Nonlinear valley Hall effect in a bilayer transition metal dichalcogenide

Jiayan Rong; Ruijing Fang; Xiao Li; Xiaoyu Wang; Zhen Zhang; Zhichao Zhou

REVIEW 3 major objections 5 minor 1 cited by

Nonlinear valley Hall effect in a bilayer transition metal dichalcogenide

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A centrosymmetric bilayer can host a valley Hall effect in the nonlinear regime.

desk verdict Solid symmetry-based prediction of strain-induced nonlinear valley Hall effect in bilayer MoS2, with quantitative numbers that are protocol-dependent because the strain model assumes zero Poisson ratio. read the letter →

arxiv 2412.19502 v1 pith:BRRBHI6L submitted 2024-12-27 cond-mat.mes-hall cond-mat.mtrl-sci

Zhichao Zhou , Ruijing Fang , Zhen Zhang , Xiaoyu Wang , Jiayan Rong , Xiao Li This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.mtrl-sci

keywords nonlinearvalleyHalleffectBerryconnectionpolarizabilitybilayerMoS2uniaxialstrainvalleytronicstransitionmetaldichalcogenidessecondharmonictransport

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Bilayer molybdenum disulfide is centrosymmetric, so it shows no linear valley Hall effect, and valley physics in it has been considered blocked. This paper shows that the block disappears in the nonlinear transport regime: under uniaxial strain, the Dirac bands at the $K_+$ and $K_-$ valleys tilt, and the Berry connection polarizability generates opposite second-order transverse currents from the two valleys. The predicted nonlinear valley Hall conductivity reaches about $0.33\,(e^3/\hbar)\,\text{Å}\,\text{meV}^{-1}$ near the conduction band edge, which is much larger than in strained graphene and comparable to the organic conductor that has been studied for the same effect. The response is tunable by strain strength and direction, chemical potential, and the interlayer van der Waals gap, and first-principles calculations confirm the model's main features. If right, the paper makes uniaxially strained centrosymmetric TMD bilayers viable valleytronic materials despite having inversion symmetry.

What carries the argument

The central object is the Berry connection polarizability (BCP), defined per band as $$$G^{{\alpha\beta}}$_n(\mathbf{k})=2\,\mathrm{Re}\sum_{m\neq n}\frac{$A^{{nm}}$_\$\alpha$(\mathbf{k})$A^{{mn}}$_\$\beta$(\mathbf{k})}{\varepsilon_{n\mathbf{k}}-\varepsilon_{m\mathbf{k}}},$$ with $A^{nm}_\alpha=\langle u_{n\mathbf{k}}|i\partial_\alpha|u_{m\mathbf{k}}\rangle$, combined with the BCP dipole $\Lambda_{\alpha\beta\gamma}=\sum_n \lambda^{\alpha\beta\gamma}_n\,\partial f/\partial \varepsilon$, where $\lambda$ involves the band velocity times $G$. The BCP is the geometric quantity that survives when both inversion and time-reversal symmetry are present, unlike the Berry curvature and Berry curvature dipole. In the model, the strain term $\epsilon[\gamma_1\sigma_0+\gamma_2\sigma_z+\gamma_3(\cos 2\theta\,\sigma_x-\tau\eta\sin2\theta\,\sigma_y)]$ tilts the low-energy Dirac bands, and the small conduction-band spin-orbit splitting $\lambda_c$ makes the energy denominator in $G$ small, strongly amplifying the response. The machinery converts an applied field squared into a valley-antisymmetric transverse current.

What would settle it

A second-harmonic transport measurement on a Hall bar of bilayer MoS2 under 2% uniaxial strain along x, with the chemical potential tuned about 3 meV above the conduction band minimum, should show a transverse current quadratic in the applied field with magnitude near $0.3\,(e^3/\hbar)\,\text{Å}\,\text{meV}^{-1}$ and with opposite signs at $K_+$ and $K_-$, while $\chi_{yxx}$ stays zero; a null transverse second-harmonic signal at that doping would contradict the central claim.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that a $\mathcal P$- and $\mathcal T$-symmetric bilayer can still have a valley Hall effect when the current is quadratic in the applied field. Symmetry analysis shows that $\mathcal P$ and $\mathcal T$ together forbid the Berry curvature dipole but not the Berry connection polarizability dipole, so the only requirement is breaking the threefold rotation $C_{3z}$. A uniaxial strain does this and tilts the Dirac cones; the tilt makes the BCP dipole $\Lambda_{xyy}$ mirror-symmetric at each valley and opposite between valleys, giving a nonzero $\chi^{\mathrm{NVH}}_{xyy}$ while $\chi^{\mathrm{NVH}}_{yxx}$ vanishes for strain along $x$. Because the conduction-band spin-orbit splitting is only a few meV, the BCP around the conduction band edge is large, and the conductivity peaks at $0.33\,(e^3/\hbar)\,\text{Å}\,\text{meV}^{-1}$ at $\mu=3$ meV. First-principles DFT gives the same qualitative behavior with a peak value of $0.28\,(e^3/\hbar)\,\text{Å}\,\text{meV}^{-1}$ at 2% strain, supporting the claim.

Load-bearing premise

The size of the predicted conductivity rests on a strain Hamiltonian derived by setting the Poisson ratio to zero, while the real material contracts in the perpendicular direction under uniaxial strain; that approximation could shift the tilts and the numerical value of the conductivity even though the symmetry-based existence of the effect would remain.

Editorial extensions

If this is right

A measurable second-harmonic transverse voltage should appear in uniaxially strained bilayer MoS2, detectable through nonlocal resistance measurements.
The predicted peak value is about 0.33 (e3/ℏ) Å meV−1 at a chemical potential of 3 meV, far exceeding the nonlinear valley Hall response of strained graphene and comparable to organic-conductor values.
Strain strength, strain direction, carrier doping, and the van der Waals gap each provide an independent tuning knob for the magnitude and sign of the nonlinear valley Hall current.
The effect is naturally largest near small conduction-band spin-orbit splittings, so the nonlinear valley Hall response can serve as a detector of those splittings.
First-principles calculations reproduce the model's sign structure and peak value at 2% strain, indicating the effect is not an artifact of the low-energy model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: because the symmetry argument depends only on $\mathcal P$, $\mathcal T$, and broken $C_{3z}$, the same mechanism should appear in other centrosymmetric 2D bilayers with gapped Dirac-like bands, not just MoS2, whenever uniaxial strain creates band tilts.
Inference: the $\cos 2\theta$/ $\sin 2\theta$ strain-direction dependence suggests a simple device control—rotating the strain axis by 45 degrees transfers the transverse response from $\chi_{xyy}$ to $\chi_{yxx}$, effectively rotating the Hall current direction without reversing the field.
Inference: since the BCP grows as the inverse cube of the interband splitting, materials with even smaller conduction-band spin-orbit splitting than MoS2 should show substantially larger nonlinear valley Hall conductivities; the paper does not calculate this, but it follows from Eq. (7).

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a good paper and the central claim holds up. The authors show that uniaxial strain breaks C3z in centrosymmetric bilayer MoS2 and generates a nonlinear valley Hall effect from Berry connection polarizability, with opposite signs for the two valleys. The symmetry analysis is convincing and the effective model matches DFT near small strain (0.33 vs 0.28 in the same units). That is real, independent cross-checking.

What is genuinely new: the BCP dipole mechanism is extended from strained graphene and organic monolayers (Ref. [19]) to bilayer TMDs. The extra ingredients — interlayer coupling, tunable van der Waals gap, and the small conduction-band SOC splitting — are not just decoration; the paper shows the conductivity is highly sensitive to the tiny conduction-band splitting, which is a useful way to detect small spin-orbit splittings, and that the van der Waals gap is a practical tuning knob. The strain-direction dependence (cos/sin 2θ) is a clean prediction.

Soft spots, in order of importance. First, the quantitative predictions rest on the strain Hamiltonian in Eq. (3), which the text explicitly says is built assuming zero Poisson ratio. That is a real limitation. Real MoS2 under uniaxial stress has ν≈0.22, which changes the pseudogauge couplings by roughly 20% and shifts the quoted peak value and its chemical-potential location. The DFT verification in Fig. 4 uses the same u_yy=0 protocol, so it does not independently validate the uniaxial-stress case. This is a quantitative concern, not an existential one: the symmetry-based existence of the NVH effect does not depend on the Poisson ratio. But the advertised peak of 0.33 (e3/ℏ) Å meV−1 is protocol-specific until this is addressed. Second, the model neglects higher-order q terms, which the paper itself acknowledges is why the model and DFT diverge for strains above 4% (Fig. 4b). That is an honest, stated limitation. Third, the analytic derivation is in the Supplementary Material, which was not available for this review, and no code or data are released. Those are addressable, not fatal.

The paper is worth a serious referee. It is a solid theoretical contribution with a clear symmetry argument, model-DFT agreement at small strain, and experimentally falsifiable predictions. I would send it to review and ask the authors to discuss the Poisson-ratio dependence and release the SM and, ideally, the calculation code.

Who is this for? Researchers in valleytronics and nonlinear Hall physics. It does not reshape the field, but it removes a symmetry constraint that was assumed necessary for valley-contrasting transport and opens a specific material class for experiments.

Referee Report

3 major / 5 minor

Summary. This paper predicts a nonlinear valley Hall (NVH) effect in centrosymmetric 2H-MoS2 bilayers under uniaxial strain. The authors first show by symmetry analysis that the simultaneous presence of inversion and time-reversal symmetry forbids the Berry curvature dipole but allows the Berry connection polarizability (BCP) to produce a nonlinear Hall current, and that C3z symmetry must be broken for the response to be nonvanishing. They then construct a k.p model with spin-orbit coupling and interlayer hopping, calculate the NVH conductivity, and find a large chi_xyy peak of 0.33 (e^3/hbar) Angstrom meV^-1 at 2% strain near the conduction-band edge, about 8.03 nm*mA/V^2. The paper demonstrates tunability of the response with strain strength, strain direction, chemical potential, SOC strength, and interlayer hopping, and corroborates the main results with first-principles calculations, which give a peak of 0.28 in the same units at the same strain.

Significance. The central symmetry argument is sound and the paper addresses an important gap: extending valley Hall physics to centrosymmetric bilayer systems. The combination of an effective model and first-principles calculations with reasonable agreement at small strain is a strength, as is the explicit identification of the BCP mechanism and the proposal of the conduction-band SOC splitting as a tunable knob and as a detectable signal. If the quantitative predictions hold for realistic experimental strain protocols, the work will be of considerable interest to the valleytronics community and could motivate experiments on strained bilayer TMDs using nonlocal resistance measurements.

major comments (3)

[Model calculations of a MX2 bilayer, Eq. (3)] The strain Hamiltonian in Eq. (3) is derived assuming a zero Poisson ratio, explicitly stated in the text. For real uniaxial stress in MoS2 with nu approximately 0.22, the pseudogauge strain combinations u_xx-u_yy and u_xy are multiplied by (1+nu) while the trace term u_xx+u_yy is multiplied by (1-nu). Because the reported NVH conductivity arises from small strain-induced band tilts and is amplified by the near-degenerate denominator in Eq. (7) associated with the small conduction-band SOC splitting, a roughly 20% change in the effective strain couplings can shift the advertised peak magnitude and its chemical-potential location, and can modify the tuning curves in Fig. 3(a). The first-principles calculation in Fig. 4 appears to use the same fixed-transverse-lattice protocol, so it does not independently validate the uniaxial-stress case. I request that the authors either incorporate a realistic Poisson ratio into the strain parameterization or clearly reframe the predictions as valid for a specific fixed-transverse-lattice-constant protocol, and discuss how the peak values and tuning curves would change under a typical experimental uniaxial-stress setup.
[Figs. 3(a) and 4(b)] The model calculation in Fig. 3(a) predicts that chi_xyy increases monotonically with strain up to 8%, while the first-principles result in Fig. 4(b) shows a decrease for strain beyond 4%. The paper attributes this discrepancy to neglecting q-related high-order terms in the Hamiltonian. Because the abstract and text emphasize that the effect is highly tunable through modulating the strength of strain, this discrepancy directly limits the predictive power of the model for the strain-strength-dependence claim. The authors should explicitly restrict their strain-tunability claim to strains of about 4% or below, or extend the effective model to include the relevant higher-order q terms.
[Eq. (5) and model calculations] The integral in Eq. (5) is written over the full Brillouin zone, but the k.p Hamiltonian in Eq. (2) is valid only in a neighborhood of the K+/- valleys. The model calculation must be restricted to a finite integration window around each valley; otherwise the numerical value depends on an uncontrolled cutoff. Please specify the integration cutoff (or matching procedure) used to produce Figs. 2 and 3, and demonstrate that the reported peak values and the strain-direction dependence are insensitive to the chosen window size.

minor comments (5)

[Near Eq. (3)] Typo: 'Possion ratio' should be 'Poisson ratio'.
[Eq. (5)] The notation (-1)^tau inside the BZ integral is ambiguous because tau is a valley index, not a function of k. It would be clearer to write the conductivity as a sum over valleys, with the valley label attached to the Hamiltonian and the integral over each valley patch.
[Fig. 2(a)] The axis label appears to list both chi_xyy and chi_yxx without subscripts; please format the labels cleanly so that the two curves are unambiguously identified.
[Fig. 1(c) caption] The phrase 'q is given in units of the reciprocal of the lattice constant, a' should read 'q is given in units of 1/a'.
[Introduction and first-principles section] When comparing with the nonlinear Hall conductivity of few-layer WTe2, give the specific value used for the comparison and the reference from which it is taken.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NVH conductivity is computed from independently fitted Hamiltonian parameters and cross-checked by first-principles DFT; no fitted input is renamed as a prediction.

full rationale

The derivation chain is self-contained: (i) symmetry analysis shows which nonlinear conductivity tensor components survive under P, T, C3z, C2x, and Mx; (ii) a k.p Hamiltonian (Eqs. 2-4) is constructed with monolayer and interlayer parameters taken from Refs. [22-24], which are external DFT-based band-structure parameters; (iii) the NVH conductivity is evaluated via the Berry connection polarizability dipole, Eqs. (5)-(7); and (iv) first-principles DFT calculations independently reproduce the qualitative and semiquantitative behavior (e.g., chi_xyy ~ 0.28 vs 0.33 from the model). No Hamiltonian parameter (v_F, Delta, lambda_c, lambda_v, t_cc, t_vv, gamma_1-3) is fitted to the target observable chi_NVH, and no equation defines chi_NVH in terms of itself. The valley-contrasting sign follows from the tau-dependent strain coupling and is verified by momentum-resolved BCP patterns rather than imposed by construction. The zero-Poisson-ratio approximation in the strain Hamiltonian is an acknowledged modeling limitation ('the strain terms are obtained by assuming a zero Possion ratio', Eq. 3), which may shift quantitative values under realistic uniaxial stress, but it is an accuracy caveat, not a circular input. Similarly, the nonmonotonic strain dependence above 4% in DFT versus the model is attributed to neglected higher-order q terms, an approximation caveat rather than a logical circularity. There is no load-bearing self-citation chain: Refs. [22-24] are independent prior works, and the first-principles calculation provides an external cross-check. Therefore the central claim is not forced by definition or by fitted inputs; it is an independent model-plus-DFT prediction.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

The effective model takes all material parameters from earlier DFT fits (Refs. [22-24]), then predicts a new observable (χNVH) that is later checked against an independent first-principles calculation. No parameter is fitted to the target observable, so the circularity burden is low, but the quantitative result inherits the accuracy of those DFT-fitted values.

free parameters (8)

vF (Fermi velocity)
Taken from DFT-fitted parameters of Refs. [22,23]; controls the Dirac cone dispersion in Eq. 3.
Δ (crystal-field band gap)
Taken from Refs. [22,23]; sets the gap in the gapped Dirac model of Eq. 3.
δ (energy shift)
Immaterial energy shift from Ref. [22]; does not affect transport qualitatively.
λc (conduction-band SOC) = -1.5 meV (MoS2 value λc,0)
Used in Eq. 3 and varied in Fig. 3(c); the smallness of λc drives the enhanced NVH response.
λv (valence-band SOC)
Used in Eq. 3; the larger valence splitting suppresses NVH from valence bands.
tcc (interlayer conduction hopping) = 70.6 meV·Å (tcc,0)
Used in Eq. 4 and varied in Fig. 3(d); models the van der Waals gap dependence.
tvv (interlayer valence hopping)
Used in Eq. 4 from Ref. [24]; sets valence band splitting.
γ1, γ2, γ3 (strain coupling parameters)
Strain parameters from Refs. [22,23] in Eq. 3; determine the strain-induced band tilt.

assumptions (5)

domain assumption The k·p Hamiltonian (Eqs. 2-4) with the given basis and parameters accurately describes the low-energy electronic structure of strained bilayer MoS2 near the K± valleys.
All model calculations of band tilts and BCP dipoles use this Hamiltonian; an inaccurate model changes the magnitude of χNVH.
domain assumption The nonlinear Hall conductivity is given by the BCP dipole formula in Eqs. 5-7, which assumes a relaxation-time approximation and the clean limit as in Ref. [19].
The paper computes χNVH from this formula without considering disorder or finite-temperature corrections.
ad hoc to paper Uniaxial strain is captured by linear terms in ε with a zero Poisson ratio (Eq. 3).
The text states the strain terms are obtained by assuming a zero Poisson ratio; real MoS2 has a nonzero one, so this affects the predicted strain dependence.
domain assumption Interlayer coupling is limited to tcc and tvv hoppings in Eq. 4.
Other interlayer orbital couplings and momentum dependencies are neglected; the van der Waals gap tuning is modeled through tcc alone.
standard math Fermi-Dirac distribution at zero temperature (Eq. 5).
The derivative ∂f/∂ε is evaluated at zero temperature with no thermal broadening.

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Pith. "Pith review of Nonlinear valley Hall effect in a bilayer transition metal dichalcogenide." pith.science (2026). https://pith.science/paper/BRRBHI6L

@misc{pith2026241219502,
  author       = {Pith},
  title        = {Pith review of: Nonlinear valley Hall effect in a bilayer transition metal dichalcogenide},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRRBHI6L}},
  note         = {Machine review of arXiv:2412.19502}
}

abstract

Valley-contrasting Hall transport conventionally relies on the inversion symmetry breaking in two-dimensional systems, which greatly limits the selection range of valley materials. In particular, while monolayer transition metal dichalcogenides have been widely utilized as a well-known class of valley materials in valleytronics, the centrosymmetric nature hinders the realization of valley-contrasting properties in the bilayer counterparts. Here, taking MoS$_{2}$ as an example, we discover valley-contrasting transport in bilayer transition metal dichalcogenides by exploring nonlinear transport regime. Using effective models and first-principles calculations, our work demonstrates that nonvanishing nonlinear valley Hall conductivities emerge in a uniaxially strained MoS$_{2}$ bilayer, owing to strain-induced band tilts of Dirac fermions. With the aid of small spin-orbit-coupling induced band splittings, the conduction bands generate much remarkable nonlinear valley Hall conductivity. Moreover, the nonlinear conductivities are highly tunable through modulating the strength and the direction of the strain, chemical potential, and interlayer gap. Our findings not only expands material choices for valleytronic applications, but also provides opportunities for designing advanced electronic devices that leverage nonlinear valley transports.

Figures

Figures reproduced from arXiv: 2412.19502 by the authors.

**Figure 2.** (a) shows calculated conductivity tensor elements, χxyy and χyxx, of the MoS2 bilayer with a 2% uniaxial strain along the x axis, as functions of the chemical potential, µ. It is seen that while χyxx is always zero, χxyy is nonvanishing. Compared with the case of the valence bands, the magnitude of χxyy at the conduction band edge is much larger. Therefore, the chemical potential is chosen around the conduction b… view at source ↗

**Figure 3.** FIG. 3. Evolutions of first peak values of NVH conductiv [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. First-principles NVH conductivities of a MoS [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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