Crystal symmetry based selection rules for anharmonic phonon-phonon scattering from a group theory formalism

Anharmonic phonon-phonon scattering serves a critical role in heat conduction in solids. Previous studies have identified many selection rules for possible phonon-phonon scattering channels imposed by phonon energy and momentum conservation conditions and crystal symmetry. However, the crystal-symmetry-based selection rules have mostly been ad hoc so far in selected materials, and a general formalism that can summarize known selection rules and lead to new ones in any given crystal is still lacking. In this work, we apply a general formalism for symmetry-based scattering selection rules based on the group theory to anharmonic phonon-phonon scatterings, which can reproduce known selection rules and guide the discovery of new selection rules between phonon branches imposed by the crystal symmetry. We apply this formalism to analyze the phonon-phonon scattering selection rules imposed by the in-plane symmetry of graphene, and demonstrate the significant impact of symmetry-breaking strain on the lattice thermal conductivity. Our work quantifies the critical influence of the crystal symmetry on the lattice thermal conductivity in solids and suggests routes to engineer heat conduction by tuning the crystal symmetry.

Figure 1

The crystal structure and the Brillouin zone of grpahene. Three phonon modes with wave vectors along the $\mathrm{\Gamma }$-M direction in the Brillouin zone of graphene are labeled ($\mathbf{q},{\mathbf{q}}^{\prime },{\mathbf{q}}^{\prime \prime }$).

Figure 2

Typical atomic vibrational patterns (eigenvectors) for phonon modes along the $\mathrm{\Gamma }$-M path and their corresponding irreducible representation as listed in Table 2. The crosses and dots represent directions into and out from the paper plane.

Figure 3

(a) Three phonon modes with wave vectors along the $\mathrm{\Gamma }$-K path, whose representations are listed in Table 3. (b) One example of three other phonons possessing the same symmetry properties as those along the $\mathrm{\Gamma }$-K path.

Figure 4

Typical atomic vibrational patterns (eigenvectors) for phonon modes along the $\mathrm{\Gamma }$-K path and their corresponding irreducible representation as listed in Table 3. The crosses and dots represent directions into and out from the paper plane.

Figure 5

The crystal structures and unit cells of (a) graphene, and (b) “skewed” graphene.

Figure 6

Phonon-phonon scattering rates for both graphene and “skewed graphene” decomposed into scattering channels. (a) and (b) show the results for scattering channels that are forbidden by in-plane symmetry in graphene, while (c) and (d) show the results for allowed scattering channels in graphene.

Figure 7

(a) Calculated phonon dispersion relations of graphene and “skewed” graphene. (b) Calculated lattice thermal conductivity of graphene and “skewed” graphene at different temperatures.

Figure 8

Typical atomic vibrational patterns (eigenvectors) for phonon modes along the 1D ${\mathrm{Ba}}_3\mathrm{N}$ chains and their corresponding irreducible representation as listed in Table 4.

Figure 9

An illustration of the graphene crystal structure, where the three atoms involved in the force constants used in Eqs. (B2) and (B3) are labeled. The vibrational modes of the three participating phonons are also illustrated.

Figure 10

An illustration of Brillouin zones of (a) simple cubic structure (b) cuboid structure and wave vectors discussed in Appendix pp3.