Valley Hall Effect in 2D Materials

• Valley Hall effect is a topological transport phenomenon in 2D materials that leverages Berry curvature to enable transverse, valley-polarized carrier flows.
• It arises from both intrinsic contributions (Berry curvature) and extrinsic mechanisms such as side-jump and skew scattering, as observed in TMDs and graphene.
• This effect underpins valleytronic devices like filters and rectifiers, with detection methods including nonlocal resistance and spatially-resolved photoluminescence.

The valley Hall effect (VHE) is a topological transport phenomenon arising in crystalline materials with multiple energy valleys in their electronic band structure. Distinguished by the flow of valley-polarized carriers transverse to applied forces, VHE requires spatial or symmetry-induced differentiation of valleys, most often realized in two-dimensional materials such as transition metal dichalcogenides (TMDs), graphene, and their van der Waals heterostructures. The effect can be observed in charge, exciton, magnon, or photonic carriers, and is underpinned by the Berry curvature—an effective momentum-space magnetic field—which is equal in magnitude and opposite in sign for time-reversed valley pairs. VHE serves as a foundational mechanism for valleytronics, enabling the generation, manipulation, and detection of the valley degree of freedom as an information carrier. This article presents a comprehensive view of the VHE, encompassing its theoretical foundation, intrinsic and extrinsic mechanisms, key experimental observations, extensions to nonlinear and excitation-specific regimes, and its role in emerging valleytronic and topologically nontrivial phases.

1. Theoretical Foundation: Berry Curvature and Valley Contrasts

The VHE fundamentally arises from the momentum-space Berry curvature $\Omega_n(\mathbf{k})$, defined for a Bloch band $n$ as

$$\Omega_n(\mathbf{k}) = i \left( \langle \partial_{k_x} u_{n,\mathbf{k}} | \partial_{k_y} u_{n,\mathbf{k}} \rangle - (x \leftrightarrow y) \right)$$

where $u_{n,\mathbf{k}}$ is the cell-periodic part of the Bloch function. In systems lacking inversion symmetry but respecting time-reversal symmetry $\mathcal{T}$, such as monolayer MoS$_2$ or hBN-aligned graphene, $\Omega_n$ is sharply peaked, with opposite sign, at inequivalent valleys (labelled $K$, $K'$ or $X$, $Y$). The semiclassical anomalous velocity acquired by a wavepacket is

$$\mathbf{v}_\mathrm{a} = -\frac{1}{\hbar} \mathbf{F} \times \Omega_n(\mathbf{k})$$

under a force $\mathbf{F}$. This velocity points in opposite transverse directions for carriers in $K$ and $K'$ valleys, yielding a net valley current $j^\mathrm{v} = j_K - j_{K'}$ without net transverse charge current.

The valley Hall conductivity is given, per spin, as

$$\sigma_{xy}^{v} = \frac{e^2}{\hbar} \sum_n \int_{\mathrm{BZ}} \frac{d^2k}{(2\pi)^2} f_n(\mathbf{k})\,\tau_z\,\Omega_n(\mathbf{k})$$

where $\tau_z$ labels the valley, and $f_n(\mathbf{k})$ is the occupation function.

Berry curvature is highly sensitive to band structure details and symmetry configuration. For example, in Bernal-stacked bilayer graphene, an interlayer potential difference introduces $\Delta$ and yields a Berry curvature

$$\Omega(\mathbf{k}) = -\tau \frac{\hbar^2 v_F^2 \Delta}{2 [\Delta^2 + (\hbar v_F k)^2 ]^{3/2}}$$

localized near each valley $K$, $K'$, and changes sign under valley inversion (Shintaku et al., 2023).

2. Intrinsic and Extrinsic Mechanisms: Disorder, Scattering, and Interactions

While the intrinsic VHE follows directly from Berry curvature, real materials often exhibit extrinsic contributions. In the presence of disorder or phonon drag, three key mechanisms contribute (Olsen et al., 2015, Glazov et al., 2020, Glazov et al., 2020):

• Intrinsic (Berry curvature) contribution: Dominates in clean samples for carrier densities near the band gap in inversion-broken 2D materials. Quantified in massive Dirac models as

$$\sigma_{xy}^{\mathrm{int}} = \frac{e^2 \Delta}{2h \sqrt{\Delta^2 + (v \hbar k_F)^2}}$$

• Side-jump contribution: Real-space displacement of carriers during scattering, independent of impurity concentration to leading order, modifies the total VHE with a correction that can have opposite sign to the intrinsic part. Exact cancellation between side-jump and Berry contributions can occur depending on the nature of the driving force.
• Skew-scattering contribution: Asymmetric scattering by impurities or phonons introduces a conductivity term scaling as $1/x$ with impurity concentration $x$, dominating in the ultra-clean limit and enabling large, sometimes divergent, valley Hall conductivity.

Table 1 summarizes the composition of the total valley Hall conductivity in monolayer MoS$_2$ (Olsen et al., 2015):

Contribution	Scaling with Impurity Concentration	Regime of Dominance
Intrinsic	$x^0$	Clean limit, low doping
Side-jump	$x^0$	Moderate disorder
Skew scattering	$x^{-1}$	Ultra-clean limit

Electron-electron and electron-hole interactions further renormalize the VHE, introducing temperature and carrier-density dependent corrections, with direct and annihilation-type e–h processes in intrinsic semiconductors (Eliseev et al., 11 Jul 2024).

3. Generalizations and Regimes: Nonlinear, Crystal, and Magnon Valley Hall Effects

Nonlinear Valley Hall Effect

The VHE can occur in higher-order response even in crystals possessing both time-reversal and inversion symmetry. In this regime, the second-order valley current is driven by the electric-field induced correction to the Berry curvature,

$j^\mathrm{(2)}_{v,a} = \chi_{a;bc} E_b E_c$

where $\chi_{a;bc}$ involves derivatives of the Berry connection polarizability and requires valley-contrasting anisotropic dispersion, e.g., in tilted Dirac cones in strained graphene (Das et al., 2023, He et al., 5 Mar 2025). Experimentally, the nonlinear VHE has been observed in hBN–graphene moiré superlattices, where its signal exceeds that of the linear VHE and is tunably gate-dependent (He et al., 5 Mar 2025).

Crystal Valley Hall Effect

When time-reversal symmetry is broken but certain crystal (spin-group) symmetries are retained—as in 2D altermagnets such as Fe$_2$WSe$_4$—opposite Berry curvature and valley degeneracy persist, enabling a crystal valley Hall effect (CVHE). Here, the valley-contrasting current arises from the combined action of nontrivial spin space operations (e.g., $M_{xy}$, $S_{4z}T$) rather than time reversal. The CVHE is robust to room temperature and is tunable via strain-induced valley splitting or topological phase transitions (Tan et al., 30 Sep 2024).

Magnon and Exciton Valley Hall Effects

Extensions of the VHE to neutral bosonic excitations have been demonstrated:

• Magnon valley Hall effect (MVHE): In van der Waals ferromagnets like CrI$_3$/MoTe$_2$, staggered magnetic anisotropies open valley-selective magnon gaps, and under a temperature gradient, the magnon thermal Hall conductivity $\kappa_{xy}$ reaches measurable values, with topologically protected edge modes traversing the domain wall (Hidalgo-Sacoto et al., 2020).
• Exciton valley Hall effect (XVHE): In TMD monolayers, neutral excitons are deflected by Berry curvature driven velocities. However, in the presence of synthetic or drag forces, the effect is controlled primarily by side-jump and skew-scattering contributions (Glazov et al., 2020). In TMD heterobilayers such as MoS$_2$/WSe$_2$, interlayer excitons show clear room-temperature valley Hall deflection, aided by their long valley lifetimes (Huang et al., 2019).

4. Measurement, Material Platforms, and Device Implications

Detection Techniques

The VHE is typically detected via nonlocal resistance measurements, transverse voltage mapping, or spatially resolved photoluminescence microscopy. Quantitative signatures such as cubic (linear VHE) or quartic (nonlinear VHE) scaling of nonlocal resistance with longitudinal resistivity, or the polarization-dependent splitting in spatial emissions, are characteristic (Shintaku et al., 2023, He et al., 5 Mar 2025, Huang et al., 2019).

Alternative detection via strain-induced pseudo-magnetic fields provides a route to quantized valley Hall conductivities, observable as quantized steps in the bulk density using the Widom–Středa formula, even in synthetic or artificial lattice systems (Jamotte et al., 2022).

Material Systems

Canonical VHE systems include:

• Monolayer TMDs (MoS$_2$, WS$_2$): Strong Berry curvature at $K/K'$ and robust optical selection rules enable opto-valleytronic devices (Mak et al., 2014, Ubrig et al., 2017).
• Bilayer/bulk TMDs: In centrosymmetric bilayers, inversion symmetry restores and the VHE vanishes; the system supports only an orbital Hall effect, distinguishable via gate-tuning (Cysne et al., 2020).
• Graphene/hBN superlattices: Alignment opens a bandgap and finite Berry curvature, with tunable VHE. In strained or twisted configurations, nonlinear effects dominate (Shintaku et al., 2023, He et al., 5 Mar 2025).

Device Implications

VHE serves as the basis for diverse device concepts:

• Valley Hall transistors and rectifiers: Nonlinear VHE enables rectification of AC charge current into DC valley current—directly demonstrated in valley rectifier devices (He et al., 5 Mar 2025).
• Valley filters, beam splitters, and switches: Gate-tunable or geometric phase–modulated structures can realize spatial separation and control of valley-polarized currents, both in charge and photonic domains (Zeng, 27 Jun 2024, Guddala et al., 2020).
• Room-temperature opto-valleytronics: The demonstration of persistent VHE in interlayer excitons at room temperature allows for practical excitonic circuits and optically addressable valleytronic memory or logic (Huang et al., 2019, Guddala et al., 2020).

5. Role of Disorder, Interactions, and Competing Topological Phenomena

The interplay of disorder (impurities, vacancies), extrinsic scattering mechanisms, and interparticle interactions strongly modulates the VHE:

• In monolayer TMDs, increasing disorder transitions the system from skew-dominated (divergent in clean limit) to side-jump or suppressed regimes, with gate-voltage tuning allowing detailed control (Olsen et al., 2015).
• Coulomb interactions—electron–electron and electron–hole, including annihilation—generate additional temperature- and density-dependent corrections to the VHE, particularly relevant in intrinsic and high-temperature regimes (Eliseev et al., 11 Jul 2024).
• Distinguishing the VHE from the orbital Hall effect (OHE) is critical in multilayer or bilayer systems: Inversion symmetry cancels the Berry curvature (and thus VHE), but an OHE persists, characterized by the orbital Chern number, and gives rise to edge states of purely orbital character (Cysne et al., 2020).

Table 2 outlines the behavior of VHE and OHE in monolayer versus bilayer TMDs:

Structure	Valley Hall Effect	Orbital Hall Effect
Monolayer	Finite (σ^v_{xy} ≠ 0)	Large (coexists with VHE)
Bilayer (centrosymmetric, unbiased)	Zero (σ^v_{xy} = 0)	Plateau (σ^L_{xy} doubles)
Bilayer (biased)	Tunable (σ^v_{xy} ∝ E_⊥)	Remains finite

6. Extensions, Topological Transitions, and Outlook

Recent advances extend the VHE landscape:

• Topological phase transitions: Strain or correlation-induced band inversion can drive transitions from trivial valley Hall insulators to half-valley-metal and quantum anomalous valley Hall (QAVHE) phases, marked by quantized Hall plateaus even in the absence of net magnetization (Hu et al., 2020).
• Fractal and tunneling-induced transmission: In engineered waveguide platforms, orientation-dependent resonant tunneling of valley-protected modes generates fractal minigap structures controlling backscattering and transmission (Shah et al., 2022).
• Optical VHE: Coupling TMD monolayers to bulk hyperbolic metamaterials enables the routing of valley-polarized exciton emission into momentum-space channels, offering room-temperature and fabrication-friendly valleytronic photonic circuitry (Guddala et al., 2020).

A unifying feature across these platforms is the persistence and tunability of valley-contrasting Berry curvature and its translation into robust, topologically protected valley or orbital flows.

7. Significance and Future Directions

The valley Hall effect, through its multifaceted theoretical framework and diverse experimental realizations, constitutes a central paradigm in the paper and application of quantum transport phenomena in two-dimensional and layered materials. Its sensitivity to Berry curvature, disorder, interaction effects, and symmetry-breaking renders it a diagnostic of band topology and a functional mechanism for valley-based electronics, optics, and spintronic analogs. Ongoing and future work leverages nonlinear, crystal-symmetry-protected, and excitation-specific variants of the VHE for room-temperature operation, all-electric or all-optical manipulation of valley currents, and integration into topologically robust device architectures (Tan et al., 30 Sep 2024, He et al., 5 Mar 2025, Huang et al., 2019). The continued convergence of theory, materials discovery, and device engineering positions the VHE as a cornerstone of next-generation quantum and valleytronic technologies.