Phonon Thermal Hall Effect

• Phonon thermal Hall effect is the emergence of a transverse temperature gradient in insulating and semiconducting solids driven solely by charge-neutral phonons under a magnetic field.
• Experimental studies using steady-state transport methods reveal a field-linear transverse conductivity, with a universal thermal Hall angle typically in the 10⁻⁴–10⁻³ range across diverse materials.
• Theoretical models debate intrinsic Berry curvature versus extrinsic skew-scattering mechanisms, highlighting the effect's sensitivity to disorder, strain, and lattice symmetry.

The phonon thermal Hall effect (THE) is the emergence of a transverse temperature gradient in an insulating or semiconducting solid subjected to a longitudinal heat current and perpendicular magnetic field, mediated entirely by charge-neutral phonons. The phenomenon is distinguished by the observation of a nonzero off-diagonal thermal conductivity, $\kappa_{xy}$, in systems lacking itinerant charge carriers or well-defined magnons. Historically considered negligible, recent discoveries reveal sizeable and universal phonon THE in a wide range of crystals, both magnetic and nonmagnetic (Xiang et al., 12 Nov 2025, Jin et al., 2024), highlighting its status as a generic and fundamental property of condensed matter. This comprehensive review addresses the key definitions, experimental realizations, mechanisms, microscopic theoretical frameworks, and unresolved issues in the field.

1. Phenomenological Definition and Quantification

The phonon thermal Hall effect is observed when a longitudinal heat current $J_Q$ along the $x$-direction, in the presence of a perpendicular magnetic field $B\parallel z$, induces a transverse temperature gradient $\nabla T_y$ (Xiang et al., 12 Nov 2025). The response is characterized by two tensors: the longitudinal thermal conductivity $\kappa_{xx}$ and the transverse (Hall) thermal conductivity $\kappa_{xy}$,

$$\kappa_{xx} = \frac{J_Q}{\nabla T_x}, \qquad \kappa_{xy} = \frac{\nabla T_y}{\nabla T_x} \kappa_{xx}.$$

The dimensionless thermal Hall angle is

$$\theta_H = \frac{\nabla T_y}{\nabla T_x} = \frac{\kappa_{xy}}{\kappa_{xx}}.$$

A nonzero $\kappa_{xy}$ exclusively due to phonons, with electrons and magnons excluded by insulation, defines the phonon thermal Hall effect.

2. Experimental Techniques and Core Observations

Phonon THE is typically measured via steady-state transport methods in high-vacuum environments, utilizing single crystal plates with embedded heater and multiple thermometers. For SrTiO$_3$, Y$_2$Ti$_2$O$_7$, quartz, MgO, and elemental Si/Ge, heat flows along a fixed axis (e.g., $x$), magnetic field is applied transverse to the plane, and longitudinal/transverse temperature gradients are recorded via antisymmetrization under field reversal to eliminate artifacts (Xiang et al., 12 Nov 2025, Jin et al., 2024, Sharma et al., 2024). The observed features are:

• Field-linear $\kappa_{xy}$ up to and often beyond 9–15 T.
• $\kappa_{xy}(T)$ and $\kappa_{xx}(T)$ peaking at the same temperature, typically 10–30 K, corresponding to the phonon-propagation optimum.
• The thermal Hall angle $\theta_H$ in the universal range $10^{-4}$–$10^{-3}$ at its maximum, with the largest absolute values $|\kappa_{xy}|$ observed to date in high-purity insulators and elemental semiconductors (Table below) (Jin et al., 2024).
• Strong suppression of the effect by disorder, strain, or chemical doping (Xiang et al., 12 Nov 2025, Li et al., 2019).
• Complete independence of $\kappa_{xy}$ from the type of electrical contact (metallic vs. insulating) used (Xiang et al., 12 Nov 2025).

Material	$T_\text{peak}$ (K)	$|\kappa_{xy}|$ (W/K·m)	$|\theta_H|_\text{max}$
SrTiO$_3$	20	0.7	$10^{-3}$
Si	30	10	$10^{-3}$
Ge	25	8	$10^{-3}$
MgO	25	0.45	$10^{-3}$
Y$_2$Ti$_2$O$_7$	15	0.96	$3\times 10^{-4}$
Cu$_3$TeO$_6$	20	1.0	$3\times 10^{-3}$

3. Disorder, Strain, and Sample Dependence

Disorder and internal strain have dramatic effects on the phonon THE. In SrTiO$_3$, only high-quality samples with long phonon mean free paths manifest the full thermal Hall angle (up to $0.3\%$ at 9 T). Disordered or strained samples show a suppressed or undetectable $\theta_H$ despite similar $\kappa_{xx}(T)$ at high $T$. Partial annealing of disordered crystals recovers the thermal Hall angle without restoring the phonon mean free path, indicating that strain textures rather than point-defect scattering critically control the effect (Xiang et al., 12 Nov 2025). Similarly, the amplitude of THE in topological insulators or pyrochlores does not simply scale with defect density, but can be restored by targeted annealing (Sharma et al., 2024, Sharma et al., 2024). This sensitivity implies a link to the lattice's microscopic symmetry, domain-structure, or strain-modulated Berry curvature.

4. Universal and Material-Specific Scaling

A quadratic scaling law $|\kappa_{xy}| \propto \kappa_{xx}^2$ is empirically established across a broad spectrum of nonmagnetic insulators, semiconductors, and even some magnetic compounds (Jin et al., 2024). This universality remains robust under variation of lattice structure, crystal purity, and field orientation (including planar Hall geometries), indicating an underlying mechanism that transcends specific bandstructure or symmetry constraints. The universal character is further reinforced by the near-constant magnitude of the “thermal Hall angle” maximum across diverse materials. Nonetheless, enhancements or suppressions related to specific structural domains (SrTiO$_3$), rare-earth ions (Dy$_2$Ti$_2$O$_7$), or anisotropic bandstructures can be observed (Xiang et al., 12 Nov 2025, Sharma et al., 2024, Li et al., 2019).

5. Microscopic Mechanisms and Theoretical Scenarios

Multiple microscopic mechanisms are advanced for the phonon THE, with ongoing debate regarding their relative importance:

(a) Intrinsic Berry Curvature of Phonons

Phonon bands in a magnetic field, or in the presence of magnetic textures, can acquire Berry curvature, giving rise to a transverse velocity in analogy to the anomalous Hall effect for electrons (Behnia, 25 Feb 2025). For both acoustic and optical modes, this curvature can arise via magneto-elastic coupling, lattice Aharonov-Bohm effects (field-induced molecular Berry phase) (Behnia, 25 Feb 2025), or time-reversal symmetry breaking by internal or external fields (Hu et al., 5 Jan 2025). The general Kubo expression is

$$\kappa_{xy} = -\frac{k_B^2 T}{\hbar V} \sum_{n} \int_{BZ} d^3k\, c_2[f_n(k)]\, \Omega_n(k),$$

where $c_2$ is a known function and $\Omega_n$ the phonon Berry curvature. In high-symmetry lattices, discrete rotational symmetry can enhance $\kappa_{xy}$ via sharply peaked Berry curvature near high-symmetry lines (Hu et al., 5 Jan 2025). However, bare Berry-curvature models typically underestimate the observed magnitudes by several orders of magnitude (Hu et al., 5 Jan 2025, Jin et al., 2024).

(b) Extrinsic Skew-Scattering by Defects and Impurities

An alternative mechanism invokes skew-scattering of phonons off charged impurities or lattice defects in a magnetic field (Flebus et al., 2021, Sharma et al., 2024, Yan et al., 16 Sep 2025). Here, Lorentz forces acting on the internal charge density of defects break time-reversal symmetry and create an antisymmetric, field-linear scattering cross-section giving

$$\kappa_{xy}/\kappa_{xx} \sim \Lambda_\omega \sim 10^{-3},$$

where $\Lambda_\omega$ is a skewness parameter empirically observed to be nearly temperature independent in the Rayleigh limit. The extrinsic scenario explains the universality and magnitude of THE in ionic crystals and topological insulators.

(c) Spin–Phonon and Spin-Chirality Coupling

In Mott insulators, paramagnets, and antiferromagnets, spin–phonon coupling enables additional mechanisms. Resonant (side-jump or skew) scattering by paramagnetic doublets produces large $\kappa_{xy}$ in Pr$_2$Ir$_2$O$_7$ (Guo, 2023, Uehara et al., 2022). In systems with scalar spin-chirality fluctuations, Berry-phase–like skew scattering is induced without requiring spin-orbit coupling, resulting in a Hall angle of $10^{-3}-10^{-2}$ (Oh et al., 2024). The derived Hamiltonians generically couple local phonon angular momentum to emergent or actual magnetic fields set by real or effective spin-chirality (Oh et al., 2024).

(d) Electron–Phonon Hall Drag in Dilute Metals

In materials like SrTiO$_3$ rendered metallic by oxygen vacancies, a large THE emerges from momentum-conserving electron–phonon collisions in the presence of a large electronic Hall angle, yielding a drag-induced phonon $\kappa_{xy}$ that far exceeds the sum of electronic and intrinsic phonon contributions (Jiang et al., 2022).

6. Theoretical Constraints and Open Questions

The experimental evidence imposes several constraints on viable theories:

• Mechanisms must reproduce the observed magnitude ($|\kappa_{xy}|/|\kappa_{xx}| \sim 10^{-3}$), scaling ($|\kappa_{xy}|\propto \kappa_{xx}^2$), and field/temperature dependence.
• The sensitivity to disorder, strain, and sample history must be physically explained—indicating possible domain-averaged Berry curvatures, or strain-induced cancellation.
• Purely Berry-curvature–driven models generally underpredict the amplitude.
• The persistence of the phonon THE in fully nonmagnetic, highly pure materials (MgO, Si, Y$_2$Ti$_2$O$_7$) supports an intrinsic contribution, yet similar signals in more disordered crystals point to a significant extrinsic component (Jin et al., 2024, Flebus et al., 2021, Sharma et al., 2024).

Despite the progress, the precise microscopic lattice Hamiltonian yielding the universal phonon THE on the observed scale remains unidentified. Both intrinsic (topological) and extrinsic (skew-scattering) channels may contribute in parallel or dominance depending on sample, temperature, and field.

7. Implications and Outlook

The identification of a ubiquitous, universal phonon thermal Hall effect in insulators, semiconductors, and certain magnetic materials recasts the interpretation of transverse thermal transport. The phononic contribution, previously neglected, must be considered as a baseline in all systems displaying $\kappa_{xy}$, including those previously ascribed to exotic neutral quasiparticles (magnons, spinons, Majorana modes). Advances in material engineering, domain/strain control, and ab initio phononic Berry curvature calculations are required to further resolve the intrinsic–extrinsic dichotomy and to harness the phonon THE for technological applications such as cryogenic heat management and nonreciprocal phononic devices (Xiang et al., 12 Nov 2025, Jin et al., 2024).