^{1}and Alan J. H. McGaughey

^{1,a)}

### Abstract

The virtual crystal (VC) approximation for mass disorder is evaluated by examining two model alloy systems: Lennard-Jones argon and Stillinger-Weber silicon. In both material systems, the perfect crystal is alloyed with a heavier mass species up to equal concentration. The analysis is performed using molecular dynamics simulations and lattice dynamics calculations. Mode frequencies and lifetimes are first calculated by treating the disorder explicitly and under the VC approximation, with differences found in the high-concentration alloys at high frequencies. Notably, the lifetimes of high-frequency modes are underpredicted using the VC approximation, a result we attribute to the neglect of higher-order terms in the model used to include point-defect scattering. The mode properties are then used to predict thermal conductivity under the VC approximation. For the Lennard-Jones alloys, where high-frequency modes make a significant contribution to thermal conductivity, the high-frequency lifetime underprediction leads to an underprediction of thermal conductivity compared to predictions from the Green-Kubo method, where no assumptions about the thermal transport are required. Based on observations of a minimum mode diffusivity, we propose a correction that brings the VC approximation thermal conductivities into better agreement with the Green-Kubo values. For the Stillinger-Weber alloys, where the thermal conductivity is dominated by low-frequency modes, the high-frequency lifetime underprediction does not affect the thermal conductivity prediction and reasonable agreement is found with the Green-Kubo values.

This work was supported by AFOSR Award FA95501010098 and by a grant of computer time from the DOD High Performance Computing Modernization Program at the US Army Engineer Research and Development Center. We thank Davide Donadio, Jivtesh Garg, Asad Hasan, Ankit Jain, Craig Maloney, and Zhiting Tian for their helpful discussions.

I. INTRODUCTION

II. THEORETICAL AND COMPUTATIONAL FRAMEWORK

A. Thermal conductivity prediction

B. Virtual crystal approximation

C. Calculation and simulation details

III. VIBRATIONAL MODE PROPERTIES IN ALLOYS

A. Density of states (DOS)

B. Dispersion and group velocity

C. Lifetimes

1. From VC-NMD and Gamma-NMD

2. From VC-ALD

D. Diffusivities

E. Discussion

IV. THERMAL CONDUCTIVITY PREDICTION

V. SW SILICON

VI. SUMMARY

### Key Topics

- Thermal conductivity
- 88.0
- Silicon
- 37.0
- Phonons
- 17.0
- Density functional theory
- 14.0
- Thermal properties
- 14.0

## Figures

(a) Explicitly disordered alloy supercell of silicon and “heavy” silicon ([100] direction into the page). 50 (b) Equivalent VC supercell with one averaged mass. The sphere size represents increasing mass only, no bond disorder is considered. The 8-atom conventional cubic unit cell is shown in(b).

(a) Explicitly disordered alloy supercell of silicon and “heavy” silicon ([100] direction into the page). 50 (b) Equivalent VC supercell with one averaged mass. The sphere size represents increasing mass only, no bond disorder is considered. The 8-atom conventional cubic unit cell is shown in(b).

Vibrational DOS for LJ alloys calculated using the VC approximation and an explicitly disordered supercell (labeled Gamma) for concentrations of (a) 0.05, (b) 0.15, and (c) 0.5. VC and Gamma show similar low-frequency behavior for all concentrations. For increasing concentrations, the frequencies of both VC and Gamma decrease, while the high frequency DOS for Gamma spreads and reaches to a higher maximum frequency because of the explicit disorder. The supercells are of size (6912 atoms).

Vibrational DOS for LJ alloys calculated using the VC approximation and an explicitly disordered supercell (labeled Gamma) for concentrations of (a) 0.05, (b) 0.15, and (c) 0.5. VC and Gamma show similar low-frequency behavior for all concentrations. For increasing concentrations, the frequencies of both VC and Gamma decrease, while the high frequency DOS for Gamma spreads and reaches to a higher maximum frequency because of the explicit disorder. The supercells are of size (6912 atoms).

Left and right panels: The structure factor for longitudinal (SL ) and transverse (ST ) polarizations along high-symmetry directions of the mass disordered LJ argon supercells ( , c = 0.05, 0.5). Center panels: The VC predicted dispersion curves (solid lines) agree well with the locations of the peaks in SL and ST (data points). The wavenumber axis in the center panel is normalized by the maximum value of the wavenumber in the given direction.

Left and right panels: The structure factor for longitudinal (SL ) and transverse (ST ) polarizations along high-symmetry directions of the mass disordered LJ argon supercells ( , c = 0.05, 0.5). Center panels: The VC predicted dispersion curves (solid lines) agree well with the locations of the peaks in SL and ST (data points). The wavenumber axis in the center panel is normalized by the maximum value of the wavenumber in the given direction.

Lifetimes predicted using VC-NMD and Gamma-NMD from MD simulations of (a) perfect LJ argon and (b)–(d) mass-disordered LJ alloys for . and scalings are observed at low to mid frequencies. For both VC-NMD and Gamma-NMD, most mode lifetimes are greater than the Ioffe-Regel limit of . 74 While there is more scatter in the Gamma-NMD data (see Sec. ??? ), the lifetime magnitudes and trends agree well, an important consideration when comparing the VC-NMD and VC-ALD lifetimes in Fig. 5(a) .

Lifetimes predicted using VC-NMD and Gamma-NMD from MD simulations of (a) perfect LJ argon and (b)–(d) mass-disordered LJ alloys for . and scalings are observed at low to mid frequencies. For both VC-NMD and Gamma-NMD, most mode lifetimes are greater than the Ioffe-Regel limit of . 74 While there is more scatter in the Gamma-NMD data (see Sec. ??? ), the lifetime magnitudes and trends agree well, an important consideration when comparing the VC-NMD and VC-ALD lifetimes in Fig. 5(a) .

(a) Predicted lifetimes using VC-NMD and VC-ALD for LJ argon (T = 10 K, , and c = 0.05). (b) Mode diffusivities compared to the high-scatter limit, DHS [Eq. (18) ], and IR limit, DIR [Eq. (19) ]. VC-NMD and VC-ALD predict a large number of high-frequency modes with . (c) Thermal conductivity frequency spectrum, which peaks at high frequency, in contrast to SW silicon [(Fig. 8(c) ].

(a) Predicted lifetimes using VC-NMD and VC-ALD for LJ argon (T = 10 K, , and c = 0.05). (b) Mode diffusivities compared to the high-scatter limit, DHS [Eq. (18) ], and IR limit, DIR [Eq. (19) ]. VC-NMD and VC-ALD predict a large number of high-frequency modes with . (c) Thermal conductivity frequency spectrum, which peaks at high frequency, in contrast to SW silicon [(Fig. 8(c) ].

AF theory predictions of disordered mode diffusivities for LJ argon alloy and amorphous phases. The amorphous phase is well-described by a mode-independent diffusivity DHS [Eq. (18) ]. The system size for the alloy is (6912 atoms), and the amorphous phase has 6912 atoms.

AF theory predictions of disordered mode diffusivities for LJ argon alloy and amorphous phases. The amorphous phase is well-described by a mode-independent diffusivity DHS [Eq. (18) ]. The system size for the alloy is (6912 atoms), and the amorphous phase has 6912 atoms.

Thermal conductivity predictions for LJ argon and alloys at T = 10 K using the VC-NMD, VC-ALD, and GK methods. The high-scatter thermal conductivity prediction kHS [Eq. (3) ] and the high-scatter adjusted VC-NMD* and VC-ALD* are also plotted.

Thermal conductivity predictions for LJ argon and alloys at T = 10 K using the VC-NMD, VC-ALD, and GK methods. The high-scatter thermal conductivity prediction kHS [Eq. (3) ] and the high-scatter adjusted VC-NMD* and VC-ALD* are also plotted.

(a) Predicted lifetimes using VC-NMD and VC-ALD for SW silicon (T = 300 K, , and c = 0.05). (b) Mode diffusivities compared to the high-scatter limit, DHS [Eq. (18) ], and the IR limit, DIR [Eq. (19) ]. VC-NMD and VC-ALD predict a large number of high-frequency modes with , as seen in the LJ argon alloys [Fig. 5(b) ]. (c) Thermal conductivity frequency spectra, which peak at low frequency, in contrast to LJ argon [Fig. 5(c) ].

(a) Predicted lifetimes using VC-NMD and VC-ALD for SW silicon (T = 300 K, , and c = 0.05). (b) Mode diffusivities compared to the high-scatter limit, DHS [Eq. (18) ], and the IR limit, DIR [Eq. (19) ]. VC-NMD and VC-ALD predict a large number of high-frequency modes with , as seen in the LJ argon alloys [Fig. 5(b) ]. (c) Thermal conductivity frequency spectra, which peak at low frequency, in contrast to LJ argon [Fig. 5(c) ].

Thermal conductivity predictions for SW silicon and alloys at a temperature of 300 K using the VC-ALD and GK methods. The high-scatter thermal conductivity prediction kHS is also plotted. The adjusted VC-ALD* is not shown since it differs by less than one percent compared to VC-ALD.

Thermal conductivity predictions for SW silicon and alloys at a temperature of 300 K using the VC-ALD and GK methods. The high-scatter thermal conductivity prediction kHS is also plotted. The adjusted VC-ALD* is not shown since it differs by less than one percent compared to VC-ALD.

The normal mode kinetic energy, , of two modes (A and B) at wavevector [0.25 0 0] calculated using VC-NMD for a mass disordered LJ FCC supercell ( and c = 0.5) is shown in the main figure. The VC dispersion-predicted peaks are labeled by . The inset shows the same mode's energy [kinetic (KE) and total (TE)] autocorrelation functions. Note the additional oscillation effects in the KE and TE autocorrelation functions for Mode B, which are due to the two peaks in . A mode lifetime can be extracted unambiguously using the integral of the TE autocorrelation function [Eq. (11) in Sec. ??? ].

The normal mode kinetic energy, , of two modes (A and B) at wavevector [0.25 0 0] calculated using VC-NMD for a mass disordered LJ FCC supercell ( and c = 0.5) is shown in the main figure. The VC dispersion-predicted peaks are labeled by . The inset shows the same mode's energy [kinetic (KE) and total (TE)] autocorrelation functions. Note the additional oscillation effects in the KE and TE autocorrelation functions for Mode B, which are due to the two peaks in . A mode lifetime can be extracted unambiguously using the integral of the TE autocorrelation function [Eq. (11) in Sec. ??? ].

## Tables

Thermal conductivity predictions using the VC-NMD, VC-ALD, and GK methods. For LJ argon alloys, the bulk extrapolation is used for all three methods. For SW silicon alloys, only VC-ALD and GK can be used to extrapolate a bulk thermal conductivity (see Sec. IV ). For VC-NMD and GK, the uncertainties are estimated by omitting independent simulations from the ensemble averaging (see Sec. II C ). For VC-ALD, the uncertainties are estimated by omitting extrapolation points used for Eq. (21) .

Thermal conductivity predictions using the VC-NMD, VC-ALD, and GK methods. For LJ argon alloys, the bulk extrapolation is used for all three methods. For SW silicon alloys, only VC-ALD and GK can be used to extrapolate a bulk thermal conductivity (see Sec. IV ). For VC-NMD and GK, the uncertainties are estimated by omitting independent simulations from the ensemble averaging (see Sec. II C ). For VC-ALD, the uncertainties are estimated by omitting extrapolation points used for Eq. (21) .

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