We report the effects of lattice dynamics on thermodynamic stability of binary $R_{1-x}$Fe$_x$ $(0<x<1)$ compounds ($R$: rare-earth elements, Y, Ce, Nd, Sm, and Dy) at finite temperature predicted by first-principles calculation based on density functional theory (DFT). We first demonstrate that the thermodynamic stability of $R_{1-x}$Fe$_x$ $(0<x<1)$ alloys cannot be predicted accurately by the conventional approach, where only the static DFT energy at $T = 0$ K is used. This issue can be overcome by considering the entropy contribution, including electronic and vibrational free energies, and we obtained convex hull plots at finite temperatures that successfully explain the thermodynamic stability of various known compounds. Our systematic calculation indicates that vibrational entropy helps stabilize various $R_{1-x}$Fe$_x$ compounds with increasing temperature. In particular, experimentally reported $R_2$Fe$_{17}$ compounds are predicted to become thermodynamically stable above $\sim$800 K. We also show that thermodynamic stability is rare-earth dependent and discuss its origin. Besides the experimentally reported structures, the stability of two new monoclinic $R$Fe$_{12}$ structures found by Ishikawa \textit{et al.} [Phys. Rev. Mater.~\textbf{4}, 104408 (2020)] based on a genetic algorithm are investigated. These monoclinic phases are found to be dynamically stable and have larger magnetization than the ThMn$_{12}$-type $R$Fe$_{12}$. Although they are thermodynamically unstable, the formation energies decrease significantly with increasing temperature, indicating the possibility of synthesizing these compounds at high temperatures.
We investigate the electronic structure and magnetic properties of four quaternary Heusler alloys ZrMnVZ and ZrCoFeZ (Z=Si, Ge) by using first-principle calculations. It is shown that ZrMnVSi, ZrMnVGe and ZrCoFeSi are half-metallic ferromagnets with considerable half-metallic gaps of 0.14, 0.18 and 0.22 eV, respectively. ZrCoFeGe is a nearly half-metallic, the changes of properties for this alloy under pressure is investigated, the spin polarization of this alloy is 98.99\% at equilibrium lattice constant.
The elastic energy of mixing for multi-component solid solutions is derived by generalizing Eshelby's sphere-in-hole model for binary alloys. By surveying the dependence of the elastic energy on chemical composition and lattice misfit, we propose a lattice strain coefficient λ*. Applying to several high-entropy alloys and superalloys, we found that most solid solution alloys are stable when λ*<0.16, analogous to the Hume-Rothery atomic-size rule for binary alloys. We also reveal that the polydispersity index δ, frequently used for describing strain in multi-component alloys, is directly relate…
First principles calculations were performed to study the structural, elastic, and bonding properties of hcp ZrxTi1-x binary alloy. The special quasi- random structure (SQS) method is employed to mimic the random hcp ZrxTi1-x alloy. It is found that Bulk modulus, B, Young's modulus, E, and shear modulus, G, exhibit decreasing trends as increasing the amount of Zr. A ductile behavior ZrxTi1-x is predicted in the whole composition range. In terms of Mulliken charge analisis, we found that the element Ti behaves much more electronegative than Zr in hcp ZrxTi1-x alloy, and the charge transfer of a…
Calculations using density functional theory (DFT) were performed to investigate the structural, dynamical, mechanical, electronic, magnetic, and thermoelectric properties of VTiRhZ (Z = Al, Ga, In) alloys. The most stable structure of these alloys was found to be the type-I configuration. Using GGA-PBE functional, VTiRhGa, and VTiRhIn alloys are predicted as half-metallic ferromagnets with a 100% spin-polarization and a total magnetic moment of 3μB, which is promising for spintronic applications. The thermoelectric properties and lattice thermal conductivity of VTiRhZ alloys were obtained usi…
Using first-principles calculations, we show that the conduction and valence band energies and their deformation potentials exhibit a non-negligible compositional bowing in strained ternary semiconductor alloys such as InGaAs. The electronic structure of these compounds has been calculated within the framework of local density approximation and hybrid functional approach for large cubic supercells and special quasi-random structures, which represent two kinds of model structures for random alloys. We find that the predicted bowing effect for the band energy deformation potentials is rather ins…
Building on the recent discovery of tough nanocrystalline Cu-Zr alloys with amorphous intergranular films, this paper investigates ternary nanocrystalline Cu-Zr-Hf alloys with a focus on understanding how alloy composition affects the formation of disordered complexions. Binary Cu-Zr and Cu-Hf alloys with similar initial grain sizes were also fabricated for comparison. The thermal stability of the nanocrystalline alloys was evaluated by annealing at 950 °C (>95% of the solidus temperatures), followed by detailed characterization of the grain boundary structure. All of the ternary alloys exhibi…
The quasistatic problem of shape memory alloys is reviewed within the phenomenological mechanics of solids without microphysics analysis. The assumption is that the temperature variation rate is small. Reissner's type of generalized variational principle is presented, and its mathematical justification is given for three-dimensional bodies made of shape memory materials.
An extensive set of first-principles density functional theory calculations have been performed to study the behaviour of He, C and N solutes in austenite, dilute Fe-Cr-Ni austenitic alloys and Ni in order to investigate their influence on the microstructural evolution of austenitic steel alloys under irradiation.
Crystalline Co$_{x}$Cu$_{1-x}$ alloys show interesting magnetic behavior over the entire concentration regime. We here present a fully self- consistent first principles electronic structure studies of the electronic structure and magnetic properties of the system.We present results for the variation of density of states, magnetic moment,spin susceptibility and Curie temperature.
Using special quasirandom structures (SQS's), we perform first-principles calculations studying the metastable bcc NiCu and FeCu alloys which occur in Fe-Cu-Ni alloy steels as precipitated second phase. The mixing enthalpies, density of state, and equilibrium lattice parameters of these alloys are reported. The results show that quasi-chemical approach and vegard rule can well predict the energetic and structural properties of FeCu alloys but fail to yield that of NiCu. The reason rests with the difference of bond energy variation with composition between NiCu and FeCu alloys induced by compet…
By performing first principles calculations combined with Boltzmann transport equations, we calculate the thermoelectric power factor (PF) of Bi-Sb-Te and Bi-Te-Se ternary alloys as a function of alloy composition ratio, carrier concentration, and temperature. The point defect formation energy calculations also perform to search potential n-type dopant candidates in ternaries.
The solubility and diffusivity of hydrogen in disordered Pd1-xCux alloys are investigated using a combination of first-principles calculations, a composition-dependent local cluster expansion (CDLCE) technique, and kinetic Monte Carlo simulations. We demonstrate that a linear CDCLE model can already accurately describe interstitial H in Pd1-xCux alloys over the entire composition range (0\leqx\leq1) with accuracy comparable to that of direct first-principles calculations. Our predicted H solubility and permeability results are in reasonable agreement with experimental measurements. The propose…
Bulk metallic glass forms when liquid metal alloys solidify without crystalization. In the search for Iron-based bulk glass-forming alloys of the metal-metalloid type (Fe-B- and Fe-C-based), crystals based on the structural prototype C6Cr23 often preempt the amorphous phase. Destabilizing this competing crystal structure could enhance glass-formability. We carry out first-principles total energy calculations of enthalpy of formation to identify third elements that can effectively destabilize C6Cr23. Yttrium appears optimal among transition metals, and rare earths also are suitable. Atomic size…
Practical applications of nanocrystalline metallic thin films are often limited by instabilities. In addition to grain growth, the thin film itself can become unstable and collapse into islands through solid-state dewetting. Selective alloying can improve nanocrystalline stability, but the impact of this approach on dewetting is not clear. In this study, two alloys that exhibit nanocrystalline thermal stability as ball milled powders are evaluated as thin films. While both alloys demonstrated dewetting behavior following annealing, the severity decreased in more dilute compositions. Ultimately…
We have used density functional theory to study the structural stability of surface alloys. Our systems consist of a single pseudomorphic layer of $M_xN_{1-x}$ on the Ru(0001) surface, where $M$ = Fe or Co, and $N$ = Pt, Au, Ag, Cd, or Pb. Several of the combinations studied by us display a preference for atomically mixed configurations over phase-segregated forms. We have also performed further {\it ab initio} calculations to obtain the parameters describing the elastic interactions between atoms in the alloy layer, including the effective atomic sizes at the surface. We find that while elast…
First-principles alloy theory is used to establish the $γ$-surface of Fe-Cr-Ni alloys as function of chemical composition and temperature. The theoretical stacking fault energy (SFE) versus chemistry and temperature trends agree well with experiments. Combining our results with the recent plasticity theory based on the $γ$-surface, the stacking fault formation is predicted to be the leading deformation mechanism for alloys with effective stacking fault energy below about 18 mJ m$^{-2}$. Alloys with SFE above this critical value show both twinning and full slip at room temperature and twinning …
In this letter, we investigate the magnetic properties, electronic structures, Slater-Pauling behaviours of some quanternary Heusler alloys with 4d and 3d transition metal elements. The energy levels of the minority-spin electronic band structures for all our calculated alloys are discussed. We have summaried 21 quanternary Heusler alloys. Our calculations indicate that the half metals ZrCoCrSi, ZrCoCrIn, ZrCoFeSi, ZrFeVAl, ZrFeVGa and NbFeCrAl have large spin-flip gaps.
The phase stability of fcc and bcc magnetic binary Fe-Cr, Fe-Ni, Cr-Ni alloys and ternary Fe-Cr-Ni alloys is investigated using a combination of density functional theory (DFT), Cluster Expansion (CE) and Magnetic Cluster Expansion (MCE). Energies, magnetic moments, and volumes of more than 500 alloy structures are evaluated using DFT, and the most stable magnetic configurations are compared with experimental data. Deviations from the Vegard law in fcc Fe-Cr-Ni alloys, associated with non-linear variation of atomic magnetic moments as functions of alloy composition, are observed. Accuracy of t…
We resolve issues that have plagued reliable prediction of relative phase stability for solid-solutions and compounds. Due to its commercially important phase diagram, we showcase Al-Li system because historically density-functional theory (DFT) results show large scatter and limited success in predicting the structural properties and stability of solid-solutions relative to ordered compounds. Using recent advances in an optimal basis-set representation of the topology of electronic charge density (and, hence, atomic size), we present DFT results that agree reasonably well with all known exper…
We present results of an extensive set of first-principles density functional theory calculations of point defect formation, binding and clustering energies in austenitic Fe with dilute concentrations of Cr and Ni solutes.
Mn doping in dilute III-V alloys has been examined as a route to enhance ferromagnetic stability. Strong valence band bowing is expected at the dilute limit, implying a strong modification of the ferromagnetic stability upon alloying, with even an increase in some cases. Using first principle electronic structure calculations we show that while codoping with a group V anion enhances the ferromagnetic stability in some cases when the effects of relaxation of the lattice are not considered, strong impurity scattering in the relaxed structure result in a reduction of the ferromagnetic stability.
Alloying is a successful strategy for tuning the phases and properties of two-dimensional (2D) transition metal dichalcogenides (TMDCs). To accelerate the synthesis of new TMDC alloys, we present a method for generating temperature-composition equilibrium phase diagrams by combining first-principles total energy calculations with thermodynamic solution models. This method is applied to three representative 2D TMDC alloys: an isostructural alloy, MoS2(1-x)Te2x, and two heterostructural alloys, Mo1-xWxTe2 and WS2(1-x)Te2x. We show that the mixing enthalpy of the entire composition range of these…
First-principles-based materials screening is systematically performed to discover new combinations of chemical elements possibly making shape-memory alloys (SMAs). The B2, D03, and L21 crystal structures are considered as the parent phases, and the 2H and 6M structures are considered as the martensitic phases. 3,384 binary and 3,243 ternary alloys (6,627 in total) with stoichiometric composition ratios are investigated by the materials screening in terms of energetic and dynamical stabilities of the martensitic phases as well as structural compatibility between the parent and the martensitic …
In order to clarify thermodynamic stability of Mg-based long-period stacking ordered (LPSO) structure, we systematically study energetic preference for alloys on multiple stacking with different composition for random mixing of constituent elements, Mg, Y, and Zn based on special quasirandom structure (SQS). Through calculation of formation free energy of SQS, Mg-Y-Zn alloy exhibits phase separation into Mg- and Y-Zn rich phase, which is consistent with previous theoretical studies. Bulk modulus of SQSs for multiple compositions, stacking sequences, and atomic configuratons ranges around 35 GP…
The present work examines the effect of alloying elements (denoted X) on the ideal shear strength for 26 dilute Ni-based alloys, Ni$_{11}$X, as determined by first-principles calculations of pure alias shear deformations. The variations in ideal shear strength are quantitatively explored with correlational analysis techniques, showing the importance of atomic properties such as size and electronegativity. The shear moduli of the alloys are affirmed to show a strong linear relationship with their ideal shear strengths, while the shear moduli of the individual alloying elements were not indicati…
The design of corrosion-resistant high entropy alloys (CR-HEAs) is challenging due to the alloys' virtually astrological composition space. To facilitate this, efficient and reliable high-throughput exploratory approaches are needed. Toward this end, the current work reports a first-principles-based approach exploiting the correlations between work function, surface energy, and corrosion resistance (i.e., work function and surface energy are, by definitions, proportional and inversely proportional to an alloy's inherent corrosion resistance, respectively). Two Bayesian CALPHAD models (or datab…
We studied the stable magnetic structure of ordered Nd$_{6}$Fe$_{14-x}$Ga$_x$ ($x = 0, 1)$ alloys, which appears in the grain-boundary (GB) phase of Nd-Fe-B permanent magnets, using first-principles techniques. Slight Ga doping ($x = 1$) was shown to contribute to the stabilization of an anti-ferromagnetic (AF) state, whereas the non-doped case ($x = 0$) was revealed to favor ferromagnetic state rather than AF state with a slight energy difference.
The Kubo-Greenwood equation, in combination with the first-principles Korringa-Kohn-Rostoker Coherent Potential Approximation (KKR-CPA) can be used to calculate the DC residual resistivity of random alloys at T = 0 K. We implemented this method in a multiple scattering theory based ab initio package, MuST, and applied it to the ab initio study of the residual resistivity of the high entropy alloy Al$_x$CoCrFeNi as a function of $x$. The calculated resistivities are compared with experimental data. We also predict the residual resistivity of refractory high entropy alloy MoNbTaV$_x$W. The calcu…
The structural, electronic, and magnetic properties of half-Heusler alloys CKMg and SiKMg are studied by using first-principle density functional theory. The calculations reveal the SiKMg alloy is a half-metallic ferromagnet with the magnetic moment of 1 $μ_B$ per formula unit at equilibrium lattice constant. The magnetic moment mainly originates from the strong spin-polarization of $p$ electrons of Si atom and partial involvement of $d$ electrons of K atom. The half-metallic gap is 0.105 eV. The robustness of half-metallic against the lattice constants for SiKMg is also calculated. CKMg alloy…
Employing first-principles calculations we study the structural, electronic and magnetic properties of the [Sc$_{1-x}$V$_x$]C and [Sc$_{1-x}$V$_x$]Si alloys. In their equilibrium rocksalt structure all alloys are non-magnetic. The zincblende and wurtzite structures are degenerated with respect to the total energy. For all concentrations the alloys in these lattice structures are half-metallic with the gap located in the spin-down band. The total spin moment follows the Slater-Pauling behavior varying linearly between the -1 $μ_B$ of the perfect ScC and ScSi alloys and the +1 $μ_B$ of the perfe…
We report on the structural, electronic and vibrational properties of the Ga$_{1-x}$In$_x$N alloy using virtual crystal approximation (VCA) from first-principles. A band gap bowing parameter of 3.85 eV is obtained with the TB09 functional. Phonon density of states shifts to lower frequency as the In content is increased. However, VCA ignores disorder effect and is therefore unable to describe the broadening of the phonon spectra upon alloying. The role of electron-phonon interaction in the temperature dependence of the band gap is also studied for GaN, InN and their alloy Ga$_{1-x}$In$_x$N. Th…
We have studied the problem of phase stability in Fe-Pt and Co-Pt alloy systems. We have used the orbital peeling technique in the conjunction of augmented space recursion based on the tight binding linear orbital method as the method for the calculation of pair interaction energies. In particular, we have generalized our earlier technique to take into account of magnetic effects for the cases where the magnetic transition is higher than the order disorder chemical transition temperature as in the case of Co$_3$Pt. Our theoretical results obtained within this framework successfully reproduce t…
We present an approach to calculate the atomic bulk solubility in binary alloys based on the statistical-thermodynamic theory of dilute lattice gas. The model considers all the appropriate ground states of the alloy and results in a simple Arrhenius-type temperature dependence determined by a {\it "low-solubility formation enthalpy"}. This quantity, directly obtainable from first-principle calculations, is defined as the composition derivative of the compound formation enthalpy with respect to nearby ground states. We apply the framework and calculate the solubility of the A specie in A-Ti all…
Using the full-potential local orbital minimum-basis method,the Ti2-based full-Heusler alloys are studied. The results show that these compounds exhibit a half-metallic behavior, however, in contrast to the conventional full-Heusler alloys,the full-Heusler alloys show a Slater-Pauling behavior and the total spin magnetic moment per unit cell(Mt) following the rule Mt=Zt-18. The origin of the gap in these half-metallic alloys are well understood. It is found that the half-metallic properties of Ti2-based compounds are insensitive to the lattice distortion and a fully spin polarization can be ob…
In order to shed light on the second harmonic generation (SHG) of new 2-D systems, first principles methods are used to calculate the second order susceptibility χ(2) for different types of layered alloys such as monolayers of transition metal dichalcogenide (TMD) alloys, TMD Haeckelite alloys, nanotubes of TMD alloys, hexagonal boron nitride (h-BN) systems which include BxNyCz alloys, BN and BNC2 nanotubes, BxNxCy Haeckelites and BN Schwarzites (porous BN). It is found that the tungsten based alloys possess higher χ(2) than Mo based at high photon energies, but at low energies, one type of Mo…
Light-weight alloys are essential pillars of transportation technologies. They also play a crucial role to achieve a more green and cost-effective aerospace technologies. Magnesium-lithium-aluminum (Mg-Li-Al) alloys are auspicious candidates due to their promising mechanical strengths at low densities. We herein present a systematic first-principles investigation of the Mg-Li-Al-based alloys to provide insights for designing ultra-strong light-weight alloys. Initial analysis indicates that the Mg-Li-Al mixtures are not thermally stabilized into random-solid solutions. Following this hint, the …
For (CoCrFeMn)$_{100-x}$Al$_{x}$ high-entropy alloys, we investigate the phase evolution with increasing Al-content (0 $\le$ x $\le$ 20 at.%). From first-principles theory, the Al-doping drives the alloy structurally from FCC to BCC separated by a narrow two-phase region (FCC+BCC), which is well supported by our experiments. We highlight the effect of Al-doping on the formation enthalpy and electronic structure of (CoCrFeMn)$_{100-x}$Al$_{x}$ alloys. As chemical short-range order (SRO) in multicomponent alloys indicates the nascent local order (and entropy changes), as well as expected low-tem…
High entropy alloys (HEAs) are multicomponent compounds whose high configurational entropy allows them to solidify into a single phase, with a simple crystal lattice structure. Some HEA's exhibit desirable properties, such as high specific strength, ductility, and corrosion resistance, while challenging the scientist to make confident predictions in the face of multiple competing phases. We demonstrate phase stability in the multicomponent alloy system of Cr-Mo-Nb-V, for which some of its binary subsystems are subject to phase separation and complex intermetallic-phase formation. Our first-pri…
We determine from first-principles the Curie temperature Tc for bulk Co in the hcp, fcc, bcc, and tetragonalized bct phases, for FeCo alloys, and for bcc and bct Fe. For bcc-Co, Tc=1420 K is predicted. This would be the highest Curie temperature among the Co phases, suggesting that bcc-Co/MgO/bcc-Co tunnel junctions offer high magnetoresistance ratios even at room temperature. The Curie temperatures are calculated by mapping ab initio results to a Heisenberg model, which is solved by a Monte Carlo method.
We develop a first-principles procedure for the individual evaluation of the intrinsic, side-jump, and skew-scattering contributions to the anomalous Hall conductivity $σ_{xy}$. This method is based on the different microscopic conductive processes of each origin of $σ_{xy}$ in the Kubo--Bastin formula. We also present an approach for implementing this scheme in the tight-binding linear muffin-tin orbital (TB-LMTO) method with the coherent potential approximation (CPA). The validity of this calculation method is demonstrated for disordered FePt and FePd alloys. We find that the estimated value…
Half-metallic Heusler alloys are amongst the most promising materials for future magnetoelectronic applications. We review some recent results on the electronic properties of these compounds. The origin of the gap in these half-metallic alloys and its connection to the magnetic properties are well understood. Changing the lattice parameter shifts slightly the Fermi level. Spin-orbit coupling induces states within the gap but the alloys keep a very high degree of spin-polarization at the Fermi level. Small degrees of doping and disorder as well as defects with low formation energy have little e…
In this work, we present the formation and characterization of a quaternary (pseudo ternary) icosahedral quasicrystal in Al65Ga5Pd17Mn13 alloy. The X ray diffraction and transmission electron microscopy confirmed the formation of icosahedral B2 type and O crystalline (orthorhombic structure) phases in as cast alloy. The icosahedral phase gets formed after annealing at 800 C for 60 hours. The formation of icosahedral phase in AlGaPdMn quaternary alloy by present technique has been studied for the first time. The Energy dispersive X-ray analysis investigations suggest the presence of Ga (5 at) i…
Alloying elements play an important role in the design of plasma facing materials with good comprehensive properties. Based on first-principles calculations, the stability of alloying element W and its interaction with vacancy defects in Ta-W alloys are studied. The results show that W tends to distribute dispersedly in Ta lattice, and is not likely to form precipitation even with the coexistence of vacancy. The aggregation behaviors of W and vacancy can be affected by their concentration competition. The increase of W atoms has a negative effect on the vacancy clustering, as well as delays th…
We present a first-principles atomistic thermodynamics framework to describe the structure, composition and segregation profile of an alloy surface in contact with a (reactive) environment. The method is illustrated with the application to a Ag3Pd(111) surface in an oxygen atmosphere, and we analyze trends in segregation, adsorption and surface free energies. We observe a wide range of oxygen adsorption energies on the various alloy surface configurations, including binding that is stronger than on a Pd(111) surface and weaker than that on a Ag(111) surface. This and the consideration of even …
The rapid degradation of the functional properties of many Ti-based alloys is due to the precipitation of the $ω$ phase. In the conventional high-temperature shape memory alloy Ti-Ta the formation of this phase compromises completely the shape memory effect and high (>100°C) transformation temperatures cannot be mantained during cycling. A solution to this problem is the addition of other elements to form Ti-Ta-X alloys, which often modifies the transformation temperatures; due to the largely unexplored space of possible compositions, very few elements are known to stabilize the shape memory e…
We present and compare three distinct atomistic models -- based on first principles and semi-empirical approaches -- of the structural and electronic properties of Ge$_{1-x}$Sn$_{x}$ alloys. Density functional theory calculations incorporating Heyd-Scuseria-Ernzerhof (HSE) and modified Becke-Johnson (mBJ) exchange-correlation functionals are used to perform structural relaxation and electronic structure calculations for a series of Ge$_{1-x}$Sn$_{x}$ alloy supercells. Based on HSE calculations, a semi-empirical valence force field (VFF) potential and $sp^{3}s^{\ast}$ tight-binding (TB) Hamilto…
The origin of the phase stability of 18$R$ Mg-Zn-Y alloys with a long-period stacking order (LPSO) is studied using first-principles calculations. We calculate the heat of formation as a function of the number of Zn vacancies to discuss the role of Zn atoms. The calculated convex hull indicates that the Zn atoms in the LPSO alloys are stable even if they number about half of the Y atoms. The bonding state with Zn $p$ orbitals leads to the stability of the LPSO structure because the partial density of states of Mg nearest to the solute cluster forms a valley structure.
The electronic structure, magnetism and phase stability of Pt2-xMn1+xGa(x=0, 0.25, 0.5, 0.75, 1) alloys are studied by first-principle calculations. The calculations reveal that a potential magnetic martensitic transformation can be expected in all the series. In addition, a large magnetic-field-induced strain is likely to appear in Pt2-xMn1+xGa(x=0, 0.25, 0.75, 1) alloys. The electronic structure calculations indicate that the tetragonal phase is stabilized upon the distortion because of the pseudogap formation at the Fermi Level. The magnetic structure is also investigated and the total magn…
Using a first-principles cluster expansion, we shed light on the solid-state phase diagram and structure of the recently discovered high-performance thermoelectrics, Pb-Ag-Sb-Te alloys. The observed nanoscale inhomogeneities are shown to be coherent precipitates of ordered (Ag)m(Sb)n(Te)m+n phases, such as AgSbTe2, all immiscible with rocksalt PbTe. The solubility is high for PbTe in AgSbTe2 and low for (Ag,Sb)Te in PbTe (8% vs. 0.6% at 850 K). The differences in the phonon spectra of PbTe and AgSbTe2 confirm that the inhomogeneities enhance the thermoelectric performance.
