Phonon thermal Hall as a lattice Aharonov-Bohm effect

TL;DR

The paper tackles the puzzling phonon thermal Hall effect in non-magnetic insulators, where a magnetic field induces a misalignment between heat flux and the temperature gradient, yielding a small but finite transverse heat flow. It advances a lattice Aharonov-Bohm mechanism: in an anharmonic crystal, a field-induced geometric (Berry) phase on atomic vibrations modifies Normal phonon–phonon collisions, producing interference that generates a finite Hall signal. The key quantitative idea is a phase scale $\delta \phi_B \approx q_e \frac{\lambda_{ph} \delta u_m}{\ell_B^2}$ with $q_e \sim 1$, predicting a Hall angle that satisfies $\tan \Theta_H \lesssim 2 \delta \phi_B$ and peaks near $T_{max}$ where Normal processes dominate. The theory aligns with experimental trends in Si, Ge, and black phosphorus, linking the amplitude to phonon wavelength and crest displacement, and suggesting ab initio tests to validate the mechanism.

Abstract

In a growing list of insulators, experiments find that magnetic field induces a misalignment between the heat flux and the thermal gradient vectors. This phenomenon, known as the phonon thermal Hall effect, implies energy flow without entropy production along the orientation perpendicular to the temperature gradient. The experimentally-measured thermal Hall angle in various insulators does not exceed a bound and becomes maximal at the temperature of peak longitudinal thermal conductivity. The present paper aims to propose a scenario providing and explanation for these two experimental facts. It begins by noticing that at this temperature, $T_{max}$, Normal phonon-phonon collisions become most frequent in comparison with Umklapp and boundary scattering events. Furthermore, the Born-Oppenheimer approximated molecular wave functions are known to acquire a phase in the presence of a magnetic field. In an anharmonic crystal, in which tensile and compressive strain do not cancel out, this field-induced atomic phase gives rise to a phonon Berry phase and generates phonon-phonon interference. The rough amplitude of the thermal Hall angle expected in this picture is set by the phonon wavelength, $λ_{ph}$, and the crest atomic displacement, $δu_m$ at $T_{max}$. The derived expression is surprisingly close to what has been experimentally found in black phosphorus, germanium and silicon.

Figures (6)

• Figure 1: a) Transverse and longitudinal thermal conductivities in three elemental insulators normalized by their peak amplitude. The two conductivities peak at the same temperature and the transverse response decreases faster at both the warmer and the colder sides of the peak. b) The peak thermal Hall conductivity divided by magnetic field in a variety of insulators as a function of their maximum longitudinal thermal conductivity. While the latter varies by four orders of magnitude, the former remains proportional to it and their ratio displays little change.
• Figure 2: a) In a thermal Hall experiment, the sample is held between a heater and a cold finger. This allows to establish a finite heat current density vector, $\overrightarrow{J_q}$, along an orientation, which is often a crystallographic axis. Three temperature sensors allow measuring the temperature gradient along two orientations, which are parallel ($\nabla_x T$) and perpendicular ($\nabla_y T$) to $\overrightarrow{q}$. b) A finite $\nabla_y T$ means that there is a misalignment between the thermal gradient vector, $\overrightarrow{\nabla} T$ and $\overrightarrow{J_q}$. c) Such a misalignment implies a heat flow without entropy production in the direction perpendicular to $\overrightarrow{\nabla} T$.
• Figure 3: a) Two kinds of three-phonon Normal collisions. A phonon with a $q_1$ wave-vector can absorb another phonon with a wave-vector $q_2$ (left) or emit a phonon with a wave-vector $q_2$ (right). These two Normal events can be transformed one to another by permuting the initial ($q_1$) and the final ($q_3$) phonons. They do not produce entropy. (b) When the sum of the wave-vector of the colliding phonons is sufficiently large, their collision is an Umklapp event, which generates entropy. Here the $q_3$ phonon is forbidden by the discrete symmetry of the crystal. (c) The relative frequency of Normal collisions respective to Umklapp and boundary scattering. At the peak temperature, the chance of a thermally excited phonon to suffer a Normal scattering event is the largest.
• Figure 4: a) In the presence of magnetic field, an atom (top), a molecule (middle) and a chain of atoms (bottom) have a gauge-dependent phase. Expressed in the symmetric gauge, it is given by the vector product of the magnetic field and the coordinate of each electron with respect to the center of mass of the molecule. The position of the nuclei is absent in this expression, because they cancel out with respect to the center of mass.
• Figure 5: a) Left: A chain of centrosymmetric square unit cells containing an atom (dark blue circles) with electron cloud filling each unit cell. The overall phase is zero because there is no mismatch between the electronic and the nuclear center of mass. Right: A collective vibration of atoms leaving their equilibrium positions (white circles) would create a local mismatch between the nuclear and the electronic centers of mass. b) The computed valence charge density in silicon along the $<111>$ crystalline orientation Lu1993. The origin of the horizontal axis is an atomic site and the units are the body diagonal of the cube. The location of four atomic Si sites are represented by green and purple circles. For each atom, a displacements leftward and rightward are not equivalent. The right panel shows the unit cell of silicon. Only Si atoms corresponding to these two specific sites are shown.