The atomistic Green’s function (25, 26) approach models the heat transfer through a finite-size device that is coupled to semi-infinite reservoirs on each end. The dynamical matrix of the entire system can be written asEmbedded Image(2)where HL, HR, and HD are the dynamical matrices of the left reservoir, right reservoir, and device region, respectively. τLD is the dynamical matrix that couples the left reservoir to the device, and τDR is the dynamical matrix that couples the device to the right reservoir. This formalism is only valid when the reservoirs are uncoupled from one another. For semi-infinite reservoirs, the dynamical matrix of Eq. 2 is infinitely large. To make this problem tractable, the interactions between the device and the reservoirs were encoded in self-energy terms Embedded Image and Embedded Image where gL,R are the surface Green’s functions obtained from a real space decimation method (49). The Green’s function of the device region were computed as GD = (ω2HD − ΣL − ΣR)− 1,where ω2 is the square of the phonon eigenfrequency. Defining Embedded Image, the transmission function for a given frequency ω and transverse wave vector k can be written asEmbedded Image(3)noting that the frequency and wave vector arguments on the right-hand side are implicit, and the results presented in Fig. 3 include all wave vectors except Fig. 3C. Defining Embedded Image as the normalized sum over Nk points in the Brillouin zone, the thermal conductance at a given temperature T can be expressed asEmbedded Image(4)where AD is the area of the device’s cross section and f(ω, T) is the Bose-Einstein distribution function.

The computation of Eq. 4 only requires the subspace of Green’s function matrix elements that connect the right reservoir and the left reservoir. We denoted the set of matrix elements of this subspace as G1N. G1N was recursively computed from the Dyson’s equation. Since G1N corresponds to the probability amplitude for a phonon to propagate across the entire device, the localization length lloc can be determined from (27)Embedded Image(5)for a system of N periods of length l. Because of computational limitations, localization lengths were extracted from an average of 10 configurations of 300-period (~1700 nm) devices, which also correspond to the thickest samples measured.

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