Surface Phonon Hall Viscosity Induced Phonon Chirality and Nonreciprocity in Magnetic Topological Insulator Films

Abstract

The surface half-quantum Hall effect, a hallmark consequence of axion electrodynamics, can be induced by gapping out the surface states of topological insulators through surface magnetization, and has led to a variety of topological response phenomena observed in experiments. In this work, we investigate phonon dynamics originating from an acoustic analog - the surface phonon Hall viscosity - that can also occur at the surface of magnetic topological insulators. This surface phonon Hall viscosity stems from the Nieh-Yan action in the strain response of topological insulators, where strain acts as the effective vierbein field for the bulk low-energy massive Dirac fermions. Crucially, this viscosity term entangles phonon dynamics with surface magnetization. In magnetic topological insulator films, we find that this interaction causes acoustic phonons to become chiral when the magnetization at the top and bottom surfaces is parallel, and nonreciprocal when it is anti-parallel. We further discuss potential experimental signatures of phonon dynamics induced by surface phonon Hall viscosity, specifically the phonon thermal Hall effect and magnon-polarons. Surface phonon Hall viscosity provides a mechanism to control phonon chirality and nonreciprocity via surface magnetization configurations in magnetic topological insulator films.

Figures (11)

• Figure 1: (a) Axion electrodynamics in 3D magnetic TI can result in surface Hall conductivity $\sigma_{xy}= e^2/2h$ when surface Dirac cone is gapped. (b) The Nieh-Yan action for the strain response of magnetic TIs can lead to surface PHV $\eta_{xy}$. (c) The connection between different classes of gravity theories. Non-zero torsion and Riemann curvature ($T, R\neq 0$) ensues Riemann-Cartan spacetime. In ordinary general relativity, the torsion is zero ($T=0$), whereas the Riemann curvature is zero in Weitzenböck spacetime ($R=0$). When both $T, R = 0$, it is flat Minkowski spacetime. (d) Magnetic TI sandwiches with surface FM configuration have equal values of surface PHV, $\eta_t = \eta_b$, at the top and bottom surfaces. Acoustic phonon modes are chiral ($L\neq 0$), but reciprocal ($\omega_\mathbf{k} = \omega_{-\mathbf{k}}$). (e) The surface AFM configuration can lead to opposite surface PHV, $\eta_t = -\eta_b$, at two surfaces, and the acoustic phonons become non-reciprocal ($\omega_\mathbf{k} \neq \omega_{-\mathbf{k}}$) but also non-chiral ($L=0$).
• Figure 2: (a) Dependence of the Nieh–Yan coefficient $\eta_0$ on the Dirac mass $|m|$ for different momentum cutoff $\Lambda_c$. (b) The surface phonon modes (green and red) with frequency below the bulk phonon mode frequencies (blue region) in the FM configuration. (c) The spatial distribution of the displacement fields for the top and bottom surface phonon modes at $k_x = k_y = 2.9$ nm$^{-1}$. (d) and (e) shows the angular momentum $L_{x,y,z}$ as a function of momentum $\mathbf{k}$ for (d) the bottom surface mode and (e) top surface mode in the FM configuration, respectively. The light blue shaded region corresponds to the momenta where the surface modes are indistinguishable from the bulk modes due to finite size effects in our numerical calculations. (f) The phonon angular momentum $L_{x,y,z}$ of the bottom surface phonon mode as a function of $\eta_1$ in units of $\Tilde{\eta} = 8 \hbar/nm^{2}$. For (a)-(e), we use $\eta_1 = \Tilde{\eta}$.
• Figure 3: (a) The phonon dispersion of the thin film slab model (N=6 sites) for the FM case (black lines) and AFM case (red lines). (b) The angular momentum $L_{x,y,z}$ of the lowest frequency acoustic phonon modes for the FM case (black lines) and the AFM case (red lines). (c) The phonon Berry curvature distribution in the $k_x-k_y$ plane for the three acoustic phonon modes in the FM configuration. (d) The phonon thermal Hall conductivity $\kappa_{xy}$ as a function of temperature $k_B T$ for the FM (black) and AFM (red) configurations. (e) The low temperature behavior of $\kappa_{xy}$ from the numerical calculations (red points) can be well described by the $T^2$ fitting (black curve).
• Figure 4: (a) The dispersion of magnon-polaron modes (black), bare phonon modes (blue) and bare magnon modes (red) in the FM configuration. (b) The zoom-in for the region indicated by the green circle in (a). The Berry curvature distribution of magnon-polaron (c) branch 1 and (d) branch 3. (e) The surface thermal Hall conductivity $\kappa_{xy}$ as a function of temperature $k_B T$ for magnon-polarons (black) and bare phonons (blue).
• Figure S1: The adiabatic evolution of (a) the Dirac complex angle $\Phi$ and the (b) magnetization $M$ from the magnetic topological insulator side to the trivial side.