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First-Principles Theory of Five- and Six-Phonon Scatterings

Yi Xia

Abstract

Higher-order phonon scatterings beyond fourth order remain largely unexplored despite their potential importance in strongly anharmonic materials at elevated temperatures. We develop a theoretical formalism for first-principles calculation of five- and six-phonon scatterings using Green's function techniques based on a diagrammatic formalism, and systematically investigate multi-phonon interactions in Si, MgO, and BaO from room temperature to near melting points. Our calculations reveal dramatically different material-dependent behaviors: while five- and six-phonon processes remain negligible in Si even at high temperatures, they become increasingly important in MgO near its melting point (3100~K) and in BaO at intermediate temperatures (1200~K). Most remarkably, five- and six-phonon scatterings surpass three- and four-phonon scattering intensity in BaO near its melting point (2100~K), reducing lattice thermal conductivity by over 50\%. We demonstrate that the strength of higher-order interactions is primarily governed by interatomic force constants, with BaO exhibiting five- and six-phonon scattering rates over one order of magnitude stronger than MgO despite identical crystal structures, due to large scattering phase space arising from softened harmonic interactions. Our work provides theoretical insights into the lattice dynamics and thermal transport in strongly anharmonic materials and at elevated temperatures.

First-Principles Theory of Five- and Six-Phonon Scatterings

Abstract

Higher-order phonon scatterings beyond fourth order remain largely unexplored despite their potential importance in strongly anharmonic materials at elevated temperatures. We develop a theoretical formalism for first-principles calculation of five- and six-phonon scatterings using Green's function techniques based on a diagrammatic formalism, and systematically investigate multi-phonon interactions in Si, MgO, and BaO from room temperature to near melting points. Our calculations reveal dramatically different material-dependent behaviors: while five- and six-phonon processes remain negligible in Si even at high temperatures, they become increasingly important in MgO near its melting point (3100~K) and in BaO at intermediate temperatures (1200~K). Most remarkably, five- and six-phonon scatterings surpass three- and four-phonon scattering intensity in BaO near its melting point (2100~K), reducing lattice thermal conductivity by over 50\%. We demonstrate that the strength of higher-order interactions is primarily governed by interatomic force constants, with BaO exhibiting five- and six-phonon scattering rates over one order of magnitude stronger than MgO despite identical crystal structures, due to large scattering phase space arising from softened harmonic interactions. Our work provides theoretical insights into the lattice dynamics and thermal transport in strongly anharmonic materials and at elevated temperatures.

Paper Structure

This paper contains 2 equations, 4 figures.

Figures (4)

  • Figure 1: Feynman diagrams illustrating (a) three-phonon (3ph), (b) four-phonon (4ph), (c) five-phonon (5ph), and (d) six-phonon (6ph) interactions. Panels (e) and (f) depict schematic representations of 5ph and 6ph scattering processes, respectively, highlighting various combinations of incoming and outgoing phonons.
  • Figure 2: (a) Calculated phonon mode-resolved scattering rates associated with three-phonon (3ph), four-phonon (4ph), five-phonon (5ph), and six-phonon (6ph) processes in silicon at 300 K. (b) Same as (a), but for MgO. (c) Same as (a), but for BaO. (d)–(f) Corresponding scattering rates at temperatures near the melting points: 1600 K for Si, 3100 K for MgO, and 2100 K for BaO. The two distinct groups of modes in BaO, observed in the scattering rates, are found to correspond to acoustic and optical modes, respectively.
  • Figure 3: (a) Temperature dependencies of three-phonon (3ph), four-phonon (4ph), five-phonon (5ph), and six-phonon (6ph) scattering rates in Si. (b) Mode-averaged 3ph, 4ph, 5ph, 6ph scattering rate ratios of BaO and MgO at 300 K. (c) Interatomic force constants-estimated 3ph, 4ph, 5ph, 6ph scattering rate ratios of BaO and MgO at 300 K. (d) Scattering phase space (as defined in Eqs. (\ref{['eq:g5ph']}) and (\ref{['eq:g6ph']}) with $|V| = 1$) associated with 5ph and 6ph scattering processes in Si, MgO, and BaO at 300 K, respectively.
  • Figure 4: Temperature-dependent lattice thermal conductivities calculated with inclusion of up to three-phonon (3ph), four-phonon (4ph), five-phonon (5ph), and six-phonon (6ph) scattering processes for Si, MgO, and BaO, respectively. Experimental measurements are shown as empty symbols (squares from Ref. [abeles1962thermal]; circles from Ref. [glassbrenner1964thermal]; diamonds from Ref. [touloukion1970thermophysical]; triangles from Ref. [cahill1998thermal]; pentagons from Ref. [hofmeister2014thermal]).