Representative HCACFs for Si (SW potential) normalized by their value at t = 0 and plotted on a logarithmic y-axis. The dashed lines are fits using Eq. (5) . (a) HCACF for 1000 K (η = 8 uc, D = 60 ns) with the short-time behavior shown in the inset. For ps the magnitude of the HCACF is comparable to the noise level calculated from the cross-correlation function. (b) Comparison with the HCACF for 500 K (η = 8 uc, D = 120 ns) with a similar noise level and ps.
Accumulated thermal conductivity as a function of the upper integration limit u at (a) 1000 K and (b) 500 K for the HCACFs of Fig. 2 . The numerical integrals of the five separate simulations each lasting 12 ns (at 1000 K) or 24 ns (at 500 K) (thin lines) and their mean value (thick line) are compared to the integral of the triple exponential fit to the HCACF (black line). The vertical lines indicate . The insets show the long-time behavior, which is dominated by noise.
(a)–(d) Dependence of the fitted relaxation times τ i and their weights A i on the total simulation duration D. (e) Dependence of κ on the total simulation duration D, broken down into the contributions from the three relaxation processes. Data are obtained using 5 independent simulations with T = 500 K and η = 5 uc for the SW potential.
Dependence of the fit parameters A i and τ i and the resulting value of κ on the size η of the simulation cell. (For η = 2 uc the parameter values are off the chosen scales.) For 500 K all the simulations are based on D ⩾ 80 ns while for 1000 K they use D = 60 ns.
Components of the autocorrelation function for the SW potential (normalized by the total HCACF). (a) and (b) Autocorrelation function of the arbitrary term S 3α and the corresponding contribution κ2 to the thermal conductivity. (c) and (d) The same for the convection term S c. (e) Contributions of two-body and three-body terms to the HCACF for long times (main panel) and short times (inset). (Data for (a)–(d) are for 500 K, while for (e) T = 1000 K, η = 5 uc, and D = 100 ns.)
T-dependence of (a) the thermal conductivity, (b) the intermediate decay time τ2, and (c) the slow decay time τ3 for the SW potential from the Green–Kubo method. Lines are power-law fits to the data. The contributions of these two decay processes to κ, which are proportional to A 2τ2 and A 3τ3, are also shown.
Evolution with time of (a) the temperature profile T(z, t) and (b) its gradient ∂T/∂z calculated from the heat diffusion equation. The source and sink have width . (c) Time-dependence of the gradient linearized by a least-squares fit in the range (thick line), compared to the approximation of Eq. (6) (black line). Reprinted with permission from Related Article(s): P. C. Howell, J. Comput. Theor. Nanosci.8, 2129 (Year: 2011)10.1166/jctn.2011.1935
. Copyright 2011, American Scientific Publishers.
Analysis of temperature profiles in the direct method. The plots on the left are for a small simulation of long duration (T = 600 K, uc, Tersoff) and those on the right are for a large simulation of relatively brief duration (T = 900 K, uc, SW). (a) and (b) Steady-state profile averaged after the decay of the initial transient (0.05 ns and 3 ns, respectively) with fits to the linear regions ( and 0.12). (c) and (d) Gradient of profiles to the left of the sink averaged over successive 0.04 ns windows. The solid line shows the one-parameter fit Eq. (6) to the gradient evolution. (e) and (f) The same for the temperature profiles to the right of the sink. (g) and (h) Dependence of κ calculated from the one-parameter fit on the size of the fitted region. If w excl is too large the fitted region is so small that the gradient is not robustly calculated.
(a) Finite-size scaling for Tersoff at 600 K and SW at 900 K, using . The straight lines show a fit with Eq. (7) . (b) Sensitivity of the extrapolated κ∞ to the size of the fitted region of the temperature profile.
Dependence of thermal conductivity κ on temperature T for the SW potential from the Green–Kubo and direct methods, and for the Tersoff potential from the direct method. Experimental data are taken from Madelung. 50 Lines show power-law fits.
(a) Fourier power spectrum of the HCACF (T = 500 K, η = 5). The inset shows the oscillations in the HCACF. (b) Dependence of scaled oscillation frequencies faη on system size η. The symbol size is proportional to the oscillation amplitude. Lines indicate v/ζ from Ref. 49 for acoustic phonons in the directions ΓX (ζ = 1), ΓK (ζ = √2), ΓL (ζ = √3) with transverse (T) or longitudinal (L) polarization.
Thermal expansion coefficient α. Experimental data are from Madelung. 50
Thermal conductivity (in W m−1 K−1) at various temperatures for the SW potential from the Green–Kubo and direct methods, and for the Tersoff potential from the direct method. β is the fitted value of the exponent in the power-law κ ∝ T −β.
Literature values for the thermal conductivity (in W m−1 K−1) of Si calculated for the SW potential using molecular dynamics. The other methods (explained in the text) are ALD-BTE: anharmonic lattice dynamics plus Boltzmann transport equation; MD-BTE: molecular dynamics plus Boltzmann transport equation; Evans: Evans' homogeneous field method. In all cases the phonon statistics are classical.
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