Spin Hall effect is the transverse flow of the electron spin in conductors under external electric field. Similarly, thermal gradient in magnetic insulators can drive a transverse flow of the spin angular momentum of magnons, which provides a thermal alternative for spin manipulation. Recently, the phonon angular momentum (PAM), which is the angular momentum of atoms as a result of their orbital motion around their equilibrium positions, has garnered attention as a quantity analogous to the magnon spin. However, can we manipulate PAM like magnon spin? Here, we show that temperature gradient generally induces a transverse flow of PAM, which we term the phonon angular momentum Hall effect (PAMHE). The PAMHE relies only on the presence of transverse and longitudinal acoustic phonons, and it is therefore ubiquitous in condensed matter systems. As a consequence of the PAMHE, PAM accumulates at the edges of a crystal. When the atoms in the crystal carry nonzero Born effective charge, the edge PAM induces edge magnetization, which may be observed through optical measurement. We believe that PAMHE provides a new principle for the manipulation of angular momenta in insulators and opens up an avenue for developing functional materials based on phonon engineering.
Berry curvature (BC) governs topological phases of matter and generates anomalous transport. When a magnetic field is applied, phonons can acquire BC indirectly through spin-lattice coupling, leading to a linear phonon Hall effect. Here, we show that polar lattice distortion directly couples to a phonon BC dipole, which causes a switchable non-linear phonon Hall effect. In a SnS monolayer, the in-plane ferroelectricity induces a phonon BC and leads to the phononic version of the non-volatile BC memory effect. As a new type of ferroelectricity-phonon coupling, the phonon Rashba effect emerges a…
Since the first experimental observation of the phonon Hall effect (PHE) in 2005, its physical origin and theoretical explanation have been extensively investigated. While spin-orbit interactions are believed to play important roles under external magnetic fields, nonmagnetic effects are also possible. Here, we propose a mechanism of PHE which is induced by electric current in a nonequilibrium system through electron-phonon interactions. The influence of the drift electrons to the phonon degrees of freedom, as a correction to the Born-Oppenheimer approximation, is represented by an antisymmetr…
Here, we propose quantum spin Hall-like phononic states in van der Waals bilayer systems with antiferromagnetic ordering. Antiferromagnetic ordering in bilayer systems with small interlayer interaction makes the total Chern number zero, where the Chern numbers for each layer are opposite to each other. In bilayer CrI3 where antiferromagnetic ordering has been experimentally demonstrated, we show that the quantum spin Hall-like states appear with spatially separated chiral edge modes.
Valley Hall effect is an appearance of the valley current in the direction transverse to the electric current. We develop the microscopic theory of the valley Hall effect in two-dimensional semiconductors where the electrons are dragged by the phonons or photons. We derive and analyze all relevant contributions to the valley current including the skew-scattering effects together with the anomalous contributions caused by the side-jumps and the anomalous velocity. The partial compensation of the anomalous contributions is studied in detail. The role of two-phonon and two-impurity scattering pro…
We generalize the notion of dissipationless, topological Hall viscosity tensor to optical phonons in thin film Weyl semimetals. By using the strained Porphyrin thin film Weyl semimetal as a model example, we show how optical phonons can couple to Weyl electrons as chiral pseudo gauge fields. These chiral vector fields lead to a novel dissipationless two-rank viscosity tensor in the effective dynamics of optical phonons whose origin is the chiral anomaly. We also compute the contribution to this two rank Hall viscosity tensor due to the presence of an external magnetic field, whose origin is th…
A mechanism for phonon Hall effect (PHE) in non-magnetic insulators under an external magnetic field is theoretically studied. PHE is known in (para)magnetic compounds, where the magnetic moments and spin-orbit interaction play an essential role. In sharp contrast, we here show that a non-zero Berry curvature of acoustic phonons is induced by an external magnetic field due to the correction to the adiabatic Born-Oppenheimer approximation. This results in the finite thermal Hall conductivity $κ_H$ in nonmagnetic band insulators. Our estimate of $κ_H$ for a simple model gives $κ_H \sim 1.0\times…
We find that in BaTiO$_3$ the phonon angular momentum is dominantly pointing in directions perpendicular to the electrical polarization. Therefore, external electric field in ferroelectric BaTiO$_3$ does not control only the direction of electrical polarization, but also the direction of phonon angular momentum. This finding opens up the possibility for electric-field control of physical phenomena that rely on phonon angular momentum. We construct an intuitive model, based on our first-principles calculations, that captures the origin of the relationship between phonon angular momentum and ele…
We provide a topological understanding on phonon Hall effect in dielectrics with Raman spinphonon coupling. A general expression for phonon Hall conductivity is obtained in terms of the Berry curvature of band structures. We find a nonmonotonic behavior of phonon Hall conductivity as a function of magnetic field. Moreover, we observe a phase transition in phonon Hall effect, which corresponds to the sudden change of band topology, characterized by the altering of integer Chern numbers. This can be explained by touching and splitting of phonon bands.
When time-reversal symmetry is broken, the low-energy description of acoustic lattice dynamics allows for a dissipationless component of the viscosity tensor, the phonon Hall viscosity, which captures how phonon chirality grows with the wavevector. In this work, we show that, in ionic crystals, a phonon Hall viscosity contribution is produced by the Lorentz forces on moving ions. We calculate typical values of the Lorentz force contribution to the Hall viscosity using a simple square lattice toy model, and we compare it with literature estimates of the strengths of other Hall-viscosity mechani…
Phonon Hall effect (PHE) has attracted a lot of attention in recent years with many theoretical and experimental explorations published. While experiments work on complicated materials, theoretical studies are still hovering around the phenomenon-based models. Moreover, previous microscopic theory was found unable to explain large thermal Hall conductivity obtained by experiments in strontium titanate (STO). Therefore, as a first attempt to bridge this gap, we implement first-principles calculations to explore the PHE in real materials. Our work provides a new benchmark of the PHE in sodium ch…
It has been known for decades that a magnetic field can deflect phonons as they flow in response to a thermal gradient, producing a thermal Hall effect. Several recent experiments have revealed ratios of the phonon Hall conductivity to the phonon longitudinal conductivity in oxide dielectrics that are larger than $10^{-3}$ when phonon mean-free-paths exceed phonon wavelengths. At the same time $κ_{H}/κ_{L}$ is not strongly temperature dependent. We argue that these two properties together imply a mechanism related to phonon scattering from defects that break time-reversal symmetry, and we show…
We derive a formula for the thermal conductivity tensor of a ballistic phonon Hall model. It is found that, although the diagonal elements of the conductivity tensor diverge to infinite, the off-diagonal elements are finite,antisymmetric, and odd in magnetic field. The off-diagonal elements are non-zero only if the dynamic matrix of the phonon system breaks mirror reflection symmetry. The results are obtained without perturbative assumptions about the spin-phonon interactions.
We show that there is a fundamental difference between spin Hall current and orbital angular momentum Hall current in Rashba- Dresselhaus spin orbit coupling systems. The orbital angular momentum Hall current has a pure topological contribution which is originated from the existence of magnetic flux in momentum space while there is no such topological nature for the spin Hall current. Moreover, we show that the orbital Hall conductance is always larger than the spin Hall conductance in the presence of both couplings. The topological part is expected to be free from the effect of disorder due t…
We study the phonon-assisted process of dissociation of a magnetoroton, in a fractional quantum Hall liquid, into an unbound pair of quasiparticles. Whilst the dissociation is forbidden to first order in the electron-phonon interaction, it can occur as a two-phonon process. Depending on the value of final separation between the quasiparticles, the dissociation is either a single event involving absorption of one phonon and emission of another phonon of similar energy, or a two-phonon diffusion of a quasiexciton in momentum space. The dependence of the magnetoroton dissociation time on the fill…
We present a systematic theory of the phonon Hall effect in a ballistic crystal lattice system, and apply it on the kagome lattice which is ubiquitous in various real materials. By proposing a proper second quantization for the non-Hermite Hamiltonian in the polarization-vector space, we obtain a new heat current density operator with two separate contributions: the normal velocity responsible for the longitudinal phonon transport, and the anomalous velocity manifesting itself as the Hall effect of transverse phonon transport. As exemplified in kagome lattices, our theory predicts that the dir…
One of the most distinctive features of graphene is its huge inter-Landau-level splitting in experimentally attainable magnetic fields which results in the room-temperature quantum Hall effect. In this paper we calculate the longitudinal conductivity induced by two-phonon scattering in graphene in a quantizing magnetic field at elevated temperatures. It is concluded that the purely phonon-induced scattering, negligible for conventional semiconductor heterostructures under quantum Hall conditions, becomes comparable to the disorder-induced contribution to the dissipative conductivity of graphen…
Based upon spin-lattice interaction, we propose a theoretical model for the phonon Hall effect in paramagnetic dielectrics. The thermal Hall conductivity is calculated by using the Kubo formula. Our theory reproduces the essential experimental features of the phonon Hall effect discovered recently in ionic dielectric Tb$_3$Ga$_5$O$_{12}$, including the sign, magnitude and linear magnetic field dependence of the thermal Hall conductivity.
A diversity of spin Hall effects in metallic systems is known to rely on Mott skew scattering. In this work its high-temperature counterpart, phonon skew scattering, which is expected to be of foremost experimental relevance, is investigated. In particular, the phonon skew scattering spin Hall conductivity is found to be practically $T$-independent for temperatures above the Debye temperature $T_D$. As a consequence, in Rashba-like systems a high-$T$ linear behavior of the spin Hall angle demonstrates the dominance of extrinsic spin-orbit scattering only if the intrinsic spin splitting is smal…
We study angular momentum of phonons in a magnetic crystal. In the presence of a spin-phonon interaction, we obtain a nonzero angular momentum of phonons, which is an odd function of magnetization. At zero temperature, phonon has a zero-point angular momentum besides a zero-point energy. With increasing temperature, the total phonon angular momentum diminishes and approaches to zero in the classical limit. The nonzero phonon angular momentum can have a significant impact on the Einstein-de Haas effect. To obtain the change of angular momentum of electrons, the change of phonon angular momentum…
The angular momentum interconversion between electron spin and other type of angular momenta is useful to develop new spintronic functionalities. The conversions from the angular momentum of photon and mechanical rotation to ferromagnetic spin moment have been well studied. While the recent studies theoretically suggested circular vibration of atoms works as angular momentum of phonon, the direct experimental demonstration of conversion to spin moments remains to be performed. Here we demonstrate that the phonon angular momentum of surface acoustic wave can control the magnetization of a ferro…
Motivated by recent experiments on the phonon contribution to the thermal Hall effect in the cuprates, we present an analysis of chiral phonon transport. We assume the chiral behavior arises from a non-zero phonon Hall vicosity, which is likely induced by the coupling to electrons. Phonons with a non-zero phonon Hall viscosity have an intrinsic thermal Hall conductivity, but Chen et al. (Phys. Rev. Lett. 124, 167601 (2020)) have argued that a significantly larger thermal Hall conductivity can arise from an extrinsic contribution which is inversely proportional to the density of impurities. We …
The quantum anomalous Hall effect, an exotic topological state first theoretically predicted by Haldane and recently experimentally observed, has attracted enormous interest for low-power-consumption electronics. In this work, we derived a Schr{ö}dinger-like equation of phonons, where topology-related quantities, time reversal symmetry and its breaking can be naturally introduced similar as for electrons. Furthermore, we proposed a phononic analog of the Haldane model, which gives the novel quantum (anomalous) Hall-like phonon states characterized by one-way gapless edge modes immune to scatte…
Recent experimental observations of unexpectedly large thermal Hall conductivities, $κ_H$, in insulating materials, including the parent compounds of the high temperature superconducting cuprates, likely reflect an extrinsic contribution from a yet to be identified extrinsic source of skew scattering of acoustic phonons. We show that resonant scattering of phonons from a certain class of three-level systems produces strong skew scattering in the presence of a modest magnetic field. We interpret this as a first step towards understanding the experiments.
The origin of phonon thermal Hall Effect (THE) observed in a variety of insulators is yet to be identified. Here, we report on the observation of a thermal Hall conductivity in a non-magnetic elemental insulator, with an amplitude exceeding what has been previously observed. In black phosphorus (BP), the longitudinal ($κ_{ii}$), and the transverse, $κ_{ij}$, thermal conductivities peak at the same temperature and at this peak temperature, the $κ_{ij}/κ_{jj}/B $ is $\approx 10^{-4}$-$10^{-3}$ T$^{-1}$. Both these features are shared by other insulators displaying THE, despite an absolute amplit…
The role of electron-phonon scattering in finite-temperature anomalous Hall effect is still poorly understood. In this work, we present a Boltzmann theory for the side-jump contribution from electron-phonon scattering, which is derived from the microscopic quantum mechanical theory. We show that the resulting phonon side-jump conductivity generally approaches different limiting values in the high and low temperature limits, and hence can exhibit strong temperature dependence in the intermediate temperature regime. Our theory is amenable to ab initio treatment, which makes quantitative comparis…
We propose a new mechanism for the thermal Hall effect in exchange spin-wave systems, which is induced by the magnon-phonon interaction. Using symmetry arguments, we first show that this effect is quite general, and exists whenever the mirror symmetry in the direction of the magnetization is broken. We then demonstrate our result in a collinear ferromagnet on a square lattice, with perpendicular easy-axis anisotropy and Dzyaloshinskii-Moriya interaction from mirror symmetry breaking. We show that the thermal Hall conductivity is controlled by the resonant contribution from the anti-crossing po…
We derive the ac spin-Hall conductivity $σ_{\rm sH}(ω)$ of two-dimensional spin-orbit coupled systems interacting with dispersionless phonons of frequency $ω_0$. For the linear Rashba model we show that the electron-phonon contribution to the spin-vertex corrections breaks the universality of $σ_{\rm sH}(ω)$ at low-frequencies and provides a non-trivial renormalization of the interband resonance. On the contrary, in a generalized Rashba model for which the spin-vertex contributions are absent, the coupling to the phonons enters only through the self-energy, leaving the low frequency behavior o…
Motivated by recent phonon spectroscopy experiments in the fractional quantum Hall regime we consider processes in which thermally excited magnetoroton excitations are scattered by low energy phonons. We show that such scattering processes can never give rise to dissociation of magnetorotons into unbound charged quasiparticles as had been proposed previously. In addition we show that scattering of magnetorotons to longer wavelengths by phonon absorption is possible because of the shape of the magnetoroton dispersion curve and it is shown that there is a characteristic cross-over temperature ab…
In monolayer hexagonal lattices, two inequivalent valleys appear in the Brillouin zone. With inversion symmetry breaking, we find chiral phonons with valley contrasting circular polarization and ionic magnetic moment. At valley centers, there is a three-fold rotational symmetry endowing phonons with a quantized pseudo angular momentum, which includes spin and orbital parts. From conservation of the pseudo angular momentum, crystal momentum and energy, selection rules in intervalley scattering of electrons by phonons are obtained. The chiral valley phonons are verified and the selection rules a…
We describe a theoretical framework for the interpretation of time-resolved phonon absorption experiments carried out in the fractional quantum Hall regime of a magnetically quantized two-dimensional electron system (2des). The only phonons which can be absorbed at low temperature are those whose energies exceed the magnetoroton gap predicted by Girvin, MacDonald and Platzman. The rate of energy transfer from the phonons to the electron liquid is entirely controlled by the creation of these collective excitations. Using simple isotropic approximations for the phonon propagation and electron-ph…
We show that the piezoelectric coupling to three-dimensional phonons in GaAs renormalizes the current-voltage exponent for tunneling of electrons into an incompressible quantum Hall edge. The leading correction is always negative, in agreement with experiments on the $ν= 1/3$ state and, depending on the precise value of the edge plasmon speed, can be as large as a few percent. We also discuss higher-order corrections, which determine the effect of the piezoelectric coupling in the extreme infrared limit.
We propose a mechanism of intrinsic spin Hall effect (SHE). In this mechanism, local orbital angular momentum (OAM) induces electron position shift and couples with the bias electric field to generate orbital Hall effect (OHE). SHE then emerges as a concomitant effect of OHE through the atomic spin-orbit coupling. Spin Hall conductivity due to this mechanism is estimated to be comparable to experimental values for heavy metals. This mechanism predicts the sign change of the spin Hall conductivity as the spin-orbit polarization changes its sign, and also correlation between the spin Hall conduc…
A two-dimensional mass-spring system with Honeycomb lattice for mimicking phononic quantum Hall effect is proposed. Its band structure shows the existence of Dirac cones and unconventional edge states that is similar to the vibrational modes in graphene. Interestingly, as the system is placed on a constantly rotational coordinate system, the Coriolis force resulted from the non-inertial reference frame provides a possibility to break the time-reversal symmetry. Thus, caused from topologically non-trivial band gaps, phononic edge states are present between bands, which are verified by the calcu…
We present computations of the thermal Hall coefficient of phonons scattering off a defect with multiple energy levels. Using a microscopic formulation based on the Kubo formula, we find that the leading contribution perturbative in the phonon-defect coupling is proportional to the phonon lifetime, and has a `side-jump' interpretation. Consequently, the thermal Hall angle is independent of the phonon lifetime. The contribution to the thermal Hall coefficient is at resonance when the phonon energy equals a defect level spacing. Our results are obtained for three different defect models, which a…
Motivated by experimental observations, Samajdar et al. [Nature Physics 15, 1290 (2019)] have proposed that the insulating Neel state in the parent compounds of the cuprates is proximate to a quantum phase transition to a state in which Neel order coexists with semion topological order. We study the manner in which proximity to this transition can make the phonons chiral, by inducing a significant phonon Hall viscosity. We describe the spinon-phonon coupling in a lattice spinon model coupled to a strain field, and also using a general continuum theory constrained only by symmetry. We find a no…
The phonon Hall effect has been observed in the paramagnetic insulator, Tb3Gd5O12. A magnetic field applied perpendicularly to a heat current induces a temperature gradient that is perpendicular to both the field and the current. We show that this effect is due to resonant skew scattering of phonons from the crystal field states of superstoichiometric Tb^3+ ions. This scattering originates from the coupling between the quadrupole moment of Tb^3+ ions and the lattice strain. The estimated magnitude of the effect is consistent with experimental observations at T=5 K, and can be significantly enh…
Thermal Hall conductivity of semimetallic graphite dominated by ambipolar phonon drag
Wuhan National High Magnetic Field Center, School of Physics, Huazhong University of Science and Technology（华中科技大学物理学院）；Laboratoire de Physique et d’Étude des Matériaux (ESPCI - CNRS - Sorbonne Université), PSL Research University（巴黎高等物理化工学院-法国国家科学研究中心-索邦大学物质物理与研究实验室）
Using an exact nonequilibrium Green's function formulism, the phonon Hall effect for paramagnetic dielectrics is studied in a four-terminal device setting. The temperature difference in the transverse direction of the heat current is calculated for two-dimensional models with the magnetic field perpendicular to the plane. We find a surprising result that the square lattice does not have the phonon Hall effect while a honeycomb lattice has. This can be explained by symmetry. The temperature difference changes sign if the magnetic field is sufficiently large.
In this study, we report first experimental evidence of spin-momentum locking in the topological phonons in an inhomogeneously strained Si thin film. The spin-momentum locking in the topological phonons lead to a longitudinal spin texture or spatially inhomogeneous spin distribution in the freestanding sample structure. The spin texture was uncovered using location dependent Hall effect and planar Hall effect measurement. The charge carrier density and anomalous Hall resistance showed a linear behavior along the length of the sample. Similarly, the planar Hall resistance related with the spin …
We study the acoustic phonon response of crystals hosting a gapped time-reversal symmetry breaking electronic state. The phonon effective action can in general acquire a dissipationless "Hall" viscosity, which is determined by the adiabatic Berry curvature of the electron wave function. This Hall viscosity endows the system with a characteristic frequency, ω_v; for acoustic phonons of frequency ω, it shifts the phonon spectrum by an amount of order (ω/ω_v)^2 and it mixes the longitudinal and transverse acoustic phonons with a relative amplitude ratio of ω/ω_v and with a phase shift of +/- π/2,…
We measure and analyze the effective spin Hall angle of platinum in the low residual resistivity regime by second harmonic measurements of the spin-orbit torques for a multilayer of Pt/Co/AlO$_x$. An angular dependent study of the torques allows us to extract the effective spin Hall angle responsible for the damping-like torque in the system. We observe a strikingly non-monotonic and reproducible temperature dependence of the torques. This behavior is compatible with recent theoretical predictions which include both intrinsic and extrinsic (impurities and phonons) contributions to the spin Hal…
It is known that phonons have angular momentum, and when the time-reversal symmetry (TRS) is broken, the total phonon angular momentum in the whole system becomes nonzero. In this paper, we propose that as an angular momentum of phonons for a crystal without TRS, we need to consider the canonical angular momentum, as opposed to the kinetic angular momentum in previous works. Next, we show that the angular momentum of phonons without TRS exhibits universal behaviors near the $Γ$ point. We focus on in-plane oscillations in two-dimensional crystals as an example. By breaking the TRS, one of the a…
We investigate the time dependent thermal relaxation of a two-dimensional electron system in the fractional quantum Hall regime where ballistic phonons are used to heat up the system to a non-equilibrium temperature. The thermal relaxation of a 2DES at $ν=1/2$ can be described in terms of a broad band emission of phonons, with a temperature dependence proportional to $T^4$. In contrast, the relaxation at fractional filling $ν=2/3$ is characterized by phonon emission around a single energy, the magneto-roton gap. This leads to a strongly reduced energy relaxation rate compared to $ν=1/2$ with o…
