Inherited Berry curvature of phonons in Dirac materials with time-reversal symmetry

Abstract

The Berry curvature of phonons is an active subject of research in condensed matter physics. Here, we present a model in which phonons acquire a Berry curvature through their coupling to electrons in crystals with time-reversal symmetry. We illustrate this effect for BaMnSb$_2$, a quasi two-dimensional Dirac insulator, whose low-energy massive Dirac fermions generate a phonon Berry curvature that is proportional to the electronic valley Chern number.

Figures (3)

• Figure 1: Layer of Sb atoms in BaMnSb$_2$. The rectangular unit cell, of dimensions $a\times b$, is shown by the dotted lines. For simplicity, we take $a=b$ in the main text. This simplification does not result in qualitative changes to our main results.
• Figure 2: Phonon dispersion along high symmetry directions indicated with blue triangle in the inset. The numerical values of the parameters are $\gamma_0=50$ N/m, $\gamma_1=75$ N/m, $\gamma_2=70$ N/m, $M_1=M_2=2 \times 10^{-25}$ kg, $\delta a=0.2$. The scale of phonon frequencies is chosen to qualitatively agree with first-principle calculations chen2024thermoelectric.
• Figure 3: Phonon dispersion and Berry curvature for optical (a) and acoustic (b) modes around the $\Gamma$ point. The phonon branches calculated using Eq. (\ref{['Eq:dynamcial_2by2']}) are shown in solid red and green lines. The electron-phonon interaction parameters are defined as $g_0^o g_x^o {\cal S} = {d^o}^2 {\cal S}_{\rm uc}$ and $g_0^a g_x^a {\cal S} = {d^a}^2 {\cal S}_{\rm uc}$ with a deformation potential $d^o=1\, {\rm eV/\AA}$ and $d^a = 5 \, {\rm eV}$ for optical and acoustic modes, respectively, and a unit cell area ${\cal S}_{uc}=a^2$. The dashed blue line shows the dimensionless Berry curvature $\widetilde{\Omega} = \Omega_-/ {\cal S}_{\rm uc}$ for the upper phonon branches (green lines). Lower phonon branches (red lines) have the opposite Berry curvature. Note the different scales for left and right axes for the two figures. For acoustic phonons, ${\cal F}(\widetilde{\Omega}) = {\rm sgn}(\widetilde{\Omega}) \ln(1+|\widetilde{\Omega}|)$ is plotted for better visibility. Also, $a q_x = 0.001$ is taken in $\Omega$ calculation for acoustic modes to circumvent the $q = 0$ divergence. Here, we have taken the Dirac velocity $v_0 = 10^6 \, {\rm m/s}$, the lattice constant $a=5 \AA$sakai2020bulk, and other parameters are same as in Fig. \ref{['fig:phonondispersion']}.