Exact Mirror-Parity Selection Rule for Electron-Phonon Coupling in Two-Dimensional Crystals

Abstract

The electron--phonon (e-ph) interaction underlies conventional superconductivity and plays a central role in transport and other many-body phenomena in solids. Here, we derive a symmetry selection rule for e-ph scattering in two-dimensional crystals possessing a horizontal mirror plane, $σ_h$. For electronic states of mirror parity $s_n(\mathbf{k})$ and $s_m(\mathbf{k}+\mathbf{q})$, and a phonon mode of parity $χ_ν(\mathbf{q})$, we show that the e-ph matrix element $g_{mn,ν}(\mathbf{k},\mathbf{q})$ vanishes identically unless $s_n(\mathbf{k})\,χ_ν(\mathbf{q})\, s_m(\mathbf{k}+\mathbf{q})=+1$. The rule follows from the covariance of the phonon-induced deformation potential under $σ_h$, constitutes the $C_s$ case of the general group-theoretical selection rule for matrix elements, and partitions the e-ph interaction throughout the Brillouin zone into symmetry-allowed and symmetry-forbidden scattering channels. Consequently, mirror-even phonons couple electronic states of the same mirror parity, either even--even or odd--odd, whereas mirror-odd phonons couple only states of opposite mirror parity, for both intra- and interband processes. We illustrate the selection rule through first-principles calculations for the two-dimensional superconductor $\mathrm{Sc_2BCH_4}$, where mirror parity is well defined throughout the entire Brillouin zone and symmetry-forbidden matrix elements vanish up to numerical precision. We then extend the analysis to bulk $\mathrm{MgB_2}$, where the rule applies exactly only on mirror-invariant planes of the three-dimensional Brillouin zone. Moving away from these planes removes the exact mirror constraint and activates scattering channels that are symmetry forbidden when the mirror eigenvalues are well defined. Our results establish horizontal mirror symmetry as an exact and practical filter of e-ph scattering processes.