Electric Valley Hall Effect Overview

• Electric Valley Hall Effect is a phenomenon where an in-plane electric field induces a transverse valley current using valley-contrasting Berry curvature and symmetry-breaking conditions.
• It encompasses multiple transport mechanisms, including Berry curvature, anisotropic Drude responses, and phase-coherent junction effects, each dependent on material symmetry and structure.
• Experimental realizations in monolayer MoS₂ and related systems highlight inversion symmetry’s role and validate the theoretical predictions of anomalous Hall responses.

Searching arXiv for recent and foundational papers on the electric valley Hall effect and closely related variants. Using arXiv search results to anchor the overview in foundational observation papers and later generalizations. The electric valley Hall effect denotes a class of transverse transport phenomena in which an applied electric field generates a response distinguished by valley index rather than by charge alone. In its canonical form, an in-plane electric field drives a longitudinal charge current and a transverse valley current in a multivalley system with valley-contrasting Berry curvature. When the two valleys are equally populated, the transverse charge contributions cancel and only a pure valley current remains; when a valley population imbalance is present, the cancellation is incomplete and a net transverse charge Hall response appears even without an external magnetic field (Mak et al., 2014, Cao et al., 14 Mar 2026). Subsequent literature has broadened the term to include strain-driven classical transport, anomalous variants tied to spontaneous valley polarization, time-reversal-invariant valleys with anisotropic Drude transport, and junction-based mechanisms that are independent of Berry curvature (Zhang et al., 2016, Tong et al., 2016, Cao et al., 14 Mar 2026, Zeng, 2 Oct 2025).

1. Canonical definition and Berry-curvature framework

For two valleys $V_1$ and $V_2$, the valley current is defined as

$$j_a^v \equiv j_a^{V_1} - j_a^{V_2} = \sum_b \sigma_{ab}^v E_b,$$

and the valley Hall angle is

$$\theta_{\mathrm{VH}} \equiv \left| \frac{j_\perp^v}{j_\parallel^c} \right|.$$

In the standard valleytronic setting, the electric valley Hall effect is therefore the generation of a transverse valley current by a longitudinal electric field (Cao et al., 14 Mar 2026).

In inversion-broken honeycomb-like crystals such as monolayer MoS$_2$, the relevant low-energy valleys lie at $K$ and $K'$, with valley index $\tau=\pm1$. The Bloch bands acquire Berry curvature $\boldsymbol{\Omega}_n(\mathbf{k})$, strongly peaked near the valleys and opposite in sign between them. Within semiclassical dynamics,

$$\dot{\mathbf r} = \frac{1}{\hbar}\nabla_{\mathbf k}\varepsilon_n(\mathbf k) + e\,\mathbf E \times \boldsymbol{\Omega}_n(\mathbf k),$$

so the Berry-curvature term produces an anomalous velocity transverse to $V_2$0. Because $V_2$1 changes sign under $V_2$2, carriers in different valleys drift in opposite transverse directions (Mak et al., 2014).

At thermal equilibrium, time-reversal symmetry enforces equal valley populations, so the two transverse charge currents cancel. The system still supports a finite valley Hall current, but no net charge Hall current. If a valley population imbalance $V_2$3 is created, the cancellation is incomplete and a measurable transverse charge Hall response appears. This is the sense in which the electric valley Hall effect connects pure valley transport and helicity- or magnetization-dependent anomalous Hall signals (Mak et al., 2014).

2. Symmetry structure and mechanism classes

The conventional electric valley Hall effect is governed by a precise symmetry pattern. In $V_2$4-type systems, the two valleys are related by time reversal and are not individually invariant under it. In that case, broken inversion symmetry is necessary for a nonzero valley Hall response, and the dominant mechanism is Berry-phase related (Cao et al., 14 Mar 2026).

Monolayer MoS$V_2$5 provides the canonical example: inversion symmetry is absent, while time-reversal symmetry is preserved. Bilayer 2H-MoS$V_2$6, by contrast, restores inversion symmetry, so the valley-contrasting Berry curvature cancels in the total crystal response and the anomalous Hall signal disappears under otherwise similar optical pumping conditions (Mak et al., 2014).

Later work identified a distinct class of time-reversal-invariant valleys (TRIVs), where each valley center is itself invariant under time reversal and the two valleys are instead related by a crystalline symmetry such as $V_2$7, $V_2$8, $V_2$9, or $$j_a^v \equiv j_a^{V_1} - j_a^{V_2} = \sum_b \sigma_{ab}^v E_b,$$0. In those systems the Berry-curvature contribution cannot generate the valley conductivity; the response is built from the time-reversal-even part of the conductivity tensor, and inversion symmetry need not be broken (Cao et al., 14 Mar 2026).

Regime	Symmetry condition	Dominant mechanism
Conventional $$j_a^v \equiv j_a^{V_1} - j_a^{V_2} = \sum_b \sigma_{ab}^v E_b,$$1 VHE	Valleys related by $$j_a^v \equiv j_a^{V_1} - j_a^{V_2} = \sum_b \sigma_{ab}^v E_b,$$2; broken inversion required	Berry curvature
TRIV eccentricity VHE	Valleys at time-reversal-invariant momenta	Anisotropic Drude response
Junction EVHE	Buckled tunnel junction with perpendicular field	Backreflection phase

This taxonomy is consequential because the same phrase, “electric valley Hall effect,” now spans several transport mechanisms. In the conventional case, the response scales with Berry curvature and therefore with band geometry near valley extrema. In TRIV systems, the response can instead be controlled by Fermi-surface eccentricity. In junction realizations, it can arise from phase-coherent scattering without any Berry-curvature contribution (Cao et al., 14 Mar 2026, Zeng, 2 Oct 2025).

3. First experimental realization in monolayer MoS$$j_a^v \equiv j_a^{V_1} - j_a^{V_2} = \sum_b \sigma_{ab}^v E_b,$$3

The first observation of the valley Hall effect was reported in monolayer MoS$$j_a^v \equiv j_a^{V_1} - j_a^{V_2} = \sum_b \sigma_{ab}^v E_b,$$4 transistors illuminated by circularly polarized light, which preferentially excites carriers into one valley and produces a finite anomalous Hall voltage whose sign is controlled by light helicity (Mak et al., 2014). The devices were fabricated on Si/300 nm SiO$$j_a^v \equiv j_a^{V_1} - j_a^{V_2} = \sum_b \sigma_{ab}^v E_b,$$5 with Hall-bar geometry, used a 657 nm diode laser on resonance with the A exciton, and employed a photoelastic modulator at 50 kHz to alternate the optical helicity.

Several signatures isolated the effect. The measured Hall voltage was linear in the longitudinal bias $$j_a^v \equiv j_a^{V_1} - j_a^{V_2} = \sum_b \sigma_{ab}^v E_b,$$6, reversed sign when the modulation changed from R–L to L–R, vanished for purely linear polarization, and appeared only in monolayers rather than bilayers. No external magnetic field was applied. These observations ruled out the ordinary Hall effect and generic photovoltage artifacts, and tied the signal to valley-selective optical pumping and Berry-curvature-driven anomalous transport (Mak et al., 2014).

Mak and co-workers further emphasized that the anomalous Hall conductivity is linear in the valley density imbalance. In the notation of the paper, the main-text expression is

$$j_a^v \equiv j_a^{V_1} - j_a^{V_2} = \sum_b \sigma_{ab}^v E_b,$$7

quoted up to numerical factors when intrinsic and side-jump terms are combined. Experimentally, the directly accessible quantity was the photogenerated carrier density $$j_a^v \equiv j_a^{V_1} - j_a^{V_2} = \sum_b \sigma_{ab}^v E_b,$$8, inferred from photoconductivity and used as an upper bound on $$j_a^v \equiv j_a^{V_1} - j_a^{V_2} = \sum_b \sigma_{ab}^v E_b,$$9. The extracted $$\theta_{\mathrm{VH}} \equiv \left| \frac{j_\perp^v}{j_\parallel^c} \right|.$$0 increased linearly with $$\theta_{\mathrm{VH}} \equiv \left| \frac{j_\perp^v}{j_\parallel^c} \right|.$$1, with magnitude of the correct order, especially at higher electron density where disorder is weaker (Mak et al., 2014).

The monolayer–bilayer contrast established the role of inversion symmetry with unusual clarity. Bilayer devices showed similar photoconductivity at the channel center but essentially zero anomalous Hall response under identical optical conditions. This comparative symmetry test became a template for later electric valley Hall experiments in other 2D systems (Mak et al., 2014).

4. Transport signatures, local readout, and nonlocal propagation

Beyond direct Hall voltages, the electric valley Hall effect is commonly diagnosed through nonlocal transport. In strained graphene, modest strain generates a pseudo-magnetic field $$\theta_{\mathrm{VH}} \equiv \left| \frac{j_\perp^v}{j_\parallel^c} \right|.$$2 that couples with opposite sign to the two valleys. Starting from the quantum Boltzmann equation, the constitutive relations take the form

$$\theta_{\mathrm{VH}} \equiv \left| \frac{j_\perp^v}{j_\parallel^c} \right|.$$3

$$\theta_{\mathrm{VH}} \equiv \left| \frac{j_\perp^v}{j_\parallel^c} \right|.$$4

so the direct and inverse valley Hall effects appear on equal footing. In Hall-bar geometry, the asymptotic nonlocal resistance is

$$\theta_{\mathrm{VH}} \equiv \left| \frac{j_\perp^v}{j_\parallel^c} \right|.$$5

with decay length controlled by the valley diffusion length $$\theta_{\mathrm{VH}} \equiv \left| \frac{j_\perp^v}{j_\parallel^c} \right|.$$6 (Zhang et al., 2016).

A related nonlocal signature appears in interaction-driven bilayer graphene. In the spontaneous antiferromagnetic state of the half-filled zero Landau level, nonlocal transport exhibits a cubic scaling relation between local and nonlocal resistance, which was identified as evidence for intrinsic Hall transport carried by a charge-neutral spin–valley current (Tanaka et al., 2019). This placed the electric valley Hall framework in direct contact with correlated symmetry-breaking phases.

Local Hall readout can also be combined with optical valley initialization. In gapped bilayer graphene, an out-of-plane electric field tunes inversion symmetry and bandgap in situ, while circularly polarized mid-infrared light creates a valley population imbalance. The resulting valley-selective Hall effect was found to be orders of magnitude larger than in MoS$$\theta_{\mathrm{VH}} \equiv \left| \frac{j_\perp^v}{j_\parallel^c} \right|.$$7, attributed to the inverse scaling of Berry curvature with bandgap (Yin et al., 2022). The same work used the measured valley-selective Hall conductivity to track the evolution of Berry curvature with bandgap.

The effect can also be imaged without relying exclusively on edge accumulation. First-principles calculations for monolayer MoS$$\theta_{\mathrm{VH}} \equiv \left| \frac{j_\perp^v}{j_\parallel^c} \right|.$$8 showed that an electric field induces a change in the electrons’ charge-density orientation and local density of states, and that this real-space response can be used to measure the energy-dependent valley and orbital Hall conductivity (Xue et al., 2020). This establishes a direct connection between momentum-space Berry curvature and electric-field-induced reorientation of the real-space electronic density.

Nonlinearity adds a further transport layer. In a graphene–hBN moiré superlattice, second-harmonic nonlocal voltages under AC drive established a nonlinear valley Hall effect with quadratic current scaling and quartic scaling with local resistance, while third- and fourth-harmonic nonlocal voltages revealed a nonlinear inverse valley Hall effect (He et al., 5 Mar 2025). This extended valley Hall transport beyond the previously dominant linear-response paradigm.

5. Anomalous, time-reversal-invariant, and perpendicular-field variants

One major extension of the electric valley Hall effect is the anomalous valley Hall effect in systems with spontaneous valley polarization. In the ferrovalley proposal based on monolayer 2H-VSe$$\theta_{\mathrm{VH}} \equiv \left| \frac{j_\perp^v}{j_\parallel^c} \right|.$$9, spin–orbit coupling and intrinsic exchange lift valley degeneracy, so the opposite valley Hall currents no longer cancel and a net transverse charge Hall current appears without external magnetic field (Tong et al., 2016). A related first-principles study of single-layer 2H-FeCl$_2$0 found a spontaneous conduction-band valley polarization of about 101 meV and predicted a spin- and valley-polarized anomalous Hall current under an in-plane electric field (Zhaos et al., 2020).

Antiferromagnetic systems provide a different route. In A-type hexagonal AFM monolayers such as Cr$_2$1CH$_2$2, the combined $_2$3 symmetry suppresses the total Berry curvature, but an out-of-plane electric field produces a layer-dependent electrostatic potential, breaks $_2$4, and induces spin splitting. The resulting response was described as a layer-locked anomalous valley Hall effect, with the sign of the Hall response reversed by reversing the electric field (Guo et al., 2023).

A more radical shift comes from time-reversal-invariant valleys. The “eccentricity valley Hall effect” reformulates electric VHE for valleys located at time-reversal-invariant momenta. For an elliptical valley Fermi surface with eccentricity $_2$5, the valley Hall angle becomes

$_2$6

a purely geometric quantity independent of chemical potential, scattering time, effective-mass scale, and gap size within the regime of validity. The effect was argued to emerge universally across all 25 layer groups supporting symmetry-connected TRIVs, and monolayer GeS$_2$7 was predicted to exhibit $_2$8 (Cao et al., 14 Mar 2026).

The phrase “electric valley Hall effect” has also acquired a more specialized meaning in junction physics. In an all-in-one tunnel junction based on a buckled 2D hexagonal Dirac material, a perpendicular electric field applied to one electrode induces a backreflection phase in the spacer, causing skew transmission in transverse momentum. In this setting the electric valley Hall effect is the generation of a transverse valley current in response to the perpendicular electric field, independent of Berry curvature. The valley Hall conductance is odd in the perpendicular field, while the spin Hall conductance is even (Zeng, 2 Oct 2025). This usage preserves the central idea of electrically generated valley transport but abandons the conventional in-plane-field, Berry-curvature mechanism.

6. Conceptual caveats, unresolved issues, and current outlook

A central conceptual correction concerns the observability of valley accumulation. In multi-valley, time-reversal-invariant insulators, valley density is not conserved in the naïve sense once the electric field driving the valley Hall effect is included. A fully gapped insulator can support a bulk valley Hall current and yet show no valley density accumulation at the edges; in that circumstance the valley Hall effect cannot be observed through edge accumulation. If the system is not fully gapped, edge accumulation becomes possible and is governed by a Fermi-surface average rather than by undergap states (Kazantsev et al., 2022). This directly qualifies the widespread assumption that any nonzero bulk valley Hall conductivity must generate observable edge polarization.

A second caution concerns excitons. For the exciton valley Hall effect in 2D crystals under synthetic electric field or phonon drag, the anomalous velocity associated with Berry curvature cancels out of the net effect. The surviving valley Hall current is governed by side-jump and skew-scattering mechanisms, not by anomalous velocity (Glazov et al., 2020). This result is specific to neutral composite excitations, but it underscores that “Berry-curvature-driven” cannot be used indiscriminately across all valley Hall settings.

Even in canonical electronic systems, open questions remain. In monolayer MoS$_2$9, the gate dependence of $K$0 did not follow the simplest intrinsic theory, and the slope of $K$1 versus photocarrier density increased with gate voltage. The paper identified possible roles for disorder, carrier-density-dependent extrinsic terms such as skew scattering and side jump, and incomplete valley polarization of photocarriers. Intervalley and spin relaxation likewise remained central unresolved quantities (Mak et al., 2014).

Practical control has nonetheless advanced substantially. In monolayer p-type WSe$K$2 transistors, electrically generated spin/valley Hall accumulation was imaged by Kerr rotation, shown to persist up to 160 K, and interpreted with a drift–diffusion model together with reflection spectra. The same analysis yielded lower-bound spin/valley lifetimes of 4.1 ns below 90 K and 0.26 ns at 160 K, demonstrating that all-electrical generation and optical readout of coupled spin–valley Hall transport is not confined to the sub-30 K regime that dominated earlier optical studies (Li et al., 2022).

Taken together, these developments define the electric valley Hall effect not as a single mechanism but as a transport family. Its canonical member is the Berry-curvature-driven transverse valley current of inversion-broken multivalley bands; its experimentally established archetype remains helicity-controlled Hall transport in monolayer MoS$K$3; and its contemporary generalizations now include spontaneous anomalous responses, TRIV eccentricity-driven transport, nonlinear nonlocal conversion, and phase-coherent junction phenomena (Mak et al., 2014, Cao et al., 14 Mar 2026, He et al., 5 Mar 2025, Zeng, 2 Oct 2025). The unifying theme is electrical generation of valley-resolved transverse transport, but the operative symmetry, observable, and microscopic mechanism depend sharply on the material class and measurement geometry.