Phonon Helicity Induced by Electronic Berry Curvature in Dirac Materials

In two-dimensional insulators with time-reversal (TR) symmetry, a nonzero local Berry curvature of low-energy massive Dirac fermions can give rise to nontrivial spin and charge responses, even though the integral of the Berry curvature over all occupied states is zero. In this Letter, we present a new effect induced by the electronic Berry curvature. By studying electron-phonon interactions in ${\mathrm{BaMnSb}}_2$, a prototype two-dimensional Dirac material possessing two TR-related massive Dirac cones, we find that the nonzero local Berry curvature of electrons can induce a phonon angular momentum. The direction of this phonon angular momentum is locked to the phonon propagation direction, and thus we refer to it as “phonon helicity” in a way that is reminiscent of electron helicity in spin-orbit-coupled electronic systems. We discuss possible experimental probes of such phonon helicity.

Figure 1

(a) Lattice structure of ${\mathrm{BaMnSb}}_2$. The green and orange dots represent two Sb atoms in each unit. Two acoustic phonons and two optical phonons are shown in (b) and (c), respectively. (d) First Brillouin zone. The two red dots label the location of two valleys $K_{\pm }$. (e) Low-energy electron bands at Dirac cones $K_{\pm }$. The up-down arrow represents the spin polarization direction, and our model only focuses on the bands within the box with the dashed lines.

Figure 2

The off-diagonal phonon self-energy ${\mathrm{\Sigma }}_{AB}$ induced by the electronic Berry curvature. We plot the real and imaginary parts of ${\mathrm{\Sigma }}_{AB}$ (in meV) as a function of $\omega /m_0$ and $q_y/q_0$ in (a) and (b), respectively. (c) shows $\mathrm{Im}[{\mathrm{\Sigma }}_{AB}]$ as a function of $q_y$ for the frequency values $\omega /m_0=0$, 0.4, 0.8, and 1.2. The dashed line gives the pure contribution from the Berry curvature. The inset shows the partial derivative of ${\mathrm{\Sigma }}_{AB}$. The parameter values are $m_0=25\mathrm{meV}$, $v_0=100\mathrm{nm}·\mathrm{meV}$, ${\omega }_{A_1}=20\mathrm{meV}$, ${\omega }_{B_1}=30\mathrm{meV}$, and $g_0=80\sqrt{20}\mathrm{meV}·{\mathrm{nm}}^{-1}$, and $g_1=80\sqrt{30}\mathrm{meV}·{\mathrm{nm}}^{-1}$. Also, $q_0=m_0/v_0$, and ${\mathrm{\Sigma }}_0=g_0{g}_1\mathcal{A}{N}_3^{+}/(v_{0}M\sqrt{{\omega }_A{\omega }_B})$.

Figure 3

(a) and (b) The spectrum and angular momentum of optical phonons with $m_0=25\mathrm{meV}$. The phonon dispersions are labeled by black dashed lines, while the color represents $\mathcal{P}$ in Eq. (7). (c) $\mathrm{Im}[{\mathrm{\Sigma }}_{AB}]$ (black line) and the phonon circular polarization (red and blue lines) as a function of $m_0$ with $q_y=m_0/2v_0$ and $\omega =10\mathrm{meV}$. The phonon circular polarization has the same sign as $\mathcal{P}$. Inset: the schematics for the elliptical vibration of phonon modes. Parameters: $v_0=100\mathrm{meV}·\mathrm{nm}$, ${\omega }_{A_1}=20\mathrm{meV}$, ${\omega }_{B_1}=30\mathrm{meV}$, $g_0=80\sqrt{20}\mathrm{meV}·{\mathrm{nm}}^{-1}$, and $g_1=80\sqrt{30}\mathrm{meV}·{\mathrm{nm}}^{-1}$. Also, $q_0=m_0/v_0$, and ${\mathrm{\Sigma }}_0=g_0{g}_1\mathcal{A}{N}_3^{+}/(v_{0}M\sqrt{{\omega }_A{\omega }_B})$.