Green’s function simulation

ML M. N. Luckyanova JM J. Mendoza HL H. Lu BS B. Song SH S. Huang JZ J. Zhou ML M. Li YD Y. Dong HZ H. Zhou JG J. Garlow LW L. Wu BK B. J. Kirby AG A. J. Grutter AP A. A. Puretzky YZ Y. Zhu MD M. S. Dresselhaus AG A. Gossard GC G. Chen

This protocol is extracted from research article:

Phonon localization in heat conduction

**
Sci Adv**,
Dec 21, 2018;
DOI:
10.1126/sciadv.aat9460

Phonon localization in heat conduction

Procedure

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 as*H*_{L}, *H*_{R}, and *H*_{D} are the dynamical matrices of the left reservoir, right reservoir, and device region, respectively. τ_{LD} is the dynamical matrix that couples the left reservoir to the device, and τ_{DR} 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 *g*_{L,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 *G*_{D} = (ω^{2} − *H*_{D} − Σ_{L} − Σ_{R})^{− 1},where ω^{2} is the square of the phonon eigenfrequency. Defining *k* can be written as*N*_{k} points in the Brillouin zone, the thermal conductance at a given temperature *T* can be expressed as*A*_{D} 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 *G*_{1N}. *G*_{1N} was recursively computed from the Dyson’s equation. Since *G*_{1N} corresponds to the probability amplitude for a phonon to propagate across the entire device, the localization length *l*_{loc} can be determined from (*27*)*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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