7 We report the role of long-and short-range order on the thermal conductivity and mode re-8 laxation times of a model Si 0.5 Ge 0.5 alloy using molecular dynamics simulation. All interactions 9 used the Stillinger-Weber potential and the Si and Ge atoms differed only by their mass. The 10 simulated alloys were generated using a Monte Carlo approach to decouple the short-range order 11 from the long-range order. The thermal conductivity is almost entirely determined by the alloy's 12

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... ghbor short-range order. Changes to the mode relaxation times between ∼3 and ∼6 13 THz upon short-range ordering, and the observed f −2 power law trend, suggests that short-range 14 ordering reduces the anharmonic scattering rate of low frequency modes. The trend of thermal 15 conductivity with short-range order may be transferred to real Si 0.5 Ge 0.5 and other semiconductor 16 alloys to the extent that scattering from mass disorder dominates their thermal conductivities. 1 Altering the composition of Si 1−x Ge x and other alloys is one route for engineering their 19 thermal conductivity, k. 1,2 In addition to numerous experimental studies, 3-6 there have been 20 many recent computational studies of the thermal properties of SiGe using classical molecu-21 lar dynamics 7-12 and density functional theory. 13,14 Studies have focused on the dependence 22 occupies the correct lattice site with reference to the ordered structure, and the probability is 33 duly normalized by the atom's concentration in the alloy. 19 The Warren-Cowley short-range 34 order parameters give the probability of an atom having the correct neighbor in a certain 35 neighbor shell with reference to the ordered structure. 20 We define a set of short-range order 36 parameters, S i , where i indexes the neighbor shells, as the square-root of the Warren-Cowley 37 short-range order parameters so that lim i→∞ S i = L. 20,23 Assuming an equimolar binary alloy 38 for which each lattice site is eligible for a disordering substitution simplifies the expressions 39 for L and S i . Then, L ≡ |R−W |/N, where R (W ) is the number of atoms occupying the right 40 (wrong) lattice site, and N is the total number of atoms. 23 Similarly and N i are the numbers of right, wrong, and total neighbor pairs in neighbor 42 shell i, respectively. 23 43 Structures with varying L and S 1 can be generated by a Monte Carlo approach. For a 44 binary alloy with the aforementioned simplifying assumptions, the seed structure is formed 45 from the definition of L and the compositional constraint. Beginning from a zincblende 46 reference structure, 24 we randomly selected and exchanged n Si atoms and n Ge atoms, 47 The fit yields τ = 1/(4πΓ) and the anharmonic linear frequency, f 0 , for each normal mode, 63 ν. At thermal equilibrium, the coefficient C is guaranteed to be 1 2 k B T from the equipartition 64 principle. The duration of data collection, t f , ought to be much greater than the maximum 65 τ (ν) for the material. 33Q is the Fourier transform of the normal mode velocity coordinate, 66q : 34Q = t f 0q exp(−2πif t)dt. 35 Due to the sharpness of the peak in |Q| 2 , we increased the 67 weighting near the base by taking the decimal logarithm of each side of Eq. 1. Only points 68 above 0.104 meV/THz (1 amu·Å 2 ·ps −1 ) were used in the fit. Because the global minimum 69 of the root-mean-square of the residuals lay in a narrow well surrounded by local maxima, 70 we found it necessary to do a grid search before regression, with 31 points linearly spaced 71 between ±0.1 THz of the peak frequency and 31 points logarithmically spaced between Γ of 72 10 −5 and 10 −1 ps −1 . 73 All simulations were performed using LAMMPS 36 with a time step of 0.5 fs. The 74 zincblende lattice constant was set to 5.43Å, and the Si and Ge atoms only differed in their 75 masses: 28.09 and 72.64 amu respectively. The Stillinger-Weber potential 37 was used for all 76 interactions since the effect of strain on k is small compared to that of mass disorder. 7 The 77 system was equilibrated at 300 K for 1 ns in a canonical ensemble enforced by a Nosé-Hoover 78 thermostat 38,39 with a coupling time of 2 ps. The system was then run for an additional 79 1 ns in a microcanonical ensemble before data were collected for 36·2 17 time steps (2.36 ns), 80 144 the phonon eigenvectors caused by short-range ordering. As S 1 increases, the eigenvectors 145 approach those of the zincblende crystal. While high frequency modes might significantly 146 contribute to k in the limit S 1 →1, most of the increase for the ordering range studied here 147 is caused by a reduction in the anharmonic scattering of the low frequency modes. The 148 reduction in anharmonic scattering may be due to fewer states that satisfy momentum 149 and energy selection rules, a reduction in the scattering cross-section, or both mechanisms. 150 The same trend of k with disorder was observed Garg et al., who saw a reduction in k 151 with greater disorder, when they went from a virtual crystal to an explicitly disordered 152 supercell. 14 They found that the change in k was due to altered mode relaxation times, 153 caused by a modification of the mode eigenvectors. 154 The findings may be cautiously generalized to other simulated and real alloys provided 155 the thermal conductivities of their disordered states arise primarily from the same mech-156 anisms as those found in the present model of Si 0.5 Ge 0.5 , namely the scattering of lattice 157 vibrations by mass disorder. That k depends almost solely on S 1 has implications for the 158 characterization and theoretical modeling of such alloys. When examining an alloy with 159 6 the purpose of understanding its thermal conductivity or predicting it, a characterization 160 technique sensitive to the short-range order must be used, e.g. diffuse X-ray scattering. 46 161 Similarly, future efforts to theoretically model thermal transport in ordered alloys should 162 focus on the short-range order or its effect on anharmonic phonon scattering. 163 In summary, we performed molecular dynamics simulations of a Si 0.5 Ge 0.5 alloy, repre-164 senting a model semiconductor alloy, and calculated the thermal conductivity as it depends 165 on the long-and short-range ordering. We found that the bulk thermal conductivity depends 166 almost wholly on the short-range order of the alloy for a fixed composition. Relaxation time 167 calculations support this dependence. Changes in the character of the mode relaxation times 168 upon ordering imply that the corresponding increase in thermal conductivity is caused by a 169 reduction in disorder-induced anharmonic phonon scattering.