Atomistic theory of the phonon angular momentum Hall effect

Abstract

The spin and orbital Hall effects convert longitudinal charge currents into transverse flows of electronic angular momentum. Here we develop an atomistic theory of the recently proposed lattice-vibrational analogue, in which a longitudinal heat current driven by a thermal gradient is converted into a transverse current of phonon angular momentum. We derive a microscopic real-space expression for this current and show that it originates from thermally induced mixing of polarized vibrational motion, leading to a characteristic edge accumulation of phonon angular momentum. We demonstrate the effect in minimal square- and honeycomb-lattice models and compute the resulting phonon angular momentum accumulations for a range of example materials using input from first-principles calculations. Our results confirm that the phonon angular momentum Hall effect is a universal response of crystalline solids and our framework is generically applicable to all materials.

Figures (7)

• Figure 1: Angular momentum Hall effects. (a) Spin Hall effect: an electric field generates a transverse current and edge accumulation of spin angular momentum SinovaSpinHallEffect. (b) Orbital Hall effect: an electric field generates a transverse current and edge accumulation of orbital angular momentum Go2018OHE. (c) Phonon angular momentum Hall effect: a temperature gradient generates a transverse current and edge accumulation of phonon angular momentum ParkPAMHE.
• Figure 2: Phonon angular momentum Hall effect in minimal models of centrosymmetric lattices. Panels (a) and (c) show the lattice geometries and spring networks used for the square- and honeycomb-lattice models, respectively. Panels (b) and (d) show the corresponding real-space nonequilibrium response under a thermal bias applied along the horizontal direction. In the square lattice, both axial and diagonal bonds are included, whereas in the honeycomb lattice nearest-neighbor bonds already produce a finite effect. The blue--red colormap shows the local phonon angular momentum $L_z(s)$ from Eq. \ref{['eq:L_def']}, and the black arrows show the PAM current $\bm j^{(L_z)}(s)$ from Eq. \ref{['eq:jsite_main']}. For the real-space maps shown here, the square lattice uses axial and diagonal spring constants of 30 N/m and 15 N/m, respectively, while the honeycomb lattice uses nearest-neighbor springs only, with isotropic and anisotropic couplings of 80 N/m and 60 N/m, respectively. In both cases, we use a damping rate $\kappa=5\,\mathrm{ps}^{-1}$ at all sites.
• Figure 3: Magnetic-field dependence of the kinetic-energy and PAM deflection angles, together with the Hall-like angle, for the square-lattice model of Fig. \ref{['fig:square-honeycomb-overview']}. Panels (a) and (b) show the deflection angles of the kinetic-energy and PAM currents, respectively; the PAM deflection angle $\theta_{L_z}$ is defined in Eq. \ref{['eq:hall_angle_main']}, with an analogous definition for the energy current. Panel (c) shows the Hall-like angle $\theta_H$, defined in Eq. \ref{['eq:mixed_angle_main']}. At zero field, the kinetic-energy current is longitudinal, whereas the PAM current is transverse. An out-of-plane magnetic field deflects the kinetic-energy current and rotates the PAM current away from the purely transverse direction. By contrast, $\theta_H$ remains unchanged over the plotted field range, indicating that the field mainly affects the longitudinal PAM and transverse kinetic-energy components rather than the conductivity components entering Eq. \ref{['eq:mixed_angle_main']}. Here the field strength is expressed through the effective scale $\Omega_B=\gamma B$, where $B$ is the applied magnetic field and $\gamma$ is the gyromagnetic ratio assigned to the lattice.
• Figure 4: Real-space edge accumulation of phonon angular momentum at zero magnetic field in representative materials: (a) graphene, (b) BaTiO$_3$, (c) MgO, and (d) silicon. The colormap shows the local phonon angular momentum $L_z(s)$ from Eq. \ref{['eq:L_def']}. The arrows indicate the applied thermal bias, and the dashed line marks the hot center of the sample. Opposite-sign accumulations appear near the upper and lower edges in each case, evidencing a transverse PAM response in finite samples. The black markers labeled a--d on the color bar indicate the corresponding values of $\max(|L_z|)$ for panels (a)--(d).
• Figure S1: Phonon angular-momentum Hall effect in minimal square and honeycomb lattices. (a--d) Square lattice: (a) lattice geometry and spring network with axial bonds of stiffness $K_{\rm ax}$ and diagonal bonds of stiffness $K_{\rm diag}$. (b) Real-space distribution of $L_z(s)$ and site current vectors $\bm j^{(L_z)}(s)$. (c) Edge accumulation of phonon angular momentum as a function of $K_{\rm ax}$ and $K_{\rm diag}$, measured by $\max |L_z|$, i.e. the maximum site-resolved magnitude of the local phonon angular momentum in the sample for each pair of stiffness parameters. (d) Deflection angle $\theta_{L_z}$ and $p_{95}(|L_z|)$ as functions of the damping rate $\kappa$, where $p_{95}(|L_z|)$ is the 95th percentile of the site-resolved magnitude $|L_z(s)|$ and is used to characterize the accumulation while reducing the influence of isolated outliers. (e--h) Honeycomb lattice: (e) lattice geometry and nearest-neighbor bond model with isotropic and anisotropic couplings $A_1$ and $B_1$. (f) Real-space distribution of $L_z(s)$ and site current vectors $\bm j^{(L_z)}(s)$. (g) Edge accumulation as a function of $A_1$ and $B_1$, again measured by $\max |L_z|$. (h) Deflection angle $\theta_{L_z}$ and $p_{95}(|L_z|)$ as functions of $\kappa$. The stars in panels (c,d,g,h) mark the parameter values used for the real-space maps in panels (b,f).