk = vF (k − kF ) + ( k − kF )2/(2mB) + O(k − kF )3
and obtain ∇2
⃗kx
ε⃗k ≈ 1/mB [this approximation is equiv-
alent to approximating the density of states (DOS) by a
constant]. ThenWK becomes πne2/(2mB) which does
not depend on temperature. The scale of the tempera-
ture dependence ofWK is then an indicator how far in
energy the changes in conductivity extend when, e.g., a
system evolves from a normal metal to a superconductor.
Because relevant energy scales increase with the interac-
tion strength, the temperature dependence ofWK is also
an indirect indicator of whether a system is in a weak,
intermediate, or strong coupling regime.
In a conventional BCS superconductor the only rele-
vant scales are the superconducting gap ∆ and the impu-
rity scattering rate Γ. Both are generally much smaller
than the Fermi energy, so the optical integral should be
almostT -independent, i.e., the spectral weight lost in a
superconducting state at low frequencies because of gap
opening is completely recovered by the zero-frequencyδ-
function. In a clean limit, the weight which goes into
aδ−function is recovered within frequencies up to 4∆.
This is the essence of FGT sum rule 2,3. In a dirty limit,
this scale is larger, O(Γ), but still WK is T -independent
and there was no “violation of sum rule”.
The issue of sum rule attracted substantial interest in
the studies of high Tc cuprates5–18,21–26 in which pairing
is without doubts a strong coupling phenomenon. From a
theoretical perspective, the interest in this issue was orig-
inally triggered by a similarity betweenWK and the ki-
netic energy K = 2 ∑ ε⃗kn⃗k.18–20 For a model with a sim-
ple tight binding cosine dispersion εk ∝ (cos kx + cos ky),
d2 ε⃗k
d k2x
∼ −ε⃗k and WK = −K. For a more complex dis-
persion there is no exact relation between WK and K,
but several groups argued 17,27,28 that WK can still be
regarded as a good monitor for the changes in the kinetic
energy. Now, in a BCS superconductor, kinetic energy
increases belowTc because nk extends to higher frequen-
cies (see Fig.2). At strong coupling, K not necessary
increases because of opposite trend associated with the
fermionic self-energy: fermions are more mobile in the
SCS due to less space for scattering at low energies than
they are in the NS. Model calculations show that above
some coupling strength, the kinetic energy decreases be-
lowTc29. While, as we said, there is no one-to-one cor-
respondence between K and WK , it is still likely that,
when K decreases, WK increases.
A good amount of experimental effort has been put into
addressing the issue of the optical sum rule in the c−axis7
and in-plane conductivities 8–16 in overdoped, optimally
doped, and underdoped cuprates. The experimental re-
sults demonstrated, above all, outstanding achievements
of experimental abilities as these groups managed to de-
tect the value of the optical integral with the accuracy
of a fraction of a percent. The analysis of the change
of the optical integral between normal and SCS is even
more complex because one has to (i) extend NS data to
T < Tc and (ii) measure superfluid density with the same
accuracy as the optical integral itself.
The analysis of the optical integral showed that in over-
doped cuprates it definitely decreases below Tc, in con-
sistency with the expectations at weak coupling 11. For
underdoped cuprates, all experimental groups agree that
a relative change of the optical integral belowTc gets
much smaller. There is no agreement yet about the sign
of the change of the optical integral : Molegraafet al.8
and Santander-Syro et al.9 argued that the optical inte-
gral increases below Tc, while Boris et al.10 argued that
it decreases.
Theoretical analysis of these results 21,22,25,28,30 added
one more degree of complexity to the issue. It is tempt-
ing to analyze the temperature dependence ofWK and
relate it to the observed behavior of the optical integral,
and some earlier works25,28,30 followed this route. In the
experiments, however, optical conductivity is integrated
only up to a certain frequencyωc, and the quantity which
is actually measured is
W (ωc) =
∫ ωc
0
Re σ(Ω) dΩ = WK + f(ωc)
f(ωc) = −
∫ ′∞′
ωc
Re σ(Ω) dΩ (4)
The Kubo formula, Eq. (3) is obtained assuming that
the second part is negligible. This is not guaranteed,
however, as typicalωc ∼ 1 − 2eV are comparable to the
bandwidth.
The differential sum rule ∆ W is also a sum of two
terms
∆ W (ωc) = ∆ WK + ∆ f(ωc) (5)
where ∆ WK is the variation of the r.h.s. of Eq. 3,
and ∆ f(ωc) is the variation of the cutoff term. Because
conductivity changes with T at all frequencies, ∆ f(ωc)
also varies with temperature. It then becomes the issue
whether the experimentally observed ∆W (ωc) is predom-
inantly due to “intrinsic” ∆ WK , or to ∆ f(ωc). [A third
possibility is non-applicability of the Kubo formula be-
cause of the close proximity of other bands, but we will
not dwell on this.]
For the NS, previous works 21,22 on particular models
for the cuprates indicated that the origin of the temper-
ature dependence ofW (ωc) is likely the T dependence
of the cutoff term f(ωc). Specifically, Norman et. al.22
approximated a fermionic DOS by a constant (in which | 1 | 1 | 1001.0764.pdf |
3
case, as we said, WK does not depend on temperature)
and analyzed the T dependence of W (ωc) due to the T
dependence of the cut-off term. They found a good agree-
ment with the experiments. This still does not solve the
problem fully as amount of theT dependence of WK in
the same model but with a lattice dispersion has not been
analyzed. For a superconductor, which of the two terms
contributes more, remains an open issue. At small fre-
quencies, ∆W (ωc) between a SCS and a NS is positive
simply because σ(Ω) in a SCS has a δ−functional term.
In the models with a constant DOS, for which ∆ WK = 0,
previous calculations 21 show that ∆ W (ωc) changes sign
at some ωc, becomes negative at larger ωc and approaches
zero from a negative side. The frequency when ∆ W (ωc)
changes sign is of order ∆ at weak coupling, but increases
as the coupling increases, and at large coupling becomes
comparable to a bandwidth (∼ 1eV ). At such frequencies
the approximation of a DOS by a constant is question-
able at best, and the behavior of ∆W (ωc) should gen-
erally be influenced by a nonzero ∆ WK . In particular,
the optical integral can either remain positive for all fre-
quencies below interband transitions (for large enough
positive ∆WK ), or change sign and remain negative (for
negative ∆ WK ). The first behavior would be consistent
with Refs. 8,9, while the second would be consistent with
Ref. 10. ∆W can even show more exotic behavior with
more than one sign change (for a small positive ∆ WK ).
We show various cases schematically in Fig.1.
0
ω c
∆ W
0
ω c
∆ W
0
ω c
∆ W 0
ω c
∆ W
(a)(a)(a) (b)
(c) (d)
FIG. 1: Schematic behavior of ∆ W vs ωc, Eq. (4). The
limiting value of ∆ W at ωc = ∞ is ∆ WK given by Eq. (3)
Depending on the value of ∆ WK , there can be either one sign
change of ∆ W (panels a and c), or no sign changes (panel b),
or two sign changes (panel d).
In our work, we perform direct numerical calculations
of optical integrals at T = 0 for a lattice dispersion ex-
tracted from ARPES of the cuprates. The goal of our
work is two-fold. First, we perform calculations of the
optical integral in the NS and analyze how rapidlyW (ωc)
approaches WK , in other words we check how much of
the Kubo sum is recovered up to the scale of the band-
width. Second, we analyze the difference between optical
integral in the SCS at T = 0 and in the NS extrapolated
to T = 0 and compare the cut off effect ∆ f(ωc) to ∆ WK
term. We also analyze the sign of ∆ W (ωc) at large fre-
quencies and discuss under what conditions theoretical
W(∞) increases in the SCS.
We perform calculations for four models. First is a
conventional BCS model with impurities (BCSI model).
Second is an Einstein boson (EB) model of fermions in-
teracting with a single Einstein boson whose propaga-
tor does not change between NS and SCS. These two
cases will illustrate a conventional idea of the spectral
weight in SCS being less than in NS. Then we con-
sider two more sophisticated models: a phenomenological
“marginal Fermi liquid with impurities” (MFLI) model
of Norman and P´ epin30, and a microscopic collective bo-
son (CB) model 31 in which in the NS fermions interact
with a gapless continuum of bosonic excitations, but in a
d−wave SCS a gapless continuum splits into a resonance
and a gaped continuum. This model describes, in par-
ticular, interaction of fermions with their own collective
spin fluctuations32 via
Σ( k, Ω) = 3 g2
∫ dω
2π
d2q
(2π)2 χ(q, ω)G(k + q, ω + Ω) (6)
where g is the spin-fermion coupling, and χ(q, ω) is the
spin susceptibility whose dynamics changes between NS
and SCS.
From our analysis we found that the introduction of
a finite fermionic bandwidth by means of a lattice has
generally a notable effect on bothW and ∆ W . We
found that for all models except for BCSI model, only
70%− 80% of the optical spectral weight is obtained by
integrating up to the bandwidth. In these three models,
there also exists a wide range ofωc in which the behavior
of ∆ W (ωc) is due to variation of ∆ f(ωc) which is domi-
nant comparable to the ∆ WK term. This dominance of
the cut off term is consistent with the analysis in Refs.
21,22,33.
We also found that for all models except for the origi-
nal version of the MFLI model the optical weight at the
highest frequencies is greater in the NS than in the SCS
(i.e., ∆W < 0). This observation is consistent with the
findings of Abanov and Chubukov 32, Benfatto et. al.28,
and Karakozov and Maksimov 34. In the original ver-
sion of the MFLI model 30 the spectral weight in SCS
was found to be greater than in the NS (∆ W > 0). We
show that the behavior of ∆ W (ωc) in this model cru-
cially depends on how the fermionic self-energy modeled
to fit ARPES data in a NS is modified when a system
becomes a superconductor and can be of either sign. We
also found, however, thatωc at which ∆ W becomes neg-
ative rapidly increases with the coupling strength and at
strong coupling becomes comparable to the bandwidth.
In the CB model, which, we believe, is most appropriate
for the application to the cuprates, ∆WK = ∆ W (∞) is
quite small, and at strong coupling a negative ∆ W (ωc)
up to ωc ∼ 1eV is nearly compensated by the optical
integral between ωc and “infinity”, which, in practice, is | 2 | 2 | 1001.0764.pdf |
4
an energy of interband transitions, which is roughly 2 eV .
This would be consistent with Refs. 8,9.
We begin with formulating our calculational basis in
the next section. Then we take up the four cases and
consider in each case the extent to which the Kubo sum is
satisfied up to the order of bandwidth and the functional
form and the sign of ∆W (ωc). The last section presents
our conclusions.
II. OPTICAL INTEGRAL IN NORMAL AND
SUPERCONDUCTING STATES
The generic formalism of the computation of the op-
tical conductivity and the optical integral has been dis-
cussed several times in the literature21–23,26,29 and we
just list the formulas that we used in our computations.
The conductivityσ(Ω) and the optical integral W (ωc)
are given by (see for example Ref. 35).
σ′(Ω) = Im
[
− Π(Ω)
Ω + iδ
]
= − Π ′′(Ω)
Ω + πδ(Ω) Π ′(Ω)
(7a)
W (ωc) =
∫ ωc
0
σ′(Ω) dΩ = −
∫ ωc
0+
Π ′′(Ω)
Ω dΩ + π
2 Π ′(0)
(7b)
where ‘X′’ and ‘X′′’ stand for real and imaginary parts
of X. We will restrict with T = 0. The polarization
operator Π(Ω) is (see Ref. 36)
Π( iΩ) = T
∑
ω
∑
⃗k
(∇⃗kε⃗k)2
(
G(iω, ⃗k)G(iω + iΩ ,⃗k) + F (iω, ⃗k)F (iω + iΩ ,⃗k)
)
(8a)
Π ′′(Ω) = − 1
π
∑
⃗k
(∇⃗kε⃗k)2
∫ 0
−Ω
dω
(
G′′(ω, ⃗k)G′′(ω + Ω ,⃗k) + F ′′(ω, ⃗k)F ′′(ω + Ω ,⃗k)
)
(8b)
Π ′(Ω) = 1
π2
∑
⃗k
(∇⃗kε⃗k)2
∫ ′ ∫ ′
dx dy
(
G′′(x,⃗k)G′′(y, ⃗k) + F ′′(x,⃗k)F ′′(y, ⃗k)
) nF (y) − nF (x)
y − x (8c)
where
∫′ denotes the principal value of the integral,∑
⃗k is understood to be 1
N
∑
⃗k,(N is the number of lat-
tice sites), nF (x) is the Fermi function which is a step
function at zero temperature, G and F are the normal
and anomalous Greens functions. given by 37
For a NS, G(ω, ⃗k) = 1
ω − Σ( k, ω) − ε⃗k + iδ (9a)
For a SCS, G(ω, ⃗k) = Zk,ωω + ε⃗k
Z2
k,ω (ω2 − ∆ 2
k,ω) − ε2
⃗k + iδsgn(ω)
(9b)
F (ω, ⃗k) = Zk,ω∆ k,ω
Z2
k,ω(ω2 − ∆ 2
k,ω) − ε2
⃗k + iδsgn(ω)
(9c)
where Zk,ω = 1 − Σ( k,ω)
ω , and ∆ k,ω, is the SC gap. Fol-
lowing earlier works 31,33, we assume that the fermionic
self-energy Σ( k, ω) predominantly depends on frequency
and approximate Σ( k, ω) ≈ Σ( ω) and also neglect the
frequency dependence of the gap, i.e., approximate ∆ k,ω
by a d−wave ∆ k. The lattice dispersion ε⃗k is taken from
Ref. 38. To calculate WK , one has to evaluate the Kubo
term in Eq.3 wherein the distribution function n⃗k, is cal-
culated from
n(ε⃗k) = −2
∫ 0
−∞
dω
2π G′′(ω, ⃗k) (10)
The 2 is due to the trace over spin indices. We show the
distribution functions in the NS and SCS under different
circumstances in Fig 2.
The ⃗k-summation is done over first Brillouin zone for a
2-D lattice with a 62x62 grid. The frequency integrals are
done analytically wherever possible, otherwise performed
using Simpson’s rule for all regular parts. Contributions
from the poles are computed separately using Cauchy’s
theorem. For comparison, in all four cases we also calcu-
lated FGT sum rule by replacing
∫
d2k = dΩ kdǫkνǫk,Ω k
and keeping ν constant. We remind that the FGT is
the result when one assumes that the integral in W (ωc)
predominantly comes from a narrow region around the
Fermi surface.
We will first use Eq 3 and compute WK in NS and SCS.
This will tell us about the magnitude of ∆ W (ωc = ∞).
We next compute the conductivity σ(ω) using the equa-
tions listed above, find W (ωc) and ∆ W (ωc) and compare
∆ f(ωc) and ∆ WK .
For simplicity and also for comparisons with earlier
studies, for BCSI, EB, and MFLI models we assumed
that the gap is just a constant along the FS. For CB
model, we used ad−wave gap and included into consid-
eration the fact that, if a CB is a spin fluctuation, its
propagator develops a resonance when the pairing gap is
d−wave. | 3 | 3 | 1001.0764.pdf |
5
FIG. 2: Distribution functions in four cases (a) BCSI model,
where one can see that forε > 0, SC >NS implying KE in-
creases in the SCS. (b) The original MFLI model of Ref. 30,
where forε > 0, SC <NS, implying KE decreases in the SCS.
(c) Our version of MFLI model (see text) and (d) the CB
model. In both cases, SC>NS, implying KE increases in the
SCS. Observe that in the impurity-free CB model there is no
jump inn(ǫ) indicating lack of fermionic coherence. This is
consistent with ARPES 39
A. The BCS case
In BCS theory the quantity Z(ω) is given by
ZBCSI (ω) = 1 + Γ
√
∆ 2 − (ω + iδ)2 (11)
and
Σ BCSI (ω) = ω (Z(ω) − 1) = iΓ ω√
(ω + iδ)2 − ∆ 2 (12)
This is consistent with having in the NS, Σ = iΓ in accor-
dance with Eq 6. In the SCS, Σ( ω) is purely imaginary
for ω > ∆ and purely real for ω < ∆. The self-energy
has a square-root singularity at ω = ∆.
It is worth noting that Eq.12 is derived from the in-
tegration over infinite band. If one uses Eq.6 for finite
band, Eq.12 acquires an additional frequency dependence
at large frequencies of the order of bandwidth (the low
frequency structure still remains the same as in Eq.12).
In principle, in a fully self-consistent analysis, one should
indeed evaluate the self-energy using a finite bandwidth.
In practice, however, the self-energy at frequencies of or-
der bandwidth is generally much smaller thanω and con-
tribute very little to optical conductivity which predom-
inantly comes from frequencies where the self-energy is
comparable or even larger thanω. Keeping this in mind,
below we will continue with the form of self-energy de-
rived form infinite band. We use the same argument for
all four models for the self-energy.
For completeness, we first present some well known
results about the conductivity and optical integral for a
constant DOS and then extend the discussion to the case
where the same calculations are done in the presence of
a particular lattice dispersion.
0.1 0.2 0.3 0.4
−4
−2
0
ω in eV
W SC −W NS
∆ W (BCSI without lattice)
Γ =70 meV
Γ =50 meV
Γ =3.5 meV
FIG. 3: The BCSI case with a dispersion linearized around the
Fermi surface. Evolution of the difference of optical integrals
in the SCS and the NS with the upper cut-off ωc Observe
that the zero crossing point increases with impurity scatte ring
rate Γ and also the ‘dip’ spreads out with increasing Γ. ∆ =
30meV
For a constant DOS, ∆ W (ωc) = WSC (ωc) − WNS (ωc)
is zero at ωc = ∞ and Kubo sum rule reduces to FGT
sum rule. In Fig. 3 we plot for this case ∆ W (ωc) as a
function of the cutoff ωc for different Γ ′s. The plot shows
the two well known features: zero-crossing point is below
2∆ in the clean limit Γ<< ∆ and is roughly 2Γ in the
dirty limit21,40 The magnitude of the ‘dip’ decreases quite
rapidly with increasing Γ. Still, there is always a point
of zero crossing and ∆W (ωc) at large ωc approaches zero
from below.
We now perform the same calculations in the presence
of lattice dispersion. The results are summarized in Figs
4,5, and 6.
Fig 4 shows conductivities σ(ω) in the NS and the SCS
and Kubo sums WK plotted against impurity scattering
Γ. We see that the optical integral in the NS is always
greater than in the SCS. The negative sign of ∆WK is
simply the consequence of the fact that nk is larger in the
NS for ǫk < 0 and smaller for ǫk < 0, and ∇2ε⃗k closely
follows −ε⃗k for our choice of dispersion 38), Hence nk is
larger in the NS for ∇2ε⃗k > 0 and smaller for ∇2ε⃗k <
0 and the Kubo sum rule, which is the integral of the
product ofnk and ∇2ε⃗k (Eq. 3), is larger in the normal
state.
We also see from Fig. 4 that ∆ WK decreases with Γ
reflecting the fact that with too much impurity scattering
there is little difference innk between NS and SCS.
Fig 5 shows the optical sum in NS and SCS in clean
and dirty limits (the parameters are stated in the fig-
ure). This plot shows that the Kubo sums are almost
completely recovered by integrating up to the bandwidth
of 1eV : the recovery is 95% in the clean limit and ∼ 90%
in the dirty limit. In Fig 6 we plot ∆ W (ωc) as a function
of ωc in clean and dirty limits. ∆ W (∞) is now non-zero,
in agreement with Fig. 4 and we also see that there is | 4 | 4 | 1001.0764.pdf |
6
0 0.5 10
0.5
1
ω in eV
σ ( ω )
Conductivities (BCSI)
NS
SC
2 ∆
0 50 100160
180
200
Γ in meV
W K in meV
BCSI
SC
NS
FIG. 4: Top - a conductivity plot for the BCSI case in the
presence of a lattice. The parameters are ∆ = 30meV , Γ =
3.5 meV . Bottom – the behavior of Kubo sums. Note that (a)
the spectral weight in the NS is always greater in the SCS, (b)
the spectral weight decreases with Γ, and (c) the difference
between NS and SCS decreases as Γ increases.
little variation of ∆W (ωc) at above 0 .1 − 0.3eV what
implies that for larger ωc, ∆ W (ωc) ≈ ∆ WK >> ∆ f(ωc).
To make this more quantitative, we compare in Fig. 6
∆ W (ωc) obtained for a constant DOS, when ∆ W (ωc) =
∆ f(ωc), and for the actual lattice dispersion, when
∆ W (ωc) = ∆ WK + ∆ f(ωc). In the clean limit there
is obviously little cutoff dependence beyond 0 .1eV , i.e.,
∆ f(ωc) is truly small, and the difference between the
two cases is just ∆ WK . In the dirty limit, the situation
is similar, but there is obviously more variation with ωc,
and ∆ f(ωc) becomes truly small only above 0 .3eV . Note
also that the position of the dip in ∆ W (ωc) in the clean
limit is at a larger ωc in the presence of the lattice than
in a continuum.
B. The Einstein boson model
We next consider the case of electrons interacting with
a single boson mode which by itself is not affected by su-
perconductivity. The primary candidate for such mode is
an optical phonon. The imaginary part of the NS self en-
ergy has been discussed numerous times in the literature.
We make one simplifying assumption – approximate the
DOS by a constant in calculating fermionic self-energy.
We will, however, keep the full lattice dispersion in the
calculations of the optical integral. The advantage of this
0 0.5 10
0.5
1
ω c in eV
W( ω c )/W( ∞ )
Normal State Optical Sum (BCSI)
Dirty Limit
Clean Limit
0 0.5 10
0.5
1
ω c in eV
W( ω c )/W( ∞ )
Superconducting State Optical Sum (BCSI)
Dirty Limit
Clean Limit
FIG. 5: The evolution of optical integral in NS(top) and
SCS(bottom) for BCSI case. Plots are made for clean limit
(solid lines, Γ = 3.5 meV ) and dirty limit (dashed lines,
Γ = 150 meV ) for ∆ = 30 meV . Observe that (a) W (0) = 0
in the NS, but has a non-zero value in the SCS because of the
δ-function (this value decreases in the dirty limit), and (b)
the flat region in the SCS is due to the fact that σ′(ω) = 0 for
Ω < 2∆. Also note that ∼ 90 − 95% of the spectral weight is
recovered up to 1 eV
approximation is that the self-energy can be computed
analytically. The full self-energy obtained with the lat-
tice dispersion is more involved and can only be obtained
numerically, but its structure is quite similar to the one
obtained with a constant DOS.
The self-energy for a constant DOS is given by
Σ( iω) = − i
2π λn
∫
dǫkd(iΩ) χ(iΩ) G(ǫk, iω + iΩ) (13)
where
χ(iΩ) = ω2
0
ω2
0 − (iΩ) 2 (14)
and λn is a dimensionless electron-boson coupling. Inte-
grating and transforming to real frequencies, we obtain
Σ ′′(ω) = − π
2 λnωo Θ( |ω| − ωo)
Σ ′(ω) = − 1
2 λnωo log
⏐
⏐
⏐
⏐
ω + ωo
ω − ωo
⏐
⏐
⏐
⏐ (15)
In the SCS, we obtain for ω < 0
Σ ′′(ω) = − π
2 λnωo Re
(
ω + ωo√
(ω + ωo)2 − ∆ 2
) | 5 | 5 | 1001.0764.pdf |
7
0 0.1 0.2 0.3
0
20
40
ω c in eV
W SC ( ω c ) − W NS ( ω c )
∆ W (BCSI−clean limit)
with lattice
without lattice
0.1 0.3 0.5
0
10
ω c in eV
W SC ( ω c ) − W NS ( ω c )
∆ W (BCSI−dirty limit)
0.5 1
−1
0
ω c in eV
∆ W
Larger ω c
FIG. 6: Evolution of ∆ W in the presence of a lattice (solid
line) compared with the case of no lattice(a constant DOS,
dashed line) for clean and dirty limits. ∆ = 30meV , Γ =
3.5 meV (clean limit), Γ = 150 meV (dirty limit)
Σ ′(ω) = − 1
2 λnωo Re
∫
dω′ 1
ω2o − ω′2 − iδ
ω + ω′
√
(ω + ω′)2 − ∆ 2
(16)
Observe that Σ ′′(ω) is no-zero only for ω < −ωo − ∆.
Also, although it does not straightforwardly follow from
Eq. 16, but real and imaginary parts of the self-energy
do satisfy Σ′(ω) = −Σ ′(−ω) and Σ ′′(ω) = Σ ′′(−ω).
Fig7 shows conductivities σ(ω) and Kubo sums WK
as a function of the dimensionless coupling λ. We see
that, like in the previous case, the Kubo sum in the NS
is larger than that in the SCS. The difference ∆WK is
between 5 and 8 meV.
Fig 8 shows the evolution of the optical integrals. Here
we see the difference with the BCSI model – only about
75% of the optical integral is recovered, both in the NS
and SCS, when we integrate up to the bandwidth of 1eV .
The rest comes from higher frequencies.
In Fig 9 we plot ∆ W (ωc) as a function of ωc. We see
the same behavior as in the BCSI model in a clean limit
– ∆W (ωc) is positive at small frequencies, crosses zero
at some ωc, passes through a deep minimum at a larger
frequency, and eventually saturates at a negative value at
the largestwc. However, in distinction to BCSI model,
∆ W (ωc) keeps varying with ωc up a much larger scale
and saturates only at around 0 .8eV . In between the dip
at 0 .1eV and 0 .8eV , the behavior of the optical integral is
predominantly determined by the variation of the cut-off
term ∆f(ωc) as evidenced by a close similarity between
the behavior of the actual ∆ W and ∆ W in the absence
0.5 1 0
0.1
0.2
Ω in eV
σ ( Ω )
σ in NS and SCS(EB model)
NS
SC
2 ∆ + ω o
ω o
0 0.2 0.4 0.6
190
200
λ (the coupling constant)
W K (meV)
Kubo Sum (EB model)
SC
NS
FIG. 7: Top- conductivities in the NS and the SCS for the EB
model. The conductivity in the NS vanishes belowω0 because
of no phase space for scattering. Bottom - Kubo sums as a
function of coupling. Observe thatWK in the SCS is below
that in the NS. We set ωo = 40 meV , ∆ = 30 meV , λ = .5
0 0.5 1
0.2
0.6
1
ω c in eV
W( ω c )/W( ∞ )
Normal State (EB model)
0 0.5 1
0.2
0.6
1
ω c in eV
W( ω c )/W( ∞ )
Superconducting State (EB model)
FIG. 8: Evolution of the optical integrals in the EB model.
Note thatW (0) has a non zero value at T = 0 in the NS
because the self-energy at small frequencies is purely real and
linear in ω, hence the polarization bubble Π(0) ̸= 0, as in an
ideal Fermi gas. Parameters are the same as in fig. 7 | 6 | 6 | 1001.0764.pdf |
8
0.2 0.4 0.6 0.8
−40
0
ω c in eV
W SC ( ω c ) − W NS ( ω c )
∆ W (EB model)
with lattice
without lattice
∆ W K
FIG. 9: ∆ W vs the cut-off for the EB model. It remains neg-
ative for larger cut-offs. Parameters are the same as before.
The dot indicates the value of ∆W (∞) = ∆ WK
of the lattice (the dashed line in Fig. 9).
C. Marginal Fermi liquid model
For their analysis of the optical integral, Norman and
P´ epin30 introduced a phenomenological model for the self
energy which fits normal state scattering rate measure-
ments by ARPES41. It constructs the NS Σ
′′
(ω) out
of two contributions - impurity scattering and electron-
electron scattering which they approximated phenomeno-
logically by the marginal Fermi liquid form ofαω at small
frequencies6 (MFLI model). The total Σ
′′
is
Σ ′′(ω) = Γ + α|ω|f
( ω
ωsat
)
(17)
where ωsat is about ∼ 1
2 of the bandwidth, and f(x) ≈ 1
for x < 1 and decreases for x > 1. In Ref 30 f(x) was
assumed to scale as 1 /x at large x such that Σ ′′ is flat at
large ω. The real part of Σ( ω) is obtained from Kramers-
Kr¨ onig relations. For the superconducting state, they
obtained Σ
′′
by cutting off the NS expression on the lower
end at some frequency ω1 (the analog of ω0 + ∆ that we
had for EB model):
Σ ′′(ω) = (Γ + α|ω|)Θ( |ω| − ω1) (18)
where Θ( x) is the step function. In reality, Σ
′′
which fits
ARPES in the NS has some angular dependence along the
Fermi surface42, but this was ignored for simplicity. This
model had gained a lot of attention as it predicted the
optical sum in the SCS to be larger than in the NS, i.e.,
∆W > 0 at large frequencies. This would be consistent
with the experimental findings in Refs. 8,9 if, indeed, one
identifies ∆W measured up to 1eV with ∆ WK .
We will show below that the sign of ∆ W in the MFLI
model actually depends on how the normal state results
are extended to the superconducting state and, moreover,
will argue that ∆WK is actually negative if the extension
is done such that at α = 0 the results are consistent with
BCSI model. However, before that, we show in Figs 10-
12 the conductivities and the optical integrals for the
original MFLI model.
0.2 0.6 1
0
0.1
0.2
ω in eV
σ ( ω )
Conductivities (Original MFLI)
NS
SC
∆ + ω 1
0 50 100120
130
140
Γ (meV)
Original MFLI
W K (meV)
SC
NS
α =0.75
FIG. 10: Top –the conductivities in the NS and SCS in the
original MFLI model of Ref.30. We set Γ = 70meV , α = 0 .75,
∆ = 32 meV , ω1 = 71 meV . Note that σ′(ω) in the SCS
begins at Ω = ∆ + ω1. Bottom – the behavior of WK with Γ.
In Fig 10 we plot the conductivities in the NS and the
SCS and Kubo sums WK vs Γ at α = 0 .75 showing that
the spectral weight in the SCS is indeed larger than in the
NS. In Fig 11 we show the behavior of the optical sums
W(ωc) in NS and SCS. The observation here is that only
∼ 75−80% of the Kubo sum is recovered up to the scale of
the bandwidth implying that there is indeed a significant
spectral weight well beyond the bandwidth. And in Fig
12 we show the behavior of ∆W (wc). We see that it does
not change sign and remain positive at all ωc, very much
unlike the BCS case. Comparing the behavior of W (wc)
with and without a lattice (solid and dashed lines in Fig.
12) we see that the ‘finite bandwidth effect’ just shifts the
curve in the positive direction. We also see that the solid
line flattens above roughly half of the bandwidth, i.e., at
these frequencies ∆W (ωc) ≈ ∆ WK . Still, we found that
∆ W continues going down even above the bandwidth
and truly saturates only at about 2 eV (not shown in the
figure) supporting the idea that there is ‘more’ left to
recover from higher frequencies.
The rationale for ∆ WK > 0 in the original MFLI
model has been provided in Ref. 30. They argued that
this is closely linked to the absence of quasiparticle peaks
in the NS and their restoration in the SCS state because
the phase space for quasiparticle scattering at low ener-
gies is smaller in a superconductor than in a normal state. | 7 | 7 | 1001.0764.pdf |
9
0 0.5 10
0.2
0.4
0.6
0.8
1
ω c in eV
W( ω c )/W( ∞ )
NS (Original MFLI)
0 0.5 1
0.2
0.6
1
ω c in eV
W( ω c )/W( ∞ )
SCS (Original MFLI)
FIG. 11: The evolution of the optical integral in the NS (top)
and the SCS (bottom) in the original MFLI model. Parame-
ters are the same as above. Note that only∼ 75 − 80% of the
spectral weight is recovered up to 1 eV .
0.2 0.4 0.6 0.8 1
0
10
20
ω c in eV
W SC ( ω c ) − W NS ( ω c )
NS and SCS ∆ W (Original MFLI)
with lattice
without lattice
∆ W K
FIG. 12: Evolution of the difference of the optical integrals in
the SCS and the NS with the upper cut-off ωc. Parameters are
the same as before. Observe that the optical sum in the SCS
is larger than in the NS and that ∆W has not yet reached
∆ WK up to the bandwidth. The dashed line is the FGT
result.
This clearly affectsnk because it is expressed via the full
Green’s function and competes with the conventional ef-
fect of the gap opening. The distribution function from
this model, which we show in Fig.2b brings this point
out by showing that in a MFLI model, atǫ < 0, nk in a
superconductor is larger than nk in the normal state, in
clear difference with the BCSI case.
We analyzed the original MFLI model for various pa-
rameters and found that the behavior presented in Fig.
12, where ∆W (ωc) > 0 for all frequencies, is typical but
0 20 40
175
185
195
Γ in meV
W K (meV)
Original MFLI in BCS limit
SC
NS
α =0.05
FIG. 13: Behavior of WK with Γ for the original MFLI model
at very small α = 0 .05. We set ω1 = ∆ = 32 meV . Observe
the inconsistency with WK in the BCSI model in Fig 4.
0.2 0.4 0.6 0.8
−0.4
0
0.4
ω c in eV
W SC ( ω c ) − W NS ( ω c )
Original MFLI−two sign changes
FIG. 14: The special case of α = 1 .5,Γ = 5 meV , other pa-
rameters the same as in Fig. 10. These parameters are chosen
to illustrate that two sign changes (indicated by arrows in the
figure) are also possible within the original MFLI model.
not not a generic one. There exists a range of parame-
tersα and Γ where ∆ WK is still positive, but ∆ W (ωc)
changes the sign twice and is negative at intermediate
frequencies. We show an example of such behavior in
Fig14. Still, for most of the parameters, the behavior of
∆W (ωc) is the same as in Fig. 12.
On more careful looking we found the problem with the
original MFLI model. We recall that in this model the
self-energy in the SCS state was obtained by just cutting
the NS self energy atω1 (see Eq.18). We argue that
this phenomenological formalism is not fully consistent,
at least for smallα. Indeed, for α = 0, the MFLI model
reduces to BCSI model for which the behavior of the self-
energy is given by Eq. (12). This self-energy evolves with
ωand Σ
′′
has a square-root singularity at ω = ∆ + ωo
(with ωo = 0). Meanwhile Σ
′′
in the original MFLI model
in Eq. (18) simply jumps to zero at ω = ω1 = ∆, and
this happens for all values of α including α = 0 where the
MFLI and BCSI model should merge. This inconsistency
is reflected in Fig 13, where we plot the near-BCS limit
of MFLI model by taking a very smallα = 0 .05. We
see that the optical integral WK in the SCS still remains
larger than in the NS over a wide range of Γ, in clear
difference with the exactly known behavior in the BCSI | 8 | 8 | 1001.0764.pdf |
10
0 0.5 10
0.4
0.8
Conductivities (Corrected MFLI)
σ ( ω )
ω in eV
NS
SC
2 ∆
0 50 100
100
120
W K (meV)
Γ in meV
Corrected MFLI
SC
NS
FIG. 15: Top – σ(ω) in the NS and the SCS in the ‘corrected’
MFLI model with the feedback from SC on the quasiparticle
damping:iΓ term transforms into Γ
√
−ω 2+∆ 2 . In the SCS σ
now begins at Ω = 2∆. The parameters are same as in Fig.
10. Bottom – the behavior of Kubo sum with Γ. Observe
thatW (ωc) in the NS is larger than in the SCS.
0.2 0.4 0.6 0.8
−10
0
10
ω c in eV
W SC ( ω c ) − W NS ( ω c )
Corrected MFLI
without lattice
with lattice
∆ W K
FIG. 16: Evolution of the difference of the optical integrals
between the SCS and the NS with the upper cut-offωc for
the “corrected” MFLI model. Now ∆ W (ωc) is negative above
some frequency. Parameters are same as in the Fig 15.
model, whereWK is larger in the NS for all Γ (see Fig.
4). In other words, the original MFLI model does not
have the BCSI theory as its limiting case.
We modified the MFLI model is a minimal way by
changing the damping term in a SCS to Γ
√
−ω2+∆ 2 to be
consistent with BCSI model. We still use Eq. (18) for
the MFL term simply because this term was introduced
in the NS on phenomenological grounds and there is no
way to guess how it gets modified in the SCS state with-
out first deriving the normal state self-energy microscop-
ically (this is what we will do in the next section). The
results of the calculations for the modified MFLI model
are presented in Figs. 15 and 16. We clearly see that the
behavior is now different and ∆WK < 0 for all Γ. This
is the same behavior as we previously found in BCSI
and EB models. So we argue that the ‘unconventional’
behavior exhibited by the original MFLI model is most
likely the manifestation of a particular modeling incon-
sistency. Still, Ref. 30 made a valid point that the fact
that quasiparticles behave more close to free fermions in
a SCS than in a NS, and this effect tends to reverse the
signs of ∆WK and of the kinetic energy 43. It just hap-
pens that in a modified MFLI model the optical integral
is still larger in the NS.
D. The collective boson model
We now turn to a more microscopic model- the CB
model. The model describes fermions interacting by ex-
changing soft, overdamped collective bosons in a partic-
ular, near-critical, spin or charge channel31,44,45. This
interaction is responsible for the normal state self-energy
and also gives rise to a superconductivity. A peculiar
feature of the CB model is that the propagator of a col-
lective boson changes belowTc because this boson is not
an independent degree of freedom (as in EB model) but
is made out of low-energy fermions which are affected by
superconductivity32.
The most relevant point for our discussion is that this
model contains the physics which we identified above as
a source of a potential sign change of ∆WK . Namely,
at strong coupling the fermionic self-energy in the NS
is large because there exists strong scattering between
low-energy fermions mediated by low-energy collective
bosons. In the SCS, the density of low-energy fermions
drops and a continuum collective excitations becomes
gaped. Both effects reduce fermionic damping and lead
to the increase ofWK in a SCS. If this increase exceeds a
conventional loss of WK due to a gap opening, the total
∆ WK may become positive.
The CB model has been applied numerous times to the
cuprates, most often under the assumption that near-
critical collective excitations are spin fluctuations with
momenta nearQ = ( π, π). This version of a CB bo-
son is commonly known as a spin-fermion model. This
model yieldsdx2−y2 superconductivity and explains in a
quantitative way a number of measured electronic fea-
tures of the cuprates, in particular the near-absence of
the quasiparticle peak in the NS of optimally doped and
underdoped cuprates39 and the peak-dip-hump structure
in the ARPES profile in the SCS 31,32,46,47. In our analy-
sis we assume that a CB is a spin fluctuation.
The results for the conductivity within a spin-fermion
model depend in quantitative (but not qualitative) way
on the assumption for the momentum dispersion of a col-
lective boson. This momentum dependence comes from | 9 | 9 | 1001.0764.pdf |
11
high-energy fermions and is an input for the low-energy
theory. Below we follow Refs. 31,33 and assume that
the momentum dependence of a collective boson is flat
near (π, π). The self energy within such model has been
worked out consistently in Ref. 31,33. In the normal
state
Σ ′′(ω) = − 1
2 λnωsf log
(
1 + ω2
ω2
sf
)
Σ ′(ω) = −λnωsf arctan ω
ωsf
(19)
where λn is the spin-fermion coupling constant, and ωsf
is a typical spin relaxation frequency of overdamped spin
collective excitations with a propagator
χ(q ∼ Q, Ω) = χQ
1 − i Ω
ωsf
(20)
where χQ is the uniform static susceptibility. If we use
Ornstein-Zernike form of χ(q) and use either Eliashberg
45 or FLEX computational schemes 48, we get rather sim-
ilar behavior of Σ as a function of frequency and rather
similar behavior of optical integrals.
The collective nature of spin fluctuations is reflected in
the fact that the coupling λ and the bosonic frequency
ωsf are related: λ scales as ξ2, where ξ is the bosonic
mass (the distance to a bosonic instability), and ωsf ∝
ξ−2 (see Ref. 49). For a flat χ(q ∼ Q) the product λωsf
does not depend on ξ and is the overall dimensional scale
for boson-mediated interactions.
In the SCS fermionic excitations acquire a gap. This
gap affects fermionic self-energy in two ways: directly, via
the change of the dispersion of an intermediate boson in
the exchange process involving a CB, and indirectly, via
the change of the propagator of a CB. We remind our-
selves that the dynamics of a CB comes from a particle-
hole bubble which is indeed affected by ∆.
The effect of a d−wave pairing gap on a CB has been
discussed in a number of papers, most recently in 31. In
a SCS a gapless continuum described by Eq. (20) trans-
forms into a gaped continuum, with a gap about 2∆ and
a resonance atω = ω0 < 2∆, where for a d−wave gap we
define ∆ as a maximum of a d−wave gap.
The spin susceptibility near ( π, π) in a superconductor
can generally be written up as
χ(q ∼ Q, Ω) = χQ
1 − i Π(Ω)
ωsf
(21)
where Π is evaluated by adding up the bubbles made
out of two normal and two anomalous Green’s functions.
Below 2∆, Π(Ω) is real (∼ Ω 2/∆ for small Ω), and the
resonance emerges at Ω = ω0 at which Π( ω0) = ωsf . At
frequencies larger than 2∆, Π(Ω) has an imaginary part,
and this gives rise to a gaped continuum inχ(Ω).
The imaginary part of the spin susceptibility around
the resonance frequency ω0 is31
χ
′′
(q, Ω) = πZoω0
2 δ(Ω − ω0) (22)
where Zo ∼ 2 ωsf χ0/ ∂Π
∂ω |Ω= ω0
. The imaginary part
of the spin susceptibility describing a gaped continuum
exists for for Ω≥ 2∆ and is
χ
′′
(q, Ω) = Im
[
χ0
1 − 1
ωsf
( 4∆ 2
Ω D(4∆ 2
Ω 2 ) + iΩ K2(1 − 4∆ 2
Ω 2 )
)
]
≈ Im
[
χ0
1 − 1
ωsf
( π∆ 2
Ω + i π
2 Ω
)
]
for Ω >> 2∆ (23)
In Eq. (23) D(x) = K1(x)−K2(x)
x , and K1(x) and K2(x)
are Elliptic integrals of first and second kind. The real
part ofχ is obtained by Kramers-Kr¨ onig transform of the
imaginary part.
Substituting Eq 6 for χ(q, Ω) into the formula for the
self-energy one obtains Σ ′′(ω) in a SCS state as a sum of
two terms 31
Σ ′′(ω) = Σ ′′
A(ω) + Σ ′′
B(ω) (24)
where,
Σ ′′
A(ω) = πZo
2 λnωo Re
(
ω + ωo√
(ω + ωo)2 − ∆ 2
)
comes from the interaction with the resonance and
Σ ′′
B(ω) = −λn
∫ |E|
2∆
dx Re ω + x
√
(ω + x)2 − ∆ 2
x
ωsf
K2
(
1 − 4∆ 2
x2
)
[
1 − 4∆ 2
xωsf
D
( 4∆ 2
x2
) ] 2
+
[
x
ωsf
K2
(
1 − 4∆ 2
x2
) ] 2 (25)
comes from the interaction with the gaped continuum. The real par t of Σ is obtained by Kramers-Kr¨ onig trans- | 10 | 10 | 1001.0764.pdf |
12
form of the imaginary part.
0 0.5 10
0.2
0.4
ω in eV
σ ( ω )
Conductivities (CB model λ =1)
NS
SC
2 ∆ + ω o
0 0.5 1
0.2
0.6
1
ω in eV
σ ( ω )
Conductivities (CB model λ =10)
NS
SC
2 ∆ + ω o
FIG. 17: Conductivities and ∆ W for a fixed λωsf . Top –
ωsf = 26 meV ,λ = 1, ωo = 40 meV ,Zo = 0 .77 Bottom –
ωsf = 2 .6 meV ,λ = 10, ωo = 13 .5 meV ,Zo = 1 .22. The zero
crossing for ∆ W is not affected by a change in λ because it
is determined only by λωsf . We set ∆ = 30 meV .
1 2 3
120
160
200
λ (coupling)
CB model ( Ω o =40 meV)
W K (meV)
SCS
NS
FIG. 18: The behavior of Kubo sums in the CB model. Note
that the spectral weight in the NS is always larger than in the
SCS. We setωsf = 26 meV ,λ = 1, and ∆ = 30 meV .
We performed the same calculations of conductivities
and optical integrals as in the previous three cases. The
results are summarized in Figs. 17 - 22. Fig 17 shows con-
ductivities in the NS and the SCS for two couplingsλ = 1
and λ = 10 (keeping λωsf constant). Other parameters
Zo and ωo are calculated according to the discussion after
Eq 21. for ωsf = 26 meV , λ = 1, we find ωo = 40 meV ,
Zo = 0 .77. And for ωsf = 2 .6 meV , λ = 10, we find
ωo = 13 .5 meV , Zo = 1 .22. Note that the conductivity
in the SCS starts at 2∆ + ωo (i.e. the resonance energy
0 0.5 1
0.2
0.6
1
ω c in eV
W( ω c )/W( ∞ )
NS Optical Sums (CB model)
0 0.5 1
0.2
0.6
1
ω c in eV
W( ω c )/W( ∞ )
SCS Optical Sums (CB model)
FIG. 19: The evolution of the optical integrals in the NS
and the SCS in the CB model. Note that about∼ 75% of
the spectral weight is recovered up to 1 eV . We set ωsf =
26 meV ,λ = 1, and ∆ = 30 meV .
0 0.5 1
−20
0
20
ω c in eV
W SC ( ω c ) − W NS ( ω c )
∆ W (CB model λ =1)
with lattice
without lattice
∆ W K
0.2 0.6 1
−20
0
20
ω c in eV
W SC ( ω c ) − W NS ( ω c )
∆ W (CB model λ =10)
with lattice
without lattice
∆ W K
FIG. 20: ∆ W (in meV) for λ = 1(top) and λ = 10(bottom).
We used ωsf = 26 meV /λ and ∆ = 30 meV . The zero crossing
is not affected because we keep λωsf constant. The notable
difference is the widening of the dip at a larger λ. | 11 | 11 | 1001.0764.pdf |
13
00
0.5
1
ε
n( ε )
NS
SC Jump
λ =1
0 0
0.5
1
ε
n( ε )
NS
SC λ =7
FIG. 21: Distribution functions n(ǫ) for CB model for λ = 1
and λ = 7 and a constant ωsf = 26 meV . We set ∆ = 30 meV .
For smaller λ (top), quasiparticles near the FS are well defined
as indicated by the well pronounced jump in n(ǫ). For λ = 7,
n(ǫ) is rather smooth implying that a coherence is almost lost.
Some irregularities is the SCS distribution function are du e
to finite sampling in the frequency domain. The irregulariti es
disappear when finer mesh for frequencies is chosen.
shows up in the optical gap), where as in the BCSI case
it would have always begun from 2∆. In Fig 18 we plot
the Kubo sumsWK vs coupling λ. We see that for all λ,
WK in the NS stays larger than in the SCS. Fig 19 shows
the cutoff dependence of the optical integrals W (ωc) for
λ = 1 separately in the NS and the SCS. We again see
that only about 73% of the Kubo sum is recovered up
to the bandwidth of 1eV indicating that there is a sig-
nificant amount left to recover beyond this energy scale.
Fig 20 shows ∆W for the two different couplings. We
see that, for both λ’s, there is only one zero-crossing for
the ∆ W curve, and ∆ W is negative at larger frequen-
cies. The only difference between the two plots is that
for larger coupling the dip in ∆W gets ‘shallower’. Ob-
serve also that the solid line in Fig. 20 is rather far away
from the dashed line atωc > 1meV , which indicates that,
although ∆ W (ωc) in this region has some dependence on
ωc, still the largest part of ∆ W (ωc) is ∆ WK , while the
contribution from ∆ f(ωc) is smaller.
0 0.5 1
0.1
0.2
ω in eV
σ ( ω )
Conductivities (CB model−larger λ ω sf )
NS
SC
2 ∆ + ω o
0.2 0.4 0.6 0.8
0
4
8
ω c in eV
W SC ( ω c ) − W NS ( ω c ) in meV
∆ W (CB model−larger λ ω sf )
with lattice
without lattice
FIG. 22: Top – conductivity at a larger value of ωsf λ (ωsf =
26 meV ,λ = 7) consistent with the one used in Ref.33). Bot-
tom – ∆ W with and without lattice. Observe that the fre-
quency of zero crossing of ∆ W enhances compared to the case
of a smaller λωsf and becomes comparable to the bandwidth.
At energies smaller than the bandwidth, ∆ W > 0, as in the
Norman- P´ epin model.
0 2 4 6
10
30
50
Self Energy prefactor
δ KE in meV
λ =10
λ =1
FIG. 23: Kinetic energy difference between the SCS and the
NS,δKE We set λ to be either λ = 1 or λ = 10 and varied ωsf
thus changing the overall prefactor in the self-energy. At w eak
coupling ( λ = 1) the behavior is BCS-like – δKE is positive
and increases with the overall factor in the self-energy. At
strong coupling (λ = 7), δKE shows a reverse trend at larger
ωsf .
The negative sign of ∆ W (ωc) above a relatively small
ωc ∼ 0.1 − 0.2eV implies that the ‘compensating’ ef-
fect from the fermionic self-energy on ∆ W is not strong
enough to overshadow the decrease of the optical inte-
gral in the SCS due to gap opening. In other words,the
CB model displays the same behavior as BCSI, EB, and | 12 | 12 | 1001.0764.pdf |
14
modified MFLI models. It is interesting that this holds
despite the fact that for largeλ CB model displays the
physics one apparently needs to reverse the sign of ∆ WK
– the absence of the quasiparticle peak in the NS and its
emergence in the SCS accompanied by the dip and the
hump at larger energies. The absence of coherent quasi-
particle in the NS at largeλ is also apparent form Fig
21 where we show the normal state distribution functions
for two differentλ. For large λ the jump (which indicates
the presence of quasiparticles) virtually disappears.
On a more careful look, we found that indifference of
δW (ωc) to the increase of λ is merely the consequence of
the fact that above we kept λωsf constant. Indeed, at
small frequencies, fermionic self-energy in the NS is Σ ′ =
λω, Σ” = λ2ω2/(λωsf ), and both Σ ′ and Σ ′′ increase
with λ if we keep λωsf constant. But at frequencies larger
than ωsf , which we actually probe by ∆ W (ωc), the self-
energy essentially depends only on λωsf , and increasing λ
but keeping λωsf constant does not bring us closer to the
physics associated with the recovery of electron coherence
in the SCS. To detect this physics, we need to see how
things evolve when we increaseλωsf above the scale of
∆ , i.e., consider a truly strong coupling when not only
λ≫ 1 but also the normal state Σ NS (ω ≥ ∆) >> ∆.
To address this issue, we took a larger λ for the same
ωsf and re-did the calculation of the conductivities and
optical integrals. The results for σ(ω) and ∆ W (ωc) are
presented in Fig. 22. We found the same behavior as be-
fore, i.e., ∆WK is negative. But we also found that the
larger is the overall scale for the self-energy, the larger is a
frequency of zero-crossing of ∆W (ωc). In particular, for
the same λ and ωsf that were used in Ref. 33 to fit the NS
conductivity data, the zero crossing is at ∼ 0.8 eV which
is quite close to the bandwidth. This implies that at a
truly strong coupling the frequency at which ∆W (ωc)
changes sign can well be larger than the bandwidth of
1eV in which case ∆ W integrated up to the bandwidth
does indeed remain positive. Such behavior would be
consistent with Refs.8,9. we also see from Fig. 22 that
∆WK becomes small at a truly strong coupling, and over
a wide range of frequencies the behavior of ∆ W (ωc) is
predominantly governed by ∆ f(ωc), i.e. by the cut-off
term.50 The implication is that, to first approximation,
∆ WK can be neglected and positive ∆ W (wc) integrated
to a frequency where it is still positive is almost compen-
sated by the integral over larger frequencies. This again
would be consistent with the experimental data in Refs.
8,9.
It is also instructive to understand the interplay be-
tween the behavior of ∆ W (ωc) and the behavior of the
difference of the kinetic energy between the SCS and the
NS,δKE . We computed the kinetic energy as a function
of λωsf and present the results in Fig. 23 for λ = 1 and
10. For a relatively weak λ = 1 the behavior is clearly
BCS like- δKE > 0 and increases with increasing λωsf .
However, at large λ = 10, we see that the kinetic energy
begin decreasing at large λωsf and eventually changes
sign. The behavior of δKE at a truly strong coupling is
consistent with earlier calculation of the kinetic energy
for Ornstein-Zernike form of the spin susceptibility43.
We clearly see that the increase of the zero crossing
frequency of ∆ W (ωc) at a truly strong coupling is cor-
related with the non-BCS behavior of δKE . At the same
time, the behavior of δW (ωc) is obviously not driven by
the kinetic energy as eventually δW (ωc) changes sign and
become negative. Rather, the increase in the frequency
range where ∆W (ωc) remains positive and non-BCS be-
havior of δKE are two indications of the same effect that
fermions are incoherent in the NS but acquire coherence
in the SCS.
III. CONCLUSION
In this work we analyzed the behavior of optical in-
tegrals W (ωc) ∝
∫ωc
o σ(ω)dω and Kubo sum rules in
the normal and superconducting states of interacting
fermionic systems on a lattice. Our key goal was to
understand what sets the sign of ∆WK = ∆ W (∞) be-
tween the normal and superconducting states and what
is the behavior ofW (ωc) and ∆ W (ωc) at finite ωc. In a
weak coupling BCS superconductor, ∆ W (ωc) is positive
at ωc < 2∆ due to a contribution from superfluid den-
sity, but becomes negative at larger ωc, and approach a
negative value of ∆ WK . Our study was motivated by fas-
cinating optical experiments on the cuprates 7–10 . In over-
doped cuprates, there is clear indication 11 that ∆ W (ωc)
becomes negative above a few ∆, consistent with BCS
behavior. In underdoped cuprates, two groups argued8,9
that ∆ W integrated up to the bandwidth remains posi-
tive, while the other group argued 10 that it is negative.
The reasoning why ∆ WK may potentially change sign
at strong coupling involves the correlation between −WK
and the kinetic energy. In the BCS limit, kinetic en-
ergy obviously increases in a SCS because of gap opening,
hence−WK increases, and ∆ WK is negative. At strong
coupling, there is a counter effect – fermions become more
mobile in a SCS due to a smaller self-energy.
We considered four models: a BCS model with impu-
rities, a model of fermions interacting with an Einstein
boson, a phenomenological MFL model with impurities,
and a model of fermions interacting with collective spin
fluctuations. In all cases, we found that ∆WK is neg-
ative, but how it evolves with ωc and how much of the
sum rule is recovered by integrating up to the bandwidth
depends on the model.
The result most relevant to the experiments on the
cuprates is obtained for the spin fluctuation model.
We found that at strong coupling, the zero-crossing of
δW(ωc) occurs at a frequency which increases with the
coupling strength and may become larger than the band-
width at a truly strong coupling. Still, at even larger
frequencies, ∆W (ωc) is negative. | 13 | 13 | 1001.0764.pdf |
15
Acknowledgements
We would like to thank M. Norman, Tom Timusk,
Dmitri Basov, Chris Homes, Nicole Bontemps, Andres
Santander-Syro, Ricardo Lobo, Dirk van der Marel, A.
Boris, E. van Heumen, A. B. Kuzmenko, L. Benfato, and
F. Marsiglio for many discussions concerning the infrared
conductivity and optical integrals and thank A. Boris, E.
van Heumen, J. Hirsch, and F. Marsiglio for the com-
ments on the manuscript. The work was supported by
nsf-dmr 0906953.
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50 In this respect, our results are consistent with the analysi s | 14 | 14 | 1001.0764.pdf |
16
of ∆ W (ωc) in a system without a lattice (Ref. 51). The
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(2004). | 15 | 15 | 1001.0764.pdf |
arXiv:1001.0770v1 [astro-ph.HE] 5 Jan 2010
2009 Fermi Symposium, Washington, D.C., Nov. 2-5 1
VERITAS Observations of Blazars
W. Benbow for the VERIT AS Collaboration
Harvard-Smithsonian Center for Astrophysics, F .L. Whippl e Observatory , PO Box 6369, Amado, AZ 85645,
USA
The VERITAS array of four 12-m diameter imaging atmospheric -Cherenkov telescopes in southern Arizona is
used to study very high energy (VHE; E >100 GeV) γ-ray emission from astrophysical objects. VERITAS is
currently the most sensitive VHE γ-ray observatory in the world and one of the VERITAS collabor ation’s Key
Science Projects (KSP) is the study of blazars. These active galactic nuclei (AGN) are the most numerous class
of identified VHE sources, with ∼ 30 known to emit VHE photons. More than 70 AGN, almost all of wh ich
are blazars, have been observed with the VERITAS array since 2007, in most cases with the deepest-ever VHE
exposure. These observations have resulted in the detectio n of VHE γ-rays from 16 AGN (15 blazars), including
8 for the first time at these energies. The VERITAS blazar KSP i s summarized in this proceeding and selected
results are presented.
1. Introduction
Active galactic nuclei are the most numerous class
of identified VHE γ-ray sources. These objects emit
non-thermal radiation across ∼ 20 orders of magnitude
in energy and rank among the most powerful particle
accelerators in the universe. A small fraction of AGN
possess strong collimated outflows (jets) powered by
accretion onto a supermassive black hole (SMBH).
VHEγ-ray emission can be generated in these jets,
likely in a compact region very near the SMBH event
horizon. Blazars, a class of AGN with jets pointed
along the line-of-sight to the observer, are of par-
ticular interest in the VHE regime. Approximately
30 blazars, primarily high-frequency-peaked BL Lacs
(HBL), are identified as sources of VHEγ-rays, and
some are spectacularly variable on time scales com-
parable to the light crossing time of their SMBH (∼ 2
min; [1]). VHE blazar studies probe the environment
very near the central SMBH and address a wide range
of physical phenomena, including the accretion and
jet-formation processes. These studies also have cos-
mological implications, as VHE blazar data can be
used to strongly constrain primordial radiation fields
(see the extragalactic background light (EBL) con-
straints from, e.g., [2, 3]).
VHE blazars have double-humped spectral energy
distributions (SEDs), with one peak at UV/X-ray en-
ergies and another at GeV/TeV energies. The ori-
gin of the lower-energy peak is commonly explained
as synchrotron emission from the relativistic electrons
in the blazar jets. The origin of the higher-energy
peak is controversial, but is widely believed to be the
result of inverse-Compton scattering of seed photons
off the same relativistic electrons. The origin of the
seed photons in these leptonic scenarios could be the
synchrotron photons themselves, or photons from an
external source. Hadronic scenarios are also plausible
explanations for the VHE emission, but generally are
not favored.
Contemporaneous multi-wavelength (MWL) obser-
vations of VHE blazars, can measure both SED peaks
and are crucial for extracting information from the
observations of VHE blazars. They are used to con-
strain the size, magnetic field and Doppler factor of
the emission region, as well as to determine the origin
(leptonic or hadronic) of the VHEγ-rays. In leptonic
scenarios, such MWL observations are used to mea-
sure the spectrum of high-energy electrons producing
the emission, as well as to elucidate the nature of the
seed photons. Additionally, an accurate measure of
the cosmological EBL density requires accurate mod-
eling of the blazar’s intrinsic VHE emission that can
only be performed with contemporaneous MWL ob-
servations.
2. VERITAS
VERITAS, a stereoscopic array of four 12-m
atmospheric-Cherenkov telescopes located in Arizona,
is used to study VHEγ-rays from a variety of astro-
physical sources [4]. VERITAS began scientific obser-
vations with a partial array in September 2006 and has
routinely observed with the full array since Septem-
ber 2007. The performance metrics of VERITAS in-
clude an energy threshold of∼ 100 GeV, an energy
resolution of ∼ 15%, an angular resolution of ∼ 0.1◦,
and a sensitivity yielding a 5 σ detection of a 1% Crab
Nebula flux object in <30 hours 1. VERITAS has an
active maintenance program (e.g. frequent mirror re-
coating and alignment) to ensure its continued high
performance over time, and an upgrade improving
both the camera (higher quantum-efficiency PMTs)
and the trigger system has been proposed to the fund-
ing agencies.
1A VERITAS telescope was relocated during Summer 2009,
increasing the array’s sensitivity by a factor ∼ 1.3.
eConf C091122 | 0 | 0 | 1001.0770.pdf |
2 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
3. VERITAS Blazar KSP
VERITAS observes for ∼ 750 h and ∼ 250 h each
year during periods of astronomical darkness and par-
tial moonlight, respectively. The moonlight observa-
tions are almost exclusively used for a blazar discovery
program, and a large fraction of the dark time is used
for the blazar KSP, which consists of:
• A VHE blazar discovery program ( ∼ 200 h / yr):
Each year ∼ 10 targets are selected to receive
∼ 10 h of observations each during astronomi-
cal darkness. These data are supplemented by
discovery observations during periods of partial
moonlight.
• A target-of-opportunity (ToO) observation pro-
gram (∼ 50 h / yr): VERITAS blazar obser-
vations can be triggered by either a VERI-
TAS blazar discovery, a VHE flaring alert (>2
Crab) from the blazar monitoring program of
the Whipple 10-m telescope or from another
VHE instrument, or a lower-energy flaring alert
(optical, X-ray or Fermi-LAT). Should the guar-
anteed allocation be exhausted, further time can
be requested from a pool of director’s discre-
tionary time.
• Multi-wavelength (MWL) studies of VHE
blazars (∼ 50 h / yr + ToO): Each year one
blazar receives a deep exposure in a pre-planned
campaign of extensive, simultaneous MWL (X-
ray, optical, radio) measurements. ToO observa-
tion proposals for MWL measurements are also
submitted to lower-energy observatories (e.g.
Swift) and are triggered by a VERITAS discov-
ery or flaring alert.
• Distant VHE blazar studies to constrain the ex-
tragalactic background light (EBL): Here dis-
tant targets are given a higher priority in the
blazar discovery program, as well as for the
MWL observations of known VHE blazars, par-
ticularly those with hard VHE spectra.
4. Blazar Discovery Program
The blazars observed in the discovery program are
largely high-frequency-peaked BL Lac objects. How-
ever, the program also includes IBLs (intermediate-
peaked) and LBLs (low-peaked), as well as flat spec-
trum radio quasars (FSRQs), in an attempt to in-
crease the types of blazars known to emit VHEγ-rays.
The observed targets are drawn from a target listcon-
taining objects visible to the telescopes at reasonable
zenith angles (− 8◦ < δ < 72◦), without a previously
published VHE limit below 1.5% Crab, and with a
measured redshiftz < 0. 3. To further the study of the
EBL a few objects having a large ( z > 0. 3) are also
included in the target list. The target list includes:
• All nearby ( z < 0. 3) HBL and IBL recom-
mended as potential VHE emitters in [5, 6, 7].
• The X-ray brightest HBL ( z < 0. 3) in the recent
Sedentary [8] and ROXA [9] surveys.
• Four distant ( z > 0. 3) BL Lac objects recom-
mended by [5, 10].
• Several FSRQ recommended as potential VHE
emitters in [6, 11].
• All nearby ( z < 0. 3) blazars detected by
EGRET [12].
• All nearby ( z < 0. 3) blazars contained in the
Fermi-LAT Bright AGN Sample [13].
• All sources ( |b| > 10◦) detected by Fermi-LAT
where extrapolations of their MeV-GeV γ-ray
spectrum (including EBL absorption; assuming
z = 0.3 if the redshift is unknown) indicates a
possible VERITAS detection in less than 20 h.
This criteria is the focus of the 2009-10 VERI-
TAS blazar discovery program.
5. VERITAS AGN Detections
VERITAS has detected VHE γ-ray emission from
16 AGN (15 blazars), including 8 VHE discoveries.
These AGN are shown in Table I, and each has been
detected by the Large Area Telescope (LAT) instru-
ment aboard the Fermi Gamma-ray Space Telescope.
Every blazar discovered by VERITAS was the sub-
ject of ToO MWL observations to enable modeling of
its simultaneously-measured SED. The known VHE
blazars detected by VERITAS were similarly the tar-
gets of MWL observations.
5.1. Recent VERITAS Blazar Discoveries
Prior to the launch of Fermi VERITAS had discov-
ered VHE emission from 2 blazars. These included
the first VHE-detected IBL, W Comae [14, 15], and
the HBL 1ES 0806+524 [16]. VERITAS has discov-
ered 6 VHE blazars since the launch of Fermi. Three
of these were initially observed by VERITAS prior to
the release of Fermi-LAT results, due to the X-ray
brightness of the synchrotron peaks of their SEDs.
VHE emission from 3C 66A was discovered by VER-
ITAS in September 2008 [17] during a flaring episode
that was also observed by the Fermi-LAT [18]. The
observed flux above 200 GeV was 6% of the Crab Neb-
ula flux and the measured VHE spectrum was very
soft (ΓVHE ∼ 4. 1). RGB J0710+591 was detected
eConf C091122 | 1 | 1 | 1001.0770.pdf |
2009 Fermi Symposium, Washington, D.C., Nov. 2-5 3
Table I VERITAS AGN Detections. The only non-blazar
object is the radio galaxy M 87. The blazars discovered
at VHE by VERITAS are marked with a dagger.
Object Class Redshift
M 87 FR I 0.004
Mkn 421 HBL 0.030
Mkn 501 HBL 0.034
1ES 2344+514 HBL 0.044
1ES 1959+650 HBL 0.047
W Comae† IBL 0.102
RGB J0710+591† HBL 0.125
H 1426+428 HBL 0.129
1ES 0806+524† HBL 0.138
1ES 0229+200 HBL 0.139
1ES 1218+304 HBL 0.182
RBS 0413† HBL 0.190
1ES 0502+675† HBL 0.341
3C 66A† IBL 0.444?
PKS 1424+240† IBL ?
VER J0521+211† ? ?
(∼ 5.5σ; 3% Crab flux above 300 GeV; Γ VHE ∼ 2. 7)
during VERITAS observations from December 2008
to March 2009. The initial announcement of the VHE
discovery [19] led to its discovery above 1 GeV in the
Fermi-LAT data using a special analysis. RBS 0413,
a relatively distant HBL (z=0.19), was observed for
16 h good-quality live time in 2008-092. These data
resulted in the discovery of VHE gamma-rays ( >270γ,
∼ 6σ) at a flux ( >200 GeV) of ∼ 2% of the Crab Neb-
ula flux. The discovery [20] was announced simultane-
ously with the LAT MeV-GeV detection. The VHE
and other MWL observations, including Fermi-LAT
data, for each of these three sources will be the sub-
ject of a joint publication involving both the VERI-
TAS and LAT collaborations.
5.2. Discoveries Motivated by Fermi-LAT
The successful VHE discovery observations by
VERITAS of three blazars was motivated primarily
by results from the first year of LAT data taking. In
particular, the VHE detections of PKS 1424+240 [21]
and 1ES 0502+675 [22] were the result of VERITAS
observations triggered by the inclusion of these objects
in the Fermi-LAT Bright AGN List [13]. The former
is only the third IBL known to emit VHE gamma-
rays, and the latter is the most distant BL Lac object
2RBS 0413 was observed further by VERITAS in Fall 2009.
(z = 0 . 341) detected in the VHE band. In addition,
VER J0521+211, likely associated with the radio-loud
AGN RGB J0521.8+2112, was detected by VERTAS
in∼ 4 h of observations in October 2009 [23]. These
observations were motivated by its identification as a
>30 GeV γ-ray source in the public Fermi-LAT data.
Its VHE flux is 5% of the Crab Nebula flux, placing it
among the brightest VHE blazars detected in recent
years. VERITAS later observed even brighter VHE
flaring from VER J0521+211 in November 2009 [24],
leading to deeper VHE observations.
6. Blazars Upper Limits
More than 50 VHE blazar candidates were observed
by VERITAS between September 2007 and June 2009.
The total exposure on the 49 non-detected candi-
dates is∼ 305 h live time (average of 6.2 h per can-
didate). Approximately 55% of the total exposure is
split amongst the 27 observed HBL. The remainder is
divided amongst the 8 IBL (26%), 5 LBL (6%), and 9
FSRQ (13%). There are no clear indications of signifi-
cant VHEγ-ray emission from any of these 49 blazars
[25]. However, the observed significance distribution is
clearly skewed towards positive values (see Figure 1).
A stacking analysis performed on the entire data sam-
ple shows an overall excess of 430γ-rays, correspond-
ing to a statistical significance of 4.8 σ, observed from
the directions of the candidate blazars. The IBL and
HBL targets make up 96% of the observed excess. Ob-
servations of these objects also comprise∼ 80% of the
total exposure. An identical stacked analysis of all
the extragalactic non-blazar targets observed, but not
clearly detected (>5σ), by VERITAS does not show
a significant excess ( ∼ 120 h exposure). The stacked
excess persists using alternate methods for estimating
the background at each blazar location, and with dif-
ferent event selection criteria (e.g.soft cutsoptimized
for sources with Γ VHE > 4). The distribution of VHE
flux upper limits is shown in Figure 1. These 49 VHE
flux upper limits are generally the most-constraining
ever reported for these objects.
7. Multi-wavelength Studies of VHE
Blazars
During the first three seasons of VERITAS obser-
vations, pre-planned extensive MWL campaigns were
organized for three blazars 1ES 2344+514 (2007-08),
1ES 1218+304 (2008-09) and 1ES 0229+200 (2009-
10 - ongoing). In addition, numerous ToO MWL-
observation campaigns were performed. These include
campaigns for every blazar/AGN discovered by VER-
ITAS, and all include Swift (XRT and UVOT) data.
All MWL campaigns on the VHE blazars discovered
eConf C091122 | 2 | 2 | 1001.0770.pdf |
4 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
σ−5 −4 −3 −2 −1 0 1 2 3 4 5
Entries
0
2
4
6
8
10
12
Crab Flux %0 2 4 6 8 10 12 14
Entries
0
2
4
6
8
10
12
14
16
18
Figure 1: (Left) The preliminary significance measured from each of the 49 non-detected candidates using standard
analysis cuts. The curve shows a Gaussian distribution, wit h mean zero and standard deviation one, normalized to the
number of blazars. A similar result is obtained using analys is cuts optimized for soft-spectrum sources. (Right) The
distribution of flux upper limits for the non-detected blaza rs in percentage of Crab Nebula flux above the observation
threshold. The time-weighted average limit is less than ∼ 2% Crab flux.
since the launch of Fermi include LAT detections. In
addition, several MWL campaigns on the well-studied
VHE blazars Mkn 421 and Mkn 501 (please see the
contributions of D. Gall and A. Konopelko in these
proceedings) were also performed. Highlights of these
campaigns include:
• 1ES 2344+514: A major (50% Crab) VHE flare,
along with correlations of the VHE and X-ray
flux were observed from this HBL. The VHE
and X-ray spectra harden during bright states,
and a synchrotron self-Compton (SSC) model
can explain the observed SED in both the high
and low states [26].
• 1ES 1218+304: This HBL flared during VER-
ITAS MWL observations. Its unusually hard
VHE spectrum strongly constrains the EBL.
The observed flaring rules out kpc-scale jet emis-
sion as the explanation of the spectral hardness
and places the EBL constraints on more solid-
footing [27, 28].
• 1ES 0806+524: The observed SED of this new
VHE HBL can be explained by an SSC model
[16].
• W Comae: This IBL, the first discovered at
VHE, flared twice in 2008 [14, 15]. Modeling of
the SED is improved by including an external-
Compton (EC) component in an SSC interpre-
tation.
• 3C 66A: This IBL flared at VHE and MeV-GeV
energies in 2008[17, 18]. Similar to W Comae
and PKS 1424+240, modeling of observed SED
suggests a strong EC component in addition to
an SSC component.
• Mkn 421: This HBL exhibited major flaring be-
havior for several months in 2008. Correlations
of the VHE and X-ray flux were observed, along
with spectral hardening with increased flux in
both bands [29].
• RGB J0710+591: Modeling the SED of this
HBL with an SSC model yields a good fit to
the data. The inclusion of an external Compton
component does not improve the fit.
• PKS 1424+240: The broadband SED of this IBL
(at unknown redshift) is well described by an
SSC model favoring a redshift of less than 0.1
[21]. Using the photon index measured with
Fermi-LAT in combination with recent EBL ab-
sorption models, the VERITAS data indicate
that the redshift of PKS 1424+240 is less than
0.66.
8. Conclusions
The first two years of the VERITAS blazar KSP
were highly successful. Highlights include the detec-
tion of more than a 16 VHE blazars with the obser-
vations almost always having contemporaneous MWL
data. Among these detections are 8 VHE blazar dis-
coveries, including the first three IBLs known to emit
VHEγ-rays. All but a handful of the blazars on the
initial VERITAS discovery target list were observed,
and the flux limits generated for those not VHE de-
tected are generally the most-constraining ever. The
excess seen in the stacked blazar analysis suggests
that the initial direction of the VERITAS discovery
program was well justified, and that follow-up obser-
vations of many of these initial targets will result in
VHE discoveries. In addition, the Fermi-LAT is iden-
tifying many new compelling targets for the VERITAS
blazar discovery program. These new candidates have
already resulted in 3 VHE blazar discoveries. The
future of the VERITAS blazar discovery program is
clearly very bright.
The MWL aspect of the VERITAS blazar KSP has
also been highly successful. Every VERITAS obser-
vation of a known, or newly discovered, VHE blazar
has been accompanied by contemporaneous MWL ob-
servations. These data have resulted in the identifica-
eConf C091122 | 3 | 3 | 1001.0770.pdf |
2009 Fermi Symposium, Washington, D.C., Nov. 2-5 5
tion of correlated VHE and X-ray flux variability, as
well as correlated spectral hardening in both the VHE
and X-ray bands. The VHE MWL observations were
performed in both ”quiescent” and flaring states for
some of the observed blazars. For the observed HBL
objects, the SEDs can be well described by a simple
SSC model in both high and low states. However, an
additional external Compton component is necessary
to adequately fit the SEDs of the IBL objects.
The Fermi-LAT is already having a significant im-
pact on the blazar KSP. In future seasons, the VER-
ITAS blazar discovery program will focus its dis-
covery program on hard-spectrum blazars detected
by Fermi-LAT, and will likely have a greater focus
on high-risk/high-reward objects at larger redshifts
(0. 3 < z < 0. 7). In addition, the number of VHE
blazars studied in pre-planned MWL campaigns will
increase as data from the Fermi-LAT will be publicly
available. In particular, the extensive pre-planned
MWL campaigns will focus on objects that are note-
worthy for the impact their data may have on under-
standing the EBL. The simultaneous observations of
blazars by VERITAS and Fermi-LAT will completely
resolve the higher-energy SED peak, often for the first
time, enabling unprecedented constraints on the un-
derlying blazar phenomena to be derived.
Acknowledgments
This research is supported by grants from the US
Department of Energy, the US National Science Foun-
dation, and the Smithsonian Institution, by NSERC in
Canada, by Science Foundation Ireland, and by STFC
in the UK. We acknowledge the excellent work of the
technical support staff at the FLWO and the collab-
orating institutions in the construction and operation
of the instrument.
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eConf C091122 | 4 | 4 | 1001.0770.pdf |
2009 Fermi Symposium, Washington, D.C., Nov. 2-5 1
Submillimeter Variability and the Gamma-ray Connection inFermi
Blazars
A. Strom
Univ. of Arizona, AZ 85721, USA
A. Siemiginowska, M. Gurwell, B. Kelly
CfA, MA 02138, USA
We present multi-epoch observations from the Submillimeter Array (SMA) for a sample of 171 bright blazars,
43 of which were detected by Fermi during the first three months of observations. We explore the correlation
between their gamma-ray properties and submillimeter observations of their parsec-scale jets, with a special
emphasis on spectral index in both bands and the variability of the synchrotron component. Subclass is de-
termined using a combination of Fermi designation and the Candidate Gamma-Ray Blazar Survey (CGRaBS),
resulting in 35 BL Lac objects and 136 flat-spectrum radio quasars (FSRQs) in our total sample. We calculate
submillimeter energy spectral indices using contemporaneous observations in the 1 mm and 850 micron bands
during the months August–October 2008. The submillimeter light curves are modeled as first-order continuous
autoregressive processes, from which we derive characteristic timescales. Our blazar sample exhibits no differ-
ences in submillimeter variability amplitude or characteristic timescale as a function of subclass or luminosity.
All of the the light curves are consistent with being produced by a single process that accounts for both low
and high states, and there is additional evidence that objects may be transitioning between blazar class during
flaring epochs.
1. INTRODUCTION
The timescales on which high-amplitude flaring
events occur in blazars indicate that much of the en-
ergy is being produced deep within the jet on small,
sub-parsec scales [1, 2]. Understanding if/how emis-
sion differs between blazar subclasses (i.e., BL Lacs
objects and flat-spectrum radio quasars (FSRQs))
may offer important insight into the similarity be-
tween blazars and, furthermore, can provide con-
straints on the formation and acceleration of the jets
themselves.
For the synchrotron component of blazar spectra,
the low-frequency spectral break due to synchrotron
self-absorption moves to higher frequencies as one
measures closer to the base of the jet [2]. This of-
ten places the peak of the spectrum in the millime-
ter and submillimeter bands, where the emission is
optically-thin and originates on parsec and sub-parsec
scales [3], allowing direct observation of the most com-
pact regions near the central engine. The high en-
ergy γ-ray emission originates as a Compton process,
typically a combination of synchrotron-self-Compton
(SSC) and external-radiation-Compton (ERC). De-
pending on the source properties, the synchrotron
photons or external photons are upscattered by the
same population of electrons that emit the millimeter
and submillimeter spectra. Therefore the submillime-
ter and γ-ray emission are closely linked and give the
full information about the source emission.
A systematic study of the submillimeter properties
of the entire sample ofFermi blazars has yet to be con-
ducted and is one of the primary goals of our work. We
present here preliminary analysis of the submillimeter
properties of Fermi blazars detected by the Submil-
limeter Array1 (SMA) at 1mm and 850 µm, including
an investigation of variable behavior and the deter-
mination of submillimeter energy spectral indices. In
addition, we consider the connection to the observed
γ-ray indices and luminosities.
2. SMA BLAZARS
The Submillimeter Array [4] consists of eight 6 m
antennas located near the summit of Mauna Kea. The
SMA is used in a variety of baseline configurations
and typically operates in the 1mm and 850 µm win-
dows, achieving spatial resolution as fine as 0.25” at
850µm. The sources used as phase calibrators for the
array are compiled in a database known as the SMA
Calibrator List2 [5]. Essentially a collection of bright
objects (stronger than 750 mJy at 230 GHz and 1 Jy
at 345 GHz), these sources are monitored regularly,
both during science observations and dedicated ob-
serving tracks.
To select our sample, we identified objects in the
calibrator list that were also classified as BL Lacs or
FSRQs by the Candidate Gamma-Ray Blazar Sur-
vey [6, CGRaBS]. Of the 243 total objects in the
calibrator list, 171 (35 BL Lacs and 136 FSRQs)
have positive blazar class identifications, although
there are three sources (J0238+166, J0428-379, and
1The Submillimeter Array is a joint project between the
Smithsonian Astrophysical Observatory and the Academia
Sinica Institute of Astronomy and Astrophysics and is funded
by the Smithsonian Institution and the Academia Sinica.
2http://sma1.sma.hawaii.edu/callist/callist.html
eConf C091122
arXiv:1001.0806v1 [astro-ph.HE] 6 Jan 2010 | 0 | 0 | 1001.0806.pdf |
2 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
Figure 1: The SMA light curves for 3C 454.3. The open
circles represent the 850µm observations, and the open
triangles represent the 1mm observations.
J1751+096) which have conflicting classifications be-
tween Fermi and CGRaBS. Some blazars found in the
calibrator list have been studied extensively (e.g., 3C
279 and 3C 454.3) but the SMA blazars have not been
studied collectively.
Forty-four of the objects in our total blazar sample
were detected by Fermi and can be found in the cata-
log of LAT Bright AGN Sources (LBAS) from Abdo et
al. [7]. J0050-094 has no redshift in either the LBAS
catalog or CGRaBS and is not included in our study.
Of the 43 remaining sources, 14 are BL Lac objects
and 29 are FSRQs, with 0 .03 ≤z≤2.19.
We examined submillimeter light curves for all of
the SMA blazars, with observations beginning in ap-
proximately 2003 (see Figure 1). Typically, the 1mm
band is much more well-sampled in comparison to the
850m band, but visual inspection reveals that the reg-
ularity and quality of observations vary greatly from
source to source. Many of the objects exhibit non-
periodic variability, either in the form of persistent,
low-amplitude fluctuations or higher amplitude flar-
ing behavior.
2.1. Submillimeter Properties
Submillimeter Luminosities. Since we are pri-
marily concerned with comparisons to Fermi observa-
tions, we note that only 129 of theSMA blazars (23 BL
Lacs and 106 FSRQs) were observed by the SMA in
either band during the three months August-October
2008. For these objects, submillimeter luminosities
are calculated in the standard way:
νeLνe = 4πD2
L
νobsFobs
1 + z , (1)
where DL is the luminosity distance, νobs is the fre-
quency of the observed band, and Fobs is the average
Figure 2: Variability index for our sample (top: 1mm,
bottom: 850 µm), with FSRQs as the hatched
distribution and BL Lacs as the solid distribution. There
is no signicant difference in the class distributions in
either band; the “tail” to the left is populated by objects
with errors larger than the intrinsic variability.
flux (in erg cm−2 s−1 Hz−1) over the three month pe-
riod. We adopt a lambda cold dark matter cosmology
with values of H0 = 71 km s −1 Mpc−1, Ω M = 0.27,
and Λ = 0.73.
Energy Spectral Indices.We derive submillime-
ter spectral energy indices from observations quasi-
simultaneous with the Fermi observations. To be con-
sistent with the use ofαγ, we define spectral energy in-
dex as νFν = ν−αS and calculate αS from the average
of the energy spectral indices over the corresponding
three months. We only calculate αS for the 16 objects
(8 BL Lacs and 35 FSRQs) with observations at both
1mm and 850µm during this time frame.
3. VARIABILITY ANALYSIS
3.1. Variability Index
We roughly characterize the level of variability of
each source using the variability index from Hovatta
et al. [8]:
V = (Fmax −σFmax ) −(Fmin + σFmin )
(Fmax −σFmax ) + (Fmin + σFmin ) (2)
Figure 2 shows the distribution for theSMA blazars.
Objects with V ≤0 are typically unsuitable for more
eConf C091122 | 1 | 1 | 1001.0806.pdf |
2009 Fermi Symposium, Washington, D.C., Nov. 2-5 3
detailed variability analysis for one of two reasons:
(1) too few data points or (2) flux measurement un-
certainties on the order of the amplitude of observed
variability. It is important to note that, due to dis-
crepancies between the sampling frequency in both
bands, the variability indices for the 850µm band may
be artificially depressed due to the fact that there are
not always corresponding measurements at higher fre-
quencies during flaring epochs.
3.2. First-Order Continuous
Autoregression
We follow the method of Kelly et al. [9], who model
quasar optical light curves as a continuous time first-
order autoregressive process (CAR(1)) in order to ex-
tract characteristic time scales and the amplitude of
flux variations. Although flaring behavior is not typi-
cally thought of as an autoregressive process, we find
that the light curves are well-fit by the models and
therefore adopt the method here to study blazar sub-
millimeter light curves.
The CAR(1) process is described by a stochastic
differential equation [9],
dS(t) = 1
τS(t) dt+ σ
√
dtϵ (t) + bdt, (3)
associated with a power spectrum of the form
PX(f) = 2σ2τ2
1 + (2πτf)2 . (4)
In equations 3 and 4, τ is called the “relaxation
time” of the process S(t) and is identified by the
break in PX(f). The power spectrum appears flat
for timescales longer than this and falls off as 1/f2 for
timescales shorter than the characteristic timescale of
the process.
Taking the logarithm of the blazar light curve (in
Jy) to be S(t), we adopt τ (in days) as the character-
istic timescale of variability, after which the physical
process “forgets” about what has happened at time
lags of greater than τ. The two other relevant pa-
rameters, σ and µ = b/a, are the overall amplitude
of variability and the logarithm of mean value of the
light curve, respectively.
In the routine, we construct an autoregressive
model for the light curves for a minimum of 100,000
iterations and calculate the value of τ from the break
in the power spectrum in each instance. Due to the
limited number of observations in the 850 µm band,
we performed this autoregressive analysis only for the
1mm light curves, which typically have more than 10
points per light curve.
This method yielded some surprising results. In
Figure 3, we see that the BL Lacs and FSRQs exhibit
virtually no difference in characteristic timescale, with
Figure 3: Characteristic timescale (days) versus
submillimeter luminosity (erg s−1) in the 1mm band for
all objects. Physically, τ represents a “relaxation
timescale”, the timescale beyond which events are no
longer correlated.
both classes extending across a large range in τ. Be-
cause of the uncertainty for objects with shorter char-
acteristic timescales, it is hard to draw any definitive
conclusions about the differences between classes. It
is important to note that τ does not necessarily rep-
resent a flaring timescale, which is a behavior that
typically operates on a scale of ∼10–100 days and not
on the longer timescales we see in τ.
4. CONNECTION WITH GAMMA-RAYS
In general, we find that in the submillimeter, we
are observing these blazars at or near the peak of the
synchrotron component ( αS ∼ 0), but that Fermi-
detected sources have more negative energy spectral
indices overall than Fermi-nondetected sources. In
Figure 4, we see that while the majority of Fermi
blazars are observed on the rising part of the syn-
chrotron component (at lower energies than the peak),
all of the objects have very steeply fallingγ-ray energy
spectral indexes, putting the γ-ray peak at lower en-
ergies than the observed Fermi band. Knowing that
we are not observing the synchrotron and γ-ray com-
ponents at analagous points in the spectrum may al-
low us to better understand the magnetic field in the
parsec-scale jet region and the population of external
photons that is being upscattered to γ-rays.
In Figure 5, the ratio between Lγ and νLν,1mm re-
flects the division between BL Lacs and FSRQs as well
eConf C091122 | 2 | 2 | 1001.0806.pdf |
4 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
Figure 4: The γ-ray index versus submillimeter index plane. The blazars fall more steeply in the γ-rays than in the
submillimeter band, where most are, in fact, rising. This LAT-detected sample contrasts with the full SMA sample,
where the blazars are more distributed around αS ∼ 0.
as the presence of SSC versus ERC. Here, we use sub-
millimeter luminosity as a proxy for jet power, which
is correlated with the integrated luminosity of the syn-
chrotron component. Elevated γ-ray luminosity with
respect to the synchrotron component (which is often
seen in FSRQs) suggests the upscattering of external
photons off the synchrotron-emitting electrons. These
objects should occupy the upper right of the ratio/jet
power plot, and BL Lacs, which generally exhibit com-
ponents with roughly comparable luminosities, should
occupy the lower left. It is clear from the figure, how-
ever, that many FSRQs exhibit ratios similar to those
of the BL Lacs and vis versa.
Sikora et al. [10] report that, during its flaring
epochs, 3C 454.3 transitions from its typical FSRQ
state to a more BL Lac-like state, where the syn-
chrotron component emits much more strongly com-
pared to the γ-ray component than during its “low
state”. 3C 454.3, which is the highest submillime-
ter luminosity FSRQ in our sample, would then shift
down and to the right in Figure 5 when it enters a
flaring period. For the first three months of the Fermi
mission, 3C 454.3 was not flaring, which may explain
its present location in Figure 5. The three objects for
which there is a type discrepancy between CGRaBS
and LBAS are all FSRQs (in CGRaBS) and exhibit
low luminosity ratios and high luminosity, which sug-
gest they may be undergoing the same changes as 3C
454.3. A possible interpretation of the elevated lumi-
nosity ratios observed in some BL Lacs objects is that
there has been a dramatic increase in γ-ray luminos-
ity due to ERC, which would not be reflected in the
synchrotron component.
5. CONCLUSIONS
The motivation for observing blazars in the sub-
millimeter is to study behavior close to the central
engine, where the jet material is presumably still be-
ing accelerated. The separate emission processes that
contribute to overall SED may present differently in
BL Lacs and FSRQs, allowing us to understand the
similarities and differences between blazar types. We
have investigated these differences between objects in
terms of submillimeter behavior and, in conclusion,
find that
•The SMA blazars exhibit submillimeter energy
spectral indexes that follow the spectral se-
quence interpretation of blazars.
eConf C091122 | 3 | 3 | 1001.0806.pdf |
2009 Fermi Symposium, Washington, D.C., Nov. 2-5 5
Figure 5: Ratio of γ-ray luminosity to submillimeter luminosity in the 1mm band. The location of an object in this
plot should be directly correlated with its blazar “state”, with FSRQs occupying the upper right and BL Lacs the lower
left. Flat-spectrum radio quasar 3C 454.3 is the object with the highest submillimeter luminosity in this plot.
•BL Lacs and FSRQs do not exhibit significant
differences in amplitude of submillimeter vari-
ability or characteristic timescale, but our sam-
ple of BL Lacs may be dominated by high-
peaked BL Lacs (HBLs), which exhibit obser-
vational similarities with FSRQs.
•Blazar submillimeter light curves are consistent
with being produced by a single process that ac-
counts for both high and low states, with char-
acteristic timescales 10 <τrest < 500 days.
•The blazars detected byFermi have synchrotron
peaks at higher frequencies, regardless of sub-
millimeter luminosity.
•FSRQs exhibit higher ratios of γ-ray to sub-
millimeter luminosity than BL Lacs (Figure 5),
but all objects inhabit a region of parameter
space suggesting transitions between states dur-
ing flaring epochs.
As Fermi continues to observe fainter sources, the
sample of objects for which we can perform this type of
analysis will increase and provide better limits on our
results. To understand the physical relevance of these
results, however, it is important to be able to distin-
guish between the difference in variability between BL
Lacs and FSRQs. One avenue for exploring this dif-
ference is to monitor changing submillimeter energy
spectral index and the ratio of γ-ray to submillime-
ter luminosity as functions of time. The full mean-
ing of the results of our autoregressive method is not
yet clear, and will require better-sampled blazar light
curves and the comparison between τrest with physical
timescales such as the synchrotron cooling timescale.
These analyses would allow us to place constraints
on the processes occurring near the base of the jet in
blazars and further understand the intimate connec-
tion between them.
Acknowledgments
This work was supported in part by the NSF
REU and DoD ASSURE programs under Grant no.
0754568 and by the Smithsonian Institution. Par-
tial support was also provided by NASA contract
NAS8-39073 and NASA grant NNX07AQ55G. We
have made use of the SIMBAD database, operated at
CDS, Strasbourg, France, and the NASA/IPAC Ex-
tragalactic Database (NED) which is operated by the
JPL, Caltech, under contract with NASA.
eConf C091122 | 4 | 4 | 1001.0806.pdf |
6 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
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eConf C091122 | 5 | 5 | 1001.0806.pdf |
2009 Fermi Symposium, Washington, D.C., Nov. 2-5 1
Observations of Soft Gamma Ray Sources> 100 keV Using Earth Occultation
with GBM
G.L. Case, M.L. Cherry, J. Rodi
Dept. of Physics & Astronomy, Louisiana State Univ., Baton Rouge, LA 70803, USA
A. Camero-Arranz
Fundaci´ on Espa˜ nola de Ciencia y Tecnolog´ ıa (MICINN), C/Rosario Pino,14-16, 28020-Madrid, Spain
E. Beklen
Middle East Technical University (METU), 06531, Ankara, Turkey
C. A. Wilson-Hodge
NASA Marshall Space Flight Center, Huntsville, AL 35812
P. Jenke
NASA Postdoctoral Program Fellow, NASA Marshall Space Flight Center, Huntsville, AL 35812
P.N. Bhat, M.S. Briggs, V. Chaplin, V. Connaughton, R. Preece
University of Alabama in Huntsville, Huntsville, AL 35899
M.H. Finger
USRA, National Space Science and Technology Center, Huntsville, AL 35899
The NaI and BGO detectors on the Gamma ray Burst Monitor (GBM) on Fermi are now being
used for long term monitoring of the hard X-ray/low energy gamma ray sky. Using the Earth
occultation technique demonstrated previously by the BATSE instrument on the Compton Gamma
Ray Observatory, GBM produces multiband light curves and spectra for known sources and transient
outbursts in the 8 keV - 1 MeV band with its NaI detectors and up to 40 MeV with its BGO. Coverage
of the entire sky is obtained every two orbits, with sensitivity exceeding that of BATSE at energies
below ∼ 25 keV and above ∼ 1.5 MeV. We describe the technique and present preliminary results
after the first ∼ 17 months of observations at energies above 100 keV. Seven sources are detected:
the Crab, Cyg X-1, Swift J1753.5-0127, 1E 1740-29, Cen A, GRS 1915+105, and the transient source
XTE J1752-223.
I. INTRODUCTION
The Gamma ray Burst Monitor (GBM) on Fermi is
currently the only instrument in orbit providing nearly
continuous full sky coverage in the hard X-ray/low
energy gamma ray energy range. The Earth occul-
tation technique, used very successfully on BATSE,
has been adapted to GBM. An initial catalog of 64
sources is currently being monitored and continuously
augmented. At energies above 100 keV, six steady
sources (the Crab, Cyg X-1, Swift J1753.5-0127, 1E
1740-29, Cen A, GRS 1915+105) and one transient
source (XTE J1752-223) have been detected in the
first year of observation. We describe the instrument,
outline the technique, and present light curves for the
seven sources.
II. GBM AND THE EARTH OCCULTATION
OBSERVATIONAL TECHNIQUE
The Gamma ray Burst Monitor is the secondary
instrument onboard the Fermi satellite [1, 2]. It con-
sists of 12 NaI detectors 5 ′′in diameter by 0.5 ′′thick
mounted on the corners of the spacecraft and oriented
such that they view the entire sky not occulted by the
Earth. GBM also contains 2 BGO detectors 5 ′′in di-
ameter by 5 ′′ thick located on opposite sides of the
spacecraft. None of the GBM detectors have direct
imaging capability.
Known sources of gamma ray emission can be mon-
itored with non-imaging detectors using the Earth oc-
cultation technique, as was successfully demonstrated
with BATSE [3, 4]. When a source of gamma rays
is occulted by the Earth, the count rate measured by
the detector will drop, producing a step-like feature.
When the source reappears from behind the Earths
limb, the count rate will increase, producing another
step. The diameter of the Earth seen from Fermi is
∼ 140◦, so roughly 30% of the sky is occulted by the
Earth at any one time. Coupled with the ±35◦slew-
ing of the pointing direction every orbit, this means
that the entire sky is occulted every two orbits. With
an altitude of 565 km, a period of 96 minutes, and
an orbital inclination of 26 .5◦, individual occultation
steps last for ∼10 seconds (Fig. 1).
eConf C091122
arXiv:1001.0955v2 [astro-ph.HE] 6 Jan 2010 | 0 | 0 | 1001.0955.pdf |
2 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
FIG. 1: Single Crab occultation step in a single GBM NaI
detector. Horizontal scale is in seconds centered on the
occultation time. Vertical scale is in measured counts.
The shape of the individual occultation steps de-
pends on energy and occultation angle. Transmis-
sion as a function of time is modeled as T(t) =
exp[−µ(E)A(h)], where µ(E) is the mass attenuation
coefficient of gamma rays at energy E in air and A(h)
is the air mass along the line of sight at a given alti-
tude h(t). Account is taken of the detector response
as it changes as a function of angle across the fit win-
dow. For each source, occultation times are predicted.
Each step is fit over a 4-minute window along with a
quadratic background and using an assumed spectrum
to determine the detector count rate due to the source.
The instrument response is used to convert the count
rate to a flux. Up to 31 steps are possible for a given
source in a day, and these steps are summed to get a
single daily average flux. The GBM occultation sensi-
tivity exceeds that of BATSE at energies below ∼ 25
keV and above ∼ 1.5 MeV [5].
This work uses the GBM CTIME data, with its
8 broad energy channels and 0.256-second resolution,
rebinned to 2-second resolution. The occultation tech-
nique relies on an input catalog of known sources.
Currently, we are monitoring 64 sources. Of these
64 sources, 6 steady sources are detected above 100
keV with a significance of at least 5σafter ∼ 490 days
of observations, and one transient source.
III. RESULTS
The results presented here are preliminary. We
have not completed the fine tuning of our algorithms,
though the average fluxes are not expected to change
much. Future work will include using the GBM
CSPEC data, with its finer energy binning, to exam-
ine the detailed spectra for these sources.
The measured 20 - 50 keV GBM light curves are
compared to Swift’s 15 - 50 keV light curves for sev-
FIG. 2: Crab light curve. Horizontal scale is in modified
Julian days over the 490 day GBM exposure period. Ver-
tical scale is in photons/cm 2/sec/keV averaged over daily
intervals. Horizontal lines show the average flux in each of
five energy bands increasing from top to bottom
eral sources over the same time intervals in ref. [2],
where it is seen that the results measured by the two
instruments compare well. At energies above the up-
per energy limit of ∼ 195 keV of the Swift 22-month
catalog [6], however, the GBM observations provide
the only wide-field monitor available of the low en-
ergy gamma ray sky.
A. Steady Sources
The sources Crab, Cyg X-1, Swift J1753.5-0127, 1E
1740-29, Cen A, and GRS 1915+105 are detected by
GBM at energies above 100 keV. We show GBM light
curves generated from the Earth occultation analysis
in several energy bands with one day resolution for
these six sources in Figures 2 - 7.
Table I gives the fluxes and significances averaged
over all the days from Aug. 12, 2008 (the beginning of
science operations) to Dec. 15, 2009, approximately
490 days.
The Crab (Fig. 2) spectrum in the hard x-ray/low
energy gamma-ray region can be described by a bro-
ken power law, with the spectrum steepening at 100
keV and then hardening at 650 keV [7, 8]. While the
GBM CTIME data do not have the spectral resolution
eConf C091122 | 1 | 1 | 1001.0955.pdf |
2009 Fermi Symposium, Washington, D.C., Nov. 2-5 3
FIG. 3: Cen A light curve. Horizontal scale is in modified
Julian days.
to observe these breaks, GBM is able to see significant
emission above 300 keV, consistent with the canonical
hard spectrum.
Cen A(Fig. 3) is a Sy 2 galaxy that is the brightest
AGN in hard x-rays/low energy gamma rays. It has
a hard spectrum (Γ = 1 .8) and has been observed at
energies > 1 MeV [9]. The GBM results are consis-
tent with this hard spectrum, though GBM does not
have the sensitivity to determine if the hard spectrum
continues beyond 300 keV or if the spectrum cuts off.
Cyg X-1 (Fig. 4) is a HMXB and one of the
first systems determined to contain a black hole. It
has been observed to emit significant emission above
100 keV including a power law tail extending out to
greater than 1 MeV [10, 11]. The GBM results show
significant emission above 300 keV, consistent with
the power law tail observed when Cyg X-1 is in its
hard state.
GRS 1915+105(Fig. 5) is a LMXB with the com-
pact object being a massive black hole. Evidence for
emission above 100 keV has been seen previously [12]
with BATSE. The GBM light curve integrated over
490 days shows significant emission above 100 keV.
1E 1740-29 (Fig. 6) is a LMXB very near the
Galactic Center. It is a microquasar, and spends most
of its time in the low/hard state. Integral observa-
tions indicate the presence of a power law tail above
200 keV [13]. The present GBM results are consis-
tent with this high energy emission. In the future, we
FIG. 4: Cyg X-1 light curve. Horizontal scale is in modi-
fied Julian days.
FIG. 5: GRS 1915+105 light curve. Horizontal scale is in
modified Julian days.
eConf C091122 | 2 | 2 | 1001.0955.pdf |
4 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
TABLE I: Fluxes and Significance in High Energy Bands
50 - 100 keV 100 - 300 keV 300 - 500 keV
Flux Error Signif. Flux Error Signif. Flux Error Signif.
(mCrab) (mCrab) (σ) (mCrab) (mCrab) (σ) (mCrab) (mCrab) (σ)
Crab 1000 3 336 1000 6 182 1000 47 21.2
Cen A 72 4 18 108 7 15 42 47 0.9
Cyg X-1 1130 4 283 1094 8 137 474 50 9.5
GRS 1915+105 121 4 30 49 7 7 41 52 0.8
1E 1740-29 113 5 23 96 10 10 97 68 1.4
SWIFT 1753.5-0127 135 5 27 151 9 17 131 64 2.0
XTE J1752-223 770 16 48 622 30 21 132 218 0.6
FIG. 6: 1E1740-29 light curve. Horizontal scale is in mod-
ified Julian days.
will use the GBM CSPEC data with their finer energy
bins to obtain a fit to the spectrum and compare the
power law index to that measured by Integral.
SWIFT J1753.5-0127 (Fig. 7) is a LMXB with
the compact object likely being a black hole. Swift
discovered this source when it observed a large flare
in July of 2005. The source did not return to qui-
escence but settled into a low intensity hard state
[14]. BATSE occultation measurements from 1991-
2000 showed no significant emission from this source
above 25 keV [15]. The GBM results show that this
source is still in a hard state, with significant emis-
sion above 100 keV. We will continue to monitor this
FIG. 7: SWIFTJ1753.5-0127 light curve. Horizontal scale
is in modified Julian days.
source while it is in the hard state, with longer obser-
vations potentially verifying significant emission above
300 keV.
B. Transient Source
The new transient black hole candidate XTE
J1752-223 rose from undetectable on 2009 October
24 to 511 ± 50 mCrab (12 - 25 keV), 570 ± 70 mCrab
(25 - 50 keV), 970 ± 100 mCrab (50 - 100 keV), and
330 ± 100 mCrab (100 - 300 keV) on 2009 November
2 [2, 16]. The light curve is variable, especially in the
eConf C091122 | 3 | 3 | 1001.0955.pdf |
2009 Fermi Symposium, Washington, D.C., Nov. 2-5 5
FIG. 8: XTEJ1752-223 light curve. Horizontal scale is in
modified Julian days.
12-25 keV band, where the flux initially rose to about
240 mCrab (2009 Oct 25-28), suddenly dropped to
non-detectable on 2009 October 29-30, then rose again
during the period 2009 October 31 to November 2. As
of mid December 2009, the source remains in a high
intensity state. The light curve is shown for the pe-
riod MJD 54700-55200, again with 1-day resolution,
in Fig. 8. The fluxes for XTE J1752-223 in Table 1
are given are for the interval of flaring activity, TJD
55130-55180.
Acknowledgments
This work is supported by the NASA Fermi Guest
Investigator program. At LSU, additional support is
provided by NASA/Louisiana Board of Regents Co-
operative Agreement NNX07AT62A.
[1] C. Meegan et al., Ap. J. 702, 791 (2009).
[2] C. Wilson-Hodge et al. (2010), these proceedings.
[3] B. A. Harmon et al., Ap. J. Suppl. 138, 149 (2002).
[4] B. A. Harmon et al., Ap. J. Suppl. 154, 585 (2004).
[5] G. L. Case et al., in The First GLAST Symposium ,
edited by S. Ritz, P. Michelson, and C. Meegan
(2007), vol. 921 of AIP Conf. Proceedings, p. 538.
[6] J. Tueller et al. (2010), ap. J. Suppl., (to be pub-
lished), astro-ph/0903.3037.
[7] J. C. Ling and W. A. Wheaton, Ap. J. 598, 334
(2003).
[8] E. Jourdain and J. P. Roques, Ap. J. 704, 17 (2009).
[9] H. Steinle et al., Astron. and Astrophys. 330, 97
(1998).
[10] M. McConnell et al., Ap. J. 523, 928 (2000).
[11] J. C. Ling and W. A. Wheaton, Chinese J. Astron.
Astrophys. Suppl. 5, 80 (2005).
[12] G. L. Case et al., Chinese J. Astron. Astrophys. Suppl.
5, 341 (2005).
[13] L. Bouchet et al., Ap. J. 693, 1871 (2009).
[14] M. C. Bell et al., Ap. J. 659, 549 (2007).
[15] G. L. Case et al. (2010), to be submitted.
[16] C. Wilson-Hodge et al., Astron. Telegram 2280
(2009).
eConf C091122 | 4 | 4 | 1001.0955.pdf |
arXiv:1001.2449v1 [cond-mat.mtrl-sci] 14 Jan 2010
Exchange bias of a ferromagnetic semiconductor by a ferromagnetic metal
K. Olejnik, 1, 2 P. Wadley,3 J. Haigh, 3 K. W. Edmonds, 3 R. P. Campion, 3 A. W. Rushforth, 3 B. L. Gallagher, 3
C. T. Foxon, 3 T. Jungwirth, 2, 3 J. Wunderlich,1, 2 S. S. Dhesi, 4 S. Cavill, 4 G. van der Laan, 4 and E. Arenholz 5
1Hitachi Cambridge Laboratory, Cambridge CB3 0HE, United Kingdom
2Institute of Physics ASCR, v.v.i., Cukrovarnicka 10, 16253Praha 6, Czech Republic
3School of Physics and Astronomy, University of Nottingham,Nottingham NG7 2RD, United Kingdom
4Diamond Light Source, Harwell Science and Innovation Campus,
Didcot, Oxfordshire, OX11 0DE, United Kingdom
5Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
(Dated: August 24, 2018)
We demonstrate an exchange bias in (Ga,Mn)As induced by anti ferromagnetic coupling to a thin
overlayer of Fe. Bias fields of up to 240 Oe are observed. Using element-specific x-ray magnetic
circular dichroism measurements, we distinguish a strongl y exchange coupled (Ga,Mn)As interface
layer in addition to the biassed bulk of the (Ga,Mn)As film. Th e interface layer remains polarized
at room temperature.
PACS numbers: 75.70.Cn, 75.50.Pp, 75.50.Bb
Ferromagnetic (FM) semiconductors offer the prospect
of combining high-density storage and gate-controlled
logic in a single material. The realization of spin-valve
devices from FM semiconductors requires the controlled
switching of magnetization in adjacent layers between
antiferromagnetic (AFM) and FM configurations. This
has motivated several theoretical investigations of inter-
layer coupling in all-semiconductor devices1, and AFM
coupling has recently been demonstrated in (Ga,Mn)As
multilayers separated byp-type non-magnetic spacers 2.
However, the Curie temperature TC of (Ga,Mn)As is
currently limited to 185 K in single layers 3, and is
typically much lower for layers embedded within a
heterostructure2, which is an obstacle to the practical
implementation of semiconductor spintronics.
The development of FM metal/FM semiconductor het-
erostructures has the potential to bring together the
benefits of metal and semiconductor based spintron-
ics, offering access to new functionalities and physi-
cal phenomena. Recent studies of MnAs/(Ga,Mn)As
and NiFe/(Ga,Mn)As bilayer films have shown FM in-
terlayer coupling and independent magnetization be-
havior, respectively4,5. Of particular interest is the
Fe/(Ga,Mn)As system, since the growth of epitaxial
Fe/GaAs(001) films is well-established6. Remarkably, a
recent x-ray magnetic circular dichroism (XMCD) study
has shown that Fe may induce a proximity polariza-
tion in the near-surface region of (Ga,Mn)As, antipar-
allel to the Fe moment and persisting even above room
temperature7. Devices incorporating Fe/(Ga,Mn)As
therefore offer the prospect of obtaining non-volatile
room temperature spin-polarization in a semiconductor.
Until now, no information has been revealed about the
coupling of Fe to (Ga,Mn)As layers away from the near-
surface region. At the surface, the (Ga,Mn)As layer may
be highly non-stoichiometric and Mn-rich, due to its non-
equilibrium nature8,9. Previously, Fe/(Ga,Mn)As layers
were produced by a process including exposure to air fol-
lowed by sputtering and annealing prior to Fe deposition,
which may further disrupt the interface order. The ori-
gin of the interface magnetism then had to be inferred by
comparison to a series of reference samples7. Demonstra-
tion of coupling between the bulk of the layers, i.e., an
exchange bias effect, would provide direct evidence of the
interface magnetic order. Moreover, such coupling would
offer new means of manipulating the FM semiconductor
spin state and utilizing the proximity polarization effect
in a spintronic device.
Here, we demonstrate an antiferromagnetic coupling
and exchange bias in Fe/(Ga,Mn)As bilayer films, by
combining element-specific XMCD measurements and
bulk-sensitive superconducting quantum interference de-
vice (SQUID) magnetometry. As with previous studies
of FM metal/FM semiconductor bilayers4,5 (and in con-
trast to AFM coupled FM metal/FM metal exchange bias
structures10,11) the layers are in direct contact without
a non-magnetic spacer in between. We distinguish in-
terface and bulk (Ga,Mn)As layers that are respectively
strongly and weakly antiferromagnetically coupled to the
Fe overlayer. In agreement with Ref.7, the interface layer
remains polarized at room temperature.
The Fe and (Ga,Mn)As layers of the present study
were both grown by molecular beam epitaxy in the same
ultra-high vacuum system, in order to ensure a clean in-
terface between them. The (Ga,Mn)As layer of thickness
10 to 50 nm was deposited on a GaAs(001) substrate
at a temperature of 260◦C, using previously established
methods3,8. A low Mn concentration of x ≈ 0. 03 was
chosen in order to avoid the formation of compensating
Mn interstitials. The substrate temperature was then
reduced to∼0◦C, before depositing a 2 nm Fe layer,
plus a 2 nm Al capping layer. In-situ reflection high
energy electron diffraction and ex-situ x-ray reflectivity
and diffraction measurements confirmed that the layers
are single-crystalline with sub-nm interface roughness.
SQUID magnetometry measurements were performed us-
ing a Quantum Design Magnetic Property Measurement
System. Mn and FeL2,3 x-ray absorption and XMCD | 0 | 0 | 1001.2449.pdf |
2
measurements were performed on beamline I06 at the
Diamond Light Source, and on beamline 4.0.2 at the Ad-
vanced Light Source. Total-electron yield (TEY) and
fluorescence yield (FY) were monitored simultaneously
using the sample drain current and the photocurrent of a
diode mounted at 90◦ to the incident beam, respectively.
SQUID magnetometry measurements were
first performed on control Fe/GaAs(001) and
(Ga,Mn)As/GaAs(001) samples, grown under the
same conditions as the bilayers, to determine the
magnetic anisotropies of the individual layers and the
Curie temperature of the (Ga,Mn)As layer. The Fe film
has a uniaxial magnetic anisotropy with easy axis along
the [110] orientation, similar to previous studies6. For
the (Ga,Mn)As control sample, there is a competition
between cubic and uniaxial magnetic anisotropies, with
the former dominant at low temperatures and favoring
easy axes along the in-plane⟨100⟩ orientations, and the
latter dominant close to TC (∼35 K) giving an easy axis
along the [1 ¯
10] orientation. Figure 1 shows [110] magne-
tization versus temperature curves and low temperature
hysteresis loops for a bilayer film containing a 20 nm
thick (Ga,Mn)As layer. The total remnant moment of
the bilayer film decreases on cooling under zero magnetic
field below theTC of the (Ga,Mn)As, indicating that
this layer aligns antiparallel to the Fe magnetization
at zero field. The hysteresis curve shows a two-step
magnetization reversal, indicating different behavior of
the Fe and (Ga,Mn)As layers, with the smaller loop
attributed to the dilute moment (Ga,Mn)As film. The
minor hysteresis loop shown in Fig. 1 clearly shows a
shift from zero field by a bias fieldHE , indicating that
the Fe layer induces an exchange bias in the magnetic
semiconductor. The shape and size of the minor loop
is in agreement with the hysteresis loop for the control
(Ga,Mn)As sample, also shown in Fig. 1. This strongly
indicates that the exchange bias affects the whole of the
(Ga,Mn)As layer in the bilayer sample.
Similar behavior is observed for bilayer samples con-
taining a 10 nm or 50 nm (Ga,Mn)As layer, with a
bias field which is approximately inversely proportional
to the thicknessd of the ferromagnetic semiconductor
layer (Fig. 1, inset). This 1/ d dependence of HE was
found previously for MnAs/(Ga,Mn)As bilayers 4, and
is generally observed in exchanged-biased thin films 12.
From this dependence it is possible to describe the ex-
change bias in terms of an interface energy per unit area,
∆E = MF SHE d = 0 . 003 erg/cm 2. This value is rather
small compared to typical exchange bias systems 12, re-
flecting the low moment density MF S of the diluted
FM semiconductor layer. However, the bias field for a
given (Ga,Mn)As thickness is larger than is observed for
MnO/(Ga,Mn)As structures13, while the reproducibility
and flexibility of the present structures is much higher
due to the single-crystalline ferromagnetic nature of the
Fe layer.
To confirm the presence of AFM interlayer coupling,
we performed XMCD measurements at the Mn and Fe
L2,3 absorption edges in order to determine the magnetic
response of the individual elements. In L2,3 XMCD, elec-
trons are excited from a 2 p core level to the unoccupied
3d valence states of the element of interest by circularly
polarized x-rays at the resonance energies of the transi-
tions. The difference in absorption for opposite polariza-
tions gives a direct and element-specific measurement of
the projection of the 3d magnetic moment along the x-
ray polarization vector. The absorption cross-section is
conventionally obtained by measuring the decay products
– either fluorescent x-rays or electrons – of the photoex-
cited core hole. The type of decay product measured
determines the probing depth of the technique. For Mn
L2,3 absorption, the probing depths for FY and TEY de-
tection are λF Y ≈ 100 nm and λT EY ≈ 3 nm. In the
current experiment, the Mn XMCD measured using FY
and TEY are thus sensitive to the bulk of the (Ga,Mn)As
film and the near-interface layers, respectively.
Figure 2(a)-(c) shows the magnetic field dependence of
XMCD asymmetry, defined as ( Il − Ir)/ (Il + Ir) where
Il(r) is the absorption for left- (right-) circularly polarized
x-rays. This is measured at the Fe and Mn L3 absorption
peaks for a Fe(2 nm)/(Ga,Mn)As(10 nm) sample at 2 K.
The external field is applied along the photon incidence
direction, which is at 70◦ to the surface normal with
an in-plane projection along the [110] axis. The XMCD
data show that the Fe film displays a square hysteresis
loop with a single magnetization switch, as expected for
a monocrystalline Fe film with strong uniaxial magnetic
anisotropy. The Mn XMCD shows a more complicated
loop due to the effect of the interlayer coupling. The pro-
jected Mn moment aligns antiparallel to the Fe moment
at remanence, and undergoes a magnetization reversal of
opposite sign to the Fe. With further increase of the ex-
ternal magnetic field, the Mn moment gradually rotates
away from antiparallel alignment with the Fe layer, and
into the field direction. Qualitatively similar behavior
is observed for the Fe(2 nm)/(Ga,Mn)As(20 nm) sam-
ple: the (Ga,Mn)As layer is aligned antiparallel to the
Fe layer at zero field, although the bias field is lower by
approximately a factor of two.
Clear differences are observed between the Mn XMCD
hysteresis loops obtained using TEY and FY detection
modes. For FY the magnitude of the XMCD is similar
(but of opposite sign) at remanence and at high mag-
netic fields, whereas for TEY at remanence it is approx-
imately a factor of two larger than at 1000 Oe. The
MnL2,3 XMCD spectra recorded at remanence and at
1000 Oe, shown in Fig. 3, confirm this result. At re-
manence the FY and TEY detected XMCD have similar
magnitudes. However, under a large external field the
XMCD is substantially smaller in TEY than in FY, con-
firming that the net magnetization of the Mn ions near
the interface is significantly less than in the bulk of the
(Ga,Mn)As film. This is the case even up to the high-
est field applied (20 kOe). By applying the XMCD sum
rules14 to the TEY data, and by comparing the spectra to
previous measurements on well-characterized (Ga,Mn)As | 1 | 1 | 1001.2449.pdf |
3
samples15, the projected Mn 3 d magnetic moments are
obtained as −1.4 µB and +0.8 µB per ion at remanence
and 1000 Oe, respectively.
The difference between these values can be understood
as being due to an interface layer which is strongly anti-
ferromagnetically coupled to the Fe layer. At zero field,
both the interfacial and bulk Mn are aligned antiparallel
to the Fe layer. At high fields, the bulk of the (Ga,Mn)As
layer away from the interface is re-oriented into the exter-
nal field direction. However, the interfacial Mn remains
antiparallel to the Fe layer and thus partially compen-
sates the XMCD signal from the bulk of the (Ga,Mn)As.
From the size of the remanent and 1000 Oe magnetic
moments, it can be estimated that around 25-30% of the
TEY XMCD signal can be ascribed to the interfacial Mn
which is strongly coupled to the Fe moments.
The interfacial Mn moments are ascribed to the prox-
imity polarization of the (Ga,Mn)As interface by the Fe
layer, such as was shown previously by XMCD as well as
ab initiotheory7. Evidence for this can be observed from
measurement of the Mn L2,3 XMCD signal at tempera-
tures above the (Ga,Mn)As TC . Similar to the previous
study7, we observe a small but not negligible signal at
room temperature (Fig. 3), with opposite sign to the Fe
L2,3 XMCD. Its spectral shape is characteristic of a local-
ized electronic configuration close to d5, similar to bulk
(Ga,Mn)As7,9,15 but in contrast to Mn in more metallic
environments such as Mn xFe1−x7 or MnAs 16. A slight
broadening is observed on the low energy side of the Mn
L3 peak, which may be due to the different screening in-
duced by proximity to the Fe layer. Since the measured
intensity is attenuated with distancez from the surface
as I = I0 exp(−z/λ T EY), the thickness of the strongly
coupled interface layer is estimated to be ∼0.7 nm or 2-3
monolayers, assuming a uniform distribution of Mn ions
and magnetic moments throughout the (Ga,Mn)As film.
This is around a factor of three thinner than in Ref.7,
which could be due to the lower Mn concentration or the
different preparation method of the present samples.
In summary, we have demonstrated antiferromagnetic
coupling between Fe and (Ga,Mn)As layers in bilayer
structures. A markedly different coupling is observed for
the bulk of the (Ga,Mn)As layer and for Mn moments
in the near-interface region. A thickness-dependent ex-
change bias field is observed to affect the whole of the
bulk (Ga,Mn)As layer, which aligns antiparallel to the
Fe layer at low fields, and switches to parallel when the
external field is large enough to overcome the bias field
and the magnetocrystalline anisotropy fields. In contrast,
the interfacial Mn moments remain aligned antiparallel
to the Fe layer even at 20 kOe, the largest field studied,
and are polarized at temperatures well above theTC of
the bulk (Ga,Mn)As layer. The latter observation con-
firms the recently reported result of Ref. 7, in which
the Fe/(Ga,Mn)As bilayers were produced by a different
method but showed qualitatively similar behavior of the
interfacial moments. Our results shed new light on the
magnetic coupling in Fe/(Ga,Mn)As hybrid layers which
are of potential interest for room temperature spintron-
ics, and also offer a means of controlling the spin orien-
tation in a FM semiconductor.
We acknowledge support from EU grants
SemiSpinNet-215368 and NAMASTE-214499, and
STFC studentship grant CMPC07100. The Advanced
Light Source is supported by the U.S. Department of
Energy under Contract No. DE-AC02-05CH11231.
We thank Leigh Shelford for help during the Diamond
beamtime.
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15 T. Jungwirth, J. Masek, K. Y. Wang, K. W. Edmonds, | 2 | 2 | 1001.2449.pdf |
4
/s48 /s52/s48 /s56/s48
/s52
/s53
/s45/s49/s48/s48/s48 /s48 /s49/s48/s48/s48
/s45/s52
/s45/s50
/s48
/s50
/s52
/s72
/s69
/s32/s40/s79/s101/s41
/s32/s65/s112/s112/s108/s105/s101/s100/s32/s102/s105/s101/s108/s100/s32/s40/s79/s101/s41
/s77/s111/s109/s101/s110/s116/s32/s40/s49/s48
/s45/s53
/s32/s101/s109/s117/s41
/s48 /s50/s48 /s52/s48
/s48
/s49/s48/s48
/s50/s48/s48
/s51/s48/s48
/s32
/s32
/s100/s32/s40/s110/s109/s41
/s72/s32/s61/s32/s48/s46/s53/s32/s107/s79/s101
/s72/s32/s61/s32/s48
/s32
/s32
/s84/s32/s40/s75/s41
/s32
FIG. 1. (color) Main figure: Major (red/black) and minor
(green) hysteresis loops along the [110] axis at 5 K, for a
Fe (2 nm)/(Ga,Mn)As (20 nm) film, and the hysteresis loop
for a control (Ga,Mn)As (20 nm) film along the same axis
(blue). Left inset: Magnetization versus temperature for the
Fe/(Ga,Mn)As film at remanence (black) and under a 500 Oe
applied field (red). Right inset: Exchange bias field versus
thicknessd of the (Ga,Mn)As film (points) and fit showing
1/d dependence (dashed line).
M. Sawicki, M. Polini, J. Sinova, A. H. MacDonald, R. P.
Campion, L. X. Zhao, N. R. S. Farley, T. K. Johal, G. van
der Laan, C. T. Foxon, and B. L. Gallagher, Phys. Rev. B
73, 165205 (2006).
16 K. W. Edmonds, A. A. Freeman, N. R. S. Farley, K. Y.
Wang, R. P. Campion, B. L. Gallagher, C. T. Foxon, G.
van der Laan, and E. Arenholz, J. Appl. Phys.102, 023902
(2007). | 3 | 3 | 1001.2449.pdf |
5
/s45/s48/s46/s48/s48/s52
/s48/s46/s48/s48/s48
/s48/s46/s48/s48/s52
/s40/s98/s41/s32/s77/s110/s32/s84/s69/s89
/s32
/s45/s48/s46/s50
/s45/s48/s46/s49
/s48/s46/s48
/s48/s46/s49
/s48/s46/s50
/s32
/s40/s97/s41/s32/s70/s101/s32/s84/s69/s89
/s32
/s88/s77/s67/s68/s32/s97/s115/s121/s109/s109/s101/s116/s114/s121
/s45/s50/s53/s48 /s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48
/s45/s48/s46/s48/s48/s52
/s48/s46/s48/s48/s48
/s48/s46/s48/s48/s52
/s70/s105/s101/s108/s100/s32/s32/s40/s79/s101/s41
/s40/s99/s41/s32/s77/s110/s32/s70/s89
/s32
/s32
FIG. 2. (color online) XMCD asymmetry versus applied field
along the [110] axis at 2 K, for a Fe (2 nm)/(Ga,Mn)As
(10 nm) film. (a) FeL3, total electron yield; (b) Mn L3,
total electron yield; (c) Mn L3, fluorescent yield. Black and
red points are data for increasing and decreasing fields resp ec-
tively; lines are to guide the eye. | 4 | 4 | 1001.2449.pdf |
6
/s54/s52/s48 /s54/s53/s48
/s48/s46/s57
/s49/s46/s48
/s40/s100/s41/s32/s72/s32/s61/s32/s50/s32/s107/s79/s101/s44/s32/s84/s32/s61/s32/s51/s48/s48/s75
/s40/s99/s41/s32/s72/s32/s61/s32/s49/s32/s107/s79/s101/s44/s32/s84/s32/s61/s32/s50/s75
/s40/s97/s41/s32/s97/s98/s115/s111/s114/s112/s116/s105/s111/s110
/s40/s98/s41/s32/s72/s32/s61/s32/s48/s44/s32/s84/s32/s61/s32/s50/s75
/s32
/s32
/s88/s45/s114/s97/s121/s32/s97/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
/s88/s45/s114/s97/s121/s32/s101/s110/s101/s114/s103/s121/s32/s32/s40/s101/s86/s41
/s88/s77/s67/s68/s32/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
FIG. 3. (color online) (a) Polarization-averaged Mn L2,3 spec-
trum for a Fe/(Ga,Mn)As film; (b) XMCD spectra measured
in remanence at 2 K; (c) XMCD spectra measured under a
1000 Oe applied field at 2 K; (d) XMCD spectrum measured
under a 2000 Oe applied field at 300 K. XMCD spectra are
obtained using TEY (thick red lines) and FY (thin blue lines)
detection. | 5 | 5 | 1001.2449.pdf |
Computational Design of Chemical Nanosensors: Metal Doped Carbon Nanotubes
J. M. Garc´ıa-Lastra1,2,∗D. J. Mowbray1,2, K. S. Thygesen 2, A. Rubio 1,3, and K. W. Jacobsen 2
1Nano-Bio Spectroscopy group and ETSF Scientific Development Centre,
Dpto. F ´ısica de Materiales, Universidad del Pa ´ıs Vasco,
Centro de F´ısica de Materiales CSIC-UPV/EHU- MPC and DIPC, Av. Tolosa 72, E-20018 San Sebasti ´an, Spain
2Center for Atomic-scale Materials Design, Department of Physics,
Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
3Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany
We use computational screening to systematically investigate the use of transition metal doped carbon nan-
otubes for chemical gas sensing. For a set of relevant target molecules (CO, NH3, H2S) and the main components
of air (N2, O2, H2O), we calculate the binding energy and change in conductance upon adsorption on a metal
atom occupying a vacancy of a (6,6) carbon nanotube. Based on these descriptors, we identify the most promis-
ing dopant candidates for detection of a given target molecule. From the fractional coverage of the metal sites
in thermal equilibrium with air, we estimate the change in the nanotube resistance per doping site as a function
of the target molecule concentration assuming charge transport in the diffusive regime. Our analysis points to
Ni-doped nanotubes as candidates for CO sensors working under typical atmospheric conditions.
PACS numbers: 73.63.–b, 68.43.–h, 73.50.Lw
The ability to detect small concentrations of specific chem-
ical species is fundamental for a variety of industrial and sci-
entific processes as well as for medical applications and en-
vironmental monitoring [1]. In general, nanostructured mate-
rials should be well suited for sensor applications because of
their large surface to volume ratio which makes them sensi-
tive to molecular adsorption. Specifically, carbon nanotubes
(CNT) [2] have been shown to work remarkably well as de-
tectors of small gas molecules. This has been demonstrated
both for individual CNTs [3–8] as well as for CNT networks
[9, 10].
Pristine CNTs are known to be chemically inert – a prop-
erty closely related to their high stability. As a consequence,
only radicals bind strong enough to the CNT to notably affect
its electrical properties [2, 5, 11–13]. To make CNTs attrac-
tive for sensor applications thus requires some kind of func-
tionalization, e.g. through doping or decoration of the CNT
sidewall [13–21]. Ideally, this type of functionalization could
be used to control not only the reactivity of the CNT but also
the selectivity towards specific chemical species.
In this work we consider the possibility of using CNTs
doped by 3d transition metal atoms for chemical gas sens-
ing. We use computational screening to systematically iden-
tify the most promising dopant candidates for detection of
three different target molecules (CO, NH 3, H2S) under typi-
cal atmospheric conditions. The screening procedure is based
on the calculation of two microscopic descriptors: the bind-
ing energy and scattering resistance of the molecules when
adsorbed on a doped CNT. These two quantities give a good
indication of the gas coverage and impact on the resistance.
For the most promising candidates we then employ a simple
thermodynamic model of the CNT sensor. In this model, the
binding energies are used to obtain the fractional coverage of
the metallic sites as a function of the target molecule concen-
tration under ambient conditions. Under the assumption of
transport in the diffusive rather than localization regime, the
change in CNT resistivity may then be obtained from the cal-
culated coverages and single impurity conductances.
We find that oxidation of the active metal site passivates
the sensor in the case of doping by Ti, V , Cr, and Mn un-
der standard conditions (room temperature and 1 bar of pres-
sure). Among the remaining metals, we identify Ni as is the
most promising candidate for CO detection. For this system
the change in resistance per active site is generally significant
(>1 Ω) for small changes in CO concentration in the relevant
range of around 0.1–10 ppm. Our approach is quite general
and is directly applicable to other nanostructures than CNTs,
other functionalizations than metal doping, and other back-
grounds than atmospheric air.
All total energy calculations and structure optimizations
have been performed with the real-space density functional
theory (DFT) code GPAW [22] which is based on the projector
augmented wave method. We use a grid spacing of 0.2 ˚A for
representing the density and wave functions and the PBE ex-
change correlation functional [23]. Transport calculations for
the optimized structures have been performed using the non-
equilibrium Green’s function method [24] with an electronic
Hamiltonian obtained from the SIESTA code [25] in a dou-
ble zeta polarized (DZP) basis set. Spin polarization has been
taken into account in all calculations.
Metallic doping of a (6,6) CNT has been modeled in a su-
percell containing six repeated minimal unit cells along the
CNT axis (dimensions: 15 ˚A×15 ˚A×14.622 ˚A). For this size
of supercell a Γ-point sampling of the Brillouin zone was
found to be sufficient. The formation energy for creating a
vacancy (VC) occupied by a transition metal atom (M) was
calculated using the relation
Eform[M@VC] = E[M@VC] + nE[C] −E[M@NT] (1)
where E[M@VC] is the total energy of a transition metal
atom occupying a vacancy in the nanotube, n is the number
of carbon atoms removed to form the vacancy,E[C] is the en-
ergy per carbon atom in a pristine nanotube, and E[M@NT]
arXiv:1001.2538v1 [cond-mat.mes-hall] 14 Jan 2010 | 0 | 0 | 1001.2538.pdf |
2
Ti V Cr Mn Fe Co Ni Cu Zn0
2
4
6
8Formation Energy E form [eV] Empty Monovacancy
Empty Divacancy II
Empty Divacancy I
Monovacancy Divacancy I Divacancy II
Carbon Nanotube Axis
FIG. 1: Structural schematics and formation energy for a 3d tran-
sition metal occupied monovacancy (black), divacancy I (gray), or
divacancy II (white) in a (6,6) carbon nanotube. Formation energies
of the empty vacancies are indicated by dashed lines.
is the total energy of the pristine nanotube with a physisorbed
transition metal atom. We have considered the monovacancy
and two divacancies shown in Fig. 1. The energy required to
form an empty vacancy is obtained from
Eform[VC] = E[VC] + nE[C] −E[NT], (2)
where E[VC] is the total energy of the nanotube with a va-
cancy of n atoms.
The calculated formation energies for the 3d transition met-
als are shown in Fig. 1. From the horizontal lines we see that
both divacancies are more stable than the monovacancy. This
may be attributed to the presence of a two-fold coordinated C
atom in the monovacancy, while all C atoms remain three-fold
coordinated in the divacancies. When a transition metal atom
occupies a vacancy, the strongest bonding to the C atoms is
through its d orbitals [26]. For this reason, Cu and Zn, which
both have filled d-bands, are rather unstable in the CNT. For
the remaining metals, adsorption in the monovacancies leads
to quite stable structures. This is because the three-fold coor-
dination of the C atoms and the CNT’s hexagonal structure are
recovered when the metal atom is inserted. On the other hand,
metal adsorption in divacancies is slightly less stable because
of the resulting pentagon defects, see upper panel in Fig. 1. A
similar behaviour has been reported by Krasheninnikov et al.
for transition metal atoms in graphene [21].
The adsorption energies for N 2, O 2, H 2O, CO, NH 3, and
H2S on the metallic site of the doped (6,6) CNTs are shown in
Fig. 2(a). The adsorption energy of a molecule X is defined
by
Eads[X@M@VC] = E[X@M@VC] −E[X] −E[M@VC],
(3)
N 2
O 2
N 2
O 2
N 2
O 2
CO
2H O
3NH
2H S
CO
CO
2H O
3NH
2H S
2H O
3NH
2H S
N 2
O 2
N 2
O 2
N 2
O 2
CO
2H O
3NH
2H S
CO
CO
2H O
3NH
2H S
2H O
3NH
2H S MonovacancyDivacancy II Divacancy I
Divacancy II Divacancy I Monovacancy
Cr Fe Co Ni Cu ZnMnTi V
0.0
(a) Adsorption Energy [eV]
−2.0 −1.5 −1.0 −0.5
Cr Fe Co Ni Cu ZnMnTi V
0.0 +0.5 +1.0
0(b) Conductance Change [G ]
−1.0 −0.5
FIG. 2: Calculated (a) adsorption energyEads in eV and (b) change in
conductance ∆G in units of G0 =2e2/h for N2, O2, H2O, CO, NH3,
and H 2S on 3d transition metals occupying a monovacancy (top),
divacancy I (middle), and divacancy II (bottom) in a (6,6) carbon
nanotube.
where E[X@M@VC] is the total energy of molecule X on
a transition metal atom occupying a vacancy, and E[X] is the
gas phase energy of the molecule.
From the adsorption energies plotted in Fig. 2(a), we see
that the earlier transition metals tend to bind the adsorbates
stronger than the late transition metals. The latest metals in
the series (Cu and Zn) bind adsorbates rather weakly in the
divacancy structures. We also note that O2 binds significantly
stronger than any of the three target molecules on Ti, V , Cr,
and Mn (except for Cr in divacancy I where H 2S is found to
dissociate). Active sites containing these metals are therefore
expected to be completely passivated if oxygen is present in
the background. Further, we find H2O is rather weakly bound
to most of the active sites. This ensures that these types of
sensors are robust against changes in humidity.
In thermodynamic equilibrium [27], the coverage of the ac-
tive sites follows from
Θ[X] = K[X]C[X]
1 + ∑
Y K[Y ]C[Y ], (4)
where K = k+/k−is the ratio of forward and backward rate
constants for the adsorption reaction,
K[X] = exp
[
−Eads[X] + TS [X]
kBT
]
. (5)
In these expressions C[X] is the concentration of species X,
S[X] is its gas phase entropy and T is the temperature. Ex-
perimental values for the gas phase entropies have been taken
from Ref. [28]. | 1 | 1 | 1001.2538.pdf |
3
10
-3
10
-2
10
-1
10
0
10
-3
10
-2
10
-1
10
0
Fractional Coverage Θ of Ni Occupied Vacancies
0.1 1 10 100
CO Concentration [ppm]
10
-4
10
-3
10
-2
10
-1
10
0
O 2
CO
COClean
O 2
O 2
Clean
(a) Monovacancy
(b) Divacancy I
(c) Divacancy II
10
2
10
3
0.1 1 10 100
CO Concentration [ppm]
-10
1
0
10
1
Monovacancy
Divacancy I
Divacancy II
Change in Resistance ∆R [Ω / Ni Occupied Vacancy]
(d)
FIG. 3: Fractional coverage Θ in thermal equilibrium of Ni in a (a)
monovacancy, (b) divacancy I, (c) divacancy II and (d) change in
resistance ∆R per dopant site as a function of CO concentration in
a background of air at room temperature and 1 bar of pressure. The
reference concentration of CO is taken to be C0 =0.1 ppm. Note the
change from linear to log scale on the y-axis at ∆R =10 Ω.
For a given background composition we may thus estimate
the fractional coverages for each available adsorbate for a
given type of doping. As an example, Fig. 3(a)-(c) shows the
fractional coverage of a Ni atom occupying a monovacancy,
divacancy I, and divacancy II, versus CO concentration in a
background of air at room temperature and 1 bar of pressure.
Due to the relatively small binding energy of N 2 and H2O as
compared to O 2 and CO, all Ni sites will be either empty or
occupied by O 2 or CO. In particular, Ni in a monovacancy
(top panel of Fig. 3) will be completely oxidized for all rel-
evant CO concentrations. For the Ni occupied divacancy II
structures we find the coverage of CO changes significantly
around toxic concentrations (∼10 ppm).
To estimate the effect of adsorbates on the electrical con-
ductance of doped CNTs, we first consider the change in con-
ductance when a single molecule is adsorbed on a metal site of
an otherwise pristine CNT. In Fig. 2(b) we show the calculated
change in conductance relative to the metal site with no ad-
sorbate. In contrast to the binding energies, there are no clear
trends in the conductances. The sensitivity of the conductance
is perhaps most clearly demonstrated by the absence of cor-
relation between different types of vacancies, i.e. between the
three panels in Fig. 2(b). Close to the Fermi level, the conduc-
tance of a perfect armchair CNT equals 2 G0. The presence
of the metal dopant leads to several dips in the transmission
function known as Fano antiresonances [20]. The position
and shape of these dips depend on the d-levels of the transi-
tion metal atom, the character of its bonding to the CNT, and
is further affected by the presence of the adsorbate molecule.
The coupling of all these factors is very complex and makes
it difficult to estimate or rationalize the value of the conduc-
tance. For the spin polarized cases, we use the spin-averaged
conductances, i.e. G = (G↑+ G↓)/2.
Next, we estimate the resistance of a CNT containing sev-
eral impurities (a specific metal dopant with different molecu-
lar adsorbates). Under the assumption that the electron phase-
coherence length, lφ, is smaller than the average distance be-
tween the dopants, d, we may neglect quantum interference
and obtain the total resistance by adding the scattering resis-
tances due to each impurity separately. The scattering resis-
tance due to a single impurity is given by
Rs(X) = 1/G(X) −1/(2G0), (6)
where G(X) is the Landauer conductance of the pristine CNT
with a single metal dopant occupied by molecule X and
1/(2G0) is the contact resistance of a (6,6) CNT.
We may now obtain the total resistance per dopant site rel-
ative to the reference background signal as a function of the
target molecule concentration
∆R
N ≈
∑
X
Rs(X)(Θ[X, C] −Θ[X, C0]), (7)
where N is the number of dopants, Θ[X, C] is the fractional
coverage of species X at concentration C of the target and C0
is the reference concentration. Notice that the contact resis-
tance drops out as we evaluate a change in resistance.
In Fig. 3(d) we show the change in resistance calculated
from Eq. (7) as a function of CO concentration for Ni occu-
pying the three types of vacancies. The background reference
concentration of CO is taken to be C0 = 0 .1 ppm. For the
monovacancy there is very little change in resistivity. This is
because most active sites are blocked by O 2 at relevant CO
concentrations, as shown in the upper panel of Fig. 3. For Ni
in the divacancies there is, however, a change in resistance on
the order of 1Ω per site. For concentrations above ∼1 ppm,
the CO coverage of Ni in the divacancy II increases dramati-
cally and this leads to a significant increase in resistance.
We now return to the discussion of the validity of Eq. (7).
As mentioned, the series coupling of individual scatterers
should be valid when lφ < d. However, even for lφ > d
and assuming that the Anderson localization length, lloc in
the system exceeds lφ, Eq. (7) remains valid if one replaces
the actual resistance R by the sample averaged resistance ⟨R⟩
[29]. At room temperature under ambient conditions, interac-
tions with external degrees of freedom such as internal CNT
phonons and vibrational modes of the adsorbed molecules
would rapidly randomize the phase of the electrons. There-
fore Eq. (7) should certainly be valid in the limit of low dop-
ing concentrations. On the other hand, the total number of
dopants, N, should be large enough for the statistical treat-
ment of the coverage to hold. Finally, we stress that Eq. (7)
represents a conservative estimate of the change in resistance.
In fact, in the regime where lφ > lloc, i.e. in the Anderson
localization regime, the resistance would be highly sensitive
to changes in the fractional coverage of active sites. Calcula-
tion of the actual resistance of the CNT in this regime would,
however, involve a full transport calculation in the presence of | 2 | 2 | 1001.2538.pdf |
4
all N impurities. At this point it suffices to see that the con-
servative estimates obtained from Eq. (7) predict measurable
signals in response to small changes in concentration of the
target molecules.
To our knowledge, controlled doping of CNTs with transi-
tion metal atoms has so far not been achieved. It has, how-
ever, been found that metal atoms incorporated into the CNT
lattice during catalytic growth are afterwards very difficult to
remove [30]. Furthermore, it has been shown that CNT vacan-
cies, which are needed for the metallic doping, may be formed
in a controlled way by irradiation by Ar ions [31]. This sug-
gests that metallic doping of CNTs should be possible.
In summary, we have presented a general model of nanos-
tructured chemical sensors which takes the adsorption en-
ergies of the relevant chemical species and their individual
scattering resistances as the only input. On the basis of this
model we have performed a computational screening of tran-
sition metal doped CNTs, and found that Ni-doped CNTs are
promising candidates for detecting CO in a background of air.
The model may be applied straightforwardly to other nanos-
tructures than CNTs, other functionalizations than metal dop-
ing and other gas compositions than air.
The authors acknowledge financial support from Span-
ish MEC (FIS2007-65702-C02-01), “Grupos Consolidados
UPV/EHU del Gobierno Vasco” (IT-319-07), e-I3 ETSF
project (Contract Number 211956), “Red Espa˜nola de Super-
computaci´on”, NABIIT and the Danish Center for Scientific
Computing. The Center for Atomic-scale Materials Design
(CAMD) is sponsored by the Lundbeck Foundation. JMG-L
acknowledges funding from Spanish MICINN through Juan
de la Cierva and Jos´e Castillejo programs.
∗ Electronic address: [email protected]
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arXiv:1001.2648v1 [physics.chem-ph] 15 Jan 2010
Models of electrolyte solutions from molecular descriptions: The example of NaCl
solutions
John Jairo Molina 1, 2, 3,∗ Jean-Fran¸ cois Dufrˆ eche1, 2, 3,† Mathieu
Salanne1, 2, Olivier Bernard 1, 2, Marie Jardat 1, 2, and Pierre Turq 1, 2
1 UPMC-Universit´ e Paris 06, UMR 7195, PECSA, F-75005 Paris, France
2 CNRS, UMR 7195, PECSA, F-75005 Paris, France
3 Institut de Chimie S´ eparative de Marcoule (ICSM),
UMR 5257 CEA–CNRS–Universit´ e Montpellier 2, Site de Marco ule,
Bˆ atiment 426, BP 17171, 30207 Bagnols-sur-C` eze Cedex, Fr ance
We present a method to derive implicit solvent models of elec trolyte solutions from all-atom
descriptions; providing analytical expressions of the the rmodynamic and structural properties of
the ions consistent with the underlying explicit solvent re presentation. Effective potentials between
ions in solution are calculated to perform perturbation the ory calculations, in order to derive the
best possible description in terms of charged hard spheres. Applying this method to NaCl solutions
yields excellent agreement with the all-atom model, provid ed ion association is taken into account.
Since the pioneering works of Debye, H¨ uckel, and
Onsager, electrolyte solutions have been commonly
described by continuous solvent models, for which
the McMillan-Mayer theory [1] provides a rigorous
statistical-mechanical foundation. Within that level of
description, simple phenomenological models such as the
primitive model (PM), for which the ions are assimi-
lated to charged hard spheres [2], can lead to explicit
formulas for the thermodynamic and structural proper-
ties (e.g., with the help of the mean spherical approxima-
tion (MSA) [3] or the binding MSA (BIMSA) [4]). These
models are the most practical to use [5], since they allow
for a direct link between the experimental measurements
and the microscopic parameters of the system. Never-
theless, they ignore the molecular structure of the sol-
vent. Consequently, they cannot properly account for
the complex specific effects of the ions, which appear in
numerous biological, chemical, and physical interfacial
phenomena [6, 7], without further developments.
An alternative procedure consists in carrying out
molecular simulations, where both the solvent and solute
are treated explicitly. After a rigorous averaging over
the solvent configurations, a coarse-grained description
of the ions, which still includes the effect of the solvent
structure, can be obtained [8–11]. However, this set of
methods is purely numeric; they do not provide any an-
alytical expression for thermodynamic quantities. They
are therefore restricted to simple geometries [12, 13] (bulk
solutions or planar interfaces). The description of com-
plex systems, such as porous or electrochemical materi-
als, is still based on continuous solvent models [14].
In this letter we present a method aimed at bridging
the gap between analytical and numerical approaches. It
is based on the application of liquid perturbation theory
(LPT) [15] to effective ion-ion potentials extracted from
∗ Electronic address: [email protected]
† Electronic address: [email protected]
molecular dynamics (MD) results. Different approxima-
tions of the PM are employed for the case of NaCl elec-
trolyte solutions: a two component model (MSA2), that
only takes free ions into account, and two different three
component models (MSA3 and BIMSA3), which include
a third species (the contact ion pair). As we proceed
to show, LPT allows us to select the best simple model
which accurately accounts for the thermodynamics and
the physical-chemistry of the system.
The first stage consists in calculating the McMillan-
Mayer effective ion-ion interaction potentials V eff
ij (r), by
inverting the radial distribution functions (RDF) gij(r)
obtained by MD. The simulations were carried out on
a box of 2000 water molecules and 48 NaCl pairs us-
ing the same interaction potentials as in reference [16].
This setup corresponds to a concentration of 0. 64 mol l− 1.
NPT ensemble sampling at standard pressure and tem-
perature was enforced, with a time step of 1 fs and a
pressure bath coupling constant of 1 ps. An equilibration
run of 0.25 ns was followed by a production run of 0.6 ns
for five different initial configurations. The averages of
the resulting RDF were then used for the potential inver-
sion via the HNC closure [15]. These effective potentials
are assumed to be concentration independent and will be
used for simulations at all concentrations.
Subtracting the long-range Coulombic potential
V LR
ij (r) (which depends on the dielectric constant of the
solvent) from V eff
ij (r), we obtain the short-range contri-
bution V SR
ij (r) to the effective potentials. These are given
in Fig.
1 (species 1 and 2 refer to Na + and Cl − free ions,
respectively). All the short-range potentials exhibit os-
cillations corresponding to the solvent layering between
the ions, but this effect is particularly important for the
cation-anion interaction: a considerable potential barrier
(≳ 2kBT ) separates the first two attractive wells. To
serve as a reference, Monte Carlo (MC) simulations were
performed with these effective potentials; a comparison
between MD and MC RDF is also provided in Fig.
1. The
excellent agreement between both sets of RDF validates
the HNC inversion procedure [17], and allows us to com- | 0 | 0 | 1001.2648.pdf |
2
r (Å)
0
1
2
3 β V 12
SR
(r)
2
4
6
8
g 12
MC
(r)
g 12
MD
(r)
4 6 8
0
1
2
3 β V 11
SR
(r)
4 6 8
0
1
2
3β V 22
SR
(r)
(a)
(b)
(d)
(c)
FIG. 1: Effective McMillan-Mayer short-range pair potentia ls
extracted from explicit solvent simulations using the HNC
closure. (a) Cation anion, (b) cation cation, (c) anion anion,
(d) cation anion RDF obtained from explicit solvent MD and
implicit solvent MC simulations.
pute all ion thermodynamic properties through implicit
solvent MC simulations.
The second stage of our coarse-graining procedure con-
sists in applying LPT, in order to deduce the best ana-
lytical model of electrolyte solutions which reproduces
this molecular description. The principle of LPT is to
describe the properties of a given system in terms of
those of a well known reference system, with the differ-
ence between them treated as a perturbation in the ref-
erence potential. Assuming pairwise additive potentials,
Vij = V (0)
ij + ∆V ij , a first-order truncated expression for
the free energy density of the system βfv is obtained,
βfv ≲ βf (0)
v + 1
2β
∑
i,j
ρiρj
∫
dr g(0)
ij (r)∆V ij (r) (1)
which depends only on the free-energy density f(0)
v and
RDF g(0) of the reference fluid, with β = ( kBT )− 1 and
ρi the concentration of species i. The Gibbs-Bogoliubov
inequality [15] ensures that the right-hand side of Eq. ( 1)
is actually a strict upper bound. Once a reference system
has been chosen, the expression on the right-hand side of
Eq. (
1) must be minimized with respect to the parameters
defining the reference. This procedure yields the best
first-order approximation to the free energy of the system
under consideration.
For a system of charged particles in solution, the nat-
ural reference is the PM, defined in terms of the charge
and diameter (σi) of each species. In this case, the per-
turbing potentials are just the short-range effective po-
tentials computed above (∆Vij = V SR
ij ). We use the
MSA [3] solution to the PM, since it provides analyti-
cal expressions for both the free energy and the RDF.
The perturbation term is evaluated using an exponential
approximation to the RDF obtained within the MSA,
g(r) = exp [ gMSA(r) − 1], which removes any unphysical
negative regions and improves the comparison with HNC
calculations.
0.9
1
1.1
1.2
1.3Φ
MC
MSA2
DHLL
Exp
0 0.5 1 1.5
c
1/2
(mol.L
-1
)
1/2
3
4
5σ (Å)
σ 1 (MSA-fit)
σ 2 (MSA-fit)
σ 1 (MSA2)
σ 2 (MSA2)
(a)
(b)
FIG. 2: (Color online) (a) Osmotic coefficient Φ in the
McMillan-Mayer frame of reference. (diamond) MC simula-
tions, (dot dashed) MSA2, (dot) Debye H¨ uckel Limiting law
(DHLL), (cross) experiments (Ref. [18] with the McMillan-
Mayer to Lewis Randall conversion). (b) Minimization diam-
eters. (dot dashed) MSA2 and (diamond) MSA-fit.
We first used LPT for a two-component system (Na +
and Cl − free ions) within the MSA (model MSA2), for
concentrations ranging from 0.1 to 2 . 0 mol l− 1. The mini-
mization leads to almost constant diameters on the whole
range of concentration:σ1 = 3 . 67 ˚
A and σ2 = 4 . 78 ˚
A.
As shown in Fig.
2, these parameters yield osmotic co-
efficients close to MC calculations only at very low con-
centration, i.e.,c ≤ 0. 1 mol l− 1 (experimental values are
given for indicative purposes only, since a perfect model
will exactly match the MC results). For molar solutions,
the LPT results differ considerably from MC calculations.
This discrepancy can easily be understood by comparing
the diameters found within the MSA2 calculation with
the effective potentials given in Fig.
1. The anion/cation
contact distance obtained within the MSA2 calculation
is 4. 2 ˚
A, which is in the region of the second minimum of
the effective potential and corresponds to the situation
where there is a single layer of water molecules between
the ions. The first minimum of the potential, which cor-
responds to the contact ion pair (CIP) is thus completely
ignored by the MSA2 calculation. If the MSA diameters
are directly fitted to reproduce the MC osmotic pres-
sure, much smaller values are obtained. These MSA-fit
hydrated diameters, which are compared to the MSA2
diameters in the bottom part of Fig.
2, are averages of
the CIP and the solvent-separated ion pair.
To overcome this difficulty, we have explicitly intro-
duced the CIP in our model (species 3). Straightforward
calculations, based on a characteristic-function formal-
ism, allow us to define an equivalent model in which
the free ions and the CIP are explicitly taken into ac-
count [19, 20]. We apply this formalism by defining a
pair as an anion and a cation at a distance less than
4˚
A, which corresponds to the position of the effective
potential maximum. The interaction between free, like
charges in this new system remains unchanged, and the
cation-anion interactions are easily approximated by ex- | 1 | 1 | 1001.2648.pdf |
3
r (Å)
0
1
2
3
β V~
12
SR
(r)
β V 12
SR
(r)
β V~
33
SR
(r)
4 6 8 10
0
1
2
3 β V~
13
SR
(r)
4 6 8 10
β V~
23
SR
(r)
(a)
(b) (c)
(d)
FIG. 3: Effective pair potentials derived for MSA3 and
BIMSA3. (a) Cation anion (dashed line: without taking the
pair into account), (b) pair cation, (c) pair anion, and (d) pair
pair. The internal potential of the pair β ˜Vint(r) is set equal
to βV eff
ij (r) for distances less than 4 ˚
A.
trapolating the original potential at the barrier separat-
ing pairs from free ions (as shown in Fig.
3). We assume
that the interaction potential is averaged over the rota-
tional degrees of freedom of the CIP and thus pairwise
additive. Hereafter, the quantities referring to such a
three-component model are written with a tilda symbol.
The short-range potentials involving the pair can be de-
rived, in the infinite dilution limit, from an average of
the contributing ion interactions. In Fourier space,
˜V SR
3i (k) = ˜w(k/ 2)
[
V SR
1i + V SR
2i
]
(k), i = 1 , 2 (2a)
˜V SR
33 (k) = ˜w(k/ 2)2[
V SR
11 + V SR
22 + 2V SR
12
]
(k) (2b)
where ˜w(r) is the pair probability distribution
˜w(r) = K− 1
0 e− β ˜Vint(r) (2c)
˜Vint(r) is the internal part of the pair potential (see
Fig.
3), and K0 is the association constant, defined as:
K0 =
∫ ∞
0
dr 4πr2e− β ˜Vint(r) = 0 . 43 L . mol− 1 (3)
The excess free-energy density of the original system
βf ex
v is that of the three component mixture β ˜fex
v plus a
correction term
βf ex
v = β ˜fex
v − ˜ρ3 ln K0, (4)
which is due to the change in standard chemical potential
between the two component and three component mod-
els. It should be noted that the fraction of pairs is now an
additional parameter in the minimization scheme, which
serves to ensure chemical equilibrium. Within this rep-
resentation, the pair can be modeled as a hard sphere
(MSA3) or as a dumbbell-like CIP (BIMSA3) [4]. Since
0 0.5 1 1.5
c
1/2
(mol.L
-1
)
1/2
-1.5
-1
-0.5
0
β f v
ex
(mol.L
-1
)
MC
MSA2
MSA3
BIMSA3
DHLL
Exp
0 0.5 1
0.1
0.2
Pair Fraction
FIG. 4: (Color online) Excess free-energy density βf ex
v as
a function of the square root of the concentration √ c. (dia-
mond) MC simulations, (dot dashed) MSA2, (dashed) MSA3,
(solid) BIMSA3, (dot) DHLL, and (cross) experiments. The
inset gives the fraction of pairs (MSA3, BIMSA3) as a func-
tion of√
c.
we have no additional information, we consider only sym-
metric dumbbells. Furthermore, since analytic expres-
sions for the RDF within BIMSA are not known, we ap-
proximate the dumbbell as a hard sphere when comput-
ing the perturbation term (this is not necessary for the
reference term, since an expression for the free energy
is available). Let˜σc be the diameter of the cation (an-
ion) within the dumbbell, the diameter of the hard sphere
representing this dumbbell is taken to be˜σ3 = 4
√
2
π ˜σc[21].
Using these two reference systems, the three-
component MSA3 and BIMSA3, we obtain results in
much better agreement with the MC simulations, as
shown in Fig.
4. The diameters obtained for species 1,
2, and 3 are 3.65, 4.79, and 5.76 ˚
A for MSA3 and 3.69,
4.75 and 6.19 ˚
A for BIMSA3. The free ion diameters are
similar for MSA2, MSA3, and BIMSA3. The pair diam-
eter is smaller when modeled as a hard sphere (MSA3)
than when modeled as a dumbbell (BIMSA3). At high
concentration (about 1 mol l− 1), the MSA3 overestimates
the free energy, because the excluded volume repulsion
becomes too important for the pairs to be represented as
hard spheres. The BIMSA3 model is the closest to the
MC simulation results. It is worth noting that even at
the lowest concentration considered, the fraction of pairs
(shown in the insert of Fig.
4), although less then 5%,
has a non-negligible effect on the thermodynamics of the
system.
This procedure also provides an accurate description of
the structure over the whole range of concentrations. A
development similar to the one that leads to Eq. (
2) de-
rives the average unpaired RDF from the corresponding
paired quantities:
ρiρj gij(k) = ˜ρ3 ˜w(k) (1 − δij ) + ˜ρi ˜ρj ˜gij(k)
+ ˜ρ3 ˜w(k/ 2)
[
˜ρi˜g3i + ˜ρj ˜g3j
]
(k) (5)
+ ˜ρ 2
3 [ ˜w(k/ 2)]2 ˜g33(k) | 2 | 2 | 1001.2648.pdf |
4
r (Å)
2
4
6
8
4 6 8
0.5
1
1.5
4 6 8
0.5 mol.L
-1
g 12 (r)
1.5 mol.L
-1
g 12 (r)
1.5 mol.L
-1
g 11 (r)
1.5 mol.L
-1
g 22 (r)
FIG. 5: (Color online) RDF obtained from MC simulations
(diamond), BIMSA3 (solid line), and MSA-fit (dot dashed)
at two concentrations.
The RDF obtained within BIMSA3 are compared with
the MC and MSA-fit results in Fig.
5. Our BIMSA3
model accounts for the strong molecular peak of the CIP
and provides the correct distances of minimal approach;
whereas the naive MSA-fit procedure ignores the former
and gives poor estimates for the latter. At larger sep-
arations, the BIMSA3 results do not reproduce the os-
cillations observed in the MC simulations, but the cor-
responding energy oscillations in the effective potentials
are less thankBT . In addition, the perturbation term
of the BIMSA3 appears to be negligible compared to the
reference term for concentrations less than 1 mol l− 1. The
perturbation can then be omitted to obtain a fully ana-
lytical theory, determined by the hard sphere diameters
and the pair fraction given by LPT; with the free energy
and the RDF given in terms of the BIMSA and MSA so-
lutions, as described above. While the procedure we have
followed uses two different approximations for the refer-
ence and perturbation terms (MSA vs BIMSA), these are
known to be accurate for the systems under consideration
and do not appear to be inconsistent with each other.
To conclude, we have combined MD simulations with
LPT to construct simple models of electrolyte solutions
which account for the molecular nature of the solvent.
The final result is fully analytical and it yields the ther-
modynamic and structural properties of the solution, in
agreement with the original molecular description. The
methodology can in principle be adapted to any molecu-
lar description of the system (MD simulations involving
interaction potentials accounting for polarization effects
or Car-Parrinello MD simulations for example) as long
as the ion-ion RDF are known. It can also be generalized
to study interfaces. The method appears to be a promis-
ing approach toward the description of the specific effects
of ions, especially for complex systems whose modeling
requires an analytic solution.
The authors are particularly grateful to Werner Kunz
for fruitful discussions.
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[21] The average contact distance between a symmetric
dumbbell and an infinite plane atβ = 0. | 3 | 3 | 1001.2648.pdf |
Modelling approaches to the dewetting of evaporating thin films of
nanoparticle suspensions
U. Thiele∗, I. Vancea, A. J. Archer, M. J. Robbins, L. Frastia
Department of Mathematical Sciences,
Loughborough University, Leicestershire LE11 3TU, UK
A. Stannard, E. Pauliac-Vaujour, C. P. Martin, M. O. Blunt, P. J. Moriarty
The School of Physics and Astronomy,
The University of Nottingham, Nottingham NG7 2RD, UK
∗ homepage: http://www.uwethiele.de, [email protected]
1
arXiv:1001.2669v1 [cond-mat.soft] 15 Jan 2010 | 0 | 0 | 1001.2669.pdf |
Abstract
We review recent experiments on dewetting thin films of evaporating colloidal nanoparticle suspensions
(nanofluids) and discuss several theoretical approaches to describe the ongoing processes including coupled
transport and phase changes. These approaches range from microscopic discrete stochastic theories to
mesoscopic continuous deterministic descriptions. In particular, we focus on (i) a microscopic kinetic
Monte Carlo model, (ii) a dynamical density functional theory and (iii) a hydrodynamic thin film model.
Models (i) and (ii) are employed to discuss the formation of polygonal networks, spinodal and branched
structures resulting from the dewetting of an ultrathin ‘postcursor film’ that remains behind a mesoscopic
dewetting front. We highlight, in particular, the presence of a transverse instability in the evaporative
dewetting front which results in highly branched fingering structures. The subtle interplay of decomposition
in the film and contact line motion is discussed.
Finally, we discuss a simple thin film model (iii) of the hydrodynamics on the mesoscale. We employ
coupled evolution equations for the film thickness profile and mean particle concentration. The model is
used to discuss the self-pinning and de-pinning of a contact line related to the ‘coffee-stain’ effect.
In the course of the review we discuss the advantages and limitations of the different theories, as well as
possible future developments and extensions.
The paper is published in: J. Phys.-Cond. Mat. 21, 264016 (2009),
in the V olume “Nanofluids on solid substrates” and can be obtained at
http://dx.doi.org/10.1088/0953-8984/21/26/264016
2 | 1 | 1 | 1001.2669.pdf |
I. INTRODUCTION
The patterns formed in dewetting processes have attracted strong interest since Reiter analysed the
process quantitatively in the early nineties. In these experiments, that proved to be a paradigm in
our understanding of dewetting, a uniform thin film of polystyrene (tens of nanometers thick) is
deposited on a flat silicon oxide substrate is brought above the glass transition temperature. The
film ruptures in several places, forming holes which subsequently grow, competing for space. As a
result, a random polygonal network of liquid rims emerges. The rims may further decay into lines
of small drops due to a Rayleigh-type instability [1–3]. The related problems of retracting contact
lines on partially wetting substrates and the opening of single holes in rather thick films have also
been studied [4, 5].
Subsequent work has mainly focused on many different aspects of the dewetting process for simple
non-volatile liquids and polymers (for reviews see Refs. [6–8]). All stages of the dewetting of a
film are studied: the initial film rupture via nucleation or a surface instability (called spinodal
dewetting) [1, 9–13], the growth process of individual holes [14–16], the evolution of the resulting
hole pattern [3, 13], and the stability of the individual dewetting fronts [17–19]. We note in
passing, that descriptions of dewetting patterns may also be found in historic papers, particularly
for the dewetting of a liquid film on a liquid substrate. Tomlinson [20, footnote 18 on p. 40]
considered turpentine on water and Marangoni [21, p. 352f] oil on water.
More recently, interest has turned to the dewetting processes of solutions and suspensions. How-
ever, these systems have not yet been investigated in any great depth. Such systems are compli-
cated because their behaviour is determined by the interplay between the various solute (or colloid)
and solvent transport processes. Furthermore, the solvents that are used often evaporate, i.e., one
has to distinguish between ‘normal’ convective dewetting and evaporative dewetting. A number
of experiments have been performed employing (colloidal) solutions of polymers [22–25], macro-
molecules like collagen and DNA [26–31] and nanoparticles [32–40]. The latter are sometimes
referred to as ‘nanofluids’. The initial focus of much of the research in the field has been on
investigating the structures that are formed which are similar to the ones observed in the ‘classi-
cal’ dewetting of non-volatile liquids. Labyrinthine structures and polygonal networks result from
spinodal dewetting and heterogeneous nucleation and growth, respectively. They are ‘decorated’
with the solute and therefore conserve the transient dewetting pattern as a dried-in structure when
all the solvent has evaporated [28, 34]. The picture is, however, not complete. The solute may
3 | 2 | 2 | 1001.2669.pdf |
also shift the spinodal and binodal lines as compared to the locations of these lines in the phase
diagram for the pure solvent [41]. As a consequence, the solute concentration influences the hole
nucleation rate. More importantly, the solute particles may also destabilise the dewetting fronts.
As a result, one may find strongly ramified structures in all three systems [23, 25, 40, 42]. A
selection of images exhibiting some of the possible structures is displayed in Fig.1.
For volatile solvents, the contact lines retract even for wetting fluids. It has been found that such
evaporatively receding contact lines may deposit very regular line or ring patterns parallel to the
moving contact line [24, 43]. The deposition of a single ring of colloids from a evaporating
drop of colloidal suspension is well known as the ‘coffee stain effect’ [44]. Detailed investiga-
tions reveal the emergence of rich structures including multiple irregular rings, networks, regular
droplet patterns, sawtooth patterns, Sierpinski carpets, and – in the case of DNA – liquid crys-
talline structures [22, 30, 45–49]. The deposition of regularly spaced straight lines orthogonal to
the moving contact line has also been reported [50]. Droplet patterns may as well be created em-
ploying solvent-induced dewetting of glassy polymer layers below the glass transition temperature
[51–53].
Note that the dewetting of pure volatile liquids has also been studied experimentally [54] and
theoretically [55–58]. In this case, different contact line instabilities have been observed for evap-
orating liquid drops [59, 60].
In the present article we review and preview the experiments and in particular the various mod-
elling approaches for dewetting suspensions of (nano-)particles in volatile partially wetting sol-
vents. After reviewing the basic experimental results in Section II, we discuss in Section III sev-
eral theoretical approaches. In particular, we present a kinetic Monte Carlo model in Section III A,
a dynamic density functional theory in Section III B, and a thin film evolution equation in Sec-
tion III C. Finally, we conclude in Section IV by discussing advantages and shortcomings of the
individual approaches and future challenges to all of them.
II. EXPERIMENT WITH NANOPARTICLE SOLUTIONS
We focus on experiments that use monodisperse colloidal suspensions of thiol-passivated gold
nanoparticles in toluene [33, 34, 37–40, 61]. The gold core of 2 – 3 nm diameter is coated by a layer
of alkyl-thiol molecules. The length of the carbon backbone of the thiol used in the experiments
ranges from 6 to 12 carbon atoms ( C6 to C12) [40]. By varying the chain length, one can control
4 | 3 | 3 | 1001.2669.pdf |
(a)
(b)
(c)
(d)
FIG. 1: (Colour online) Images of strongly ramified dewetting structures obtained using Atomic Force
Microscopy in the case of (a) an aqueous collagen solution on graphite (courtesy of U. Thiele, M. Mertig
and W. Pompe; see also Ref. [42]. Image size:5µm×5µm); (b) poly(acrylic acid) in water spin-coated onto
a polystyrene substrate (reprinted with permission of John Wiley & Sons, Inc. from Ref. [23]; copyright
John Wiley & Sons, Inc. 2002; Image size: 2.5µm×2.5µm); and in both (c) and (d), a solution of gold
nanoparticles in toluene, spin-coated onto native oxide terminated silicon substrates (scale bars given in
panels). In all the images the lighter areas correspond to the deposited solute and the dark areas to the
empty substrate.
5 | 4 | 4 | 1001.2669.pdf |
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