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10.48550_arXiv.0708.1447	###### Abstract The non-specific adhesion of spherical particles to a cell substrate is analyzed in a parallel plate flow chamber, addressing the effect of the particle size. Differently from other experiments, the total volume of the injected particles has been fixed, rather than the total number of particles, as the diameter $d$ of the particles is changed from 500 nm up to 10 $\upmu$m. From the analysis of the experimental data, simple and instructive scaling adhesion laws have been derived showing that (i) the number of particles adherent to the cell layer per unit surface decreases with the size of the particle as $d^{1/2}$; and consequently (ii) the volume of the particles adherent per unit surface increases with the size of the particles as $d^{1.3}$. These results are of importance in the 'rational design' of nanoparticles for drug delivery and biomedical imaging. F. Gentile ${}^{a}$, A. Granaldi ${}^{b}$, M. Ferrari ${}^{c}$, P. Graecia, Catanzaro - ITALY ${}^{b}$ Center of Excellence in Computational Mechanics (CEMEC), Politecnico di Bari, Bari - ITALY ${}^{c}$ The Univeristy of Texas Health Science Center at Houston, Houston, Texas - USA ## 1 Introduction Advances in nanosciences and micro/nanofabrication techniques have led to novel strategies and devices for drug delivery and diagnosis in biomedical applications. In particular, biomimetic artificial carriers are emerging as powerful tools for cancer and heart diseases treatment and molecular imaging. These are particulates carrying therapeutic and/or imaging agents administered at the systemic level and delivered towards a specific biological target (diseased cell or microenvironment). The surface of these particulates is generally coated by ligand molecules that can recognize countermolecules (receptors) expressed at the biological target. The power of intravascularly injectable particles over free molecules administration lies in their multifunctionality and engineerability. The use of such particles as intravascular delivery vectors affords substantial advantages including (i) the specific biomolecular targeting through one or more conjugated antibodies or other recognition molecules; (ii) the ability to carry one or more therapeutic agents; (iii) the signal amplification for imaging through co-encapsulated contrast agents; (iv) the tailoring of the physico-chemical and geometrical properties favoring avoidance of the biological and physiological barriers and improve the recognition of the target cells or microenvironment. A broad spectrum of particulates have been presented in the literature which can be used both in medical therapy and imaging. These have different sizes, ranging from few tens of nanometers (dendrimers) to hundreds of nanometers (polymer particles and liposome) up to few microns (silicon and silica based particles); different shapes from the classical spherical to spheroidal and even more complex shapes; and with different compositions and surface chemico-physical properties. Differently from freely administered drugs, these particles can be engineered and it is of fundamental importance to understand if, in the great variety of shapes, sizes and compositions, there is any 'optimal particle' with the largest selectivity and delivery efficiency. In this work, the adhesive performance of the particles with a size ranging between 500 nm and 10 $\upmu$m is assessed in terms of the number of particles adhering non-specifically to a confluent layer of endothelial cells under fixed hydrodynamic conditions. Goetz and colleagues have already analyzed the effect of the size of spherical particles on their adhesive performance. However, since they were interested in analyzing the interaction between leukocytes and endothelial cells, employed 5-, 10-, 15-, and 20 - $\upmu$m diameter microspheres. Also, the experimental analysis was carried out using a biological substrate consisting of a glass slide covered with P-selectin molecules, whereas in this work the biological substrate is made up of a confluent layer of endothelial cells. Nonetheless, they showed that the adhesive strength of the particles, expressed in terms of the critical shear stress and rolling velocity, is significantly influenced by their diameter. ## 2 Materials and Methods A Glycotech (Rockville, MD) parallel plate flow chamber has been used as described in. The flowchamber is connected to a Harvard Apparatus syringe pump through the inlet hole and a silastic tubing; and to a reservoir through the outlet hole and a silastic tubing. The channel within the flow chamber is defined by a silicon rubber gasket sitting between the flow deck and the microscope 35-mm dish. The gasket used in the present experiments has a thickness of 0.01 in and a width of 1 cm. The volumetric flow rate Q has been fixed equal to 50 ml/min for all the experiments. After assembling the flow chamber on the stage of an inverted microscope, a region at the center of the channel is chosen with a sufficiently large number of confluent cells. This is the region of interest (ROI). The microscope objective is focused on this region and after a short rinse, the flow with the particles enters the chamber. Using the fluorescent module of the microscope, the flow of the particles in the ROI is recorded during the whole experiments. Each acquired frame is then analyzed in MatLab(r) (Matlab6.5) using a self-developed imaging analysis code. The number of particles adherent to the whole substrate ntot within the ROI and the number of particles adherent to the sole cells nc within the ROI are measured as a function of the frame number, which is easily converted in time by knowing the acquisition rate. Human umbilical vein endothelial cells (HUVECs), purchased from Cambrex, Inc. (East Rutherford, NJ), were plated on a borosilicate glass with a 0.2 mg/cm2 substratum of type A gelatine (Sigma-Aldrich Corporation, MO). Fluoresbrite(r) Microspheres from Polysciences were used. ## 3 Results The number of adherent particles within the region of interest (ROI) has been monitored as a function of the size during the whole experiment. Figure1 shows, as an example, the variation with time of the particles nc adhering solely to the cells (lower curve) and the total number ntot of adherent particles within the ROI (upper curve), for the 1 mm, 750 and 500 nm particles. An image of the fluorescent particles adhering to the culture dish and to the cells is shown in Figure2, for the 1 mm particle. From the analysis of the experimental data, the number of particles ntot and nc at the end of each experiment can be plotted as a function of the particle diameter $d$ (Figure3). \[n_{tot} = 627.96d^{-1.696}\] \[n_{c} = 164.754d^{-1.143}\] Differently, in Figure4, the number of particles adherent per unit surface is shown as a function of their diameter: the total number of adherent particles ntot is normalized by the area of the ROI, whereas the number of particles adherent to the sole cells nc is normalized by the area occupied by the cells within the ROI, which depends on the level of cell confluency. \[\widetilde{n}_{tot} = 1116.39d^{-1.696}\] \[\widetilde{n}_{c} = 2784.34d^{-1.746}\] ## 4 Discussion and Conclusions The non-specific adhesion under flow of spherical particles onto a confluent layer of endothelial cells is analyzed in-vitro using a parallel plate flow chamber. Differently from other experimental analysis, the total number of particles injected in the flow chamber has been changed to keep fixed the total volume of the particles with their size. Since the focus is on drug delivery and biomedical imaging, by keeping fixed the total volume of the particles, the total volume of therapeutic or imaging agents delivered to the cell layer are fixed in the analysis. The experimental results, which have been derived under fixed hydrodynamic conditions, lead to conclude that (i) the surface density of particles adhering to the cells $\widetilde{n}_{c}$ decreases with the diameter $d$ following the scaling law = _d_-1.7; (ii) the volume per unit surface of the particles adhering to the cells $\widetilde{V}( = \pi\widetilde{n}_{c}d^{3}/6)$ increases with the diameter $d$ following the scaling law = _d_+1.3. These scaling laws have been also justified by theoretical reasonings, as described in details in. Since in drug delivery, it is desirable to increase the amount of drug that can be released per unit surface on a cell layer, the above results would suggest to use the largest possible particles in drug delivery. Notice however, that the following suggestion is derived by experimental observations of non-specific adhesion at sufficiently small shear stresses at the wall. In specific adhesion, Decuzzi and Ferrari have already shown that there is an optimal particle size corresponding to the largest strength of adhesion. Differently in biomedical imaging, it is desirable to increase the number of particles adherent per unit surface to have the largest possible resolution of the cell layer. As a consequence, the above results would suggest to use the smallest possible particles in biomedical imaging.	10.48550/arXiv.0708.1447	On the adhesion of particles to a cell layer under flow	F. Gentile, A. Granaldi, P. Decuzzi	5,795
10.48550_arXiv.0807.5119	###### Abstract We have carried out density functional theory based calculations of the size dependent formation energy and geometry of Pt islands on Ru, to model Pt-Ru nanocatalysts which have been recently proposed as fuel cell anode. The Pt islands are found to prefer two-dimensional structures. Furthermore, a monotonic decrease in the formation energy per Pt atom suggests a propensity of Pt atoms to wet the Ru surface. Calculated energy barriers for the diffusion of Pt monomers and dimers on the facets and through the edges of a superstructure modeling a Ru nanoparticle indicate that these edges help reduce considerably the diffusion rates across them such that the Pt atoms prefer to remain in the facet on which they were adsorbed originally and form 2D islands. pacs: Introduction Direct methanol fuel cells (DMFC) are considered a promising means for energy conversion in "hydrogen-based economy" because they work at low temperatures ($\sim$ 350-400 K) and use liquid methanol as fuel, which is easy to deliver and store. In DMFC, the same anode is used as a catalyst for both methanol reforming and for the oxidation of hydrogen obtained from the reforming. Although the carbon monoxide released in the course of this reaction is expected to be oxidized by hydroxyl radicals obtained from admixed water, experiments find that it still severely poisons the commonly used Pt anode by blocking the reactive Pt sites and, thereby, reducing the rate of hydrogen electro-oxidation. Similarly, in proton exchange fuel cells operating with pre-reformed gas, the anode is poisoned by carbon monoxide molecules, inevitably present in hydrogen obtained from hydrocarbons. It is known that PtRu alloys are more tolerant to CO poisoning than pure Pt, though their tolerance is still unsatisfactory. The high content of expensive platinum in these alloys also speaks against their choice as cost effective catalysts. It is thus encouraging to note that nanoclusters of Ru exposed to spontaneous Pt deposition are much more tolerant to CO than commercial PtRu catalysts, particularly since the content of Pt on these novel materials is significantly smaller than that in Ru-Pt alloys. For example, Brankovic _et al_. deposited Pt on $\sim$2.5 nm size Ru nanoparticles and found that the 1:20 ratio (PtRu${}_{20}$), which corresponds to $\sim$ 0.1 monolayer (ML) coverage, surpass substantially the catalytic performance of PtRu and Pt${}_{2}$Ru${}_{3}$ in the presence of CO. Assuming that Pt atoms form two dimensional (2D) islands on the facets of the Ru nanoparticles, the CO tolerance was attributed and explained on the basis of the _spillover_ effect, which refers to the tendency of CO to leave the Pt island and diffuse towards Ru. As noted above, Pt coverage is critical for the catalytic properties of the Ru nanoparticles. Island size effects - tuned by Pt coverage - have also been reported for methanol electro-oxidation. Earlier calculations have indicated that the CO adsorption energies on binary Pt-Ru systems, ranging from Pt to surfaces of ordered Pt-Ru alloys to 1ML of Pt on Ru, are reduced upon lowering Pt content. Moreover, our recent calculation supporting the idea of CO spillover relies on the presence of small 2D Pt islands on the facets of Ru nanoparticles. All of the above suggest that the correlation between catalytic activity and coverage has implications on how Pt arranges itself on the Ru nanoparticles, as we shall see later. Indeed, the first step in obtaining a systematic understanding of the enhanced reactivity of this catalyst is the determination of the geometry and relative stability of the Pt islands on the nanoparticle facets. Towards this end we have adopted in this work a theoretical approach based on _ab initio_ electronic structure calculations. Ever since the work by Stranski in 1927 about crystal growth, a large amount of effort has been dedicated to understanding the microscopic processes that control nucleation and epitaxial growth. It is now recognized that the resulting structure when adatoms are adsorbed onto a surface is thermodynamically and kinetically controlled. The energetic barriers for the various processes to occur, the surface and/or post-growth annealing temperature, along with the deposition rate and technique, essentially determine the morphology of the deposited adatom structure at a given coverage. Three major types of growth modes have been observed in experiment. The Frank van der Merwe mode refers to the two dimensional (2D) layer-by-layer growth, the Volmer-Weber growth describes three-dimensional (3D) clustering of adatoms on the bare substrate, while the Stranski-Krastanov mode consists of a mixture of the above two growth modes, i.e., 3D clustering occurs after one or a few complete adlayers are formed. Theoretical models of heteroepitaxial growth suggest that the growth mode is determined by the competition of factors such as the surface energies of the bare substrate and the heteroepitaxial layer, the interface free energy, and the strain energy introduced by the lattice mismatch of the two species. For example, a mismatch in the respective bulk lattice constants strains the interface and may set off the 3D clustering growth mode. On the contrary, growth of adlayers with lower surface energy than the substrate may favor the 2D layer-by-layer growth mode. Considerations along these lines, however, lead to ambiguities in predicting Pt growth on Ru. While the higher cohesive energy of Ru (relative to Pt) may imply that the surface free energy of Ru is higher than that of Pt and point to 2D layer-by-layer growth, the stress caused by the Ru-Ru, Ru-Pt, and Pt-Pt bond length misfit (The bond lengths in bulk Ru and Pt are 2.706 A and 2.775 A, respectively.) may lead to 3D clustering. Of course, there may be a competition between the above two factors, leading to a critical Pt island size at which there is crossover between 2D and 3D growth mode or island-substrate atom exchange. One of the goals in this work is to determine whether there is indeed a critical size beyond which 2D Pt islands are no longer stable on Ru. Turning to the case of the Ru nanoparticles described in the experiments in question, it is important to note that, while the experimental determination of the structure and chemical ordering of Pt-Ru nanoalloys represented a challenge until the recent work of Maillard _et al._, the dominant features observed via surface X-ray scattering, scanning tunneling microscopy, Fourier transform infrared spectroscopy, and high resolution transmission electron microscopy techniques, in 2-3 nm pure Ru nanoparticles on a carbon substrate have been consistent with the hexagonally close-packed (hcp) Ru single crystal structure. The Ru nanoparticles present well-defined facets with the lowest-surface-energy geometries, such as, ($\overline{1}$101), etc. In the present study, as a first step in the modeling of the Pt-decorated Ru nanoparticles, we focus our attention on the formation of Pt islands as a function of size on Ru, one of the most predominant and stable facet orientations. To this end, we carry out first principles calculations of the system's total energy to determine the geometry and formation energy of Pt islands, as well as that of 1ML of Pt on Ru. Clearly, the main drawback of such calculations is that the predictions are relevant to zero temperature and samples relaxed in _infinite time_. To partially include the system dynamics, we have taken into account the diffusivity of Pt adatoms since, particularly at temperatures well below room temperature, the three aforementioned growth modes can be understood in terms of diffusion barrier differences between on-step and step-descending hopping, Schwoebel barriers. It is argued that if the energy barrier at the step edge is higher (reflective barrier) than that of adatom diffusion on the step - positive Schwoebel barrier - there is a probability that adatoms will be trapped on the step terraces and 3D clustering will be favored. On the contrary, if the Schwoebel barrier is zero or negative (non-reflective barrier), adatoms that happen to lie on the step terrace are more likely to hop onto the substrate and favor the 2D growth. Moreover, in heteroepitaxial growth, the mixed Stranki-Krastanov mode dominates if the Schwoebel barrier is reflective for homo-steps (formed by the adatoms), but non-reflective for hetero-steps. Furthermore, since Ru nanoparticles present facets with various geometries, it is reasonable to assume that the equilibrium shape is a polyhedron and, therefore, there are edges delimiting the facets. Theoretical and experimental studies of the shape of Pt nanoparticles supported on carbon, for instance, have consistently concluded that they are in the form of cubo-octahedra, which is a shape characteristic of fcc packing. For hcp metals, in turn, one of the proposed structures for sufficiently large nanoparticles (with more than 500 atoms) is the so-called anticubo-octahedron or the truncated hexagonal bipyramid. The essential point is that, since the above polyhedra have several facets (14 for the cubo- and anticubo-octahedron and 20 for the truncated hexagonal bipyramid), one expects the morphology of the adsorbed Pt to be sensibly influenced by the diffusivity of Pt adatoms/islands through the edges of the Ru nanoparticles. To assess the possibility of diffusion of Pt monomers and dimers, we simulate two edges of a Ru nanoparticle using a superstructure described in the next section and calculate diffusion barriers of Pt monomers and dimers on its and ($\overline{1}$101) facets and across its edges. Note that our model for the facet geometries coincide with some of those detected in experiment and are also proper to the truncated hexagonal bipyramid. The rest of this article is organized as follows: Section II presents the computational details, Section III contains our results and provides some discussion about Pt islands on Ru (subsection III.1) and of Pt diffusion on the facet and across the edges of our Ru nanostructure (subsection III.2). Section IV summarizes our results and conclusions. ## II Computational details Periodic supercell calculations have been carried out within the density functional theory (DFT) framework, using the plane wave and pseudopotential methods as embodied into VASP (Vienna _ab initio_ Simulation package), and with ultrasoft pseudopotentials. We have used a kinetic energy cutoff of 400 eV for the wave functions and 700 eV for the charge density to obtain convergent results with sufficient computational accuracy of the lattice constant of bulk Ru and Pt. Brillouin zones were sampled with either the ($4\times 3\times 1$) or the ($3\times 3\times 1$) Monkhorst-Pack k-point meshes, depending on the size of the supercell, as we will see. Since the main uncertainty of DFT comes from the exchange-correlation potential, we have used two different approximations for the exchange-correlation functional: the Perdew and Wang generalized gradient approximation (GGA) and the Perdew, Burke, and Ernzerhof (PBE) modified GGA, and compared some of the results obtained using these two approximations. To achieve force relaxation of the studied structures, the total energy of the system and the forces acting on each atom are obtained after each self-consistent electronic structure and minimized by the conjugated-gradient algorithm. At equilibrium, forces on each atom are required to be below 0.02 eV/ A. The diffusion barriers for monomers and dimers on the Ru superstructure are obtained by the direct dragging method. The 3Dgraphics presented in this work were generated by the _Xcrysden_ program. We note that in the calculations of the geometry and formation energy of Pt islands the usage of supercells may introduce contributions from island-island interactions, as a result of the imposed periodicity of the system. This is particularly true for the largest islands in the $3\times 4$ and $4\times 4$ supercells in which edge-atoms of neighboring islands are as close as third nearest neighbors (NN) of the Ru surface. A simple way to estimate this spurious interaction is to consider the interaction energy, $E^{int}$, between two Pt atoms adsorbed on Ru as their separation varies and defined as $E^{int}=E(2$Pt/Ru$)-2E(1$Pt/Ru$)$, where $E(2$Pt/Ru$)$ is the adsorption energy of 2 Pt atoms on Ru in a $4\times 4$ supercell and $E(1$Pt/Ru$)$ is the adsorption energy of 1 Pt atom on the same surface and supercell. We find that $E^{int}=$ -0.203, +0.074, and -0.018 eV, as the separation between the two Pt atoms increases from $1^{st}$ to $2^{nd}$, and $3^{rd}$ NN bond length, respectively. The interaction between $3^{rd}$ NN is thus expected to be 10 times smaller than that of $1^{st}$ NN. Accordingly, $E_{form}$, as calculated in this work, can be reliable up to $\pm$ 0.02 eV for the largest islands. ### 2D Pt islands on Ru The facets of Ru nanoparticles are first modeled by a 5 layer Ru slab, which is the surface known to have the lowest energy. Pt adatoms are placed on only one side of the slab. To avoid the interaction between surfaces and Pt adatoms of neighboring periodic supercells we have imposed a 15 A vacuum layer, whereas to reduce the interaction between deposited Pt islands, the surface unit cells is extended to either ($3\times 4$) or ($4\times 4$) structures depending on the island size. The ($3\times 4$) and ($4\times 4$) supercells contained 60 and 80 Ru atoms, respectively, plus Pt atoms forming the island. Their corresponding surface Brillouin zone is sampled with a ($4\times 4\times 1$) and ($3\times 4\times 1$) k-point mesh, respectively. ### Monomer and dimer on faceted superstructure To model Pt diffusion through the Ru nanoparticles edges we have taken into consideration edges formed by facets of and ($\overline{1}101$) geometry, which are among the most stable Ru surfaces. We consider a periodic 3D superstructure containing 116 Ru atoms and made of a 4-atom wide Ru facet and two Ru($\overline{1}101$) facets (see Fig. 1). The con struction of this Ru supercell, which has $7\times 4$ in-plane periodicity, is achieved by stacking five Ru layers: two of $7\times 4$, one of $6\times 4$, one of $5\times 4$, and one of $4\times 4$ atoms. The so obtained edges, on each side of the Ru have different local geometry which for convenience are labeled as $A$ and $B$. Atoms forming edge $A$ (edge $B$) are contiguous to hcp (fcc) hollow sites of the facet. The bottom two layers (see Fig. 1) were not allowed to relax to guarantee the stability of the superstructure. We impose a 15 A vacuum layer between periodic superstructures along the direction perpendicular to the surface, as in the system described previously. The Brillouin zone is sampled with a $(2\times 3\times 1)$ k-point mesh. The adsorption energy and diffusion barriers of Pt monomers and dimers are calculated on the and the ($\overline{1}$101) facets. ## III Results and discussion To determine in-plane slab periodicity, we have calculated bulk lattice parameters using PW and PBE approximation for the exchange-correlation functional. The bond lengths of bulk Ru and Pt are found to be 2.706(2.744PBE) and 2.77(2.82PBE) A, respectively, while the c/a ratio of bulk Ru is found to be 1.585 (PBE) A. ### Pt islands on Ru We have calculated the optimized geometric structure and energetics of 1 to 5 Pt atom islands adsorbed on Ru using the $(3\times 4)$ supercell and that of 1 to 9 Pt atom island on Ru using the $(4\times 4)$ supercell. The relaxed structure and total energy of one Pt monolayer on Ru has been also obtained. To characterize the energetic stability of a given Pt island, we obtain its formation energy, which is defined as: $E_{form}=E(\text{Ru+Pt})-E(\text{Ru})-nE(\text{Pt}_{at})$. Here, $E(\text{Ru+Pt})$ is the total energy of a Ru slab adsorbed with a $n$-atom Pt island, while $E(\text{Ru})$ and $E(\text{Pt}_{at})$ denote the total energies of the clean Ru slab and a free Pt atom, respectively. Note that the formation energy of stable structures is negative. The structure with lowest average formation energy per Pt atom, $E_{form}/n$, will thus be distinguished as the energetically most favorable one. Fig. 2(a) presents $E_{form}/\text{n}$ as a function of the size of the island, $n$, for $n$ = 1-5 on the $(3\times 4)$ supercell, as provided by PBE and GGA. Fig. 2(b) displays the dependencies of $E_{form}/n$, for $n$ = 1-4, 6, 7 and 9, on the $(4\times 4)$ supercell. Note that the values $n$ = 12 and $n$ = 16 represent full monolayer coverage of Pt for the $(3\times 4)$ and $(4\times 4)$ supercells, respectively. We find that both PBE and GGA provide similar qualitative dependencies: the larger island, the stronger the bonds. Such a trend culminates and is confirmed at full monolayer Pt coverage, which provides lowest $E_{form}/n$. The effect of atom detachment (from Pt islands) on $E_{form}$ has been studied as well. shows two configurations considered for the 7-atom Pt island adsorbed on Ru. We find that the detachment (transition from the left to the right configuration in the figure) causes an increase in $E_{form}$ from -38.55 eV to -37.52 eV. Similar results have been obtained for the islands of other sizes. The increment of energy per Pt atom upon detachment, however, does not depend significantly on the island size and vary in the range of 0.11 - 0.14 eV. For instance, for a 2-atom Pt island, detachment leads to an increase in $E_{form}$ from -10.46 eV to -10.24 eV while, for a 3-atom island, the energy increases from -15.88 eV to -15.52 eV upon the detachment. We thus obtain a clear trend: the larger a two-dimensional Pt island (up to 1 ML) is, the lower its formation energy per atom. Thus, assuming that the free energy of the system in consideration is dominated by its DFT total energy, we conclude that Pt tends to wet Ru. We have also performed calculations for some 3D Pt islands on Ru and found that their 2D isomers have lower energy. For example, two configurations of a 9-atom Pt island with 2D and 3D structures (see Fig. 4) were found to have $E_{form}$ = -50.06 eV and -48.54 eV, respectively. The above results indicate that Pt atoms overpower the stress effects derived from the Pt-Ru bond length misfit and have a propensity to increase their local coordination on Ru regardless of the chemical nature of the bonds, thus forming 2D islands up to $\sim$ 0.56 ML and, presumably, up to 1 ML (see Fig. 2). Of course our calculations do not include the case of larger Pt islands (more than 10 atoms) and leave open the question of formation of 3D structures in such cases. Nevertheless, there is experimental evidence that Pt may form 2D pseudomorphic islands up to full coverage through vapor deposition. In turn, from the standpoint of _ab initio_ calculations, our conclusion that Pt grows pseudomorphologically is justified only at low temperatures and after a long relaxation time. To take into account the kinetics of the system, we have looked at the diffusivity of a Pt adatom in the event that it be adsorbed on top of Pt islands. Consider, for example, the case presented in The question is thus whether the Pt adatom will be trapped on top of the Pt island (as shown in the right inside of Fig. 4) or will it descend at the kink site on the Ru surface (left inside of Fig. 4). Our calculated activation barrier of the Pt monomer diffusion on the 8-atom Pt island from the hcp to fcc sites is $\sim 0.23$ eV and from the fcc to hcp is $\sim 0.09$ eV. The activation barrier for the step descent for a Pt monomer from the fcc to the kink site on the Ru is $\sim 0.39$ eV (and $\sim 1.92$ eV for the inverse process). Therefore, the Schwoebel barrier of this heteroepitaxial step descent of Pt monomers is $\sim$ +0.30 eV, pointing to 3D clustering growth. These opposing results hint why the growth mode of Pt on Ru sensibly depends on experimental conditions. Spontaneous deposition of Pt, for example, has led to the conclusion that 3D structures follow the 2D pseudomorphic islands if the Ru nanoparticles are immersed for sufficiently long periods of time (not specified) into a 0.1M HClO${}_{4}$ solution containing 0.1mM H${}_{2}$PtCl${}_{6}$, whereas spontaneous deposition by two-minute immersion into 0.1 mM [PtCl${}_{6}$]${}^{2}-$ and subsequent rinsing with 0.1 M H${}_{2}$SO${}_{4}$ and ultra pure water lead to Pt nanoparticles with columnar shape of 10-15 MLs and 3-5 nm. For the purpose of our analysis, however, we shall notice that both vapor deposition below 0.6 ML and spontaneous deposition via _short_ immersion into Pt-containing solutions find that the probability for Pt adatom to arrive on top of a Pt island is low. Indeed, the Ru nanoparticles that we are trying to model are decorated with Pt at very low coverage ($\sim$ 0.1 ML) via spontaneous deposition by immersion in a [PtCl${}_{6}$]${}^{2-}$ and rinsed with 0.1 M H${}_{2}$SO${}_{4}$ solution at room temperature, therefore, only formation of 2D islands is expected. Besides, there is indication that, even in the case of positive Schwoebel barriers, as long as the atom deposition rate is low and the diffusion speed fast, the reflective property of steps is valid only at temperatures well below room temperature for fcc metals. There still remains the question: why, despite the misfit, 2D configurations are favorable? In order to grasp further understanding of the issue, one must realize that a number of counter-trends must be taken into account. Firstly, the 2D growth is not all that surprising for the reason that the bond-length misfit between Pt and Ru amounts only to a few percent. Secondly, in the 2D Pt islands under consideration, the number of NN fluctuates from 3 (single atom) to 9 (full monolayer). Decrease in the number of NN usually causes reduction of the equilibrium interatomic distances. Indeed, we find that for a free standing Pt monolayer, in which every Pt atom has only 6 nearest neighbors, the equilibrium Pt-Pt NN distance is much shorter ($\sim$2.6 A) than that in bulk Pt ($\sim$2.8 A) and even shorter than the Ru-RuNN distance in bulk Ru ($\sim$2.71 A). The misfit in low dimensional structures is thus not a well defined quantity because of the dependency on the coordination number of the atoms in question. Correspondingly, the issue of stress induced by the bond-length misfit between the Pt nanostructures and the Ru surface atoms, neither one of which is expected to be at the bulk value given the diversity of their local geometric environment, is thus not clear for these heteroepitaxial system. In addition, Barcaro _et al_. have pointed out that heteroepitaxial Volmer-Weber (3D) growth is favored for those systems in which the adsorbed metal interacts more weakly with the substrate than with atoms of the same species. In our particular case, however, we obtain that the Ru-Pt bond may in fact be stronger than that of Pt-Pt since our calculations show that the average formation energy per atom of a Pt monolayer on Ru ($\sim$ 5.89 eV) is higher than both the cohesive energy of Pt bulk ($\sim$ 5.85 eV) and the average cohesive energy per atom of the Pt surface ($\sim$ 5.6 eV). For the Pt atoms on the top of a hcp metal such as Ru, there is also an incommensurability in bulk structure. We find that the bulk NN bond length of Pt atoms certainly decreases when they arrange in an hcp structure. In that case, the bulk bond length misfit between Pt and Ru decreases from $\sim$ 2.8 to $\sim$ 1.4 %. Furthermore, the surface _interlayer_ distance in Pt expands to 2.49 A ($\sim$ 1.0 % with respect to bulk), while that of the hypothetical Pt contracts to 2.39 A notwithstanding that _intralayer_ NN distances are 1.8 % smaller than in the fcc bulk. Therefore, both the reduced coordination in adsorbed Pt clusters and the Pt-Ru bond strength may quench the anticipated effect of bulk bond-length misfit in such a way that Pt atoms adsorbed on Ru surface tend to form 2D islands as large as possible up to one monolayer. ### Modeling Pt diffusion on the facets and through the edges of Ru nanostructures From the previous section, we have gained an understanding of the tendency of Pt atoms to form 2D islands, wetting the Ru surface rather than clustering in multiple 2D or 3D structures. The experimental evidence, however, suggests that Pt islands maintain small sizes on Ru nanoparticles, say $\sim$0.5 nm (5 to 7 atoms), for 0.1 ML coverage. The difference in the characteristic of Pt surface alloys on Ru and on Ru nanoparticle is part of the reason for the substantial reactivity of Pt adatoms on Ru nanoclusters and not on Ru surfaces. If Pt atoms were to make as many bonds as possible on the Ru nanoparticle as they do on the Ru surface, one would expect that, even for low coverage ($\sim$0.1 ML), a large island should totally cover one of the facets of the Ru nanoparticles. For example, a hcp Ru nanoparticle of 2.5 nm with the proposed anticubo-octahedral structure could hold roughly $\sim$7 ($\sim$4) Pt atoms per squared (triangular) facet for homogeneous coverage of 0.1 ML ($\sim$ 70 Pt atoms), whereas the same coverage coalesced into a single island could totally cover one of the squared facets, which seems not to be the case. Note that a similar argument would follow for a truncated hexagonal bipyramid. One main difference between the _infinite_ surface and the nanoparticle is that the latter exhibits edges dividing its facets. It is natural to assume that they prevent Pt coalescence on the Ru nanoparticles. If this is true, 2D islands are formed on each facet, but they do not join together into a large unique island because the edges prevent those initial small islands from diffusing to other facets, thus persisting as few-atom 2D islands. Support for the aforesaid reasoning will be attained below by comparing the barrier for Pt atoms to diffuse on a facet with that for Pt to diffuse across the edges towards a ($\overline{1}$101) facet. #### iii.2.1 Pt Monomers The Ru nanostructure used for the above purpose is displayed in It possesses 3 hcp (hcp1, hcp2, hcp3) and 3 fcc (fcc4, fcc5, fcc6) non-equivalent hollow sites on the facet, as shown in Fig. 5(a). The calculated adsorption energies ($E_{ads}$) of Pt monomers on these sites are listed in Table 1. We find that for all hcp sites $E_{ads}$ is higher than for any fcc site. Table 1 shows in addition that Pt monomers preferably sit on sites surrounded by 2 edge atoms (fcc4 and hcp3), rather than on those surrounded by only one (fcc6 and hcp1) or none (fcc5, hcp2) edge atom. Note also that the adsorption energy of a monomer on the facet is lower than on the infinite surface (5.13 eV). Across edge $A$, the first ($\overline{1}$101) available site is a 4-fold hollow site, denoted by "1" in Fig. 5(b), whose adsorption energy, 5.66 eV, is substantially higher than that on hcp sites of the facet. Across edge $B$, the first ($\overline{1}$101) available site is a 3-fold hollow site, denoted by "2" in Fig. 5(b), whose adsorption energy is 4.92 eV. The above results, incidentally, suggest that there may be a propensity of Pt to deposit on ($\overline{1}$101) facets in the long range. The calculated diffusion barriers, $\Delta E$, through edge $A$, appear to be highly asymmetric (see Fig. 6(a)): $\Delta E$(hcp3 $\rightarrow$ "1") = 0.49 eV and $\Delta E$("1" $\rightarrow$ hcp3) = 1.10 eV. On the other hand, the shortest path for diffusion through the edge B connects the hcp1 site on the facet with the three-fold hollow site on the ($\overline{1}$101) facet (fcc4 $\rightarrow$ "2"). The barrier for diffusion along this path is found to be 0.22 eV, which is just slightly higher than the barriers for diffusion on the facet. However, our calculations indicate that the initial state for this process (hcp1 site) is metastable since its total energy is 80 meV higher than that for the monomer adsorbed on the neighboring fcc4 site. The diffusion of the Pt monomer through the edge B is thus a two-step process with diffusion from the hcp1 to fcc4 site on and then from that to the tree-fold site on ($\overline{1}$101). For the above reason, the overall probability of diffusion through the edge B is substantially reduced. Even at room temperature, the probability to find a Pt adatom on the fcc4 site is $\sim$ 24 times lower than that on the hcp1 site. Moreover, the hcp1 $\leftrightarrow$ fcc4 barrier is asymmetric (0.20 eV for hcp1 $\rightarrow$ fcc4 and 0.12 eV to diffuse back), which further reduces the diffusion probability. In order to estimate the probability of Pt diffusion predicted by these barriers, we have calculated roughly the diffusion rate of Pt monomers. The latter, dominated by the exponential of the energy barrier, is given by $R=D_{0}e^{-\frac{\Delta E}{k_{B}T}}$, where $k_{B}$ is the Boltzmann constant, $T$ is the temperature, and $D_{0}$ is the diffusion prefactor whose typical values are of the order of $\sim$ 10${}^{12}s^{-1}$. By noting from above that the diffusion rate through the edge $B$ is reduced by a factor of $\sim$ 24 with respect to that from fcc4 $\rightarrow$ "2", we obtain that the former is effectively $\sim$ 6$\times$10${}^{6}$$s^{-}$1, thus significantly smaller than that on the facet ($\sim$ 4$\times$10${}^{8}$ and $\sim$1$\times$10${}^{10}s^{-1}$, for hcp1 $\rightarrow$ fcc4 and fcc4 $\rightarrow$ hcp1, respectively). On the other hand, the diffusion rate through edge A ($\sim$ 3$\times$10${}^{-7}$ and $\sim$ 6$\times$10${}^{3}$$s^{-}$1, for "1" $\rightarrow$ hcp3 and hcp3 $\rightarrow$ "1", respectively) is found to be three orders of magnitude lower than that through edge $B$ and at least five orders of magnitude lower than that on the facet. Notice that since the barriers to diffuse back from the ($\overline{1}$101) facet to the facet through edge $B$ ("2" $\rightarrow$ fcc4) is only slightly larger (0.28 eV) than the barrier for diffusion in the opposite direction, diffusion across edge B may be actually inefficient. The rate at which Pt monomers return to the facet can nevertheless be expected to be lower than that by which they diffuse to the neighboring four-fold hollow site on the ($\overline{1}$101) facet, since the barriers for the latter process are significantly lower. The diffusion from the three-fold hollow site to the four-fold hollow site on the ($\overline{1}$101) facet is found to be a two-step process. It involves the diffusion to an intermediate local minimum posing a 0.12 eV barrier and a subsequent step with a barrier of only 70 meV. The barriers to return to the three-fold hollow site are, correspondingly, 0.20 eV and 0.6 eV. These values confirm that, at least energetically, monomers may have a propensity to remain on the ($\overline{1}101$) facet once they occupy it. However, exact implications of these rates and the impact of competing processes can only be visualized after kinetic effects are properly included, such as in kinetic Monte Carlo simulations which we leave as a further task. In summary, our results indicate that edges compel Pt monomers to remain on the facet where they are initially adsorbed, thus preventing the formation of large Pt islands on the Ru nanoparticles, although Pt monomers may diffuse across some edges more efficiently than across others. Of course, since the work reported here has not considered facets of the Ru nanoparticles that exhibit other geometries, the relative adsorption energies and diffusion barriers of these may expand and tune the occupancy landscape from the one we have inferred. #### iii.2.2 Pt dimers The results of previous sections indicate that clustering of 2D Pt islands on the Ru facets occurs readily and that, although the Ru edges hinder the inter-facet diffusivity of Pt monomers, they do not unquestionably prevent it at room temperature. Yet, in order for Pt islands to coalesce and form larger islands, they would have to diffuse through the edges. We therefore turn to the calculation of the energy barrier for a dimer to diffuse from hcp to fcc sites (on the facet) and through edge $B$, the _easy_ edge for monomers to cross. Some of the sites that Pt dimers may adopt on the and the ($\overline{1}101$) facets, as well as the corresponding average formation energies per atom, are shown in We find that $E_{form}/n$ of the dimer increases (by $\sim$ 0.12 eV) with respect to that of the monomer, suggesting that dimers would preferably form rather than diffuse as monomers through the _easy_ edges. As in the case of monomers, dimers prefer to sit on hcp sites on the facet (see Fig. 7(a) and (b)); similarly, the adsorption energy of Pt dimers at hcp sites near edge $B$ (only one edge-atom neighbor) and in the middle of the Ru stripe is almost the same (see Fig. 7(a) and (c)). On the facet, when one of the atoms in the dimer comes closer to the edge and its coordination is reduced from 5 to 4 (compare Fig. 7(c) and (d)), $E_{form}/n$ drops $\sim$ 0.16 eV, suggesting that there is a higher barrier for Pt dimers to approach the edges to the point where its atoms become more undercoordinated. On the ($\overline{1}101$) facet, analogously, $E_{form}/n$ is 0.52 eV lower for a dimer across the and ($\overline{1}101$) edges (see Fig. 7(e)) than for a dimer sitting on the ($\overline{1}101$) facet close to the edge (see Fig. 7(f)). As shown in Fig. 8, the barrier for the dimer to diffuse from fcc to hcp sites has a height comparable to that for monomers while it is 2.5 times smaller for the inverse process, from hcp to fcc. The diffusion rates (1.4$\times 10^{9}$ and 8.8$\times 10^{4}\;s^{-1}$, correspondingly) thus differ by four orders of magnitude, indicating that dimers are most of the time at hcp sites. This presents another indication that Pt dimers (or larger islands) would not leave the facet since, for it to diffuse across edge $B$, it must be on fcc sites. As shown in Fig. 9, the diffusion across edge $B$ comprises two stages. The initial state of the first stage corresponds to the configuration shown in Fig. 7(d) in which one atom is on a hcp1 site and the other is slightly beyond the edge whereas, in the final state, the atom at the hcp1 site diffuses to a fcc4 site and the other moves to a type "2 " site (see Fig. 5(b)) of the ($\overline{1}101$) facet, as shown in Fig. 7(e). The energy barriers for the dimer to diffuse back and forth from the initial and final states are shown in while the corresponding diffusion rates for these processes are low ($\sim 10^{4}$). Indeed, the energy barriers of the first stage are very similar to those for dimer diffusion from hcp to fcc sites (see Fig. 8) and for monomer diffusion through edge $A$ (see Fig. 6). Notice also that the final state described above is only an intermediate stage of the diffusion towards the ($\overline{1}101$) facet. For the second stage, the initial state is naturally the configuration shown in Fig. 7(e) while the final state corresponds to that shown in Fig. 7(f), in which both atoms sit on the ($\overline{1}101$) facet. The energy barrier to move from the initial to the final state (see Fig. 9) produces also a low diffusion rate of 6$\times 10^{3}\;s^{-1}$, while the inverse process, whose barrier is 3 times larger, provides a diffusion rate of $\sim$1.4$\times 10^{-14}\,s^{-1}$, indicating that dimers like monomers on the ($\overline{1}101$) facets will most likely remain there. In short, edge $B$, which offers a low-barrier diffusivity path to monomers, renders to Pt dimmers two energy barriers for facet-to-facet diffusion, each representing a diffusion rate at least three orders of magnitude lower than those of the monomer. We expect the diffusion rates of trimers and other n-mers of Pt on Ru nanoparticle facets to be even lower than that found here for dimers. Summary We have calculated from first principles the energetics and geometry of Pt islands deposited on Ru, as well as the energy barriers for diffusion of Pt monomers and dimers through the edges intersecting the and ($\overline{1}$101) facets of a superstructure modeling a Ru nanoparticle. We find that the low coordination of Pt atoms composing the islands, the strong Pt-Ru interaction, and the hcp structure of the substrate promote formation of increasingly large Pt islands on Ru, possibly up to 1ML, and avoid the 2D/3D crossover. On the other hand, the scenario is quite different when Pt atoms are deposited on Ru nanoparticles. In a simple model for facctes with edges connecting to ($\overline{1}$101) orientations, we concur with experimental predictions that Pt atoms arrange homogeneously over the facets of Ru nanoparticles by spontaneous deposition and form 2D islands. We also predict that these islands do not coalesce into a large unique island because the edges of the Ru nanoparticles prevent monomers and dimers from diffusing to other facets. Our calculated barriers indicate that there may be some edges in the Ru nanoparticles for which the diffusion rate _across-edge_ is several orders of magnitude lower than the diffusion rate _on-facet_, even for monomers. For those edges that may offer relatively low diffusion barriers to monomers, our calculated barriers for dimers suggest that dimers or larger islands, whose formation is more probable than the monomer diffusion, remain in the facet where they were formed since the diffusion rate _across-edge_ is several orders of magnitude lower than that of monomers.	10.48550/arXiv.0807.5119	Formation of Pt islets on facets of Ru nanoparticles: a first-principles study	Marisol Alcántara Ortigoza, Sergey Stolbov, Talat Rahman	4,712
10.48550_arXiv.0711.4041	###### Abstract We study electromagnetic properties of periodic composite structures, such as photonic crystals, involving lossy components. We show that in many cases a properly designed periodic structure can dramatically suppress the losses associated with the absorptive component, while preserving or even enhancing its useful functionality. As an example, we consider magnetic photonic crystals, in which the lossy magnetic component provides nonreciprocal Faraday rotation. We show that the electromagnetic losses in the composite structure can be reduced by up to two orders of magnitude, compared to those of the uniform magnetic sample made of the same lossy magnetic material. Importantly, the dramatic absorption reduction is not a resonance effect and occurs over a broad frequency range covering a significant portion of the respective photonic frequency band. ## 1 Introduction Magnetic materials play a crucial role in microwave technology and optics. They are absolutely essential in numerous nonreciprocal devices such as isolators, circulators, phase shifters, etc. They can also provide tunability, miniaturization, better impedance matching, and other important features. A major obstacle for broader applications of magnetic materials is tied up with the issue of absorption. Many magnetic materials with otherwise perfect physical characteristics have been rejected because of strong losses at frequency range of interest. In this paper we explore the idea of composite magnetic structures having desired physical properties associated with magnetism but, at the same time, significantly suppressing the effects of absorption. In other words, we want to take advantage of the useful characteristics of a particular magnetic material, while drastically reducing its contribution to the energy dissipation. Magnetic photonic crystals are spatially periodic composite structures with one of the components being a magnetic material, such as a ferromagnet or a ferrite. Extensive information on the subject and numerous references can be found in a recent review article. Similarly to other photonic crystals, magnetic photonic crystals display strong spatial dispersion, resulting in appearance of electromagnetic band-gap structure. But in addition, magnetic photonic crystals can provide tunability and better impedance matching than regular non-magnetic photonic crystals. In comparison to uniform magnetic materials, magnetic photonic crystals can display much stronger nonreciprocal properties, such as magnetic Faraday rotation. Strong nonreciprocal effects are essential in isolators, circulators, and other microwave and optical devices. The possibility of appreciable enhancement of Faraday rotation is particularly important at infrared and optical frequencies, where the non-reciprocal effects in uniform magnetic materials are very weak. In addition, the use of periodic structures instead of uniform magnetic materials can dramatically reduce the size of non-reciprocal and other microwave and optical devices. The subject of our investigation is another important aspect of electrodynamics of periodic composite structures. Namely, we show that properly design periodic array can dramatically reduce the losses associated with individual constitutive components. In the particular case of magnetic photonic crystals, the broadband suppression of losses can be achieved in a combination with the enhancement of nonreciprocal properties, such as Faraday rotation, linear magnetoelectric response, etc. The possibility of the reduction of losses is related to the fact that in most cases the absorption and the useful functionality of the particular magnetic material are related to different components of its permittivity and/or permeability tensors $\hat{\varepsilon}$ and $\hat{\mu}$. \[\hat{\varepsilon}^{\prime\prime}=-\frac{i}{2}\left(\hat{\varepsilon}-\hat{ \varepsilon}^{\dagger}\right),\ \hat{\mu}^{\prime\prime}=-\frac{i}{2}\left(\hat{ \mu}-\hat{\mu}^{\dagger}\right), \tag{1}\] By contrast, the nonreciprocal circular birefringence responsible for the Faraday rotation is usually determined by the Hermitian skew-symmetric part of the respective tensors \[\hat{\varepsilon}_{a}=\frac{1}{2}\left(\hat{\varepsilon}-\hat{\varepsilon}^{T }\right),\ \hat{\mu}_{a}=\frac{1}{2}\left(\hat{\mu}-\hat{\mu}^{T}\right), \tag{2}\]where the subscript $T$ denotes matrix transposition. The relations and suggest that the rate of energy absorption by the lossy material can be functionally different from its useful functionality (nonreciprocal circular birefringence in our case). Such a difference allows us to adjust the physical and geometric characteristics of the periodic structure so that the electromagnetic field distribution inside the photonic crystal suppresses the energy dissipation by the lossy magnetic component, while preserving or even enhancing its useful functionality. Moreover, in some cases, a sufficiently strong absorption can affect the electromagnetic field distribution in such a way that it suppresses its own contribution to the total rate of absorption of the composite material. In the latter situation, the stronger absorption coefficient of the lossy component is, the less it contributes to the total rate of absorption of the composite structure. The way to address the problem of absorption suppression in a periodic composite structure essentially depends on the following three factors. 1. The useful functionality of the lossy material. In our example, it will be the nonreciprocal circular birefringence producing the Faraday rotation. 2. The dominant physical mechanism of absorption. For instance, energy dissipation caused by electric conductivity requires a different approach, compared to the situation where the losses are associated with the dynamics of magnetic domains, or some other physical mechanisms. In each individual case, the structure of the anti-Hermitian part of the permittivity and/or permeability tensors can be different, and so can be the optimal configuration of the composite material. 3. The frequency range of interest. The same periodic array can significantly reduce losses at some frequencies, while enhancing losses at different frequencies. In other words, the same periodic structure can be either effective or counterproductive, depending on the frequency range and the dominant physical mechanism of electromagnetic energy dissipation. The possibility of a significant absorption reduction is not limited to magnetic composites. Similar approach can be applied to other heterogeneous structures with lossy components. A key requirement is that the useful functionality of the lossy material should be functionally different from its contribution to the energy dissipation. In such a case, the periodic structure can be engineered so that at frequency range of interest, the electromagnetic field distribution inside the composite medium suppresses the losses, while preserving the useful functionality of the lossy component. This can always be achieved if the components of the tensors $\hat{\varepsilon}$ and $\hat{\mu}$ related to the useful functionality of the lossy material are different from those dominant in the anti-Hermitian tensors $\hat{\varepsilon}^{\prime\prime}$ and $\hat{\mu}^{\prime\prime}$. In any event, in order to suppress the energy dissipation caused by lossy component of the periodic structure we have to take into account the physical nature of absorption, the useful functionality of the lossy material, and the frequency range of interest. In the next section we produce a specific numerical example of a periodic structure that includes a lossy magnetic material. The amount of Faraday rotation produced by this periodic structure is comparable to that of a uniform slab made of the same lossy magnetic material. But the absorption rate of the composite structure is 20 to 100 times lower than that of the uniform magnetic slab. Importantly, such a dramatic absorption reduction is achieved in a broad frequency range covering almost an entire photonic frequency band. Another practical advantage is that in some cases the size of the composite structure can be much smaller compared to that of the uniform magnetic sample with similar characteristics. Note, though, that a significant size reduction usually comes at the expense of the bandwidth. ## 2 Absorption suppression in a periodic layered structure Consider a monochromatic plane wave normally incident on a uniform magnetic slab, as shown in Suppose that the useful functionality of the slab is the nonreciprocal Faraday rotation. This is the case with almost all nonreciprocal microwave and optical devices, such as the isolators, circulators, etc. The statement of the problem is as follows. On the one hand, we have a plane-parallel uniform magnetic slab characterized by certain Faraday rotation and absorption. This uniform slab is shown in On the other hand, we have a stack of layers made of the same lossy magnetic material alternating with some other layers. Such a stack is shown in We expect the properly designed periodic stack in to be superior to the uniform magnetic slab in The superiority can be defined by the following set of requirements. 1. The stack produces similar or larger Faraday rotation, as compared to that of the uniform magnetic slab. This means that the stack and the uniform slab have comparable useful functionality. 2. The stack has much lower absorption than the uniform magnetic slab. This requirement reflects our prime objective. 3. The stack dimensions do not exceed those of the uniform magnetic slab. 4. The stack displays all the above properties within a reasonably broad frequency range. The above set of requirements corresponds to a broadband absorption suppression. Alternatively, we can impose a slightly different set of requirements. 1. The stack produces similar or larger Faraday rotation in comparison to the uniform magnetic slab. 2. The stack has much lower absorption than the uniform magnetic slab. 3. The stack has much smaller dimensions than the uniform magnetic slab. Notice, that the requirement 2.3 of much smaller dimensions comes at the expense of the bandwidth (the requirement 1.4). In this section we provide numerical examples proving the effectiveness of the photonic crystal approach to absorption suppression. \[\hat{\varepsilon}_{F}=\left[\begin{array}{ccc}\varepsilon_{1}+i\gamma_{e}&i \alpha&0\\ -i\alpha&\varepsilon_{1}+i\gamma_{e}&0\\ 0&0&\varepsilon_{3}\end{array}\right],\ \ \hat{\mu}_{F}=\left[\begin{array}{ccc} \mu_{1}+i\gamma_{m}&i\beta&0\\ -i\beta&\mu_{1}+i\gamma_{m}&0\\ 0&0&\mu_{3}\end{array}\right], \tag{3}\] The quantities $\alpha$ and $\beta$ in are responsible for nonreciprocal circular birefringence. If the direction of magnetization is changed for the opposite, the parameters $\alpha$ and $\beta$ will also change sign and so will the sense of Faraday rotation. Usually, at microwave frequencies, $\|\beta\|\gg\|\alpha\|$, while at optical frequencies, $\|\beta\|\ll\|\alpha\|$. The positive parameters $\gamma_{e}$ and $\gamma_{m}$ in are responsible for absorption. We can assume, for example, that the dominant physical mechanism of absorption is the electric conductivity $\sigma$, which is often the case at microwave frequencies. In a transverse electromagnetic wave, the effect of electric conductivity reduces to the following anti-Hermitian contribution to the electric permittivity tensor $\hat{\varepsilon}_{F}$ \[i\gamma_{e}=4\pi\sigma i/\omega. \tag{4}\] Different absorption mechanisms could also result in the dominance of the parameter $\gamma_{m}$ in, rather than $\gamma_{e}$. In our numerical examples we only consider the case of $\gamma_{e}\gg\gamma_{m}$. The uniform magnetic slab in has the thickness $D_{U}$ and the permittivity and permeability tensors. In we show a setting similar to that of Fig. 1, but the uniform magnetic slab is now replaced with the periodic layered structure composed of alternating magnetic and dielectric layers. The magnetic $F$ layers are made of the same lossy magnetic material as the uniform slab in The respective permittivity and permeability tensors are given in. \[\hat{\varepsilon}_{A}=\left[\begin{array}{ccc}\varepsilon_{0}&0&0\\ 0&\varepsilon_{0}&0\\ 0&0&\varepsilon_{0}\end{array}\right],\;\;\hat{\mu}_{A}=\left[\begin{array}{ cccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right]. \tag{5}\] The role of the $A$ layers is to create the proper electromagnetic field distribution, which would suppress the energy absorption in the lossy $F$ layers, while preserving or even enhancing the non-reciprocal effects associated with magnetism. The total number of the double layers $L$ in the periodic stack is $N$. The thicknesses $d_{A}$ and $d_{F}$ of the $A$ and $F$ layers satisfy the relation $d_{A}+d_{F}=L$. The total thickness $D_{S}$ of the periodic stack is the product $NL$. The key to the possibility of absorption suppression in the composite structure is that the nonreciprocal effects on the one hand, and the energy dissipation on the other hand, are determined by different components of the material tensors. This allows to design the periodic array so that at the frequency range of interest, the electromagnetic field distribution inside the composite structure will favor the nonreciprocal components of the permittivity/permeability tensors responsible for the desired Faraday rotation, while not engaging with the anti-Hermitian components responsible for absorption. Computations of electromagnetic field distribution inside the stratified and uniform media, as well as the amplitude and polarization of the transmitted and reflected waves are based on the time-harmonic Maxwell equations \[\nabla\times\vec{E}\left(z\right)=i\frac{\omega}{c}\hat{\mu}\left(z\right)\vec {H}\left(z\right),\,\nabla\times\vec{H}\left(z\right)=-i\frac{\omega}{c}\hat{ \varepsilon}\left(z\right)\vec{E}\left(z\right), \tag{6}\] The electric conductivity $\sigma$ is included in the definition of the respective permittivity tensor $\hat{\varepsilon}_{F}$, as specified in. In all cases, the incident wave $\Psi_{I}$ propagates along the $z$ direction normal to the layers and has linear polarization with $\vec{E}_{I}\parallel x$. Due to the nonreciprocal circular birefringence of the magnetic material, the transmitted and reflected waves $\Psi_{P}$ and $\Psi_{R}$ will be elliptically polarized with the ellipse axes being at an angle with the $x$ direction. The magnitude of nonreciprocal effects can be characterized by the $y$ component $\left(\vec{E}_{P}\right)_{y}$ of the transmitted wave $\Psi_{P}$. Indeed, in the absence of magnetism, the parameters $\alpha$ and $\beta$ in vanish and the transmitted wave will be linearly polarized with $\vec{E}_{P}\parallel x$. The transmission and reflection coefficients of the slab (either uniform, or layered) are defined as follows \[t=\frac{S_{P}}{S_{I}},\ r=-\frac{S_{R}}{S_{I}}, \tag{7}\] The slab absorption is \[a=1-t-r. \tag{8}\] If the incident wave polarization is linear, the coefficients $t$, $r$, and $a$ are independent of the orientation of vector $\vec{E}_{I}$ in the $x-y$ plane. \[\rho=\frac{\left(E_{P}\right)_{y}}{\left(E_{I}\right)_{x}},\ \ \mbox{where}\ \ \ \|\rho\|<1. \tag{9}\] Generally, the transmitted wave polarization in Figs. 1 and 2 is elliptical, rather than linear. Therefore, the quantity $\rho$ in is not literally the sine of the Faraday rotation angle. The nonreciprocal effects in the scattering problem of Figs. 1 and 2 are more complicated than a simple Faraday rotation. Let us elaborate on this point. The electromagnetic eigenmodes of the uniform magnetic slab in and the eigenmodes of the layered structure in are all circularly polarized. This implies that if the polarization of the incident wave is circular, the transmitted and reflected waves will also be circularly polarized, both in the case of a uniform slab and in the case of a layered stack, with or without absorption. On the other hand, due to the nonreciprocal (magnetic) effects, the transmission/reflection coefficients for the right-hand circular polarization are different from those for the left-hand circular polarization. This is true regardless of the presence or absence of absorption. Consider now a linearly polarized incident wave. It can be viewed as a superposition of two circularly polarized waves with equal amplitudes. Sine the transmission/reflection coefficients for the right-hand and left-hand circular polarizations are different, the transmitted and reflected waves will be elliptically polarized. Such an ellipticity develops both in the case of a uniform slab and in the case of a layered stack, with or without absorption. Note, though, that at optical frequencies, the dominant contribution to the ellipticity of transmitted wave is usually associated with absorption, which is largely responsible for circular dichroism. Without absorption, the ellipticity of the wave transmitted through magnetic the slab in would be negligible. This is not the case, though, at microwave frequencies, where the ellipticity of transmitted and reflected waves can be significant even in the absence of absorption. To avoid confusion, note that a linear polarized wave propagating in a uniform, lossless, unbounded, magnetic medium will not develop any ellipticity. Instead, it will display a pure Faraday rotation. But the slab boundaries in and the layer interfaces in will produce some ellipticity even in the case of lossless magnetic material. The absorption provides an additional contribution to the ellipticity of transmitted and reflected waves. The latter contribution is referred to as circular dichroism. The following numerical examples illustrate some of the above statements. For simplicity, in further consideration we will often refer to the quantity $\rho$ in as the amount of (nonreciprocal) Faraday rotation, although, due to the ellipticity, it is not exactly the sine of the Faraday rotation angle. In all plots, the frequency $\omega$ and the Bloch wave number $k$ are expressed in dimensionless units of $cL^{-1}$ and $L^{-1}$, respectively. In our computations we use a transfer matrix approach identical to that described in Ref.. ### Broadband absorption suppression We start with the following set of numerical values \[\begin{array}{c}d_{A}=0.8L,\;d_{F}=0.2L,\;N=8,\;D_{S}=8L,\;D_{U}=10L,\\ \varepsilon_{1}=2.89,\;\mu_{1}=32.49,\;\varepsilon_{0}=26.01,\\ \alpha=0,\;\beta=4.0,\;\;\gamma_{e}=0.1,\;\gamma_{m}=0.\end{array} \tag{10}\] The value $\gamma_{e}=0.1$ corresponds to a relatively strong absorption which can be associated with the electric conductivity of the magnetic material. According to, the thickness $D_{S}$ of the periodic stack is somewhat smaller than the thickness $D_{U}$ of the uniform magnetic slab. In addition, that the combined thickness $Nd_{F}$ of all magnetic layers in the stack is 6 times smaller than the thickness $D_{U}$ of the uniform magnetic slab. The latter implies that not only the periodic stack in is thinner than the uniform magnetic slab in Fig. 1, but the actual amount of magnetic material used in the composite structure is just a one sixth of that used in the uniform slab. In we present a fragment of the $k-\omega$ diagram of the periodic array of layers described in; only the lowest photonic frequency band is shown. The split of the two spectral branches is due to the strong circular birefringence $\beta=4$ in. In we show the transmission dispersion of the periodic magnetic stack composed of 8 unit cells $L$. The strong polarization dependence of the stack transmission is also caused by the large value of the nonreciprocal parameter $\beta$ in. Let us now compare the performance of the uniform magnetic slab in and the periodic stack in Fig. 5(a) shows that the magnitude $\|\rho\|$ of the Faraday rotation produced by the periodic stack is comparable to that of the uniform slab. In fact, the periodic stack produces even stronger nonreciprocal effect. At the same time, Fig. 5(b) shows that within the same broad frequency range, the energy dissipation in the layered structure is 20 to 50 times lower, compared to the uniform slab. Hence, the composite structure does indeed dramatically reduce the losses while enhancing the useful functionality of the magnetic material and reducing the overall dimensions. In other words, by all accounts, the layered structure is by far superior to the uniform magnetic slab. The numerical parameters used in our computations are hypothetical, although realistic. We did not try to optimize the configuration of the periodic layered structure, or to see if the periodic arrays with two or three dimensional periodicity can produce even better results. Our goal here is to prove that even a simple periodic array can dramatically improve the situation with losses. A key is the proper configuration of the composite structure, which, in turn, essentially depends on the physical nature of absorption and the frequency range of interest. Also, we would like to emphasize that the dramatic absorption reduction seen in is not a resonance effect. This is why the suppression of losses is achieved within a relatively broad frequency range and the results are not particularly sensitive to the number of layers in the periodic structure. By contrast, the possibility of a significant size reduction discussed in the next subsection is associated with the transmission band edge resonance, which is usually characterized by a relatively narrow bandwidth and strong dependence on the number of layers in the periodic stack. ### Resonant absorption suppression Under what circumstances can we not only suppress the absorption but also have the size of the periodic composite structure much smaller than that of the uniform (magnetic) slab with similar performance? When considering this question we should keep in mind that within the framework of the photonic crystal approach the characteristic length $L$ of the periodic array is always comparable to that of the electromagnetic wavelength in the medium. Therefore, for a given frequency range and for a given set of the constitutive materials, we cannot significantly change the structural period $L$. Nor can we substantially reduce the number $N$ of unit cells without loosing all the effects of coherent interference. For instance, in our numerical example we have chosen $N=8$, and there is very little room for further reduction in size. All we can achieve by adjusting the configuration of the periodic array comprising as few as several periods is to suppress the losses and/or to enhance the useful functionality, such as Faraday rotation. The real question is: what is the thickness $D_{U}$ of the uniform slab with the useful functionality comparable to that of the optimized layered structure? Indeed, if such a uniform slab turns out to be much thicker than the layered structure, then we can claim that not only the periodic array dramatically reduces the losses, but it also has much smaller dimensions. The latter is only possible if the thickness $D_{U}$ of the uniform slab with desired functionality is much greater than the electromagnetic wavelength in the medium. Otherwise, all we can achieve by introducing periodic inhomogeniety is a significant reduction of losses which, by the way, has been our primary objective. Let us turn back to the periodic stack in with the permittivity and permeability tensors of the $F$ and $A$ layers given in and. In the numerical example, we assume a relatively strong circular birefringence $\beta=4.0$ of the magnetic material. The resulting Faraday rotation in a uniform magnetic slab is of the order of unity even if the magnetic slab thickness is equal to just several electromagnetic wavelengths. Such a small thickness is comparable to the thickness of a photonic structure (a periodic stack) comprising as few as several unit cells $L$. Therefore, in the case of a strong circular birefringence of the magnetic material, the periodic stack cannot have significantly smaller size in comparison to the uniform magnetic slab producing similar nonreciprocal Faraday rotation. The composite structure in this case is only useful for absorption reduction, as we have demonstrated in the previous subsection. A strong circular birefringence usually occurs in the vicinity of a ferromagnetic resonance at microwave frequency range. If the frequency $\omega$ is much higher or much lower than that, the gyrotropic parameter $\beta$ is much smaller. For instance, at infrared and optical frequencies, in order to produce Faraday rotation $\rho$ of the order of unity, the thickness of the uniform magnetic slab should exceed the electromagnetic wavelengthby at least two or three orders of magnitude. By contrast, the periodic magnetic stack incorporating the same magnetic material can be much thinner (of the order of several electromagnetic wavelengths), while producing Faraday rotation $\rho$ of the order of unity and dramatically reducing the absorption. \[\beta\ll\mu_{1}.\] To illustrate this point, let us consider the following set of numerical values describing the uniform magnetic slab in and the periodic magnetic stack in \[\begin{array}{c}d_{A}=d_{F}=0.5L,\ N=8,\ D_{S}=8L,\ D_{U}=40L,\\ \varepsilon_{1}=2.89,\ \mu_{1}=32.49,\ \varepsilon_{0}=26.01,\\ \alpha=0,\ \beta=0.1,\ \ \gamma_{e}=0.01,\ \gamma_{m}=0.\end{array} \tag{11}\] Again, the $F$ layers of the periodic array are made of the same lossy magnetic material as the uniform magnetic slab. The Bloch dispersion relation of the respective periodic structure is shown in There are two major differences between the numerical values in and. In the latter case, the value $\beta$ of specific Faraday rotation of the magnetic material is 40 times smaller. To partially offset the weaker circular birefringence and still have the circular birefringence $\rho$ of the order of unity, the thickness $D_{U}$ of the uniform magnetic slab is set to be much larger ($40L$). By contrast, properly designed periodic array can have just several unit cells $L$ and sill produce the desired nonreciprocal effect. All the above can be achieved if the frequency is close enough to one of the transmission band edge resonances, also known as slow wave Fabry-Perot resonances. Two of such resonances are shown in As we shall see below, in the case we can suppress the losses and reduce the size at the same time, but all this comes at the expense of the bandwidth. Specifically, the bandwidth of the desired effect of strong Faraday rotation in a combination with a significant absorption suppression is now limited by the width of the respective transmission band edge resonance. By contrast, in the case described in Fig. 5, the dramatic reduction of absorption occurs within a broad frequency range spanning a significant portion of the lowest photonic band. A graphic illustration of the effect of resonance absorption suppression is presented in A sharp peak in the frequency dependence of the Faraday rotation $\rho$ in Fig. 8(a) coincides with the transmission band edge resonance seen in Observe that the dramatic enhancement of the Faraday rotation in Fig. 8(a) only occurs in the vicinity of the transmission band edge resonance and is characterized by a relatively narrow bandwidth. Away from the resonance, the stack performance is now mixed. This situation is typical for different resonance and slow-wave phenomena, such as the frozen mode regime. By contrast, in our previous example, the absorption suppression occurred within a broad frequency range and it was not related to any resonance. ## 3 Conclusion We have shown that periodic composite structures, such as photonic crystals, can be used to dramatically suppress the energy dissipation associated with the presence of lossy component. In most cases, the useful functionality of the lossy component can be preserved or even enhanced by the presence of spatially periodic inhomogeniety. In our first example described in Fig. 5, the absorption suppression by the periodic structure is not a resonance effect. As a consequence, it occurs in a relatively broad frequency range and it is not particularly sensitive to the size and shape of the photonic crystal. The optimal parameters of the periodic structure depend on: (i) the useful functionality of the lossy material, such as Faraday rotation, nonlinearity, etc., (ii) the physical nature of absorption, and (iii) the frequency range of interest. In addition to a significant reduction of losses, the properly designed periodic array can also have much smaller dimensions, compared to the uniform sample with similar functionality.	10.48550/arXiv.0711.4041	Absorption suppression in photonic crystals	A. Figotin, I. Vitebskiy	2,543
10.48550_arXiv.1605.06548	###### Abstract The adiabatic elastic modulus is often useful in the high frequency response of materials. Unfortunately, it can be much more difficult to directly measure the adiabatic elastic modulus of material than the isothermal elastic modulus. We derive the relationship between the adiabatic and isothermal elastic tensors from the first law of thermodynamics. ## 1 Notation $\sigma_{kl}=$ Stress Tensor $\varepsilon_{ij}^{o}=$ Strain Tensor, includes both stress-induced and temperature-induced strain $T=$ Temperature $S=$ Entropy $U=$ Internal Energy $H=$ Enthalpy $p=$ Pressure $V=$ Volume $\alpha_{ij}=$ Coefficient of Thermal Expansion $C_{ijkl}=$ Stiffness Tensor $S_{ijkl}=$ Compliance Tensor $c_{\sigma_{ij}}=$ Heat Capacity at Constant Stress $c_{\varepsilon_{ij}^{o}}=$ Heat Capacity at Constant Strain ## 2 Definitions ### Heat Capacity With liquids, there are two possible heat capacities which can be defined, one at constant pressure and one at constant volume. We are defining an analogous pair of heat capacities at constant stress and at constant strain. The differential strain of a material is defined by changes in temperature and stress state. \[d\varepsilon_{ij}^{o}\equiv\left(\frac{\partial\varepsilon_{ij}^{o}}{\partial T }\right)_{\sigma_{kl}}dT+\left(\frac{\partial\varepsilon_{ij}^{o}}{\partial \sigma_{kl}}\right)_{T}d\sigma_{kl} \tag{3}\] The thermal expansion coefficient is defined at constant stress state. \[\left(\frac{\partial\varepsilon_{ij}^{o}}{\partial T}\right)_{\sigma_{kl}} \equiv\left(\alpha_{ij}\right)_{\sigma_{kl}} \tag{4}\] ### Isothermal Stiffness and Compliance The isothermal stiffness tensor is defined as: \[\left(\frac{\partial\sigma_{ij}}{\partial\varepsilon_{kl}^{o}}\right)_{T} \equiv C_{ijkl} \tag{5}\] Correspondingly the isothermal compliance is defined as: \[\left(\frac{\partial\varepsilon_{ij}^{o}}{\partial\sigma_{kl}}\right)_{T} \equiv S_{ijkl} \tag{6}\] ### Differential Form of Internal Energy and Free Energies Our differential definition of internal energy with stress and strain work is: \[dU=TdS+\sigma_{ij}d\varepsilon_{ij}^{o} \tag{7}\] The internal energy relations to our enthalpy, Helmholtz free energy, and Gibbs free energy are: \[H =U-\sigma_{ij}\varepsilon_{ij}^{o} \tag{8}\] \[A =U-TS\] \[G =U-TS-\sigma_{ij}\varepsilon_{ij}^{o} \tag{10}\]In differential form: \[dH =TdS-\varepsilon_{ij}^{o}d\sigma_{ij} \tag{11}\] \[dA =-SdT+\sigma_{ij}d\varepsilon_{ij}^{o}\] \[dG =-SdT-\varepsilon_{ij}^{o}d\sigma_{ij} \tag{13}\] ## 3 Introduction Pressure waves such as sound waves and seismic waves are generally well understood phenomena but over many years and scientific fields have increasingly become useful as a means of probing the structure and behavior in a wide range system sizes from Brillouin light scattering techniques to asteroseismology. The basic physical model for any wave is the wave equation. Pressure waves are mechanical and their wave equations can be derived purely by applying Lagrange's second equation to the thermodynamic equation of state and a continuity equation. The results of which for isotropic media look something like this: \[\frac{\partial^{2}p}{\partial t^{2}}=\left(\frac{\partial p}{\partial\rho} \right)_{S}\nabla^{2}p,\] In an ideal gas, it is trivial to show that this only depends on the temperature. in an anisotropic material using Einstein notation, they look like this: \[\partial_{tt}u_{i}=\frac{1}{\rho}\left(\frac{\partial\sigma_{ij}}{\partial \varepsilon_{kl}^{o}}\right)_{S}\partial_{j}\partial_{l}u_{k}\] The adiabatic stiffness tensor in the parenthesis can be difficult to measure so we seek in this derivation to related it to the more readily available isothermal stiffness tensor. ### Derivation We start a triple product rule that relates the adiabatic stiffness tensor to two other quantities: \[\left(\frac{\partial\sigma_{kl}}{\partial\varepsilon_{ij}^{o}}\right)_{S}\left( \frac{\partial\varepsilon_{ij}^{o}}{\partial S}\right)_{\sigma_{kl}}\left( \frac{\partial S}{\partial\sigma_{kl}}\right)_{\varepsilon_{ij}^{o}}=-1 \tag{14}\] $\left(\frac{\partial\sigma_{kl}}{\partial\varepsilon_{ij}^{o}}\right)_{S}$ is the adiabatic stiffness tensor. The goal of this derivation is to find an expression that relates this term to the isothermal stiffness tensor. To this end, we must find expressions for the other terms in Equation 14. We first find an expression for the second term in Equation 14, $\ \left(\frac{\partial\varepsilon_{ij}^{o}}{\partial S}\right)_{\sigma_{kl}}$. This term represents the change in strain due to a change in entropy at constant stress. We can expand this term using the chain rule:\[\left(\frac{\partial\varepsilon_{ij}^{o}}{\partial S}\right)_{\sigma_{kl}}=\left( \frac{\partial\varepsilon_{ij}^{o}}{\partial T}\right)_{\sigma_{kl}}\left(\frac {\partial T}{\partial S}\right)_{\sigma_{kl}} \tag{15}\] The first term on the right hand side of Equation 15 is simply the coefficient of thermal expansion at constant stress as defined in Equation 4: \[\left(\frac{\partial\varepsilon_{ij}^{o}}{\partial T}\right)_{\sigma_{kl}}\equiv \left(\alpha_{ij}\right)_{\sigma_{kl}}\] We must find an expression for the second term on the right hand side of Equation 15, $\left(\frac{\partial T}{\partial S}\right)_{\sigma_{kl}}$. This is the change in temperature due to change in entropy at constant stress. In order to obtain this expression, we use the chain rule to expand the change in enthalpy due to a change in entropy at constant stress: \[\left(\frac{\partial H}{\partial S}\right)_{\sigma_{kl}}=\left(\frac{\partial H }{\partial T}\right)_{\sigma_{kl}}\left(\frac{\partial T}{\partial S}\right)_ {\sigma_{kl}} \tag{16}\] In order to solve Equation 16 for $\ \left(\frac{\partial T}{\partial S}\right)_{\sigma_{kl}}$, we must find expressions for the first two terms in Equation 16. We start with an expression for enthalpy in differential form: \[dH=TdS-\varepsilon_{kl}^{o}d\sigma_{kl} \tag{17}\] Since we are under the condition of constant stress, Equation 17 can be solved assuming $d\sigma_{kl}=0$: \[\left(\frac{\partial H}{\partial S}\right)_{\sigma_{kl}}=T \tag{18}\] The first term on the right hand side of Equation 16 is the heat capacity at constant stress as defined in Equation 2. \[\left(\frac{\partial H}{\partial T}\right)_{\sigma_{kl}}\equiv c_{\sigma_{kl}}\] Substituting Equations 2 and 18 back into Equation 16: \[T =c_{\sigma_{kl}}\left(\frac{\partial T}{\partial S}\right)_{ \sigma_{kl}}\] \[\left(\frac{\partial T}{\partial S}\right)_{\sigma_{kl}} =\frac{T}{c_{\sigma_{kl}}} \tag{19}\] We can now substitute Equations 4 and 19 into Equation 15 to obtain the second term in Equation 14: \[\left(\frac{\partial\varepsilon_{ij}^{o}}{\partial S}\right)_{\sigma_{kl}}= \left(\alpha_{ij}\right)_{\sigma_{kl}}\left(\frac{T}{c_{\sigma_{kl}}}\right) \tag{20}\]Next we must find an expression for the third term in Equation 14, $\left(\frac{\partial S}{\partial\sigma_{kl}}\right)_{\varepsilon_{ij}^{o}}$, which represents the change in entropy due to a change in stress at constant strain. An alternative expression for this term can be found using the Maxwell relation from internal energy: \[\left(\frac{\partial T}{\partial\varepsilon_{ij}^{o}}\right)_{S}=\left(\frac{ \partial\sigma_{ij}}{\partial S}\right)_{\varepsilon_{ij}^{o}} \tag{21}\] We use again use the triple product rule, this time including the change in temperature due to a change in strain at constant entropy: \[\left(\frac{\partial T}{\partial\varepsilon_{ij}^{o}}\right)_{S}\left(\frac{ \partial S}{\partial T}\right)_{\varepsilon_{ij}^{o}}\left(\frac{\partial \varepsilon_{ij}^{o}}{\partial S}\right)_{T}=-1 \tag{22}\] Next we find an expression for the second term in Equation 22, $\ \left(\frac{\partial S}{\partial T}\right)_{\varepsilon_{ij}^{o}}$, which represents the change in entropy due to a change in temperature at constant strain. We start by using the chain rule to expand, $\ \left(\frac{\partial U}{\partial S}\right)_{\varepsilon_{ij}^{o}}$, which is the change in internal energy due to a change in entropy at constant strain: \[\left(\frac{\partial U}{\partial S}\right)_{\varepsilon_{ij}^{o}}=\left(\frac {\partial U}{\partial T}\right)_{\varepsilon_{ij}^{o}}\left(\frac{\partial T} {\partial S}\right)_{\varepsilon_{ij}^{o}} \tag{23}\] We use an expression for internal energy in differential form: \[dU=TdS-pdV \tag{24}\] Since strain is held constant, we assume that volume is also constant ($dV=0$): \[dU =TdS\] \[\left(\frac{\partial U}{\partial S}\right)_{\varepsilon_{ij}^{o}}=T \tag{25}\] The first term on the right hand side of Equation 23, $\ \left(\frac{\partial U}{\partial T}\right)_{\varepsilon_{ij}^{o}}$, was defined in Equation 1 as the heat capacity at constant strain: \[\left(\frac{\partial U}{\partial T}\right)_{\varepsilon_{ij}^{o}}\equiv c_{ \varepsilon_{ij}^{o}}\] Substituting Equations 1 and 25 into Equation 23 gives us an expression for the second term in Equation 22:\[T = c_{\varepsilon^{o}_{ij}}\left(\frac{\partial T}{\partial S}\right)_{ \varepsilon^{o}_{ij}}\] \[\left(\frac{\partial T}{\partial S}\right)_{\varepsilon^{o}_{ij}} = \frac{T}{c_{\varepsilon^{o}_{ij}}}\] \[\left(\frac{\partial S}{\partial T}\right)_{\varepsilon^{o}_{ij}} = \frac{c_{\varepsilon^{o}_{ij}}}{T} \tag{26}\] Next we find an expression for the third term in Equation 22. ### 5 Appendices ### 5.1 Maxwells Relations The Maxwell relation for the internal energy $U$ (Equation 7) is: \[\left(\frac{\partial T}{\partial\varepsilon_{ij}^{o}}\right)_{S,\varepsilon_{ kl\neq ij}^{o}}=\left(\frac{\partial\sigma_{ij}}{\partial S}\right)_{\varepsilon _{kl}^{o}} \tag{34}\] On the left, all strains are held constant except $\varepsilon_{ij}^{o}$. Likewise, the Maxwell relation for enthalpy $H$ (Equation 11) is: \[\left(\frac{\partial T}{\partial\sigma_{ij}}\right)_{S,\sigma_{kl\neq ij}}=- \left(\frac{\partial\varepsilon_{ij}^{o}}{\partial S}\right)_{\sigma_{kl}} \tag{35}\] The Maxwell relation for Helmholtz free energy $A$ (Equation 12) is: \[-\left(\frac{\partial S}{\partial\varepsilon_{ij}^{o}}\right)_{T,\varepsilon_ {kl\neq ij}^{o}}=\left(\frac{\partial\sigma_{ij}}{\partial T}\right)_{ \varepsilon_{kl}^{o}} \tag{36}\]The Maxwell relation for Gibbs free energy $A$ (Equation 13) is: \[\left(\frac{\partial S}{\partial\sigma_{ij}}\right)_{T,\sigma_{kl\neq ij}}=\left( \frac{\partial\varepsilon_{ij}^{o}}{\partial T}\right)_{\sigma_{kl}} \tag{37}\]	10.48550/arXiv.1605.06548	Isothermal and Adiabatic Elastic Tensors	Michael J. Waters, Andrew W. Bielawski	4,751
10.48550_arXiv.1610.09804	###### Abstract Dirac-like electronic states are the main engines powering the tremendous advances in research of graphene, topological insulators and other materials with these states. Zero effective mass, high carrier mobility and numerous applications are some consequences of linear dispersion that distinguishes Dirac states. Here we report a new class of linear electronic bands in two-dimensional materials with zero effective mass and sharp band edges never seen in solid state matter before, and predict stable materials with such electronic structure utilizing symmetry group analysis and ab initio approach. We make a full classification of completely linear bands in two-dimensional materials and find that only two classes exist: Dirac fermions on one hand and pyramidal-like and cootie catcher-like states on the other hand. The new class supports zero effective mass and hence high carrier mobility similar to that of graphene, anisotropic electronic properties like that of phosphorene, and robustness of states with respect to electronic correlations. ## Introduction Electrons can move in certain materials as if they have no mass. Massless fermions in solid state materials have played an increasingly important role since the discovery of graphene (ref.), a material where zero electron effective mass is caused by linear Dirac-like dispersion. While the first mapping of electronic structure of graphene to the Dirac equation was an interesting theoretical curiosity (ref.), true significance of the Dirac-like states in solid state systems became apparent upon identification of many physical, measurable consequences of the linear dispersion (ref.). For instance, the existence of massless fermions in graphene yields the extraordinary high electron and hole mobilities (ref.), with revolutionary implications in electronics. Other implications include and are not limited to the Klein tunneling in single- and bi-layer graphene (ref.), quantum Hall (ref.) and fractional quantum Hall (ref.) effects at room temperature. Two-dimensional (2D) nature of graphene and related materials bring numerous additional advantages including mechanical flexibility, optical transparency (ref.), possibilities for engineering heterostructures with desired properties by stacking of two or more 2D materials (ref.). There is a whole plethora of various 2D materials beyond graphene with the linear dispersion in their band structure (ref.), but also in topological insulators (ref.) and semimetals (ref.). Dirac cones are intricately linked to symmetry, as exemplified with cone engineering by symmetry manipulation (ref.), whereas new Dirac cones have been generated in graphene under external periodic potential (ref.). The symmetry-electronic dispersion connection has been utilized in theoretical search of new materials with Dirac fermions. Manes has used space group representation to find sufficient conditions for the existence of Weyl points (3D analogue of Dirac points) in the Brillouin zone (BZ) of three-dimensional single crystals (ref.). Recently, a set of symmetry conditions that guarantees Dirac-like dispersion in the vicinity of high symmetry points in the BZ of any non-magnetic 2D material has been reported (refs.,). Also, the existence of Dirac fermions in bilayer non-honeycomb crystals using symmetry analysis has been predicted (ref.). New band hourglass-like dispersion has been theoretically predicted in ref., while fermionic excitations in electronic structure of three-dimensional materials have been theoretically classified in. Surprisingly, the existence of other classes of massless fermions in 2D materials has not been addressed yet. Here we report that combined time-reversal (TRS) and certain crystal non-symmorphic symmetries of 2D materials lead to the emergence of peculiar massless linearly dispersive bands. The geometries of these states in reciprocal space are pyramidal-like and cootie catcher-like, which have never been seen in solid state matter before. Our analysis indicates that these states and the Dirac cones are the _unique_ possibilities for non-accidental linear dispersive bands in all diperiodic directions of non-magnetic 2D materials without spin-orbit coupling. Finally, we predict a number of 2D materials with these massless fermions using density functional theory and our own-developed software. ## Classification of linear states in 2D The bands carrying zero effective mass are possible only in the vicinity of points in the Brillouin zone where the electron energy is (orbitally) degenerate. Energy dispersion of non-degenerate bands is smooth, hence second derivative is finite and its inverse, being proportional to the effective mass (ref.), is nonzero. First we classify all possibilities for linear dispersions in the band structure of 2D materials. In order to achieve this aim we define a set of parameters, which values determine possible existence of linear dispersions in 2D crystals. If $G\big{(}\vec{k}_{0}\big{)}$ is the group of the wave vector $\vec{k}_{0}$ and $R$ is allowed (ref.) (relevant (ref.), small (ref.)) irreducible representation (irrep) of $G\big{(}\vec{k}_{0}\big{)}$, then the set of parameters consists of $\begin{array}{l}\succ\\ \succ\\ \succ\end{array}$ equivalence of $\vec{k}_{0}$ and its inverse $-\vec{k}_{0}$, $\begin{array}{l}\succ\\ \succ\\ \end{array}$ dimensionality of representation $R$, $\begin{array}{l}\succ\\ \succ\\ \end{array}$ reality of representation $R$. Diveriodic groups have only 1D or 2D allowed irreps (ref.), while they can be real on one hand or pseudo-real or complex on the other hand. (color online) Full classification of linearly dispersive electronic bands in non-magnetic 2D materials based on symmetry conditions. Panel (a) corresponds to the case $\vec{k}_{0}\Leftrightarrow-\vec{k}_{0}$ and panel (b) to the case $\vec{k}_{0}\Leftrightarrow-\vec{k}_{0}$. For single crystals the corresponding theory was developed in 1937 (ref.). We first consider the case $\vec{k}_{0}\Leftrightarrow-\vec{k}_{0}$. The wave vector $\vec{k}_{0}$ must have a locally maximal symmetry, otherwise linear dispersion cannot appear, due to either too many band contacts (ref.) or none at all. If $R$ is two-dimensional, and the irrep $R_{in}$ of the whole diperiodic group $G$, which is obtained by induction from $R$ ($R_{in}=R\dagger G$) is real, Dirac-like dispersion appears (ref.) (orange upper left section). If $R_{in}$ is not real, $-\vec{k}_{0}$ is not in the star of $\vec{k}_{0}$ (ref.) and the additional degeneracy due to TRS does not appear. This case also leads to Dirac dispersion (ref.) (orange upper right section). In the last two cases double degeneracy at Dirac point is caused by the crystal symmetry. For $R$ one-dimensional and $R_{in}$ real (left panel blue section), $E_{\theta}$ is non-degenerate preventing Dirac dispersion in the vicinity of $\vec{k}_{0}$. For $R$ one-dimensional and $R_{in}$ pseudoreal or complex (blue-orange section) there are two possibilities. If $-\vec{k}_{0}$ is not in the star of $\vec{k}_{0}$, $E_{\theta}$ is non-degenerate while in the opposite case TRS causes $E_{\theta}$ to be double degenerate. In this case, there is a complete Dirac-like dispersion around $\vec{k}_{0}$ (ref.). Next we consider the case $\vec{k}_{0}\Leftrightarrow-\vec{k}_{0}$. If $R$ is one-dimensional and real, the energy level $E_{\theta}$ at $\vec{k}_{0}$ is non-degenerate and linear dispersion cannot appear (right panel blue section). If $R$ is one-dimensional and not real (right panel green section down), TRS causes $E_{\theta}$ to be double degenerate, but complete linear Dirac-like dispersion is not possible since the TRS causes $u_{2}$=$0$ (ref.). The same statement holds for two-dimensional, real $R$ (right panel green section up). The remaining case in which $R$ is two-dimensional and pseudoreal or complex will be treated in more detail here (right panel red section). ### Pyramidal and cootie catcher states In following, the functional form of the new linear dispersion relation that corresponds to the red section of will be presented and compared to Dirac-like dispersion. In the following $\vec{q}$ is a wave vector of small modulus, $\vec{t}$ a real 2D vector, $u_{1},u_{2}$ positive quantities and $q_{1}$, $q_{2}$ projections of $\vec{q}$ along mutually orthogonal directions. If $\vec{k}_{0}$ is a point that hosts a pair of Dirac cones, then the Taylor expansion of the electron energy around this point reads: \[E_{1,2}\big{(}\vec{k}_{0}+\vec{q}\big{)}\approx E_{0}+\vec{t}\cdot\vec{q} \pm\sqrt{u_{1}q_{1}^{2}+u_{2}q_{2}^{2}}. \tag{1}\] For $u_{1}=u_{2}$ Dirac cones are isotropic and for $\vec{t}\neq 0$ the cones are tilted (ref.). The new electronic dispersion presented in this paper is: \[E_{1,2,3,4}\big{(}\vec{k}_{0}+\vec{q}\big{)}\approx E_{0}\pm\big{\|}u_{1}\|q_{1} \|\pm u_{2}\|q_{2}\big{\|}\big{\|}. \tag{2}\] Note that energy level $E_{\theta}$ is double orbitally degenerate in and 4-fold degenerate in. While the Dirac band has the geometric form of simple cone with circular or elliptical cross section, the geometry of dispersion consists of two different geometric forms. The plus sign under the absolute value yields the geometry of a four-sided pyramid (PY), whereas the minus sign corresponds to a more complex geometry, which looks like the paper origami called cootie catcher (CC) (Fig. 3(b)). Now we show briefly how the linear dispersion is connected to TRS and crystal symmetry, whereas more details are given in supplementary materials. A general matrix form of Taylor expansion of Hamiltonian around a given $\vec{k}_{0}$ point of BZ of non-magnetic 2D material is $\tilde{\beta}\big{(}\vec{k}_{0}+\vec{q}\big{)}\approx E_{0}I_{4}+\tilde{H}^{ \prime}$, where $\tilde{H}^{\prime}=\tilde{W}\big{(}\hat{I}_{4}\otimes\vec{q}\big{)}$, $\tilde{W}=\langle\frac{\partial}{\partial\vec{q}}\,\|\tilde{H}\big{(}\vec{k}_ {0}+\vec{q}\big{)}\|_{\vec{q}=0}$, $\hat{I}_{4}$ is four-dimensional unit matrix, while $\tilde{W}$ is four-by-eight matrix. Diagonalization of such a Hamiltonian leads to a general functional form of dispersion relation: \[E_{1,2,3,4}=E_{0}\pm \tag{3}\] \[\pm\frac{1}{\sqrt{2}}\sqrt{\sum_{j=1}^{6}\bigl{(}\overrightarrow{\overline{v_{j}} \cdot\tilde{q}}\bigr{)}^{2}\pm\sqrt{\left[\sum_{j=1}^{6}\bigl{(}\overrightarrow {\overline{v_{j}}\cdot\tilde{q}}\bigr{)}^{2}\right]^{2}-4[(\overrightarrow{ \overline{v_{1}}}\cdot\tilde{q})(\overrightarrow{\overline{v_{6}}}\cdot\tilde{ q})-(\overrightarrow{\overline{v_{2}}}\cdot\tilde{q})(\overrightarrow{\overline{v_{5}}} \cdot\tilde{q})+(\overrightarrow{\overline{v_{3}}}\cdot\tilde{q})( \overrightarrow{\overline{v_{4}}}\cdot\tilde{q})\bigr{]}^{2}}\] Now we introduce conditions, which the allowed representation must satisfy in order to have PY and CC states (red section in Fig. 1). * $O_{l}$: $\vec{k}_{0}$ is equivalent to its inverse $-\vec{k}_{0}$, * $O_{2}$: $R$ is two-dimensional, * $O_{3}$: $R$ is pseudo-real or complex. It turns out that only 3 out of 80 diperiodic groups, Dg33, Dg43 and Dg45, have allowed representations satisfying the conditions $O_{l}$ - $O_{3}$. For all of them we have found the form that symmetry imposes on the vectors $\overrightarrow{\overline{v_{j}}}$ using Wigner's method of group projectors. Upon their insertion into equation it reduces to the linear dispersion for the three groups. The groups allowing the dispersion are listed in the Table 1. All three groups are non-symmorphic and belong to the rectangular system. The component $q_{1}$ can be chosen as projection of $\vec{q}$ along direction that is parallel to any screw axis $2_{1},q_{2}$ is projection along the perpendicular direction. The points $\vec{k}_{0}$ hosting the dispersion are located at the corners of the rectangle that presents the BZ border. The corresponding space groups from Table 1 denote the space groups that are obtained by periodic repetition of diperiodic groups' elements along the axis perpendicular to the diperiodic plane. Due to the Bloch theorem, orbital wave functions must belong to an allowed irrep at a given point in the Brillouin zone. This statement is valid irrespectively of strength of electronic correlations, since the Coulomb repulsion between electrons has the same transformation properties as the rest of the Hamiltonian. Allowed irreps of groups listed in Table 1 are the only ones at these points of the BZ so the electronic correlations cannot change the dispersion. Symmetry of the crystal lattice is responsible for (an)isotropy of single crystals (ref.). For example, isotropy of the electric susceptibility tensor in silicon is caused by the cubic symmetry, while the in-plane isotropy of graphene is caused by the hexagonal symmetry. In our cases, crystal axes are maximally of the second order, irreps of corresponding point groups are all one-dimensional and the materials belonging to diperiodic groups listed in Table 1 are expected to be _anisotropic_. \begin{table} \begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{Dperiodic group} & \multicolumn{2}{c}{Corresponding space group} & \multicolumn{2}{c}{Dperiodic plane} & $R$ \\ \hline Dg33 & _p b 2${}_{1}$a_ & 29 & _P c a 2${}_{1}$_ & $C_{2w}^{5}$ & _y = 0_ & $U_{1}$ \\ Dg43 & _p 2/b 2${}_{1}$a 2/a_ & 54 & _P 2${}_{1}$/c 2/c 2/a_ & $D_{2h}^{8}$ & _y = 0_ & $U_{1},U_{2}\Leftrightarrow U_{1}^{}$ \\ Dg45 & _p 2/b 2${}_{1}$/m 2/a_ & 57 & _P 2/b 2/c 2${}_{1}$/m_ & $D_{2h}^{11}$ & _x = 0_ & $T_{1},T_{2}\Leftrightarrow T_{1}^{}$ \\ \hline \hline \end{tabular} _Notes:_ The notations for diperiodic and space groups are according to Kopsky and Litvin (ref.) and Hahn (ref.), respectively. The _x_-, _y_- and _z_-axes are along a-, b- and c-directions of the orthorhombic 3D unit cell, respectively. The notation for allowed representation in the last column is according to the Bilbao Crystallographic Server (ref.). $\Leftrightarrow$ denotes equivalence between representations. \end{table} Table 1: Dperiodic groups hosting the dispersion in the vicinity of BZ corners.The effective mass for dispersions and is zero, which can be simply shown from $\left[\hat{\mathrm{m}}_{\mathrm{eff}}{}^{-1}\right]_{Jl}=\hbar^{-2}\left[\frac{ \partial^{2}}{\partial q_{J}q_{l}}E(q_{1},q_{2})\right]\bigg{\|}_{q_{1}=q_{2}=0}$. Details of derivation are included in the supplementary material. ### Ab initio search for realistic materials Next we report (meta)stable 2D materials, which are predicted using ab initio calculations. We have developed and utilized software that automatically searches materials with a given group, analyzes their stability and band structure. The outline of algorithm is listed in the supplementary materials. We limited our search only to four elements of the periodic table that are known to build stable 2D materials of various crystal symmetries. We expect that many more stable materials consisting of other elements will be found in further research. The stable and metastable structures are listed in Table II, together with their structural and electronic parameters. Note that symmetry arguments presented above do not determine position of PY and CC states on the energy scale. Positioning of PY & CC states eventually at $\mathrm{E}_{\mathrm{F}}$ is of ultimate importance for their future experimental confirmation by e.g. angle resolved photoemission spectroscopy, but also because many physical phenomena originate only from states around $\mathrm{E}_{\mathrm{F}}$. We have applied recently reported theory (refs.,) to Dg33, Dg43 and Dg45 groups and found that the number of electrons in valence states of a material must be 8$n$+4 per unit cell, where $n$ is a positive integer, in order to be a zero-gap semiconductor. This is a necessary condition for PY & CC states to touch exactly at $\mathrm{E}_{\mathrm{F}}$ and that no other bands cross the Fermi level. More specifically, elemental crystals with listed groups with 4-fold multiplicity of Wyckoff positions can be such zero-gap semiconductors only if they contain elements with an odd number of valence electrons, including IA, IIIA, VA, VIIA and some other groups of the periodic table. Another option, 8-fold multiplicity of Wyckoff positions, guarantees metallic systems. Groups Dg33 and Dg43 do not have Wyckoff positions with 4-fold multiplicity, which remain in the same group when singly occupied. Therefore we present in Table II the elemental crystals with 4 atoms per unit cell, which includes only group Dg45. Note that compounds of 2 or more elements with any of Dg33, Dg43 and Dg45 symmetries can obey the condition for touching of PY & CC states at $\mathrm{E}_{\mathrm{F}}$. Importantly we have found a (meta)stable structure with PY & CC contact points positioned exactly at the Fermi level: P with Dg45 symmetry group. The structure of P (Dg45) is shown in Fig. 2(a). It consists of zig-zag chains of P atoms placed alternately at two parallel planes. The elementary unit cell contains 4 atoms, 2 per plane, with lattice parameters $b=3.22$ A and $c=5.24$ A. The potential energy surface of P (Dg45) has multiple local minima (many allotropes have been recently predicted (ref.)) so there are numerous (meta)stable phases of phosphorus with different symmetries. \begin{table} \begin{tabular}{l\|c c c c c c c} \hline & \multicolumn{6}{c}{Dg45} \\ Element & $\mathrm{E}_{\mathrm{at}}$(eV/at) & b(Å) & c(Å) & coordinates (Å) & $\Delta\mathrm{E}_{\mathrm{f}}$(eV) & $\mathrm{v}_{b}$(10${}^{6}$m/s) & $\mathrm{v}_{c}$(10${}^{6}$m/s) \\ \hline B & -6.51 & 3.12 & 2.97 & (0.390 0.387 0.743) & -0.65 & 1.22 & 1.52 \\ C & \multicolumn{6}{c}{Not found a stable structure with 4-fold multiplicity of Wyckoff positions} \\ Si & -5.56 & 3.74 & 4.64 & (0.765 0.583 1.160) & -2.30 & 0.91 & 0.79 \\ ## P & -6.03 & 3.22 & 5.24 & (0.780 0.708 1.310) & 0.00 & 1.08 & 0.40 \\ \hline \end{tabular} _Notes:_$\mathrm{E}_{\mathrm{at}}$ is atomization energy, b and c are lattice parameters. Coordinates of only one atom are given for each element. Other coordinates can be obtained from Wyckoff positions. $\Delta E_{F}=E_{PY-CC}-E_{F}$ – energy difference between Fermi level and nearest PY & CC states. Group velocities $\mathrm{v}_{\mathrm{b}}$ and $\mathrm{v}_{\mathrm{c}}$ are calculated using $\mathrm{v}_{i}=\frac{1}{\mathrm{h}}\frac{\partial E(k_{1},k_{2})}{\partial k_{i}}$, where index $i$ corresponds to $b$, or $c$ lattice directions. \end{table} Table 2: (Meta)stable 2D crystals with Dg45 group and 4-fold Wyckoff multiplicity.minimum shown in Fig. 2(b). Its stability is further confirmed by molecular dynamics simulations at 100K. The band structure of P (Dg45) along lines between high symmetry points is shown in Fig. 3(a). At _X-R-Z_ section of the BZ four states touch at $\rm E_{F}$ (both upper and lower states are doubly degenerate), yielding 4-fold degeneracy at the point of contact ($\rm E_{F}$). The bands around $R$ point obey the electron-hole symmetry. The Fermi velocities of these states in _X-R_ and _R-Z_ directions are 1.08 and 0.46$\cdot$10${}^{6}$ m/s, which are in the range of the Fermi velocity of graphene. Therefore, we expect high electron and hole mobilities like that in graphene. The Fermi velocity of P (Dg45) is highly anisotropic; therefore anisotropic electronic properties are expected similar to those of phosphorene. The 4-fold degeneracy of bands at $R$ is lifted along the diagonal direction ($\varGamma$-_R_). Note another set of bands between $Z$ and $\varGamma$, with accidental 2-fold degeneracy below $\rm E_{F}$. More geometry details of the states around $R$ are visible in Fig 3(b). These states look exactly as the symmetry analysis predicted above: two pyramid-like and two cootie catcher-like bands touch at their tips and two lines, respectively. Due to the difference in Fermi velocities, coefficients _u1_ and _u2_ in eq. are different, and the two lines intersect at an acute angle. Sharp edges of the dispersion are unique among electronic structure of any known crystal, and particularly in contrast to smooth features of Dirac cones. In conclusion, we have established for the first time a full classification of states with linear dispersions in 2D materials based on group theory analysis, and found that only one additional class to Dirac states is possible. These states have not only unique and interesting geometric forms, but they can open new horizons for both fundamental research and applications. For instance, these fermions do not have an counterpart in elementary particle physics, i.e. they go beyond known Dirac, Mayorana and Weyl excitations. The sharp edges in electronic bands have been neither predicted nor measured before, so their existence in PY and CC states may spawn new phenomena in solid state materials. Possibly, very high mobilities on par with the mobility of graphene can be of uppermost interest for applications. (color online) Top and side views of optimized geometry of P (Dg45) is shown in panel (a). An elementary unit cell is marked with a green rectangle together with lattice vectors. Potential energy surface (atomization energy given in eV/atom) with respect to lattice parameters is presented in panel (b). The lattice parameters are given in angstroms. Calculations are not done in white region of panel (b) since the search algorithm predetermined instability of the crystal in this region. The robustness of PY-PY and CC-CC contacts with respect to electron correlations may play a role in analogy to topological protection of states in topological insulators. Our unified classification of linearly dispersive bands paves the way to engineer new materials with Dirac and PY & CC states. We hope that findings presented here will be of large motivation for experimental groups to bring these materials into existence.	10.48550/arXiv.1610.09804	Pyramids and cootie catchers: new massless fermions in 2D materials	Vladimir Damljanovic, Rados Gajic, Igor Popov	2,087
10.48550_arXiv.1101.4248	###### Abstract We present an _ab initio_ theory of core- and valence resonant inelastic x-ray scattering (RIXS) based on a real-space multiple scattering Green's function formalism and a quasi-boson model Hamiltonian. Simplifying assumptions are made which lead to an approximation of the RIXS spectrum in terms of a convolution of an effective x-ray absorption signal with the x-ray emission signal. Additional many body corrections are incorporated in terms of an effective energy dependent spectral function. Example calculations of RIXS are found to give qualitative agreement with experimental data. Our approach also yields simulations of lifetime-broadening suppressed XAS, as observed in high energy resolution fluorescence detection experiment (HERFD). Finally possible improvements to our approach are briefly discussed. RIXS,XAS,XES pacs: 78.70.Dm,78.70.En,78.70.Ck Introduction Resonant inelastic x-ray scattering (RIXS) is a powerful tool for probing occupied and unoccupied densities of states at high resolution. Moreover, the RIXS signal contains valuable information about the many-body excitations of a system, e.g., those beyond the primary quasi-particle excitation. However, because the RIXS signal is described by the Kramers-Heisenberg equation rather than Fermi's golden rule and is sensitive to many-body excitations, theoretical calculations of RIXS are more difficult than those of related core-level spectroscopies such as x-ray absorption (XAS), x-ray emission (XES), and electron energy loss (EELS). Even so, models of the RIXS spectrum based on single particle band structure arguments can be quite useful for systems with weak electron correlations, i.e., $sp$-electron systems. In order to account for particle-hole interactions, methods based on the Bethe-Salpeter equation have been employed. In addition, there have been works modeling $d$-electron systems within the single particle picture. Also, for systems with strong electron correlations, the Anderson impurity model and atomic multiplet theories have been used to explain RIXS spectra of localized $d-$ and $f$-state systems. For more detailed reviews see e.g., Ref.. In this paper we introduce a theoretical treatment of RIXS based on an approximation to the Kramers-Heisenberg equation which uses a real space multiple-scattering Green's function (RSGF) formalism to describe the single particle spectrum and a quasi-boson model Hamiltonian to account for multi-electron (e.g. shake-up and shake-off) excitations. Although extensions are possible, our approach is currently limited to systems with weak correlations. Our derivation is similar in some respects to that of Ref., which also uses a multiple-scattering formalism and a related treatment of inelastic losses. The main differences are: i) our expression does not rely on a single site approximation for the single particle Green's function; ii) we include quasi-particle self-energy corrections based on a many-pole model of the dielectric function; and iii) we approximate the many-body losses via a convolution with an effective spectral function that includes intrinsic and extrinsic losses and interference effects. In addition, with several simplifying assumptions, we demonstrate that the RIXS cross section can be approximated as a convolution of XAS and XES signals. This simplified formula is analogous to an expression in terms of appropriate joint densities of states. However, our result also explicitly includes the energy dependence of the dipole matrix elements. In order to calculate our approximation to the RIXS spectrum efficiently, we have implemented the theory in an extension of the real space multiple-scattering Green's function code FEFF9. This RSGF technique has already been used to calculate several other core-level spectroscopies, including XAS, XES and EELS, as well as VIS-UV spectra. The RSGF method has been particularly beneficial for the core-level spectroscopies of complex systems, since it does not rely on periodic symmetry, which is generally broken by the presence of the core hole. Moreover, the approach is applicable over a broad range of energies. Illustrative examples are presented which yield reasonable agreement with available experimental RIXS. Our theory also yields simulations for related spectra, e.g., lifetime broadening suppressed x-ray absorption spectra, as observed in high energy resolution fluorescence detection (HERFD) experiments. The remainder of this article is ordered as follows. We begin by introducing the theory of RIXS based on the Kramers-Heisenberg equation and then summarize our key results. In particular we derive an approximate formula for the RIXS cross section in terms of a convolution of XES and effective XAS signals. We then present a more detailed derivation of RIXS in terms of quasi-particle Green's functions within the RSGF formalism. Subsequently, we present several illustrative calculations and compare with experimental data in a number of weakly correlated systems. Although our treatment is in principle more general, we restrict our attention in this work to systems for which the quasi-particle approximation is reasonable. Finally, we make a number of concluding remarks. Technical details are relegated to the Appendices. ## II Theory ### RIXS in terms of XAS and XES Below we briefly outline the basic theory of RIXS and describe the key expressions used in our calculations. All quantities are expressed in Hartree atomic units ($e=\hbar=m=1$) unless otherwise noted. \[\frac{d^{2}\sigma}{d\Omega d\omega}= \frac{\omega}{\Omega}\sum_{F}\left\|\frac{\sum_{M}\langle F\|\Delta_{ 2}^{\dagger}\|M\rangle\langle M\|\Delta_{1}\|\Psi_{0}\rangle}{E_{M}-\Omega-E_{0}+i \Gamma_{M}}\right\|^{2}\] \[\times\delta(\Omega-\omega+E_{0}-E_{F}). \tag{1}\] Here $\Omega$ and $\omega$ are the energies of the incoming and outgoing photons; $\Delta_{1}$ and $\Delta_{2}$ are the many-body transition operators; and $\|\Psi_{0}\rangle$, $\|M\rangle$ and $\|F\rangle$ are many-body electronic ground, intermediate, and final states with corresponding energies $E_{0}$, $E_{M}$, and $E_{F}$. This formula for the cross section can be expressed in terms of effective one particle Green's functions [cf. Ref. and our derivation in Sec. (II.3)] corresponding to the intermediate and final many-body states \[\frac{d^{2}\sigma}{d\Omega d\omega} =-\frac{1}{\pi}\frac{\omega}{\Omega}\|\langle b\|d_{2}Q\|c\rangle\|^{ 2}\mbox{Im}\left[\langle b\|d_{1}^{\dagger}Pg^{b}(\Omega+E^{\prime})\right.\] \[\times\left.g^{c}(\Omega-\omega+E_{c})g^{b}(\Omega+E_{b})^{\dagger }Pd_{1}\|b\rangle\right]\!. \tag{2}\] Here the one-particle Green's functions $g^{b}$ and $g^{c}$ are given by \[g^{b}(\omega) =\langle\Phi_{0}^{b}\|\frac{1}{\omega-h_{p}^{b}-V_{pv}+i\Gamma_{b }}\|\Phi_{0}^{b}\rangle\] \[\equiv\frac{1}{\omega-h_{p}^{b}-\Sigma_{p}(\omega)+i\Gamma_{b}}\] \[g^{c}(\omega) =\frac{1}{\omega-h_{p}^{c}-\Sigma_{p}(\omega)+i\Gamma_{c}}\, \tag{3}\] As shown in Appendix A, we can rewrite this expression in terms of a non-local transition operator $T$ \[\frac{d^{2}\sigma}{d\Omega d\omega} =-\frac{1}{\pi}\frac{\omega}{\Omega}\frac{\|\langle b\|d_{2}Q\|c \rangle\|^{2}}{\|\omega+E_{b}-E_{c}+i\Gamma_{b}\|^{2}}\] \[\times\,\mbox{Im}\left[\langle b\|T^{\dagger}(\Omega)g^{c}( \Omega-\omega+E_{c})T(\Omega)\|b\rangle\right], \tag{4}\] \[\frac{d^{2}\sigma}{d\Omega d\omega}=\frac{\omega}{\Omega}\int d\omega_{1}\ \frac{\mu_{e}(\omega_{1})\bar{\mu}(\Omega,\Omega-\omega-\omega_{1}+E_{b})}{\|\omega-\omega_{1}+i\Gamma_{b}\|^{2}}. \tag{5}\]where the effective absorption coefficient $\bar{\mu}$ is \[\bar{\mu}(\Omega,\Omega-\omega)=-\frac{1}{\pi}\mbox{Im}\left[\langle b\|T^{\dagger} (\Omega)g^{c}(\Omega-\omega+E_{c})T(\Omega)\|b\rangle\right]. \tag{6}\] The quantity $\bar{\mu}$ differs from normal x-ray absorption coefficient since the dipole transition operator in XAS is replaced by $T(\Omega)$. If the matrix elements of $g^{\prime}\Delta V$ are much smaller than unity, which is the case for all but localized excitations, we may take the leading order approximation, and thus relate the RIXS to the usual x-ray absorption coefficient $\mu(\omega)$, i.e., \[\frac{d^{2}\sigma}{d\Omega d\omega}\propto\frac{\omega}{\Omega}\int d\omega_{1 }\ \frac{\mu_{e}(\omega_{1})\mu(\Omega-\omega-\omega_{1}+E_{b})}{\|\omega- \omega_{1}-i\Gamma_{b}\|^{2}}. \tag{7}\] Thus we obtain a relatively simple expression for the RIXS cross section in terms of the x-ray absorption, x-ray emission, and a resonant denominator. Moreover, the terms in either expression [Eq. or] can be calculated within the RSGF framework, as in the FEFF codes. It should be noted that the above expressions (Eq. 5 and 7) are similar to those given in the pioneering work of Tulkki and Aberg, in which a derivation of electronic resonant Raman spectra is given in terms of multichannel scattering states, and applied to the K-alpha RIXS of KMnO${}_{4}$. ### Multiple Scattering Theory We now turn our attention to the application of the multiple-scattering RSGF formalism to Eq.. Within this formalism the single particle Green's function can be expanded about the absorbing atom (see Ref.) \[G(\mathbf{r},\mathbf{r}^{\prime},E)= -2k\left[\ \sum_{LL^{\prime}}\|R_{L}(E)\rangle G_{L0L^{\prime}0}(E) \langle R_{L^{\prime}}(E)\|\right.\] \[+\ \left.\delta_{L,L^{\prime}}\|H_{L}(E)\rangle\langle R_{L}(E)\| \right]. \tag{8}\] Calculations of the RIXS cross section require both the single particle XES signal, which is relatively simple to calculate with FEFF9, and the effective absorption cross-section $\bar{\mu}$. In order to calculate the latter we begin by rewriting Eq. in terms of the one-electron density matrix $\rho^{c}$, \[\bar{\mu}(\Omega,\Omega-\omega)\propto\langle b\|T^{\dagger}(\Omega)\rho^{c}( \Omega-\omega+E_{c})T(\Omega)\|b\rangle. \tag{9}\]Note again that $\|b\rangle$ and $\|c\rangle$ signify states calculated in the presence of the deep or shallow core hole respectively. \[\bar{\mu}(\Omega,\Omega-\omega) = -2k\sum_{LL^{\prime}}\langle b\|T^{\dagger}(\Omega)\|R_{L}^{c} \rangle\,[\delta_{LL^{\prime}}+ \tag{10}\] \[\rho_{0L0L^{\prime}}^{c}(\Omega-\omega+E_{c})]\,\langle R_{L^{ \prime}}^{c}\|T(\Omega)\|b\rangle,\] Note that the energy arguments of the bras and kets have been omited above for the sake of brevity. \[T_{Lb}(\Omega)=\langle R_{L}^{c}\|T(\Omega)\|b\rangle=\langle R_{L}^{c}\|\left[ \Delta Vg^{b}(\Omega)^{\dagger}+1\right]d_{1}\|b\rangle, \tag{11}\] approximated $P$=1. Then rewriting the Green's function $g^{b}$ in spectral representation, and again inserting the MS expression for the Green's function in Eq. \[T_{Lb}(\Omega)=\langle R_{L}^{c}\|d_{1}\|b\rangle+\pi\int d\omega_ {1}\frac{2k_{1}}{\omega_{1}+i\Gamma_{b}}\times \tag{12}\] \[\sum_{L_{1}}\langle R_{L}^{c}\|\Delta V\|R_{L}^{b}\rangle\left[ \delta_{LL_{1}}+\rho_{LL_{1}}^{b}(\Omega-\omega_{1})\right]\langle R_{L_{1}}^{ b}\|d_{1}\|b\rangle.\] Thus in addition to the usual dipole matrix elements, we have a second term which depends on both energies in the problem, the incoming photon frequency $\Omega$ and the energy loss $\Omega-\omega$. ### Many-Body Effects and the Quasi-Boson Model In this subsection we discuss the application of the quasi-boson model to calculations of inelastic loss effects in RIXS. Assuming that the absorption occurs from a deep core level $\|b\rangle$ and employing the dipole approximation for the transition operators in Eq. \[\Delta_{1} = \sum_{k}\langle k\|d_{1}\|b\rangle c_{k}^{\dagger}b+{\rm h.c.}\] \[\Delta_{2} = \sum_{k}\langle b\|d_{2}\|k\rangle b^{\dagger}c_{k}+{\rm h.c.}. \tag{13}\]If we neglect exchange terms between the particle and hole, or at least assume that they are dealt with via an effective single particle potential, we can write the many-body ground state as \[\|\Psi_{0}\rangle=\|\Phi_{0}\rangle\|b\rangle\|k_{2}\rangle, \tag{14}\] Note that this approximation is only justified if $k_{2}$ denotes a core electron or a high energy photo-electron, although we will use the approximation for valence electrons as well. \[\Delta_{1}^{k_{1}}\|\Psi_{0}\rangle=M_{1}^{k_{1}b}\|\Phi_{0}\rangle \|k_{2}\rangle\|k_{1}\rangle\theta(E_{k_{1}}-E_{\rm F})\] \[\Delta_{2}^{k_{2}}\|\Phi_{0}\rangle\|k_{2}\rangle\|k_{1}\rangle=M_{2 }^{bk_{2}}\|\Phi_{0}\rangle\|b\rangle\|k_{1}\rangle\theta(E_{\rm F}-E_{k_{2}}), \tag{15}\] where $M_{i}^{kb}=\langle k\|d_{i}\|b\rangle$ and \[H\|\Psi_{0}\rangle=E_{0}\|\Psi_{0}\rangle=(\epsilon_{b}+\epsilon_{k_{2}}+E_{0}^{ 0})\|\Psi_{0}\rangle. \tag{16}\] Note that $E_{\rm F}$ is now the Fermi energy. \[\frac{d^{2}\sigma}{d\Omega d\omega}=-\frac{1}{\pi}\frac{\omega}{ \Omega}{\rm Im}\left[\sum_{k_{1}k_{2}}^{\rm unocc}\sum_{k_{3}k_{4}}^{\rm occ} M_{2}^{bk_{3}}(M_{2}^{k_{4}b}M_{1}^{bk_{1}})^{}M_{1}^{k_{2}b}\right.\] \[\left.\langle k_{1}\|\langle k_{3}\|\langle\Phi_{0}\|K_{k_{3}k_{4}}( \xi_{1},\xi_{2})\;\|\Phi_{0}\rangle\|k_{4}\rangle\|k_{2}\rangle\right], \tag{17}\] where $\xi_{1}=\Omega+E_{0}$, $\xi_{2}=\Omega+E_{0}-\omega$, and $K$ is given by \[K_{kk^{\prime}}(\omega,\omega^{\prime})=G(\omega)c_{k}^{\dagger}bG(\omega^{ \prime})b^{\dagger}c_{k^{\prime}}G(\omega)^{\dagger}, \tag{18}\] We now introduce a quasi-boson approximation to the Hamiltonian following the treatment of Ref.. \[H=H_{0}^{N-2}+h_{p}+h_{h}+V_{hv}+V_{pv}+V_{ph}, \tag{19}\]where $h_{h}$ and $h_{p}$ are the one-particle Hamiltonians for the hole and particle respectively, $V_{hv}/V_{pv}$ describes the interaction of the hole/particle with the valence electrons, \[h_{p} = \sum_{k}\epsilon_{k}c_{k}^{\dagger}c_{k};\quad h_{h}=-\sum_{k} \epsilon_{k}c_{k}c_{k}^{\dagger}, \tag{20}\] \[V_{pv} = \sum_{n,k_{1}k_{2}}\left[V_{k_{1}k_{2}}^{n}a_{n}^{\dagger}+(V_{k_{ 1}k_{2}}^{n})^{}a_{n}\right]c_{k_{1}}^{\dagger}c_{k_{2}},\] \[V_{hv} = \sum_{n,k_{1}k_{2}}\left[V_{k_{1}k_{2}}^{n}a_{n}^{\dagger}+(V_{k_{ 1}k_{2}}^{n})^{}a_{n}\right]c_{k_{1}}c_{k_{2}}^{\dagger}, \tag{22}\] This last interaction term should in principle be treated via the Bethe-Salpeter equation; however, here we will approximate it using either a self-consistent final state rule approximation for deep core holes (i.e., with the screened core-hole potential of the deep core-hole), or by neglecting it altogether, as in the initial state rule (independent particle approximation) for valence holes. Experience with such models in the FEFF code shows that these approximations are reasonable. We now define $\|\Phi_{0}^{b}\rangle$ as the ground state of the $N-2$ electron system in the presence of the deep core hole $\|b\rangle$, and $\|\Phi_{0}^{c}\rangle$ as the ground state of the $N-2$ electron system in the presence of the second core hole $\|c\rangle$ so that \[H^{b}\|\Phi_{0}^{b}\rangle = E_{0}^{b}\|\Phi_{0}^{b}\rangle;\;\;H^{b}=H_{0}^{N-2}+V_{hv}^{b}\] \[H^{c}\|\Phi_{0}^{c}\rangle = E_{0}^{c}\|\Phi_{0}^{c}\rangle;\;\;H^{c}=H_{0}^{N-2}+V_{hv}^{c}i\, \tag{23}\] with core level energies \[E_{b} = \epsilon_{b}-E_{0}^{0}+E_{0}^{b},\] \[E_{c} = \epsilon_{c}-E_{0}^{0}+E_{0}^{c}. \tag{24}\] The transition matrix elements corresponding to emission ($d_{2}$) may be pulled outside the imaginary part, and serve as an amplitude factor, i.e., \[\frac{d^{2}\sigma}{d\Omega d\omega} = -\frac{1}{\pi}\frac{\omega}{\Omega}\sum_{c}\|\langle b\|d_{2}Q\|c \rangle\|^{2} \tag{25}\] \[\times \mbox{Im}\left[\sum_{k_{1}k_{2}}^{\rm unocc}\langle b\|d_{1}^{ \dagger}PF(E_{1},E_{2})Pd_{1}\|b\rangle\right].\]Here $E_{1}=\Omega+E_{b}$, $E_{2}=\Omega+E_{c}-\omega$, $P$ is a projector onto unoccupied states of the ground state Hamiltonian, $Q$ is a projector onto occupied states of the intermediate state Hamiltonian, \[F(E_{1},E_{2})=G^{b}(E_{1})G^{c}(E_{2})G^{b\dagger}(E_{1}); \tag{26}\] and finally, the Green's functions are calculated in the presence of the deep ($b$) or shallow ($c$) core hole, i.e., \[G^{b}(\omega)=\frac{1}{\omega-(H_{b}-E_{0}^{b})-h_{p}^{b}-V_{pv} +i\Gamma_{b}}\] \[G^{c}(\omega)=\frac{1}{\omega-(H_{c}-E_{0}^{c})-h_{p}^{c}-V_{pv} +i\Gamma_{c}}, \tag{27}\] Next we derive an expression for the effects of multi-electron excitations in terms of an effective spectral function. \[\|\Phi_{0}\rangle=e^{-S_{b}}\|\Phi_{0}^{b}\rangle;\ \ \ S_{b}= \frac{a_{b}}{2}-\sum_{n}\frac{V_{bb}^{n}}{\omega_{n}}a_{bn}^{\dagger};\] \[\|\Phi_{0}^{b}\rangle=e^{-\Delta S}\|\Phi_{0}^{c}\rangle;\ \ \Delta S = \frac{\Delta a}{2}-\sum_{n}\frac{\Delta V^{n}}{\omega_{n}}a_{cn}^{\dagger};\] \[\Delta a=\sum_{n}\left(\frac{\Delta V^{n}}{\omega_{n}}\right)^{2},\ a_{b}=\sum_{n}\left(\frac{V_{bb}^{n}}{\omega_{n}}\right)^{2}. \tag{28}\] Here $\Delta V^{n}=V_{cc}^{n}-V_{bb}^{n}$ is the difference between the intermediate and final state core hole potentials. \[\|\Phi_{n}^{b}\rangle=\left[a_{cn}^{\dagger}-\frac{\Delta V^{n}}{\omega_{n}} \right]e^{-\Delta S}\|\Phi_{0}^{c}\rangle, \tag{29}\] Ignoring the off-diagonal terms in $V_{pv}$ we obtain \[\frac{d^{2}\sigma}{d\Omega d\omega}=-\frac{1}{\pi}\frac{\omega}{ \Omega}\mbox{Im}\left\{\sum_{n_{1}n_{2}}\langle b\|d_{1}^{\dagger}P\langle\Phi_ {0}^{b}\|e^{-{S_{b}}^{\dagger}}G^{b}(E_{1})\|\Phi_{n_{1}}^{b}\rangle\right.\] \[\left.\langle b\|d_{2}Q\langle\Phi_{0}^{c}\|e^{-\Delta S^{\dagger} }\left[a_{cn_{1}}-\left(\frac{\Delta V^{n_{1}}}{\omega_{n_{1}}}\right)^{}\right]\right.\] \[\left.G^{c}(E_{2})\left[a_{cn_{2}}^{\dagger}-\frac{\Delta V^{n_{ 2}}}{\omega_{n_{2}}}\right]e^{-\Delta S}\|\Phi_{0}^{c}\rangle Qd_{2}^{\dagger}\|b\rangle\right.\] \[\left.\langle\Phi_{n_{2}}^{b}\right\|\left[G^{b}(\Omega+E_{c}) \right]^{\dagger}e^{-S_{b}}\|\Phi_{0}^{b}\rangle Pd_{1}\|b\rangle\right\}. \tag{30}\]Note that in the case of valence emission (valence hole), we are assuming that the core hole potential is negligible, hence $E_{c}=\epsilon_{c}$, and $\Gamma_{c}=0$. Expanding to second order in the amplitudes to create and annihilate bosons, and neglecting off resonant terms, gives the total cross section in terms of a convolution with an effective spectral function $A_{\rm eff}$. \[\frac{d^{2}\sigma}{d\Omega d\omega}= \int\,d\omega_{1}d\omega_{2}\,A_{\rm eff}(\Omega,\Omega-\omega, \omega_{1},\omega_{2})\] \[\times\,\left[\frac{d^{2}\sigma}{d\Omega d\omega}\right]_{\rm sp} \Bigg{\|}_{\Omega=\Omega-\omega_{1},\omega=\omega-\omega_{1}+\omega_{2}}\,, \tag{31}\] where $\left[d^{2}\sigma/d\Omega d\omega\right]_{\rm sp}$ is the single particle cross section as given in Eq., and the spectral function is given by \[A_{\rm eff}(E_{1},E_{2},\omega_{1},\omega_{2})=e^{-a_{c}}\left\{ \delta(\omega_{1})\delta(\omega_{2})\right.\] \[+\,\left.\sum_{n}\left[\beta_{cn}(E_{2})\alpha_{cn}(E_{2})\delta( \omega_{1})\delta(\omega_{2}-\omega_{n})\right.\right.\] \[+\,\left.\left\|\beta_{bn}(E_{1})\right\|^{2}\delta(\omega_{1}- \omega_{n})\delta(\omega_{2}-\omega_{n})\right]\,\right\}. \tag{32}\] Here $\alpha_{n}$, $\beta_{n}$ are the amplitudes to create or annihilate a bosonic excitation, respectively, and include extrinsic as well as intrinsic amplitudes (see Appendix B). It should be noted that our current formalism for the spectral function is not suited for highly correlated materials, although an extension of the quasi-boson model Hamiltonian is possible, as suggested in Ref. and. The application of the spectral function to RIXS is similar to that of Ref. and, where a convolution was applied to XAS. In this paper, however, we will restrict our calculations to the quasiparticle approximation, i.e., with the spectral function replaced by a $\delta$-function, \[A_{\rm eff}(E_{1},E_{2},\omega_{1},\omega_{2})=\delta(\omega_{1})\delta(\omega _{2}-\omega_{n}). \tag{33}\] The use of this approximation is expected to cause the calculated spectral line-shapes to be more symmetric than experimental results, since the main quasiparticle peak is modeled in the above as a Lorentzian, while in general multi-electron excitations lead to asymmetric peaks so that Eq. has a Fano type main lineshape. Thus satellite peaks due to multi-electron excitations are also neglected. ## III Experiment The experiments described here were performed at beamline ID26 of the European Synchrotron Radiation Facility (ESRF). The incident energy was selected by means of a pair of cryogenically cooled Si crystals in reflection with an energy bandwidth of 0.2 eV (0.3 eV) at 4.9 keV (6.5 keV). The incident flux on the sample was $1\times 10^{13}$ photons/second using the fundamental peak of the undulator radiation. The beam size on the sample was 0.2 mm vertical by 1.0 mm horizontal. Higher harmonics were suppressed by three Si mirrors operating in total reflection. The resonantly scattered x-rays were analyzed using the and reflection of spherically bent Ge single crystal wafers for the Ti $K_{\beta}$ and $K_{\alpha}$ emission, respectively. The Ge reflection was used for Mn $K_{\alpha}$. Sample, analyzer crystals and an avalanche photo diode were arranged in a vertical Rowland geometry ($R=1$ m) at $90\pm 3$ deg scattering angle. The combined instrumental energy bandwidth was $0.8-1.0$ eV. All samples were purchased from Aldrich and used as is. Self-absorption effects distort the spectral shape and let the K absorption pre-edge region appear stronger relative to the edge jump. These effects are negligible in the K absorption pre-edge region and the samples were not diluted for the measurements. (color online) Calculated (left) Mn K${}_{\alpha}$ RIXS of MnO based on Eq. compared to experiment (right). (right). ## IV Results and Discussion ### Rixs Calculations of RIXS were carried out for several materials based on the present theoretical approach using an extension of the RSGF FEFF9 code applied to Eq.. All results were calculated using self-consistent potentials and a full multiple scattering (FMS) treatment of the Green's functions for suitably large clusters centered at the core absorption site. (color online) Calculated RIXS (left) of TiO${}_{2}$ based on Eq. compared to experiment (right). for (top to bottom) Ti K${}_{\alpha}$, K${}_{\beta}$, and $K_{\rm valence}$ RIXS. The core-hole screening was calculated using the random phase approximation (RPA), and non-spherical parts of the core-hole potential were neglected. In order to simplify the calculations the Green's functions are restricted to include only the site and angular momentum diagonal elements. We find that for the cases presented here, the angular momentum diagonal approximation is reasonable for all but the lowest energy peaks since the overlap with $\Delta V$ is then small, and our approximation then becomes equivalent to Eq. in terms of XES and XAS where the angular momentum diagonal elements dominate due to dipole selection rules. To obtain better agreement with the the experimental threshold energy, we allowed a small shift of the calculated Fermi energy, which is typically too high by about 1 eV in the self-consistent FEFF9.0 calculation. In addition, overall energy shifts were added in each axis in order to align the calculation with the energy scale in the experiment. In the case of K${}_{\alpha}$ RIXS, atomic values were used for the splitting between the P${}_{1/2}$ and P${}_{3/2}$ emission energies. In addition the amplitudes were taken (from simple counting arguments) to have a ratio $A_{3/2}/A_{1/2}=2$, although this ratio is not generally accurate, since the particle-hole interaction mixes the hole states. presents a comparison of our calculated (right) Mn K${}_{\alpha}$ RIXS of MnO and experimental data (left). The overall agreement is qualitatively satisfactory: all main features of the experiment are reproduced including both the dipole as well as quadrupole pre-edge peaks. The main edge is also at about the correct energy, although the asymmetry caused by multi-electron excitations is absent in our calculation which is restricted to the quasiparticle level where the spectral function is given by Eq.. We expect that going beyond this quasi-particle approximation for the spectral function as in Eq. 32 would capture some of the asymmetry since the Lorentzian spectral shape currently used for the quasiparticle peak in these calculations would be replaced by a Fano type lineshape. In addition, new features could arise due to satellite peaks in the spectral function. The main diagonal structure in our calculation appears to be sharper that that of the experiment; this is possibly due to self-absorption effects in the experimental data. Note that the pre-edge peak for this case is basically on the diagonal; i.e., the emission energy is the same for the pre-edge peak as for the main edge. Core-hole effects are important for a variety of excited state spectoscopies, including EELS and XAS as well as RIXS. RIXS spectra in particular however, can give us insight into these effects since the intermediate and final states have different core-holes. In order to illustrate the effect of different final state core-holes, we calculated the RIXS for Ti K${}_{\alpha}$, K${}_{\beta}$and Kvalence RIXS of TiO${}_{2}$ Anatase. In order to obtain reasonable results for the quadrupole peak in the K-edge absorption, we increased the strength of the core-hole potential by using 95% screened core-hole and 5% bare core-hole, and kept the same ratio for all core-hole calculations. Note that our calculation again reproduces all peaks, although the intensity of the calculated quadrupole peaks is weak compared to that observed in the experiment. There is also a noticeable effect on the spectrum due to changes in final state core-hole. For the K${}_{\alpha}$ RIXS, the intermediate (1s) and final (2p) core-holes are both quite localized and the difference $\Delta V$ is small, thus peaks should occur roughly on the diagonal, as seen by Eq.. For the K$-\beta$ spectrum the final state has a 3p core hole, and the core-hole potential has a vastly different shape than the $1s$ core-hole potential. This causes $\Delta V$ to be large, and we expect off diagonal peaks to be present. This is indeed the case, although only the quadrupole peaks are off diagonal. This is due to the fact that the dipole pre-edge peaks are caused by $p-d$ hybridization between the absorbing atom and neighboring Ti atoms, and thus are relatively unaffected by the core-hole potential. The quadrupole peaks, however, are due to a direct transition to the Ti $d-$states which are localized around the absorbing atom and are effected by the core-hole potential to a greater extent than the hybridized $p$-states. The effect is also present in the valence spectrum. In addition, the valence spectrum has multiple peaks due to the fact that the emission is from a broad valence band which is split due to solid state effects. The qualitative structure of the valence band is also correct in the calculation, which reproduces the double peak structure with the correct splitting. The intensities are also qualitatively correct with the lower energy-transfer peaks being less intense than the higher energy-transfer peaks. The gap is too small in our calculation, however, this could be accounted for via a GW gap correction. Note that we have not included the elastic scattering contribution in our calculation of the valence RIXS, which could effect the overall asymmetry of the signal. Finally, for all three spectra, the main edge occurs at a larger energy in the calculated results than in the experiment. This could be due to strong correlation effects, which would be expected to shift the Ti $d$-states closer to the $p$-states. Another possible explanation which is important in the case of Ti K pre-edge XAS of Rutile TiO${}_{2}$ is the failure of the spherical muffin-tin approximation. ### Lifetime Broadening Suppressed XAS In addition to the RIXS planes, there are also several methods for obtaining lifetime broadening suppressed (LBS) XAS. In high energy resolution fluorescence detected (HERFD) XAS, an approximate absorption spectrum is found by partial fluorescence yield using a detector with resolution higher than the natural width due to core-hole lifetime effects. This corresponds to viewing the spectra of constant emission energy in the RIXS plane. Another method of obtaining LBS XAS is to set the incident energy at a point well below the edge while scanning the emission energy. Under certain assumptions, the spectrum obtained in this way is approximately proportional to the XAS signal multiplied by a Lorentzian with the width of the intermediate state core-hole. In we show a comparison of experimental Cr-K edge HERFD XAS of K${}_{2}$CrO${}_{4}$ with our calculated results. We find reasonable qualitative agreement, with the exception of the peak just above 6000 eV, which is not seen in the calculation. We note however, that the size of this peak is sensitive to distortions. In addition, the amplitude of the main peak after the rising edge is too small. This could be due to the approximate treatment of the core-hole interaction or corrections to the spherical muffin-tin potentials used in FEFF9. Here we see that all of the features are well reproduced although the broadening is too large at high energies, and the higher energy peaks are also red shifted toward the edge in comparison to the experimental result. ## V Conclusions We have presented a theory of resonant inelastic x-ray scattering (RIXS) which is amenable to practical calculations as an extension of current x-ray-absorption and -emission codes. Starting from the Kramers-Heisenberg equation, we derive an expression for the RIXS cross-section which can be calculated using the real-space Green's function approach in the FEFF9 code. Inelastic losses and quasi-particle effects are included in terms of an effective spectral-function that is obtained from a quasi-boson model Hamiltonian. These many-body effects are incorporated into a single-particle approximation via a convolution with an effective spectral function. Quasi-particle self-energy effects are included based ona many-pole model of the dielectric function. Approximation of the many-body states as a product of an N-2 electron state with either two core electronic states (i.e. the ground state), or a core and photo-electron state (intermediate and final states) gives the cross section in terms of effective single particle Green's functions. The further approximation that the intermediate and final photo-electron states are orthogonal with identical energies (valid at high photo-electron energies) gives the signal in terms of a convolution of the XAS and XES spectra. In addition, we have derived a formulation of the core-core RIXS spectrum that, due to the localized nature of $\Delta V$, depends primarily on the Green's functions evaluated close to the absorbing atom. The extent of this localization is yet to be thoroughly investigated, although the degree of agreement between our calculations and experimental data suggests that the on-site approximation is valid for the systems shown here. In addition, the on-site approximation provides good qualitative agreement for the core-valence RIXS, although the approximation is less justifiable. The theory is implemented in an efficient program which is based on the $\Delta V$-matrix approach. The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in The results of the calculations are shown in Fig. Several illustrative calculations were presented within the quasi-particle approximation where the spectral function is replaced by a $\delta$-function, which appears to be a reasonable approximation for these cases. Calculated results for MnO and for Anatase TiO${}_{2}$ based on this quasi-particle approximation are found to agree qualitatively with experimental spectra: the results reproduce both pre-edge and main edge features, the behavior of pre-edge features with varying core-hole interaction strength, and peak structure due to solid state effects in valence RIXS. Further investigations including treatments beyond the quasi-particle approximation of Eq. will be reserved for the future. It should be noted that our current formalism for the spectral function is not suited for highly correlated materials, although an extension of the quasi-boson model Hamiltonian is possible as suggested in Ref. and. (color online) Pt L${}_{3}$ edge normal XANES (crosses) compared with HERFD XAS (x’s) and our calculated result (solid).	10.48550/arXiv.1101.4248	Real Space Green's Function Approach to RIXS	J. J. Kas, J. J. Rehr, J. A. Soininen, P. Glatzel	2,328
10.48550_arXiv.1512.07791	###### Abstract In this work, the elastic and thermodynamic properties of Pt${}_{3}$Al under high pressure are investigated using density functional theory within the generalized gradient approximation. The results of bulk modulus and elastic constants at zero pressure are in good agreement with the available theoretical and experimental values. Under high pressure, all the elastic constants meet the corresponding mechanical stability criteria, meaning that Pt${}_{3}$Al possesses mechanical stability. In addition, the elastic constants and elastic modulus increase linearly with the applied pressure. According to the Poisson's ratio $\nu$ and elastic modulus ratio ($B/G$), Pt${}_{3}$Al alloy is found to be ductile, and higher pressure can significantly enhance the ductility. Those indicate that the elastic properties of Pt${}_{3}$Al will be improved under high pressure. Through the quasi-harmonic Debye model, we first successfully report the variations of the Debye temperature $\Theta_{\rm D}$, specific heats $C_{P}$, thermal expansion coefficient $\alpha$, and Gruneisen parameter $\gamma$ under pressure range from 0 to 100 GPa and temperature range from 0 to 1000 K. first-principles, elastic properties, thermodynamic properties 61.82.Bg, 62.20.dc, 71.20.Be, 71.15.Mb ## 1 Introduction The high-temperature and ultra-high temperature structural materials, platinum group metal alloys (PGMS) are of great interest due to the high melting point, high strength and exceptional environmental resistance. They have been applied to many important industrial fields, such as catalysts, high temperature structural materials, special solder and shape memory alloys. In recent years, they have been receiving increasing concerns from researchers and have been extensively investigated by experiments and theoretical calculations. Wu et al. have studied the phase diagram of Al-Pt system using the CALPHAD method. Then, J. Feng et al. employed the density functional theory (DFT) method to investigate the stability, thermal and mechanical properties of Pt-Al intermetallic compounds. They found that their Poisson's ratio varies from 0.26 to 0.39 and the bonds in the compounds are mainly of metallic and covalent type, which is the same as in Zr-Al alloys. As one of the most valuable intermetallic compounds, the L1${}_{2}$ phase has been also extensively investigated. Chauke et al. have performed DFT calculations to examine the heats of formation, elastic modulus and the phonon dispersion curves of four different structure-types of Pt${}_{3}$Al at absolute zero pressure. The tetragonal $\rm{DO}_{c}$ structure is found to collapse to the cubic L1${}_{2}$ structure. Norihiko et al. have discussed the single crystals of Pt${}_{3}$Al with the L1${}_{2}$ structure from 77 to 1073 K. Gornostyrev et al. have employed first-principles electronic structure and total energy calculations of the phase stability and dislocation properties of Pt${}_{3}$Al to reveal the origins of its yield stress low temperature anomaly (LTA). In addition, Yan et al. have researched the phase transition and formation enthalpies of Pt${}_{3}$Al under high pressure. The results show that the cubic structure is stable compared to the tetragonal structure up to the pressure of 100 GPa and has excellent resistanceto volume deformation under high pressure. Despite the above investigations, there have been no systemic experimental or theoretical reports on the elastic and thermodynamic properties of L12 phase Pt3Al alloys under high pressure. As we know, high pressure leads to the phase transition and changes the physical and chemical properties of a solid, such as mechanical and thermodynamic properties, which are essential for a profound understanding the application of Pt3Al alloy. The elastic constants determine the response of a crystal to external forces and provide important information on the brittleness, ductility, anisotropy, and the resistance to deformation. A comprehensive understanding of the bulk modulus $B$, shear modulus $G$, Young's modulus $Y$ and Poisson's ratio plays an important role in determining the mechanical properties of solid materials. Furthermore, to better understand the thermodynamic properties of Pt3Al under high pressure, the quasi-harmonic Debye model was adopted. Then, we discuss the obtained thermodynamic parameters including Debye temperature $\Theta_{\text{D}}$, specific heats $C_{P}$, thermal expansion coefficient $\alpha$, and Gruneisen parameter $\gamma$. Therefore, it is highly desirable to understand the physical, mechanical, and thermal properties of L12 phase Pt3Al. In this paper, we performed a systematic investigation of the structural, elastic, and thermodynamic properties on the Pt3Al alloy with the L12 structure by first-principle calculations, and the calculated results were discussed in comparison with the available theoretical and experimental data. ## 2 Methods In the present work, all the calculations were performed based on the plane wave pseudopotential density-function theory method as implemented in CASTEP package. The exchange correlation energy is described in the generalized gradient approximation (GGA) for the exchange correlation functional. Pt 5$d^{9}4s^{1}$ and Al 3$s^{2}3p^{1}$ were treated as valence electrons. A plane wave cutoff energy of 400 eV was employed. The Brillouin zone was sampled by a $14\times 14\times 14$ uniform $k$-point mesh according to the Monkhorst-Pack scheme grids. In this work, the quasi-harmonic Debye model implemented in the Gibbs program is used to obtain the thermodynamic properties of Pt3Al. This model is sufficiently flexible in giving all thermodynamic quantities by incorporating the obtained results of energy and volume. The non-equilibrium Gibbs function $G^{}(V;P,T)$ is described in the following form: \[G^{}(V;P,T)=E(V)+PV+A_{\text{vib}}(\Theta,T). \tag{1}\] Here, $E(V)$ represents total energy/formula of Pt3Al, $P$ is the hydrostatic pressure, $A_{\text{vib}}(\Theta,T)$ is used to represent lattice vibration Helmholtz free energy and is taken as: \[A_{\text{vib}}(\Theta,T)=nk_{\text{B}}T\left[\frac{9\Theta}{8T}+3\ln\left(1- \text{e}^{-\Theta/T}\right)-D(\Theta/T)\right], \tag{2}\] where $D(\Theta/T)$ stands for the Debye integral, $n$ is the number of atoms per formula unit, and $\Theta$ is expressed by \[\Theta=\frac{h}{k_{\text{B}}}\left(6\pi^{2}V^{\frac{1}{2}}n\right)^{\frac{1}{3 }}f(\nu)\sqrt{\frac{B_{\text{S}}}{M}}. \tag{3}\] In relation, $M$ is the molecular mass per formula unit, $B_{\text{S}}$ is a representative for adiabatic bulk modulus, which is estimated in terms of static compressibility by using the following relation: \[B_{\text{S}}=V\left(\frac{\text{d}^{2}E(V)}{\text{d}V^{2}}\right). \tag{4}\] And $f(\nu)$ is defined as follows: \[f(\nu)=\left\{3\left[2\left(\frac{21+\nu}{31-\nu}\right)^{3/2}+\left(\frac{1 }{3}\frac{1+\nu}{1-\nu}\right)^{3/2}\right]^{-1}\right\}^{1/3}, \tag{5}\]where $\nu$ is Poisson's ratio. Hence, the non-equilibrium Gibbs function $G^{}(V;P,T)$ as a function of $(V;P,T)$ can be minimized with respect to the volume as follows: \[\left(\frac{\text{d}G^{}(V;P,T)}{\text{d}V}\right)_{P,T}=0. \tag{6}\] In order to obtain the thermal equation of states, we should solve the equation. After the equilibrium state of a given $V(P,T)$ has been obtained, the isothermal bulk modulus and other thermodynamic properties, such as the heat capacity, vibrational internal energy, and thermal expansion can be evaluated using the relations given as below: \[C_{V}=3nk_{\text{B}}\left[4D(\Theta/T)-\frac{3\Theta/T}{\text{e}^{\Theta/T}-1} \right], \tag{7}\] \[C_{P}=3nk_{\text{B}}\left[4D(\Theta/T)-\frac{3\Theta/T}{\text{e}^{\Theta/T}-1} \right](1+\alpha\gamma T), \tag{8}\] \[\alpha=\frac{\gamma C_{V}}{BV}, \tag{9}\] \[\gamma=-\frac{\text{d}\ln\Theta(V)}{\text{d}\ln V}, \tag{10}\] This method has already been successfully used to investigate the thermodynamic properties of a series of compounds. ## 3 Results and discussions ### Structure property As we know, Pt3Al has two kinds of structures including cubic phase and tetragonal phase. For cubic phase, Pt3Al alloy has a Cu3Au-type structure (space group: Pm3m, No: 221), with lattice parameters: $a=b=c=3.876$ A. The Pt and Al atoms are located at the site and (0.5, 0, 0), respectively. Each Al atom is surrounded by twelve Pt atoms. For tetragonal phase, Pt3Al has a space group: P4/mmm (No: 123) with experimental lattice parameters: $a=b=3.832$ A and $c=3.894$ A. There are two types of Pt: 1c (0.5, 0.5, 0) and 2e (0, 0.5, 0.5), respectively. The Al atom is in the site 1a. We have calculated the formation enthalpies of tetragonal and cubic phase as the pressure increasing from 0 GPa to 100 GPa. Our results show that the formation enthalpies of cubic structure are lower than that of tetragonal structure below 100 GPa. This means that cubic Pt3Al is stable under high pressure which is consistent with the result of Liu. To obtain equilibrium structural parameters, the atom position and structure of Pt3Al were optimized. At 0 GPa, the calculated lattice parameters of cubic phase $a$ is 3.86 A. We note a very good agreement between our results and experimental data. This offers the reliability and accuracy to our further investigation. ### Elastic property To the best of our knowledge, the elastic properties define the behavior of a solid under different stress and strain conditions. The elastic stiffness parameters can describe the bonding characteristics, mechanical deformations, and structural stability. To obtain the elastic constants, a small strain should be loaded to the crystal. They can be got by calculating the total energy as a function of appropriate lattice deformation, which are expanded as the Taylor expansion for a system with respect to a small strain $\delta$ and volume $V_{0}$. The elastic strain energy $E(V)$ is expressed as follows: \[E(V)=E(V_{0},0)+\frac{1}{2}\sum_{i}^{6}\sum_{j}^{6}C_{ij}\delta_{i}\delta_{j}. \tag{11}\] Here, $C_{ij}$ are elastic constants, $\delta_{i}$ and $\delta_{j}$ are related to the strain on the crystal. For cubic symmetry, there are three independent elastic constants, that are $C_{11}$, $C_{12}$, $C_{44}$. The calculated elastic constantsof Pt3Al are shown in At 0 GPa, the calculated elastic constants of Pt3Al ($C_{11}$ = 400.8, $C_{12}$ = 205.27, $C_{44}$ = 131.71, and $B$ = 270.46) are consistent with the experimental values ($B$ = 277) and other theoretical results ($C_{11}$ = 395, $C_{12}$ = 210, $C_{44}$ = 118). In general, the requirements of mechanical stability in a cubic crystal lead to the following restrictions on the elastic constants: $C_{11}$ > 0, $C_{12}$ > 0, $C_{11}$ - $C_{12}$ > 0, $C_{11}$ + 2$C_{12}$ > 0. Obviously, our results in (a) show that all the elastic constants satisfy the stabilities criteria up to 100 GPa. This clearly indicates that Pt3Al under high pressure possesses mechanical stabilities. There is no doubt that the elastic constants of a solid are strongly affected by the pressure. It should be noted that the elastic constants $C_{11}$, $C_{12}$, $C_{44}$ increase linearly with the pressure increase because the lattice parameters of Pt3Al become shorter under pressure. It is acknowledged that bulk modulus $B$ and shear modulus $G$ can measure the hardness in an indirect way. The calculated bulk modulus $B$, shear modulus $G$, and Young's modulus $Y$ under different pressures are shown in (b). It is found that bulk modulus $B$, shear modulus $G$, and Young's modulus $Y$ of Pt3Al gradually increase as pressure increases, indicating that Pt3Al becomes more and more difficult to be compressed as the pressure increases. In addition, all the elastic modulus can be used as a measure of the average bond strength of atoms for a given crystal. A larger bulk modulus $B$ and Young's modulus $Y$ respond to the more covalent and stronger bond strength. Hence, it can be expected that Pt and Al atom can form covalent bonds under high pressure. The shear modulus $G$ is the relationship between the resistance to reversible deformations and the shear stress. A high shear modulus $G$ is mainly due to the elastic constants $C_{44}$, because a large $C_{44}$ implies a stronger resistance to shear in the plane. For a further analysis, the deformation behavior of Pt3Al, the value of $B/G$ and Poisson's ratio $\nu$ which are related with the brittleness and hardness of the materials are shown in table 1. Generally, the $B/G$ ratio is used to predict the brittle or ductile behavior of materials. The critical value which separates ductile and brittle material is 1.75. The material exhibits a ductile behaviour when the value $B/G$ > 1.75; otherwise, the material behaves in a brittle manner. We found that the $B/G$ is 2.89 at 0 GPa and increases with the pressure increase. It means that the Pt3Al belongs to a ductility material and the pressure can improve the ductility of Pt3Al. Another important property is the Poisson's ratio $\nu$ which is defined as the absolute value of the ratio of transverse strain to longitudinal strain. It is used to quantify the stability of the crystal against shear. The larger is the Poisson's ratio $\nu$, the better is the plasticity. \begin{table} \begin{tabular}{c c c c c c c c c c c c} \hline \hline Pressure & 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 & 100 \\ (GPa) & & & & & & & & & & & \\ \hline $B/G$ & 2.289 & 2.356 & 2.410 & 2.436 & 2.462 & 2.480 & 2.506 & 2.527 & 2.543 & 2.577 & 2.592 \\ $\nu$ & 0.309 & 0.314 & 0.318 & 0.319 & 0.321 & 0.322 & 0.324 & 0.325 & 0.326 & 0.328 & 0.329 \\ \hline \hline \end{tabular} \end{table} Table 1: The Poisson’s ratio $\nu$ and $B/G$ of Pt3Al under high pressure. The calculated elastic constants $C_{ij}$ and the elastic modulus of cubic Pt3Al under pressure from 0 GPa to 100 GPa. This usually refers to ductile compounds with a large value (> 0.26). It is noted that the Poisson's ratio $n$ increases from 0.309 to 0.329 with the pressure increase, indicating that Pt3Al is ductile and the pressure can enhance the stability and the ductility of Pt3Al. This result is consistent with that of the above elastic modulus ratio (_B_/_G_). ### Thermodynamic properties To our knowledge, Debye temperature _th_D is one of the most important parameters describing the thermal characteristics of compounds. The Debye temperature correlates with many physical properties of material, which are derived from elastic properties under pressures. Some detailed information of a solid, such as the melting temperature and specific heat can be found by calculating the Debye temperature. We obtain the thermodynamic properties of Pt3Al at various temperatures and pressures from the energy-volume relations using the quasi-harmonic Debye model. The Debye temperature _th_D as a function of temperature at different pressures is shown in (a). It can be clearly seen that _th_D in the range of temperatures from 0 to 1000 K approximately remains unaltered with the temperature increase, meaning that it is insensitive to temperature. (b) shows the Debye temperature as a function of pressure at different temperatures of $T$ = 0, 300, 600, and 900 K. It is noted that _th_D linearly increases and further compression slows down the increase. As the pressure goes higher, the decreased magnitude of Debye temperature _th_D becomes small. This is because the Debye temperature is related to the volume $V$ and adiabatic bulk modulus. (b) shows that when the temperature is constant, the Debye temperature _th_D increases non-linearly with the applied pressures, indicating the change of the vibration frequency of particles under pressure. Hence, the temperature has a more significant effect on the Debye temperature _th_D than pressure, and the temperature exhibits a smaller and smaller effect on the Debye temperature with an increase of pressure. To describe the thermal properties of a material, the volume thermal expansion coefficient $a$ is another essential parameter. The dependence of the volume thermal expansion coefficient $a$ of Pt3Al on the temperature and pressure is illustrated in We noted that $a$ increases rapidly with $T$3 at zero or low pressure when the temperature is below 200 K and gradually approaches a very low linear increase above 400 K for a given pressure. Moreover, we can also see that the values of $a$ at zero pressure are much greater than those at other pressures. (b) gives $a$ as a function of pressure at different temperatures of $T$ = 0, 300, 600, 900 K. It can be seen that for a given temperature, the thermal expansion coefficient $a$ is zero at 0 K and rapidly decreases with the pressure increase, and it becomes flat under high pressure. Moreover, the higher is the temperature, the faster the $a$ decreases. There is observed a larger thermal expansion at a higher temperature and at a lower pressure, and it provides less sensitivity of $a$ at high temperature and high pressure for Pt3Al. (Color online) The Debye temperature as a function of temperature and pressure. (a) $P$ = 0, 25, 50, and 75 GPa, respectively; (b) $T$ = 0, 300, 600, and 900 K, respectively. As another important thermodynamic parameter of solids, the heat capacity $C_{P}$ is of key importance for linking thermodynamics with microscopic structures and dynamics. Moreover, the knowledge of the heat capacity of a substance not only provides an essential insight into its vibrational properties but also is mandatory for many applications. shows the calculated heat capacity $C_{P}$ as a function of temperature and pressure. It is obvious that $C_{P}$ follows the relationship of the Debye model [$C(T)\propto T^{3}$] up to 200 K. Then, it monotonously increases with the temperature increase and converges to a constant Dulong-Petit limit, which is common to all solids at high temperatures. We note that the heat capacity $C_{P}$ slowly decreases with the pressure increase, and the high temperature will slow down this trend seen in (b). implies that temperature and pressure have an opposite effect on the heat capacity, while the temperature has a greater effect on the heat capacity than the pressure. In the quasi-harmonic Debye model, the Gruneisen parameter $\gamma$ is of a great significance. It describes the anharmonic effects of the crystal lattice thermal vibration and has been widely used to characterize the thermodynamic behavior of a material at high pressure. (Color online) The heat capacity as a function of temperature and pressure. (a) $P$ = 0, 25, 50, and 75 GPa, respectively; (b) $T$ = 0, 300, 600, and 900 K, respectively. (Color online) The volume thermal expansion coefficient as a function of temperature and pressure. (a) $P$ = 0, 25, 50, and 75 GPa, respectively; (b) $T$ = 0, 300, 600, and 900 K, respectively. This is because the Gruneisen parameter $\gamma$ is as function of the volume which is affected by the pressure in the quasi-harmonic model. And, there is a larger thermal expansion at low pressure. Those results suggest that the effect of the temperature on the Gruneisen parameter $\gamma$ is not as significant as that of the pressure $P$. Furthermore, the Gruneisen parameter $\gamma$ increases more slowly at high pressure than at low pressure. ## 4 Conclusions First principles calculations are performed to investigate the elastic and thermodynamic properties of L12 phase Pt3Al alloy under high pressure and high temperature. The elastic constants, bulk modulus $B$, shear modulus $G$, and Young's modulus $Y$ as a function of the pressure have been systematically investigated. The results show that all the elastic constants meet the corresponding mechanical stability criteria and the elastic modulus increases linearly with the applied pressure. The Poisson's ratio $\nu$ and the elastic modulus ratio ($B/G$) show that L12 phase Pt3Al alloy is found to be ductile and higher pressure can significantly enhance the ductility. This means that the elastic properties of Pt3Al will be improved under high pressure. To study the thermal and vibrational effects, the quasi-harmonic Debye is used. The dependence of Debye temperature $\Theta_{\text{D}}$, specific heats $C_{P}$, thermal expansion coefficient $\alpha$, and the Gruneisen parameter $\gamma$ are systematically explored in the ranges of 0-100 GPa and 0-1000 K. We find that the temperature has a more significant effect on the Debye temperature $\Theta_{\text{D}}$ and the heat capacity $C_{P}$ than pressure. Furthermore, the thermal expansion coefficient $\alpha$ becomes insensitive to high temperature and high pressure.	10.48550/arXiv.1512.07791	Theoretical study of the elastic and thermodynamic properties of Pt$_{3}$Al with the L1$_{2}$ structure under high pressure	N. Wei, Ch. Zhang, S. Hou	5,663
10.48550_arXiv.1005.1115	###### Abstract The electronic structure and magnetic coupling properties of rare-earth metals (Gd, Nd) doped ZnO have been investigated using first-principles methods. We show that the magnetic coupling between Gd or Nd ions in the nearest neighbor sites is ferromagnetic. The stability of the ferromagnetic coupling between Gd ions can be enhanced by appropriate electron doping into ZnO:Gd system and the room-temperature ferromagnetism can be achieved. However, for ZnO:Nd system, the ferromagnetism between Nd ions can be enhanced by appropriate holes doping into the sample. The room-temperature ferromagnetism can also be achieved in the _n_-conducting ZnO:Nd sample. Our calculated results are in good agreement with the conclusions of the recent experiments. The effect of native defects (V${}_{\rm Zn}$, V${}_{\rm O}$) on the ferromagnetism is also discussed. pacs: 75.50.Pp, 71.55.Gs, 71.70.-d ## I Introduction The diluted magnetics semiconductors (DMSs) such as 3$d$ transition metals (TMs) doped ZnO have attracted a lot of attention due to their great potential applications in the spintronic devices. Recently, rare earth (RE) ions doped DMSs have also invoked great interests since the colossal magnetic moment of Gd in GaN was reported by Dhar _et al._. They reported that the average value of the moment per Gd is as high as 4000 $\mu_{B}$ which can be explained in terms of a long-range spin polarization of the GaN matrix by Gd. First principles studies have been carried out by different teams to explain ferromagnetism in GaN:Gd. Dalpian and Wei reported that the coupling between Gd atoms is antiferromagnetic and the electrons can stabilize the ferromagnetic phase because of the coupling between the Gd $f$ and host $s$ states introduced by the same symmetry. However, Liu _et al._ argued that the room-temperature ferromagnetism in GaN:Gd can be explained by the interaction of Gd 4$f$ spins via $p$-$d$ coupling involving holes introduced by intrinsic defects and holes are more effective than electrons in contributing to the observed colossal magnetic moment of Gd ions. Experimental studies for the ferromagnetism of ZnO:Gd systems have also produced controversial conclusions. Potzger _et al._ found ferromagnetism in Gd-implanted ZnO single crystals. They noted that when sufficient density of Gd ions is present, then annealing is necessary to release enough charge carriers to establish the ferromagnetic coupling of the diluted Gd ions. However, Ungureanu _et al._ reported that there are no exchange interaction between the RE ions and a large negative magnetoresistance is obtained which can be interpreted as a paramagnetic response of the system to the applied magnetic field. Compared with 3$d$ transition metals (TMs), 4$f$ rare earth (RE) metals have larger magnetic moments. Furthermore, the electrons may mediate the ferromagnetic coupling between the RE ions, due to the coupling between $f$ electrons (with $a_{1}$ symmetry) and host $s$ electrons. This conclusion may be good for ZnO based DMSs, because grown ZnO films or single crystals always exhibit $n$-type conductivity. Until now, no theoretical studies about RE ions doped ZnO systems are found. Therefore, it is interesting to investigate the electronic structures and magnetic couplings properties of ZnO:RE. In this work, we have systematically studied the magnetic coupling between the RE ions with different electron and hole concentrations doped into the ZnO:RE samples. We find that the coupling between Gd ions in ZnO is ferromagnetic. Furthermore, the electrons of appropriate concentration can enhance the ferromagnetic coupling between them. However, for Nd ions, the holes of appropriate concentration can enhance the ferromagnetic coupling between them. The effect of native defects (V${}_{\rm Zn}$, V${}_{\rm O}$) on the ferromagnetism is also discussed. We note that although the oxygen vacancy can contribute electrons to the system, the coupling between Gd ions is anti-ferromagnetic. Maybe this is because the concentration of electrons introduced by oxygen vacancy is too high in our studied supercell. ## II Details of calculation Our first-principles calculations are based on the density functional theory (DFT) and the Vienna ab initio simulation package using the generalized gradient approximation (GGA) of PBE functional for the exchange correlation potential. The electron and core interactions are included using the frozen-core projected augmented wave (PAW) approach. The Zn 3$d$ andrare earth metals $5p4f$ electrons are explicitly treated as valence electrons. The electron wave function is expanded in plane waves up to a cutoff energy of 400 eV. For the Brillouin zone integration, a 2$\times$2$\times$4 Monkhorst-Pack $k$-point mesh is used for the supercell containing 32 atoms and a good convergence is obtained. All the geometries are optimized until the quantum mechanical forces acting on the atoms are smaller than 0.01 eV/A. In our present work, we calculate the ferromagnetic properties for the RE ions doped in the zinc-blende structure ZnO and we expect our conclusions are similar to those for RE ions doped in ground phase ZnO with wurtzite structure since the band structures of the ZB and WZ alloy are very similar near the band edge at $\Gamma$. In our magnetic calculations, we substitute two Zn atoms with two RE ions in the nearest neighbor (NN) sites, corresponding to a concentration of 12.5% for RE ions. ## III Results and Discussions ### Electronic structure of hypothetical zinc-blende phase GdO and NdO In order to obtain a clear understanding of the magnetic coupling of the RE ions, we first study the electronic structures of the hypothetical zinc-blende (ZB) phase GdO and NdO binary alloys. The present calculated lattice constants for ZB structure ZnO, GdO, and NdO are 4.626, 5.375, and 5.444 A, respectively. Our calculated results show that both GdO and NdO are more stable in the ferromagnetic phases and the energy differences between anti-ferromagnetic (AFM) and ferromagnetic (FM) coupling states are 16 meV and 291 meV, respectively. Based on the crystal field theory, in a tetragonal substitutional site of ZB structure, the $d$ orbitals are split into one triply degenerate $t_{2}$ state and one doubly generate $e$ state; the $t_{2}$ orbitals are split into two triply degenerate $t_{2}$, $t_{1}$ states and one singly $a_{1}$ state. The spin-resolved band structures of GdO and NdO are plotted in and Fig. 2, respectively. The labels in these two figures represent the characters of Bloch wave functions at $\Gamma$ point. In Fig. 1, due to the large electronegativity of oxygen, the 4$f$ majority spin channels are above the oxygen $p$ states, unlike the GdN in which the 4$f$ majority spin channels are below the nitrogen $p$ states. At $\Gamma$ point, the O $p$ states with $t_{2}$ symmetry in spin up channels are below those in spin down channels. This is because at $\Gamma$ point the coupling between the Gd $t_{2f}$ and O $t_{2p}$ states is larger in the spin up channel, which pushes down the O $p$ states. We also notice that at $\Gamma$ point, the Gd $s$ state is located below the Fermi energy in both spin up and spin down channels. Furthermore, for GdN, Gd is isovalent with Ga because of the 4$f^{\prime}$5$d^{\prime}$6$s^{2}$ valence configuration, therefore, GdN is semiconducting. For GdO, however, Gd is not isovalent with Zn, thus GdO is not semiconducting. From the density of states (DOS) of GdO plotted in Fig. 3(a), we can also conclude that GdO is not semiconducting because there are majority spin states at the Fermi-energy position and the density of states are not symmetrical for the majority and minority spins. In Fig. 2, the band structure of the ferromagnetic phase NdO with ZB structure is showed. At $\Gamma$ point, part of the 4$f$ states are occupied in the spin up channel below the Fermi energy, while the 4$f$ states in the spin down channel are fully empty. According to the DOS plotted in Fig. 3(b), we see that the 4$f$ bands of majority spin are located between -0.5 eV and 0.4 eV, and the peak in the DOS of 4$f$ states of minority spin occurs around 2.3 eV. Overall, the DOS of 4$f$ states are broadened due to the $f$-$s$ hybridization. ### Ferromagnetic coupling of Gd and Nd ions doped in ZnO In the following, we systematically study the properties of ferromagnetic coupling between Gd (Nd) ions in different charged states. Experimentally, Potzger _et al._ found ferromagnetism in Gd-implanted ZnO single crystals. They further reported that when sufficient density of Gd ions is present, then annealing is necessary to release enough charge carriers to establish the ferromagnetic coupling of the diluted Gd ions. In our present study, we substitute two Zn atoms with two RE ions (Gd or Nd) in the nearest neighbor sites. We simulate the effect of donors (acceptors) by introducing electrons (holes) into the ZnO:RE systems. Band structure of the ferromagnetic phase GdO with ZB structure. The labels represent the symmetry character of the band at $\Gamma$ point. We have also investigated whether the intrinsic defects (V${}_{\rm Zn}$, V${}_{\rm O}$) play an important role in the magnetic properties of ZnO:RE systems. Our results show that the ground states of Gd and Nd doped ZnO systems are all ferromagnetic coupling states in the neutral state, and the energy differences $\Delta$E between the anti-ferromagnetic coupling state (AFM) and the ferromagnetic coupling state (FM) are 7 and 94 meV, respectively. The direct coupling between 4$f$ electrons is very weak compared with the energy difference $\Delta$E for 3$d$ transition metals doped in ZnO. This is because the orbitals of 4$f$ electrons are very localized. The band structure of FM phase for ZnO:Gd and ZnO:Nd systems are plotted in and Fig. 5, respectively. In for ZnO:Gd system, the 4$f$ bands of the majority spin (lying at about $-20$ eV) are not plotted because they are far from the Fermi energy. However, in for ZnO:Nd system, the 4$f$ bands of the majority spin are located near the Fermi energy due to the partially occupied 4$f$ orbitals. Due to the $s$-$f$ coupling, electrons may mediate the ferromagnetism in the vicinity of the Fermi energy ($\Delta$E). The 4$f$ bands of the majority spin are located near the Fermi energy due to the partially occupied 4$f$ orbitals. Band structure of ferromagnetic Zn${}_{14}$Gd${}_{2}$O${}_{16}$ Band structure of ferromagnetic Zn${}_{14}$Nd${}_{2}$O${}_{16}$ Density of states of (a) GdO and (b) NdO. The Fermi energy is taken to be zero and the green line is to make it more clear. Band structure of the ferromagnetic phase NdO with ZB structure. The labels represent the symmetry character of the band at $\Gamma$ point. Experimentally, Ungureanu _et al._ also reported that the presence of ferromagnetism in ZnO:Gd films might indicate electron mediated interior exchange. In order to test this speculation, we insert different amounts of electrons into the ZnO:Gd supercell. Remarkably, we find that even when 0.15 electron per Gd is inserted into the ZnO:Gd system, the stability of the FM phase is enhanced with $\Delta$E=44 meV ($\Delta$E=E${}_{\rm AFM}$-E${}_{\rm FM}$), much larger than the neutral case of $\Delta$E=7 meV. We also calculate the cases of 0.25, 0.50, 0.75, and 1.0 electron per Gd. The corresponding $\Delta$E are 67, 61, $-$8, and $-$37 meV. We note that the ZnO:Gd system will favor AFM phase if 0.75 or 1.0 electron per Gd is inserted into the system. In practice, however, even for 0.5 electron per Gd, such high doping corresponding to 1.314 $\times$ 10${}^{21}$ cm${}^{-3}$ is hard to achieve experimentally. For hole doping, the calculated $\Delta$E is 4 meV after creating 0.5 hole per Gd in the system, thus indicating that the stability of the FM phase is weakened by the hole doping. In particular, when the hole doping is as high as 1.0 hole per Gd, we find that the energies of FM and AFM phases of ZnO:Gd are degenerate and thus the system favors paramagnetic phase. Since the intrinsic defects are important as well for the magnetic properties, we have also investigated how the zinc and oxygen vacancies affect Gd-Gd magnetic interaction. For zinc vacancies V${}_{\rm Zn}$, in our studied supercell one V${}_{\rm Zn}$ contributes 2 holes to the system, and our calculated $\Delta$E is 0 meV, i.e., the ZnO:Gd system favors paramagnetic phase. This is consistent with the case of 1.0 hole/Gd, both can lead the ZnO:Gd system to favor paramagnetic phase. However, one V${}_{\rm O}$ introduces two electrons to the system and leads the system to anti-ferromagnetic phase with $\Delta$E=$-$8 meV. The effect of one V${}_{\rm O}$ on the system is similar to that of the case of 1.0 electron per Gd. Our above calculated results are listed in table 1. According these results we can conclude that electrons can effectively mediate the ferromagnetism of the ZnO:Gd system through controlling its concentrations. Ungureanu _et al._ also reported that the presence of ferromagnetism at 300 K in ZnO:Gd films with around 1% Gd co-doped with 0.2% Al might indicate electron mediated exchange in ZnO:Gd systems. In their samples, Al servers as _n_-type dopant. The concentration of electrons introduced by Al (1% Gd co-doped with 0.2% Al) is in the range of our studied cases of 0.15$\sim$0.25 electron per Gd, which, as predicted from Table 1, also leads the ZnO:Gd system to favor ferromagnetic phase. As for the Curie temperature (_T${}_{c}$_), our calculated energy differences $\Delta$E in Table 1 suggest that for the present calculated ZnO:Gd system with the electron concentration comparable with that in attainable experiment, the derived _T${}_{c}$_ is higher than room temperature (RT). This prediction, which is really consistent with the experiment, is based on the established fact that RT ferromagnetism can only be achieved when $\Delta$E is larger than about 30 meV. Note again that for the ZnO:Gd system, the _s-f_ coupling is much larger than the _f-f_ and _f-p_ couplings, which makes it be the main factor for the electron-mediated ferromagnetism in ZnO:Gd system. For ZnO:Nd system, the holes can also mediate the ferromagnetism according to our calculated results listed in table 1. In the neutral case, the calculated energy difference $\Delta$E for the ZnO:Nd system is 94 meV. In the hole-doped cases, the energy differences $\Delta$E are 128 and 134 meV for 0.25 and 0.50 hole per Nd, respectively. Thus the FM coupling with low or intermediate hole concentration is more stable than that of the neutral system. However, when the concentration of holes is as high as 0.75 (1.0) hole per Nd, the energy difference $\Delta$E is 69 meV, which indicates that overdoping of holes to the ZnO:Nd system will lower $\Delta$E and the subsequent stability of the FM phase. For electron doping, according to our calculated $\Delta$E in Table 1, the stability of FM ordering of ZnO:Nd system is weakened. In general, grown ZnO sample is always _n_-type conducting. Based on our calculated results, if the electron concentration is relatively low (0.25 electron/Nd for example shown in Table 1), the RT ferromagnetism can also be achieved for ZnO:Nd system. Ungureanu _et al._ also reported that the ferromagnetism is present in ZnO:Nd system similar to that of ZnO:Gd system at the same doping level. For vacancies, both V${}_{\rm Zn}$ and V${}_{\rm O}$ in ZnO:Nd system result in the FM phase with $\Delta$E=52 and 67 meV, respectively. Due to the strong on-site Coulomb repulsion among the localized 4$f$ electrons, the traditional DFT with GGA (PBE) or LDA scheme can not accurately described the strong correlation. Therefore, several methods are adopted to overcome the drawback mentioned above, such as LDA+_U_, and Heyd-Scuseria-Ernzerhof (HSE) hybrid-functional. \begin{table} \begin{tabular}{c c c} RE & charge state & $\Delta$E (meV) \\ \hline \multirow{8}{}{Gd} & neutral & 7 \\ & 0.15e/Gd & 44 \\ & 0.25e/Gd & 67 \\ & 0.50e/Gd & 61 \\ & 0.75e/Gd & -8 \\ & 1.00e/Gd & -37 \\ & 0.50h/Gd & 4 \\ & 1.00h/Gd & 0 \\ & V${}_{\rm Zn}$ & 0 \\ & V${}_{\rm O}$ & -8 \\ \hline \multirow{8}{}{Nd} & neutral & 94 \\ & 0.25h/Nd & 128 \\ & 0.50h/Nd & 134 \\ & 0.75h/Nd & 69 \\ & 1.00h/Nd & 20 \\ & 0.25e/Nd & 46 \\ & 0.50e/Nd & 16 \\ & V${}_{\rm Zn}$ & 52 \\ & V${}_{\rm O}$ & 67 \\ \end{tabular} \end{table} Table 1: Calculated energy differences $\Delta$E ($\Delta$E=E${}_{\rm AFM}$-E${}_{\rm FM}$) between anti-ferromagnetic and ferromagnetic configurations for ZnO:Gd and ZnO:Nd in different charged states. Therefore, we may expect that the main trends of carries mediated ferromagnetism discussed in our paper will not change if the Hubbard $U$ correlation is taken into account. ## IV Summary In summary, we have systematically investigated the magnetic properties of ZnO:RE (RE=Gd, Nd) systems in different charged states. Because of the _s-f_ coupling between the Gd ions $f$ and the host $s$ states, the electrons can mediate the ferromagnetism in the ZnO:Gd system. We present that the RT ferromagnetism can be achieved in ZnO:Gd system if electrons of appropriate concentration are doped in the sample. However, for ZnO:Nd system, the holes can enhance the stability of the FM ordering, and the RT ferromagnetism can also be achieved in the _n_-type ZnO:Nd. Our calculated results agree well with the recent experimental observation.	10.48550/arXiv.1005.1115	Magnetic coupling properties of rare-earth metals (Gd, Nd) doped ZnO: first-principles calculations	Hongliang Shi, Ping Zhang, Shu-Shen Li, Jian-Bai Xia	3,426
10.48550_arXiv.1404.1658	## Edge step normalized Co-K edge XANES spectra of ceramic CTO sample. Co metal is used for photon energy calibration. Cobalt oxide (CoO and CoF${}_{3}$) standards are shown together to obtain Co valency in CTO. ## LCF fit of CTO raw data with CoO and CoF${}_{3}$ standard samples and linear dependence of oxidation states as a function of energy obtained from derived formula, as in text (inset). ## Figure 4: High magnetic field DC magnetization in ZFC/FC protocol indicates the same antiferromagnetic behavior as that of a single crystal and Curie-Weiss fit of the same dc magnetization in FC mode (inset). $\mu_{\rm eff}$ has been calculated through $\mu_{\rm eff}$ = $\surd$(8C) $\mu_{\rm B}$ u. ## Synchrotron powder X-Ray diffraction patterns recorded at room temperature for ceramic CTO sample. Open red circles represent the raw data, solid black line is the fit obtained by the Rietveld refinements using the monoclinic structure with C2/$c$ space group. The vertical bar lines show Bragg reflections. Zigzag line beneath the pattern shows the difference between the observed and calculated Intensity. SXRD pattern have been taken using photon energy of wavelength $\lambda$ = 0.94805Å. ## Edge step normalized Co-K edge XANES spectra of ceramic CTO sample. Co metal is used for photon energy calibration. Cobalt oxide (CoO and CoF${}_{3}$) standards are shown together to obtain Co valency in CTO. ## LCF fit of CTO raw data with CoO and CoF${}_{3}$ standard samples and linear dependence of oxidation states as a function of energy obtained from derived formula, as in text (inset). ## Table 1 \begin{tabular}{c c c c c c c} \hline ## Sr. & Configuration & Co${}^{2+}$ & Co${}^{3+}$ & Spin state & Total relative \\ ## No. & & (\%) & (\%) & Co${}^{2+}$ & Co${}^{3+}$ & Energy (meV) \\ \hline 1. & AFM & 100 & 0.0 & High &... & 0 \\ & Co${}^{2+}$-O-Co${}^{2+}$ & (Co${}_{1}$,Co${}_{2}$,Co${}_{3}$,Co${}_{4}$, & & & & \\ & & Co${}_{5}$) & & & & \\ \hline 2. & AFM & 78 & 22 & High & Interme- & -267.16 \\ & Co${}^{2+}$-O-Co${}^{2+}$ & (Co${}_{1}$,Co${}_{2}$,Co${}_{3}$,Co${}_{4}$) & (Co${}_{5}$) & & & \\ \hline 3. & AFM & 78 & 22 & High & Low & -316.31 \\ & Co${}^{2+}$-O-Co${}^{2+}$ & (Co${}_{1}$,Co${}_{2}$,Co${}_{3}$,Co${}_{4}$) & (Co${}_{5}$) & & & \\ \hline 4. & AFM & 78 & 22 & High & High & -305.18 \\ & Co${}^{2+}$-O-Co${}^{2+}$ & (Co${}_{1}$,Co${}_{2}$,Co${}_{3}$,Co${}_{4}$) & (Co${}_{5}$) & & & \\ \hline 5. & Co${}^{2+}$-O-Co${}^{2+}$ & 67 & 33 & High & Interme- & -381.51 \\ & AFM+GKA & (Co${}_{2}$,Co${}_{3}$Co${}_{4}$) & (Co${}_{5}$,Co${}_{1}$) & & & \\ \hline 7. & Co${}^{2+}$-O-Co${}^{2+}$ & 67 & 33 & High & High & -433.00 \\ & AFM+GKA & (Co${}_{2}$,Co${}_{3}$Co${}_{4}$) & (Co${}_{5}$,Co${}_{1}$) & & & \\ \hline 8. & Co${}^{2+}$-O-Co${}^{2+}$ & 58 & 42 & High & Interme- & -117.19 \\ & AFM+GKA & (Co${}_{1}$,Co${}_{4}$,Co) & (Co${}_{2}$,Co${}_{3}$) & & & \\ \hline 9. & Co${}^{2+}$-O-Co${}^{2+}$ & 58 & 42 & High & Low & -450.32 \\ \hline \end{tabular} \begin{tabular}{c c c c c c c} & AFM+GKA & (Co${}_{1}$,Co${}_{4}$,Co${}_{5}$ & (Co${}_{2}$,Co${}_{3}$) & & & \\ 10. & Co${}^{2+}$-O-Co${}^{2+}$ & 58 & 42 & High & High & -529.89 \\ & AFM+GKA & (Co${}_{1}$,Co${}_{4}$,Co${}_{5}$) & (Co${}_{2}$,Co${}_{3}$) & & & \\ \end{tabular} ## Fig.3 Fig.3	10.48550/arXiv.1404.1658	Coexistence of Co3+ and Co2+ in ceramic Co3TeO6; XANES, Magnetization and first principle study	Harishchandra Singh, Haranath Ghosh, T. V. Chandrasekhar Rao, A. K. Sinha	4,831
10.48550_arXiv.1005.3344	"###### Abstract\n\nWe have investigated plasma-surface interactions with molecular dynamics (MD) si(...TRUNCATED)	10.48550/arXiv.1005.3344	Extension of the simulation code ACAT to treat real atomic positions	Arimichi Takayama, Seiki Saito, Atsushi M. Ito, Takahiro Kenmotsu, Hiroaki Nakamura	2,127

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ChemPile-Paper

A comprehensive collection of scientific literature spanning academic papers and preprints focused on chemistry and related fields

📋 Dataset Summary

ChemPile-Paper serves as a resource for cutting-edge applications of chemical knowledge and reasoning, containing curated papers from diverse repositories. This dataset represents a comprehensive collection of scientific literature spanning academic papers and preprints, all focused on chemistry and related fields.

📊 Dataset Statistics

Subset	Tokens	Documents	Description
ArXiv Cond-Mat Materials Science	35.9M	5,94K	Materials science papers
ArXiv Physics Chemical Physics	62.3M	6,75K	Chemical physics papers
bioRxiv	82.6M	60,3K	Biology preprints
medRxiv	451M	15,8K	Health sciences preprints
ChemRxiv	210M	28,9K	Chemistry community preprints
EuroPMC Chemistry Abstracts	3.3B	10.4M	Scientific literature abstracts
EuroPMC Chemistry Papers	10B	1.2M	Scientific literature full articles
Total	~13.9B	~11.7M	Chemical scientific literature

🗂️ Dataset Configurations

The dataset includes different subsets available as Hugging Face configurations:

arxiv-cond-mat.mtrl-sci_processed-default
arxiv-physics.chem-ph_processed-default
biorxiv_processed-default
chemrxiv_processed-default
euro_pmc_chemistry_abstracts-default
euro_pmc_chemistry_papers-default
medrxiv_processed-default

📜 License

All content is released under the CC BY-NC-ND 4.0 license, which allows for:

✅ Non-commercial use
✅ Sharing and redistribution
⚠️ Attribution required
❌ No derivatives allowed

📖 Dataset Details

📚 ArXiv Subsets

Sources:

Condensed Matter > Materials Science (arxiv-cond-mat.mtrl-sci_processed-default)
Physics > Chemical Physics (arxiv-physics.chem-ph_processed-default)

Coverage: Academic papers from ArXiv in materials science and chemical physics

Extraction Method: Articles filtered by field using PaperScraper package for PDF download and processing

Fields:

fn: ArXiv identifier (e.g., 10.48550_arXiv.0708.1447)
text: Parsed text of the article
doi: DOI of the article (if available)
title: Article title
authors: Article authors
index: Document identifier

Statistics:

Materials Science: 35.9M tokens across 5,940 documents
Chemical Physics: 62.3M tokens across 6,750 documents

🧬 bioRxiv and medRxiv

Sources:

bioRxiv - Biology preprint repository
medRxiv - Health sciences preprint repository

Coverage: Preprints in biology and health sciences with chemistry relevance

Extraction Method: PaperScraper package for DOI-based retrieval, processed with Nougat for text extraction

Fields:

fn: Unique identifier (e.g., 014597_file10)
text: Full text content extracted via Nougat

Statistics:

bioRxiv: 82.6M tokens across 60,300 documents
medRxiv: 451M tokens across 15,800 documents

⚗️ ChemRxiv

Source: ChemRxiv - Preprint server for the global chemistry community

Coverage: Chemistry preprints from the community

Extraction Method: PaperScraper for DOI-based retrieval, processed with Nougat for text extraction

Fields:

fn: Unique identifier (e.g., 10.26434_chemrxiv-2022-cgnf5)
text: Full text content extracted via Nougat
doi: DOI of the article (if available)
title: Article title
authors: Article authors
license: Preprint license (e.g., CC BY-NC 4.0)
published_url: Publication URL
index: Document identifier

Statistics: 210M tokens across 28,900 documents

🔬 EuroPMC Filtered Papers

Source: EuroPMC - 27 million abstracts and 5 million full-text articles

Coverage: Chemistry-related scientific papers filtered from comprehensive medical literature

Extraction Method:

BERT-based multilabel classifier trained on CAMEL datasets (20,000 examples per discipline)
Validated against FineWebMath annotations (F1-score ~0.77 on 150 manually annotated entries)
Analysis of first five 512-token chunks per document with 50-token overlaps

Quality Control:

Postprocessing to remove non-chemical content (authors, acknowledgments, page numbers)
Chemistry-specific content identification and filtering

Fields:

pmcid: PubMed Central identifier
pmid: PubMed identifier
topic: Main classification topic (e.g., "Chemistry", "Physics", "Biology")
confidence: Classification confidence score
class_distribution: Multilabel classification distribution
text: Full article text content

Statistics:

Abstracts: 3.3B tokens across 10.4M documents
Full Papers: 10B tokens across 1.2M documents

🚀 Quick Start

from datasets import load_dataset, get_dataset_config_names

# List all available configurations
configs = get_dataset_config_names("jablonkagroup/chempile-paper")
print(f"Available configs: {configs}")
# ['arxiv-cond-mat.mtrl-sci_processed-default', 'arxiv-physics.chem-ph_processed-default', 
#  'biorxiv_processed-default', 'chemrxiv_processed-default', 'euro_pmc_chemistry_abstracts-default',
#  'euro_pmc_chemistry_papers-default', 'medrxiv_processed-default']

# Load a specific subset
dataset = load_dataset("jablonkagroup/chempile-paper", name="arxiv-cond-mat.mtrl-sci_processed-default")

print(dataset)
# DatasetDict({
#     train: Dataset({
#         features: ['fn', 'text', 'doi', 'title', 'authors', '__index_level_0__'],
#         num_rows: 5899
#     })
#     test: Dataset({
#         features: ['fn', 'text', 'doi', 'title', 'authors', '__index_level_0__'],
#         num_rows: 328
#     })
#     val: Dataset({
#         features: ['fn', 'text', 'doi', 'title', 'authors', '__index_level_0__'],
#         num_rows: 328
#     })
# })

# Access a sample
sample = dataset['train'][0]
print(f"Sample ID: {sample['fn']}")
print(f"Sample text: {sample['text'][:200]}...")

🎯 Use Cases

🤖 Language Model Training: Pre-training or fine-tuning models for chemistry domain with cutting-edge research
🔬 Research Intelligence: Building systems for scientific literature analysis and discovery
🔍 Information Retrieval: Advanced chemistry knowledge base construction from research literature
📝 Content Generation: Automated scientific writing and research synthesis
🧠 Domain Adaptation: Adapting models to cutting-edge chemical research and terminology

⚠️ Limitations & Considerations

Language: Primarily English (monolingual dataset)
Scope: Focused on published research; may include technical jargon and advanced concepts
Quality: Variable quality across sources; some OCR errors possible in older papers
Bias: Reflects biases present in scientific publishing and academic literature
License: No derivatives allowed due to CC BY-NC-ND 4.0 license
Recency: Content reflects publication dates; cutting-edge developments may not be included

🛠️ Data Processing Pipeline

Collection: Automated scraping from academic repositories and databases
Filtering: BERT-based classification for chemistry relevance
Extraction: PDF processing with PaperScraper and Nougat OCR
Quality Control: Automated filtering and expert validation
Standardization: Consistent formatting and metadata extraction
Validation: Train/validation/test splits and quality checks

🏗️ ChemPile Collection

This dataset is part of the ChemPile collection, a comprehensive open dataset containing over 75 billion tokens of curated chemical data for training and evaluating general-purpose models in the chemical sciences.

Collection Overview

📊 Scale: 75+ billion tokens across multiple modalities
🧬 Modalities: Structured representations (SMILES, SELFIES, IUPAC, InChI), scientific text, executable code, and molecular images
🎯 Design: Integrates foundational educational knowledge with specialized scientific literature
🔬 Curation: Extensive expert curation and validation
📈 Benchmarking: Standardized train/validation/test splits for robust evaluation
🌐 Availability: Openly released via Hugging Face

📄 Citation

If you use this dataset in your research, please cite:

@article{mirza2025chempile0,
  title   = {ChemPile: A 250GB Diverse and Curated Dataset for Chemical Foundation Models},
  author  = {Adrian Mirza and Nawaf Alampara and Martiño Ríos-García and others},
  year    = {2025},
  journal = {arXiv preprint arXiv:2505.12534}
}

👥 Contact & Support

Paper: arXiv:2505.12534
Website: ChemPile Project
Dataset: Hugging Face
Issues: Please report data issues or questions via the Hugging Face dataset page

Part of the ChemPile project - Advancing AI for Chemical Sciences

Downloads last month: 636

Size of downloaded dataset files:

31.6 GB

Size of the auto-converted Parquet files:

31.6 GB

Number of rows:

11,737,032

Collection including jablonkagroup/chempile-paper

ChemPile

Collection

The ChemPile is a dataset with over 75 billion curated multimodal tokens about chemistry. For more information, visit https://chempile.lamalab.org/. • 9 items • Updated 21 days ago • 13