Ferron Hall effect: Transverse accumulation of polarization driven by thermal gradients in ferroelectrics

Abstract: The phonon Hall effect describes the generation of a transverse heat current in response to a longitudinal thermal gradient in a magnetic field. Here, we theoretically demonstrate that, when the lattice excitations deflected by the Hall effect carry electric dipole moments, their transverse motion produces an accumulation of electric polarization in ferroelectric materials. This accumulation is driven by lattice excitations that carry polarization, known as ferrons, and we therefore call the mechanism the ferron Hall effect. Using atomistic lattice dynamics with parameters obtained from density functional theory, we illustrate the effect in the prototypical ferroelectric BaTiO3. Our results identify ferrons as the electric-polarization analogues of magnons in transverse transport and provide a route toward thermal and magnetic manipulation of ferroic order.

• The paper demonstrates theoretically that a longitudinal thermal gradient and magnetic field generate opposite, transverse polarization accumulations at ferroelectric edges, reaching an estimated 16 μC/m² in BaTiO₃.
• The authors use first-principles-parametrized atomistic Langevin dynamics and a gyroscopic coupling to show that field-odd kinetic-energy redistribution, combined with anharmonicity, produces the ferron Hall signal.
• The predicted response reverses with both magnetic field and thermal-gradient direction, distinguishing it from pyroelectricity and thermopolarization while suggesting detection through piezoresponse force microscopy, second-harmonic generation, or lock-in measurements.

The Hall effect family has expanded well beyond its electronic origins to encompass bosonic quasiparticles: magnons, photons, and phonons all exhibit transverse deflection under broken time-reversal symmetry. The paper by Bustamante Lopez, Brehm, and Juraschek (2606.29765) extends this hierarchy to a previously unaddressed degree of freedom—electric polarization—by demonstrating theoretically that a longitudinal thermal gradient applied to a ferroelectric in a magnetic field produces a transverse accumulation of polarization at the sample edges. Because the transported excitations are ferrons, the dipole-carrying excitations of the ferroelectric order parameter, the authors term the mechanism the ferron Hall effect. The demonstration is carried out with atomistic Langevin dynamics on a first-principles-parametrized local-mode model of BaTiO$_3$.

Theoretical formalism

The low-energy lattice dynamics are reduced to the polar soft mode of the ferroelectric, represented by a local-mode coordinate $\mathbf{u}_i$ per unit cell. This single projected degree of freedom carries three observables simultaneously: electric polarization via the mode effective charge tensor $\mathbf{Z}$, kinetic energy via the projected mass matrix $\mathbf{M}$, and phonon angular momentum via antisymmetric angular-momentum matrices $\boldsymbol{\Lambda}^{(k)}$. The energy-conserving Hamiltonian combines an anharmonic on-site double-well potential along the ferroelectric coordinate, short-range intercell couplings obtained by projecting DFT interatomic force constants onto the soft-mode basis, and a long-range screened dipole-dipole interaction evaluated in reciprocal space with the high-frequency dielectric tensor $\boldsymbol{\epsilon}_\infty$; the $q=0$ component is set to zero, corresponding to no imposed macroscopic depolarizing field.

Time-reversal symmetry is broken phenomenologically through a gyroscopic force $\mathbf{G}_{\Omega_B}\dot{\mathbf{u}}_i$, where $$\mathbf{G}_{\Omega_B}=\Omega_B\mathbf{M}^{1/2}\mathbf{C}(\hat{\mathbf{B}})\mathbf{M}^{1/2}$$ is antisymmetric. Because $$\dot{\mathbf{u}}_i^T\mathbf{G}_{\Omega_B}\dot{\mathbf{u}}_i=0$$, this Zeeman-like coupling rotates the local-mode velocity without doing work—it deflects transport but neither injects nor dissipates energy. Nonequilibrium driving is imposed through a spatially varying Langevin bath satisfying the fluctuation-dissipation relation in the projected mass coordinates, and the equations are integrated with a symmetric BAOAB splitting in which the gyroscopic step is solved exactly as a velocity rotation and the Langevin step as an exact Ornstein–Uhlenbeck update.

First-principles parametrization of BaTiO$\mathbf{u}_i$0

The model parameters derive from PBEsol DFT calculations (VASP, PAW potentials, 600 eV cutoff) with frozen-phonon force constants from $\mathbf{u}_i$1 supercells. The cubic soft-mode basis yields a nearly isotropic mode effective charge $\mathbf{u}_i$2, dominated by the anomalous Ti charge ($\mathbf{u}_i$3) and the longitudinal oxygen charge ($\mathbf{u}_i$4), reproducing the established dynamical-charge pattern of perovskite ferroelectrics. The on-site double-well potential is calibrated from a fixed-cell Berry-phase switching path between tetragonal variants: the barrier is 38.43 meV/f.u., close to the 34.63 meV PBEsol value reported independently, and the model polarization $\mathbf{u}_i$5 C/m$\mathbf{u}_i$6 matches the Berry-phase value of 0.3466 C/m$\mathbf{u}_i$7 to within 3.35%. The authors candidly note two modeling concessions: the projection residual of the tetragonal endpoint onto a single cubic soft-mode coordinate is substantial ($\mathbf{u}_i$8), so the coordinate is a reduced representation rather than a full reconstruction; and the transverse confinement $\mathbf{u}_i$9 is not fitted to any first-principles path but chosen as a weak simulation parameter.

The ferron Hall response

Simulations on a $\mathbf{Z}$0 slab, periodic along the longitudinal and out-of-plane directions, use a hot central band at 50 K between cold regions near zero temperature, producing oppositely directed gradients on either side. Field-odd responses are isolated as $\mathbf{Z}$1 and averaged over 200 noise realizations. Three channel-resolved results emerge from the same driven soft-mode dynamics:

Channel	Observable	Response
Angular momentum	$\mathbf{Z}$2	Edge accumulation $\mathbf{Z}$3/cell, present even at zero field
Thermal	$\mathbf{Z}$4	Phonon-Hall-type field-odd edge redistribution
Polarization	$\mathbf{Z}$5	Ferron Hall accumulation, odd in both $\mathbf{Z}$6 and $\mathbf{Z}$7

The central result is the polarization map: $\mathbf{Z}$8 shows opposite-sign shifts of the longitudinal ferroelectric polarization at the two transverse edges, reversing sign across the hot band where the thermal gradient reverses. For BaTiO$\mathbf{Z}$9 the estimated edge accumulation reaches 16 μC/m²—about four orders of magnitude smaller than the equilibrium spontaneous polarization (~0.35 C/m$\mathbf{M}$0), but comparable to weak polar signals in magnetically induced multiferroics, which the authors argue places it within experimental reach.

A useful analytic result connects the polarization shift to the energy channel: expanding the steady-state force balance about the double-well minimum shows that cubic anharmonicity converts fluctuation variance into a mean displacement,

$\mathbf{M}$1

so regions where the magnetic field enhances longitudinal soft-mode kinetic energy exhibit reduced polarization, and vice versa. The ferron Hall signal is thus the anharmonic-well polarization response to a phonon-Hall-like redistribution of kinetic energy—an internally consistent microscopic mechanism rather than a purely phenomenological assertion.

Relation to other thermally driven polar responses

The paper distinguishes the effect carefully from neighboring phenomena: unlike the Seebeck effect (longitudinal voltage), pyroelectricity (polarization change under temperature modulation), or thermopolarization (gradient-induced bulk polarization), the ferron Hall effect produces transverse, edge-localized polarization shifts that are odd under reversal of both magnetic field and thermal gradient. This signature makes it cleanly separable experimentally via lock-in detection against simultaneous reversal of both drives.

Limitations and open questions

The authors are explicit that the work is a demonstration of principle. The gyroscopic coupling strength $\mathbf{M}$2 THz is treated as a model parameter; material-specific values require microscopic magnetophononic couplings that remain to be computed or measured for BaTiO$\mathbf{M}$3. Real samples introduce complications absent from the model: domain structures and domain walls, surface electrostatic boundary conditions, mobile screening charges, and defects could all modify or mask the transverse polarization profile. Whether the predicted 16 μC/m$\mathbf{M}$4 survives these effects—and whether materials with stronger magnetophononic coupling yield substantially larger signals—are open questions the paper leaves to future combined theory-experiment effort. Proposed detection routes include piezoresponse force microscopy, spatially resolved second-harmonic generation, or side-electrode lock-in measurements in a thermal Hall-bar geometry adapted from phonon Hall experiments.

Conclusion

This work completes a hierarchy of lattice Hall effects in which the transverse channel is heat, phonon angular momentum, or electric polarization, and extends ferronics from longitudinal polarization transport to Hall-type transverse responses. By establishing ferrons as the electric-polarization analogues of magnons in transverse transport, the paper identifies a concrete route toward thermal and magnetic manipulation of ferroic order, and suggests that other local-double-well lattice orders—ferroaxiality and chirality—may support analogous transverse responses.