Quantum mechanical modeling of anharmonic phonon-phonon scattering in nanostructures

Abstract

The coherent quantum effect has become increasingly important in the heat dissipation bottleneck of semiconductor nanoelectronics with the characteristic size recently shrinking down to a few nanometers scale. However, the quantum mechanical model remains elusive for anharmonic phonon-phonon scattering in extremely small nanostructures with broken translational symmetry. It is a long-term challenging task to correctly simulate quantum heat transport including anharmonic scattering at a scale relevant to practical applications. In this article, we present a clarified theoretical formulation of anharmonic phonon nonequilibrium Green's function (NEGF) formalism for both one-and three-dimensional nanostructures, through a diagrammatic perturbation expansion and an introduction of Fourier's representation to both harmonic and anharmonic terms. A parallelized computational framework with first-principle force constants input is developed for large-scale quantum heat transport simulation. Some crucial approximations in numerical implementation are investigated to ensure the balance between numerical accuracy and efficiency. A quantitative validation is demonstrated for the anharmonic phonon NEGF formalism and computational framework by modeling cross-plane heat transport through a silicon thin film. The phonon-phonon scattering is shown to appreciably enhance the thermal resistance or conductance in extremely small homogeneous or heterogeneous thin films. The present methodology provides a robust platform for the device quantum thermal modeling, as well as a study on the transition from coherent to incoherent heat transport in nanophononic crystals. This work thus paves the way to understand and to manipulate heat conduction via the wave nature of phonons.

Anharmonic phonon-phonon scattering at the interface between two solids by nonequilibrium Green's function formalism

“…The anharmonicity is included only in the interface device region, whereas the contacts are harmonic. The harmonic or anharmonic contacts have negligible influence on the thermal transport properties, as verified in previous works [30,31]. In principle, we have to consider a sufficiently long device region to determine the TBC more seriously by extracting the heat flux and temperature jump across the interface, as done in a previous empirical probe approach [25,26].…”

“…In this work, we present a theoretical model to extract and decompose the spectral energy exchange due to phononphonon scattering at an interface from the anharmonic NEGF formalism developed in our recent work [30]. As a result, we demonstrate a clear understanding of how lattice anharmonicity contributes to phonon mode conversion and energy exchange at the interface.…”

“…1. For simplicity, we assume that Si and Ge have the same lattice and force constants, with an only atomic mass difference, which is an acceptable approximation as usually done in previous NEGF studies [22,27,30]. The realistic lattice constants (5.43 Å and 5.66 Å) of Si and Ge have a tiny (∼4%) difference, which would introduce strain and slightly reduce the TBC.…”

“…The second-and third-order force constants are computed by first-principle method, with the details given in Ref. [30]. The anharmonicity is included only in the interface device region, whereas the contacts are harmonic.…”

“…The retarded Green's function of the interface region is computed in matrix notation as [30][31][32][33]…”