Abstract
The concept of quasiparticles forms the theoretical basis of our microscopic understanding of emergent phenomena associated with quantummechanical manybody interactions. However, the quasiparticle theory in disordered materials has proven difficult, resulting in the predominance of meanfield solutions. Here, we report firstprinciples phonon calculations and inelastic Xray and neutronscattering measurements on equiatomic alloys (NiCo, NiFe, AgPd, and NiFeCo) with forceconstant dominant disorder—confronting a key 50yearold assumption in the Hamiltonian of all meanfield quasiparticle solutions for offdiagonal disorder. Our results have revealed the presence of a large, and heretofore unrecognized, impact of local chemical environments on the distribution of the speciespairresolved forceconstant disorder that can dominate phonon scattering. This discovery not only identifies a critical analysis issue that has broad implications for other elementary excitations, such as magnons and skyrmions in magnetic alloys, but also provides an important tool for the design of materials with ultralow thermal conductivities.
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Introduction
Quasiparticle elementary excitations, including electron quasiparticles, phonons, magnons, plasmons, excitons, etc., along with skyrmions,^{1,2} Majorana fermions,^{3} and their mutual interactions, represent remarkably successful theoretical descriptions of emergent phenomena associated with quantummechanical manybody interactions. For example, the seminal discovery of the quantized thermal Hall effect in the spin liquid candidate RuCl_{3}^{4,5} can be explained by the coupling of quasiparticles (phonons and chiral Majorana edge modes) in Kitaev’s nonAbelian spin liquid.^{6,7} As such, quasiparticle physics has provided a microscopic understanding of the underlying science of phenomena ranging from the novel properties of quantum materials to the electronic and vibrationalbased physics in solids. In contrast to fully ordered crystals that can be described within the Bloch theorem, the broken translational symmetry in alloys associated with substitutional chemical disorder has long challenged the development of analogous robust quasiparticle theories of configurationally averaged observables that inherently lead to the finite quasiparticle lifetimes measured experimentally—even at zero temperature. Therefore, describing both the spectral dispersion and lifetimes of quasiparticles in disordered materials is fundamentally important for understanding the underlying science of condensed matter and the design of technological materials with targeted properties.
Historically, a landmark advance in treating compositional disorder was the coherent potential approximation (CPA).^{8,9,10} As an analytic “singlesite” theory of the configurationally averaged observables that restores the translational invariance of a selfconsistently determined effective medium, the CPA formalism was initially designed to treat onsite (sitediagonal) fluctuations associated with the different alloying elements—different onsite orbital energies for the case of electrons, different atomic masses for the case of phonons. Effective as the original CPA was, it was quickly realized that extending the singlesite CPA to include fluctuations in twosite quantities—intersite hopping integrals for electrons, pairwise force constants for phonons—is nontrivial.^{11,12,13,14} This problem is particularly acute for phonons where forceconstant (offdiagonal) disorder is crucial and entangled with the mass (diagonal) disorder, as illustrated in Fig. 1a. For electron quasiparticles, it was possible to account for both diagonal and offdiagonal fluctuations by reformulating CPA within the language of multiple scattering theory^{15} and ab initio density functional theory.^{16,17,18} For phonon quasiparticles, no similar “simple” transformation of the underlying theory has been possible.
For phonons, the formulation of itinerant CPA (ICPA) by Ghosh et al.^{19} provided a substantial advance in analytic meanfield theories of phonon quasiparticle physics in disordered materials based on the augmented space formalism of Mookerjee.^{20} The theory was demonstrated initially for the cases of NiPd and NiPt that contain large mass disorder as well as forceconstant disorder,^{19,21} and has been applied subsequently to multiple binary alloys with similarly dominant mass disorder.^{22,23,24,25} Remarkably, with the exception of a limited study of facecentered cubic (fcc) Ni_{88}Cr_{12} and bcc Fe_{53}Cr_{47} using an augmented space recursion (ASR) approach,^{26,27} neither systematic theoretical calculations of phonon dispersions and linewidths nor the corresponding experimental measurements have been performed for materials exhibiting strong forceconstant disorder with small mass disorder. This unsatisfactory situation regarding experimental validation of the theory is further complicated by the form of the underlying Hamiltonian upon which it is based. The Hamiltonian for the ICPA and all previous meanfield theories (including the CPA) for offdiagonal as well as diagonal disorder make use of the simplifying assumption that forceconstant disorder between individual AA, BB, and ABtype pairs can be approximated by their global average in AB binary alloys.^{13,28} Similar to lattice vibrations (phonons), the magnetic excitations (e.g., magnons) of disordered alloys are also challenging to describe due to sitediagonal (local moment) and offdiagonal (exchange interactions) disorder, as depicted in Fig. 1b. The crucial exchange interactions between individual AA, BB, and ABtype pairs are usually approximated by their ensemble averages as calculated from linear response theory.^{29} Unaccountably, although this 50yearold assumption within the Hamiltonian is at the core of the analytic meanfield formalisms, it has so far not been questioned.
In this work, we address this knowledge gap for the first time through a combined firstprinciples theory and experimental measurement investigation of the phonon quasiparticle physics (dispersions and linewidths) of concentrated disordered alloys, NiCo, NiFe, AgPd, and NiFeCo, with strong forceconstant disorder but minimal mass disorder. Remarkably, we have discovered from ab initio supercell phononunfolding (SPU) simulations^{30,31,32} and their comparison with ICPA and experimental measurements, that forceconstant fluctuations considering each individual AA, BB, and ABtype species pairs far surpass that of the usual global average forceconstant fluctuations. Moreover, we have shown that the source of the enhanced fluctuations is the inherent random variations in local chemical environments surrounding individual AA, BB, and ABtype pairs. Accordingly, we have demonstrated that the longstanding approximation of replacing individualpair forceconstant fluctuations with their global averages in the Hamiltonian of quasiparticle meanfield theories for disordered materials must be reconsidered.
Results and discussion
Phonon properties of NiCo and NiFe
Phonon dispersion and linewidth measurements for NiCo and NiFe samples were performed at room temperature by using high resolution inelastic Xray scattering (IXS) and inelastic neutronscattering (INS) measurements along the [001] and [011] reciprocal lattice directions. In addition, density functional theory (DFT) was employed to calculate the force constants of NiCo, NiFe, AgPd, and NiFeCo alloys with chemical disorder modeled by the supercell method. Phonon spectral functions and the corresponding linewidths for equiatomic NiCo, NiFe, AgPd, and NiFeCo alloys were extracted using both the ICPA^{19} and the SPU approach.^{32} We note that the analytical meanfield ICPA approach employs the symmetryaveraged forceconstant matrix for a pair of atoms, i.e., one single matrix for each of Φ^{AA}, Φ^{BB}, and Φ^{AB} in binary A–B alloys, while the nonanalytical SPU approach uses the full forceconstant matrix of the supercell. The SPU maps phonon bands within the Brillouin zone (BZ) of large supercells to those of the primitive BZ. While the phonon dispersion curves for an Natom supercell form 3N continuous (sharp) phonon dispersion bands, when unfolded into the primitive BZ they map into three effective acoustic bands with finitewidth distributions of discrete eigenvalues. This is illustrated in Fig. 1c, d. More experimental and calculation details can be found in the “Methods” section.
Figure 2 provides a colorcontour overview of the phonon dispersions and phonon linewidths as obtained by SPU and ICPA calculations, along with the corresponding IXS and INS measurements for disordered solidsolution NiCo and NiFe, respectively. These plots present results as a function of the wavevector q = [ζ_{x}, ζ_{y}, ζ_{z}] in reciprocal lattice units (rlu) of 2π/a_{0} (a_{0} = cubic lattice parameter) for longitudinal (LA) and transverse (TA) acoustic phonons. The color contours in Fig. 2 give the simulated phonon spectral functions overlaid by discrete phonon dispersion measurements (black solid circles). We note general agreement between firstprinciples phonon dispersions (SPU and ICPA) and the corresponding measurements for both NiCo and NiFe, albeit small discrepancies occuring near the [0, 0, ζ] BZ boundary X point, and for TA_{1} phonons along the [0, ζ, ζ] direction. Due to different normalizations used in the SPU and ICPA approaches, the magnitude of spectral functions is given in arbitrary units, and is therefore not directly comparable (as indicated in the different color bars). While this makes it difficult to compare visualizations, the different normalizations do not affect the peak position and disorder broadening of the spectral function, which are the focus of this work.
More important for disordered materials, though, are the phonon linewidths that provide a direct probe of forceconstant disorder, when mass disorder is minimal. Figure 3 shows plots of the full width at half maximum (FWHM) linewidths, Δ, of the SPU and ICPA spectral functions associated with the color contours in Fig. 2. These were obtained by fitting the spectral functions with Lorentzian profiles using negligible (0.05 meV) resolution. Similarly, the linewidths corresponding to the measurements in Fig. 2 (open circles with error bars denoting uncertainties) were obtained by fitting the (constant q) IXS and INS spectral data with Lorentzian profiles convoluted with the qresolution functions for the respective IXS and INS instruments. The IXS instrument resolution was determined directly by measurements of the elastic linewidth (1.6 meV), and the qdependent INS resolution was determined by RESLIB using the Popovici method.^{33}
We observe first of all, significant phonon broadening for both NiCo and NiFe in Fig. 3, especially for q > 0.7 rlu. This broadening implies that forceconstant disorder alone (in the absence of mass disorder) can cause significant phonon scattering, and hence, lead to shorter phonon lifetimes corresponding to substantial linewidth broadening. Furthermore, the measured linewidths are not only larger for NiFe than for NiCo, but dramatically so for q < ~0.5 rlu. The linewidths for NiCo tend to fall below ~0.5 meV for q < 0.5 rlu, while the linewidths for NiFe range from 1.5 to 2 meV down to q as low as ~0.2 rlu, the importance of which will be discussed below.
Focusing on the simulations, we observe in Fig. 3a that (within the measurement uncertainties) both the SPU and ICPA simulations agree with the measured linewidths for NiCo, albeit ICPA tends to overestimate those of LA phonons in the [0,0,ζ] direction near the BZ boundary, and that the SPU simulations tend to underestimate the linewidths in the same region. In Fig. 3b, it is demonstrated that the linewidth simulations for NiFe using SPU are in excellent agreement with the measured linewidths for all wavevectors sampled. Conversely, the ICPA linewidths significantly underestimate the measured linewidths for NiFe—lower by more than a factor of two for the LA phonons and lower by factors of five to ten for the TA phonons.
Decomposition of disorderinduced phonon linewidths
To provide physical insight into the impact of forceconstant disorder on phonon linewidths, we show by SPU simulations in Fig. 4a, b the total and pairresolved spectral functions for selected phonon modes for NiCo and NiFe. For small q, [0, 0, 0.25] rlu, the individualpair resolved and the total spectral functions for NiCo are quite sharp and symmetric, while for NiFe both the pairresolved and the total spectral functions are more than twice as broad. For larger q, [0, 0, 0.75] rlu, the spectral functions for NiCo and NiFe are both broader and tend to be less symmetric, particularly for NiFe. For q close to the Xpoint [0, 0, 0.95] rlu, neither the NiCo nor the NiFe total spectral functions exhibit symmetric Lorentzian behavior. Instead, they comprise a sharp peak and a diffuse lowenergy tail. The extended tails (together with the reduced intensity) on the lowenergy sides for both NiCo and NiFe are analogous to those from ICPA investigations of alloys possessing both large mass and forceconstant disorder.^{19,22,23,24,25} The tails result from the fact that the softerlikespecies bonds—such as Co–Co bonds in NiCo and Ni–Ni bonds in NiFe (shown later)—do not vibrate resonantly at the high frequencies that occur for the other bonds at large wavevectors. Physically, the speciespairdependent bond strengths can be understood within the context of the local electronic structure, the strength of electron hybridization, and the occupations of the bonding and antibonding states, as discussed in Supplementary Note1.3 and Supplementary Fig.4.
Comparison between the SPUcalculated and the measured linewidths for NiFe and NiCo provides a direct validation of the SPU approach for predicting vibrational properties in disordered binary alloys where forceconstant disorder dominates. This success now raises the question as to why the ICPA so significantly underestimates the linewidths in NiFe, given that the ICPA approach^{19} was specifically designed to incorporate forceconstant disorder within an analytic meanfield theory.
Importance of offdiagonal randomness
To examine offdiagonal disorder, we have used SPU calculations to test the fundamental approximation within the present ICPA Hamiltonian for AB binary alloys,^{19} which includes only the interspecies disorder between the “global averaged” force constants associated with individual species pairs (<Φ^{AA}>, <Φ^{AB}>, <Φ^{BB}> ). That is, ICPA averages over the actual distribution of force constants for individual species pairs, i.e., intraspecies forceconstant disorder. While this approximation (in general use for meanfield quasiparticle theories) greatly simplifies the ICPA formalism and has underpinned attempts to generalize the original CPA to include offdiagonal randomness,^{13,28} it tacitly assumes that the local atomic environment surrounding interacting pairs does not impact the force constants for individual atomic pairs significantly. In contrast, however, we show in Fig. 5a, b that the impact of local chemical environments on the speciespair force constants is significant for NiCo and dramatically so for NiFe. Only the Φ_{xy} component of the forceconstant matrix is shown in the figure, with the other components shown in Supplementary Note 1.1. Figure 5a, b gives the individual (nearest neighbor) speciespair force constants for NiCo and NiFe as a function of the fractional distortion, δr, from the perfect crystal bond lengths. The individual force constants vary in two important aspects: (1) they tend to become softer for increasing bond lengths δr according to the expected bond length proportionality,^{23,34,35} and (2) the force constants vary strongly in strength (in a statistical manner) for each bond length δr due to a direct effect of random local chemical environment. We note that the variation of the bond lengths, as a result of atomic relaxation, is influenced by the local chemical environment. Therefore, the expected bond length proportionality represents an indirect effect of the local environment on the force constants. The histograms along the left vertical axes of Fig. 5a, b show the statistical distributions of the Φ^{αβ} for each species pair (α, β denote species), underscoring graphically the wider distributions of force constants for NiFe than for NiCo. Therefore, the local chemical environment clearly plays a critical role in determining speciespair force constants and their fluctuations, which in turn determine the linewidth behavior. In Fig. 5a, b the average force constants <Φ^{αβ}> (denoted by the yellow symbols) that represent the forceconstant input to the ICPA Hamiltonian are also shown.
The thick gray SPUA curves in Fig. 3 explicitly demonstrate the consequence of using averaged speciespair force constants. SPUA replaces the individual AA, AB, and BB speciespair force constants of the full SPU with their averages (<Φ^{AA}>, < Φ^{AB}>, <Φ^{BB}>), as detailed in Supplementary Note 1.2. Therefore, SPUA is a supercell phononunfolding calculation that mimics the forceconstant averaging of the ICPA Hamiltonian. We note that the SPUA, the full SPU, and the ICPA linewidth simulations are quite similar for NiCo. While the forceconstantaveraged SPUA calculations for NiFe depart strongly from those of SPU, they reproduce almost exactly the (anomalously low) ICPA linewidth results. These observations demonstrate conclusively that it is indeed the intraspecies forceconstant averaging associated with the present ICPA Hamiltonian that is responsible for the breakdown in the ICPA linewidth predictions for NiFe. Conversely, the agreement between the SPUA and the ICPA results indicates directly that the present ICPA properly accounts for the effects of offdiagonal disorder when it is limited to interspecies (averaged) randomness. It is important to note that the underestimated ICPA linewidths are due solely to the oversimplified Hamiltonian rather than the ICPA methodology itself.
This limitation was not apparent in previous ICPA alloy studies^{19,22,23,24,25,36} because the impact of the large (100–150%) mass disorder on phonon dispersion curves tended to mask the impact of forceconstant disorder on phonon linewidths. In any case, the results presented above indicate that the incorporation of (individual) intraspecies forceconstant fluctuations in the ICPA phonon Hamiltonian would lead to an accurate description of phonons in disordered materials. While outside the scope of this study, in principle the inclusion of individualpair forceconstant disorder in ICPA could be achieved by discretizing intraspecies forceconstant distributions through the introduction of additional, appropriately weighted, ICPA components. This is similar to the discretization of the distribution of the relevant random variables (atomic displacements, spin disorder) by introducing more CPA components for configurational averaging in the electronic structure methodology.^{37}
Phonon linewidths for other alloys with forceconstant dominant disorder
To demonstrate that the importance of retaining full forceconstant distributions in NiFe is not an isolated case, we performed SPU and SPUA phonon linewidth calculations for the equiatomic 4d binary AgPd and the 3d ternary NiFeCo alloys, both of which have negligible mass disorder (few percent). Figure 6a compares the SPUA, SPU, and ICPA linewidths for AgPd, demonstrating that the linewidth behaviors of SPUA and ICPA are again similar, both of which differ strongly from the full SPU simulations for all phonon modes. Qualitatively different from NiCo and NiFe, the speciesaveraged force constants in AgPd are nearly equal (see Fig. 5d), and the intraspecies forceconstant fluctuations vary drastically depending on the bond lengths (see Supplementary Fig. 1). We emphasize that the force constants in AgPd depend mainly on the bond lengths and are relatively insensitive to the direct impact of the chemical environment. This was demonstrated by evaluating the speciespairdependent forceconstant distribution for the unrelaxed structure with all atoms fixed at their ideal lattice sites, i.e., making all bond lengths identical. This resulted in narrow forceconstant distributions in which the magnitude of the force constants depends only on the bond length, as seen in Supplementary Fig. 2. Moreover, even though the linewidth magnitudes predicted by the full SPU simulations are much smaller than those for either NiCo or NiFe, the fractional disagreement of the speciesaveraged SPUA linewidth is similar to that for NiFe. As a more complex example we show in Fig. 6b SPU and SPUA linewidth simulations for the ternary 3d equiatomic alloy, NiFeCo, for which ICPA is not yet formulated. The large discrepancy between the SPU and SPUA linewidth simulations predicts that the present ICPA Hamiltonian will again break down for linewidth predictions of NiFeCo, analogous to those observed for NiFe. The SPU and SPUA phonon spectral functions of AgPd and NiFeCo are shown in Supplementary Fig. 3.
Analyzing further, the magnitude and the wavevector, q, dependence of the simulated and measured linewidths for NiCo and NiFe (see Fig. 3) provides direct quantitative insight into the forceconstant disorderinduced phonon linewidth Δ averaged over a length scale l = 2π/q. In particular, linewidths of Δ = 1 meV correspond to phonon lifetimes of 0.66 ps, and q = 0.25 rlu corresponds to l ~15 Å. Accordingly, at low q = 0.25 rlu, the phonon broadening is expected to be small (such as that observed for LA phonons for NiCo in Fig. 3) since longerwavelength phonons are expected to be less sensitive to local force constant disorder. Surprisingly, however, for NiFe at q = [0, 0, 0.25] rlu—corresponding to l ~15 Å—the broadening of the LA mode is substantial and more than five times that of NiCo. This result is consistent with the observations in Fig. 5 of the remarkably broader forceconstant distributions in NiFe than in NiCo. Moreover, by comparing SPU linewidths for NiFe with SPUA simulations in Fig. 3, the intraspecies forceconstant disorder (included in SPU but not in SPUA) plays the dominant role in the NiFe phonon broadening at low q. At large q (near the X point), where forceconstant disorder is resolved on the scale of the fcc cubic cell, the measured and simulated phonon broadening for both NiCo and NiFe tend to increase (see Fig. 3), albeit not necessarily monotonically.
The results of this work have farreaching implications. The discovery that local chemical environments play a dominant role in determining the impact of forceconstant fluctuations in disordered alloys provides both fundamental insight on the underlying science of disordered alloys and technological insight toward potential atomiclevel engineering—by tuning the types of alloying elements—to manipulate the microscopic vibrational physics in alloys. That is, the combination of theory and experimental measurements has demonstrated that the ab initio supercell approach combined with band unfolding provides an accurate description of the vibrational physics (phonon dispersion and lifetimes) of alloys with forceconstant dominant disorder. Hence, SPUcalculated and experimentally measured linewidths can be used as predictors of phonon scattering and latticemediated macroscopic thermal transport in the design of novel materials such as that for new thermoelectric materials.
Finally, we emphasize that the physics of quasiparticles in disordered materials revealed in this study is applicable far beyond the scope of phonon excitations. For example, similar effects can be expected in magnetic alloys with broad, localenvironment induced distributions of exchange interactions. This was demonstrated in NiFeCoCr random solid solutions,^{38} where the broad distributions of species–pairwise Heisenberg exchange interactions calculated for this alloy could not be adequately represented by speciespair averages. In addition, the impact of random (offdiagonal) interatomic exchange (e.g., Heisenberg exchange, Dzyaloshinskii–Moriya interaction^{39,40}) can be important for magnon dispersion and lifetimes, for lowenergy control and magnon excitations in magnon spintronics,^{41} and for the helical period and dynamics of skyrmions in disordered helical magnets.^{42} Accordingly, the physics of forceconstant disorder revealed here within the context of phonons is transferrable not only to random alloys with both mass and forceconstant disorder, but also to the physics of other types of quasiparticles in disordered materials in which offdiagonal disorder is dominant, e.g., magnons or skyrmions in magnetic alloys. Thus, the potential for dominant, localenvironment disorder effects, such as those reported for phonons in this study, provides a compelling case for further investigation across the rich variety of quasiparticle physics.
Methods
Experimental
Phonon dispersion and linewidth measurements were performed on singlephase, singlecrystal, solidsolution NiCo and NiFe samples (with negligibly small mass differences of 0.4% and 4.8%, respectively) grown in an optical floating zone furnace. The measurements were performed at room temperature using highresolution inelastic Xray scattering (IXS) and inelastic neutron scattering (INS) measurements along the [001] and [011] reciprocal lattice directions. For NiCo, IXS measurements were performed on the HERIX spectrometer at beamline 30IDC at the Advanced Photon Source at Argonne National Laboratory, and for NiFe INS measurements, were performed on the HB3 tripleaxis spectrometer at the High Flux Isotope Reactor at Oak Ridge National Laboratory (ORNL).
Computational
To explore the effect of forceconstant disorder on the phonon properties of considered equiatomic alloys, we employed the projectoraugmented wave method (PAW)^{43} implemented in the Vienna ab initio Simulation Package (VASP)^{44} to evaluate the force constants using the standard supercell technique with the help of PHONOPY software.^{45} Spinpolarized calculations were performed in the spin colinear state except for AgPd. Exchange and correlation were treated using the generalized gradient approximation (GGA) parameterized by Perdew, Burke, and Ernzehof.^{46} The electron wave functions were expanded in a planewave basis set with cutoff kinetic energy set to 400 eV for all calculations. Moreover, the disordered local environment was simulated explicitly using supercells constructed from special quasirandom structure (SQS)^{47} without considering shortrange order. A Γcentered 4 × 4 × 4 (3 × 3 × 3) Monkhorst–Pack kpoint mesh^{48} was used for Brillouin zone (BZ) integrations in multiple 64atom (108atom) SQSs. The equilibrium atomic positions were obtained by optimizing the cell volume, and all internal degrees of freedom until the Hellmann–Feynman force on each atom are lower than 5 meV/Å. The cubic shape of the supercell is maintained while relaxing. The lattice parameters (a_{0}) of NiCo, NiFe, NiFeCo, and AgPd are 3.52, 3.57, 3.55, and 4.03 Å, respectively. This is consistent with previous experiments.^{49} Force constants of the supercells were calculated based on the optimized theoretical structures. The influence of spin noncolinearity on the force constants of ferromagnetic NiFe, NiFeCo, and NiCo alloys at finite temperature is neglected here. This is justified by the fact that all the ferromagnetic alloys considered here have high Curie temperatures as compared with room temperature,^{49} and therefore, the spin noncolinearity and its effect on the force constants are likely negligible. The phonon spectral functions of the random alloys were calculated using the itinerant coherent potential approximation (ICPA) and supercell phononunfolding (SPU) methods, as discussed separately later. For direct comparison with experiments, theoretical phonon spectral functions were convoluted with the E and Q resolution of the measurements. Then the same procedure that was used to extract linewidths from the experimental data was used to extract the theoretical linewidths from the convoluted theoretical phonon spectra.
The itinerant coherent potential approximation (ICPA)
The itinerant coherent potential approximation (ICPA)^{19} is a Green’s functionbased formalism to calculate the phonon spectra in substitutionally disordered alloys. It extends the singlesite coherent potential approximation (CPA) by considering scattering from more than one site embedded in an effective medium. Hence, all relevant disorders, i.e., the mass, the force constant, and the environmental disorders, are appropriately addressed within the ICPA. While calculating the configurationaveraged quantities, the method also ensures selfconsistency, sitetranslational invariance, and analyticity of the Green’s function. The phonon frequencies and the disorderinduced linewidths were determined, respectively, from the peaks and from the widths of the coherent scattering structure factor defined as
where λ is the normalmode branch index, c_{s} is the coherent scattering length for species s, and \(\frac{1}{\pi }{\mathrm{Im}} \, {\ll} \, G_\lambda ^{s,s^{\prime} }(\vec q,\omega ^2) \, {\gg}\) is the configurationaveraged and thermodynamicsaveraged spectral function associated with the species pair s, \(s^{\prime}\).
Supercell phononunfolding (SPU) method
In the supercell method for alloys, a particular finitesize supercell with defects breaks the space group symmetry and leads to a shrinking BZ in reciprocal space. To recover the phonon spectra within the BZ of the primitive cell, stateoftheart bandunfolding methods have been developed for electronic problems^{50,51} as well as for phonon problems.^{30,31,32} Here we use the unfolding program developed by Ikeda et al. to carry out the phonon band unfolding. According to the formula derived by Ikeda et al.,^{32} the phonon spectral function A(k, ω) at the wavevector k and frequency ω is given as \(A\left( {{\boldsymbol{k}},\omega } \right) = \mathop {\sum }\nolimits_J  \, \hat p(k)v(K,J)^2\delta [\omega  \omega (K,J)]\). In this expression \(\hat p(k)\) is the projection operator for wavevector k in the primitive BZ, v(K, J) is the eigenvector of the dynamical matrix of the supercell for phonon mode J at the K point (defined in the reduced BZ), and ω(K, J) is the corresponding eigenvalue. See ref. ^{32} for further details.
Finally, the SPUA calculations were performed for a single 256atom supercell to capture aspects of the configurational averaging inherent to ICPA, whereas each of the full SPU phonon spectral function calculations presented (except for AgPd) represent averages over several (i.e., six for 64 and three for 108 atoms) supercells. The SPU phonon spectral function for AgPd was obtained from a single 108atom supercell.
Connection between ICPA and SPU
The connection of the SPU with the ICPA can be established through the postulated ansatz that configurational averaging for an infinite random system can be approximated by manually averaging the observables (PDOS and spectral functions) over many finite SQS cells. A particular advantage of the SPU approach, however, is that it couples straightforwardly with density functional theory (DFT)computed force constants without the need for any special averaging procedure. Hence, SPU has the required capability to provide complete information on the impact of forceconstant disorder, i.e., the fluctuations between different atomic pairs as well as their full environment dependence. However, we note that so far, neither ICPA nor SPU has been tested for random alloys dominated by forceconstant disorder.
Data availability
The authors declare that the data supporting this study are available from the corresponding author upon request.
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Acknowledgements
This work was supported as part of the Energy Dissipation and Defect Evolution (EDDE), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, and Basic Energy Sciences under contract number DEAC0500OR22725. This research used resources of the Advanced Photon Source, a DOE Office of Science User Facility operated by Argonne National Laboratory under Contract No. DEAC0206CH11357. Beamline support by Ayman Said of the Advanced Photon Source and Songxue Chi of the High Flux Isotope Reactor is acknowledged. This research also used resources at the High Flux Isotope Reactor, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory. This research used resources of Oak Ridge National Laboratory’s Compute and Data Environment for Sciences (CADES) and the Oak Ridge Leadership Computing Facility, which is a DOE office of Science User Facility supported under Contract DEAC0500OR22725. Work at MPI was supported by Deutsche Forschungsgemeinschaft (Germany) within the priority programme SPP 1599. L.L. and T.B. acknowledge support from the U.S. Department of Energy, Office of Science, Basic Energy Sciences, and Materials Sciences and Engineering Division. B.D. and S.M. acknowledge Dr. Fritz Körmann and Dr. Yuji Ikeda for fruitful discussions. S.M. is grateful to Dr. Xin Huang and Dr. Lisha Fan for graphic support.
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S.M., R.O. and B.D. contributed equally to this work. R.O. performed the experimental measurements; H.B. and K.J. grew the single crystals and characterized the samples; S.M. carried out the firstprinciples calculations and phononunfolding simulations; B.D. performed ICPA simulations; S.M., B.C.L. and G.M.S. wrote the initial draft and all authors participated in discussions and contributed materially in finalizing the paper.
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Mu, S., Olsen, R.J., Dutta, B. et al. Unfolding the complexity of phonon quasiparticle physics in disordered materials. npj Comput Mater 6, 4 (2020). https://doi.org/10.1038/s4152402002713
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DOI: https://doi.org/10.1038/s4152402002713
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