Suppression of Intrinsic Hall Effect through Competing Berry Curvature in Cr$_{1+\delta}$Te$_2$

Jhuma Sannigrahi; Jyotirmay Sau; Manoranjan Kumar; Matthias Gutmann; Mohamad Numan; Prasanta Chowdhury; Saurav Giri; Subham Majumdar

REVIEW 4 major objections 5 minor 1 cited by

Suppression of Intrinsic Hall Effect through Competing Berry Curvature in Cr$_{1+\delta}$Te$_2$

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Rival Berry curvature contributions cancel the intrinsic anomalous Hall effect in Cr1+δTe2

desk verdict Solid experiment-plus-DFT study; the Berry-curvature-cancellation story is plausible but not yet proven at the measured composition, and the paper's own cited δ-dependence is the load-bearing loose end. read the letter →

arxiv 2411.14045 v1 pith:ZFJFGFST submitted 2024-11-21 cond-mat.mtrl-sci

Prasanta Chowdhury , Jyotirmay Sau , Mohamad Numan , Jhuma Sannigrahi , Matthias Gutmann , Saurav Giri , Manoranjan Kumar , Subham Majumdar This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords Cr1+δTe2anomalousHalleffectBerrycurvatureskewscatteringself-intercalatedvanderWaalsferromagnetdensityfunctionaltheoryspin-orbitcouplingmagnetocrystallineanisotropy

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

This paper tries to explain why the self-intercalated ferromagnet Cr1+δTe2 (δ≈0.33) shows a sizable anomalous Hall effect that is nevertheless not intrinsic. Magnetization, magnetocaloric, and magnetotransport measurements on single crystals show strong uniaxial anisotropy, a first-order-like hysteresis, and a Hall response dominated by skew scattering according to the modified scaling relation. Density functional theory with spin-orbit coupling finds gapped nodal points near the Fermi energy with substantial Berry curvature of opposite signs along the Γ–M and Γ–K directions, and these contributions nearly cancel in the anomalous Hall conductivity. The authors conclude that the intrinsic Berry-curvature channel is effectively zero, and that extrinsic skew scattering accounts for the observed AHE.

What carries the argument

The load-bearing object is the momentum-space Berry curvature $\Omega^z_n(\mathbf{k})$ computed from a tight-binding model derived from DFT+U+SOC bands. Opposite-sign curvature contributions at the gapped Γ–M (Cr d–d) and Γ–K (Cr d–Te p) nodal points cancel in the linear-response integral for the intrinsic anomalous Hall conductivity, reducing it to near zero. The same machinery predicts a sign change in the intrinsic Hall response upon shifting the Fermi level, which is why the authors suggest electron doping as a route to switch on a topological Hall signal.

What would settle it

Recompute the Berry curvature and intrinsic anomalous Hall conductivity for a supercell that exactly matches the measured Cr1.33Te2 composition and intercalation pattern; if the integrated value is non-negligible at the Fermi level, the cancellation argument is falsified. Alternatively, electron-dope the crystal to move the Fermi level onto the conduction-band node at K; if a large intrinsic Hall signal does not appear, the predicted node contribution is wrong.

Watch

Extended reading notes

Core claim

The central discovery is that the intrinsic anomalous Hall contribution in Cr1+δTe2 is suppressed by cancellation of Berry curvature from different momentum-space nodal points. In the Cr10Te16 supercell used to approximate the experimental composition, density functional theory plus spin-orbit coupling opens gaps at band crossings along Γ–M and Γ–K; the resulting Berry curvature is positive near one node and negative near the other. Integrating this Berry curvature over occupied states yields an intrinsic anomalous Hall conductivity that is effectively zero, even though individual Berry-curvature values are large. The paper therefore attributes the measured anomalous Hall effect to the extrinsic skew-scattering mechanism, and notes that electron doping could move the Fermi level to a conduction-band node where a large intrinsic Hall effect would reappear.

Load-bearing premise

The theoretical model approximates the measured Cr1.33Te2 crystal with a Cr10Te16 (Cr1.25Te2) supercell, and the conclusion of near-zero intrinsic Hall effect assumes that this substitution preserves the near-cancellation of Berry curvature.

Editorial extensions

If this is right

The observed anomalous Hall effect in Cr1+δTe2 (δ≈0.33) is extrinsic, dominated by skew scattering, rather than Berry-curvature-driven.
Nontrivial band topology with gapped nodal points can coexist with a vanishing intrinsic Hall response when opposite-sign Berry curvature cancels.
Electron doping of Cr1+δTe2 should shift the Fermi level toward the conduction-band node at K and produce a measurable intrinsic anomalous Hall effect.
The near-zero thermal expansion between roughly 115 and 150 K is coupled to the magnetic hysteresis, indicating magneto-structural coupling in the same temperature window.
The absence of a topological Hall effect in this centro-symmetric crystal contrasts with earlier reports on Cr1.33Te2, suggesting the topological Hall signal may require a non-centrosymmetric environment.

Reading between the lines

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

Because the DFT model uses Cr1.25Te2 instead of the measured Cr1.33Te2, and because earlier work shows the Berry curvature changes sign with Cr concentration, the exact cancellation may be a composition-specific accident; a slightly different doping could restore a finite intrinsic AHE.
The symmetry and orbital-hybridization picture at the conduction-band K node suggests a design rule for centrosymmetric intercalated chalcogenides: the sign of Berry curvature is controlled by which orbitals hybridize at a crossing, so tuning orbital character could balance or amplify the two contributions.
The staircase features in magnetization and Hall resistivity, interpreted here as spin-flop transitions, could be tested directly by neutron diffraction on a single crystal in a magnetic field; if they are spin-flop, the magnetic structure should reorient without forming a skyrmion phase.
The prediction that electron doping should turn on a large intrinsic Hall effect is testable by electrostatic gating or chemical intercalation; observing such a signal would confirm both the K-node origin and the cancellation picture.

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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

Here's the short version: this is a careful experiment-plus-DFT paper on AHE in Cr1.33Te2, and the central idea—that opposing Berry curvature at two gapped nodal points cancels the intrinsic AHC—is plausible and nicely worked out. The catch is that the DFT is computed for Cr1.25Te2, not the measured δ=0.33, and the paper's own reference [37] shows that δ tunes the sign of the Berry curvature. So the theory demonstrates a mechanism at a nearby composition, but it doesn't yet prove that this is why the measured sample has no intrinsic AHE.

What I like: the experimental side is thorough. SCXRD, PXRD, EDXS, magnetization, MCE, resistivity, Hall—they characterize the crystal well. The AHE scaling fit to Eq. 5 is standard and the conclusion that skew scattering dominates is a reasonable read of the data. The DFT is a genuine first-principles calculation: they identify the Γ-M (Cr-d/Cr-d) and Γ-K (Cr-d/Te-p) crossings, show SOC opens gaps, compute Berry curvature with opposite signs, and show the integrated AHC is near zero at EF. The predicted unoccupied node at K that could produce large AHE under electron doping is a nice falsifiable claim. They also argue against the previously reported THE, attributing the staircase to spin-flop instead.

Soft spots, in order of severity. First, the composition issue. Cr10Te16 is δ=0.25, but the crystals are δ≈0.33. The authors call it a 'close approximation', but [37] shows δ-driven sign flips in BC. A calculation at the real composition, or a scan over δ and U, would close the loop. Without that, the theory is suggestive, not demonstrative. Second, the scaling fit has no error bars or goodness-of-fit; the α′, α″, β values are reported without uncertainties, and the decomposition into linear vs quadratic terms is model dependent. Minor, but it would help. Third, the absence of THE is asserted rather than quantitatively shown—no attempt to extract a topological Hall contribution in the field range where it might appear.

Who should read this? People working on AHE in Cr-Te and related layered magnets. It's a useful data point and the doping prediction is testable. I'd send it to a serious referee, but with a request that the authors either compute at δ=0.33 or explicitly test sensitivity to δ and U. The paper is honest and well cited; the gap is a validation gap, not a fabrication. My verdict: conditional, but worth engaging.

Referee Report

4 major / 5 minor

Summary. The manuscript presents a combined experimental and ab initio study of self-intercalated Cr1+δTe2 single crystals with δ ≈ 0.33. Magnetization, magnetocaloric effect, synchrotron X-ray diffraction, resistivity, and Hall resistivity measurements characterize a ferromagnet with TC ≈ 191 K and strong c-axis anisotropy. The anomalous Hall data are analyzed with a modified scaling law (Eq. 5) to conclude that skew scattering dominates the AHE. DFT + GGA+U calculations on a Cr10Te16 (Cr1.25Te2) supercell find gapped nodal points along Γ-M and Γ-K with Berry curvature of opposite signs; the Kubo-formula intrinsic AHC is argued to be near zero because of this cancellation.

Significance. If confirmed, the paper offers a plausible resolution of the puzzle of why a compound with sizable Berry curvature shows no intrinsic AHE. The theoretical part is a genuine first-principles calculation of the Berry curvature and AHC via the Kubo formula, not a fit to the measured Hall data; the near-cancellation is computed rather than assumed. The paper also provides a useful experimental dataset on the magnetocaloric effect and thermal expansion, and it makes a falsifiable prediction that electron doping should induce a large intrinsic AHE when the Fermi level reaches the conduction-band nodal point. The main weakness is the gap between the measured composition (δ ≈ 0.33) and the calculated one (δ = 0.25), which is not tested for sensitivity.

major comments (4)

[Sec. IV, Fig. 7] The DFT calculation is performed for Cr10Te16, i.e., Cr1.25Te2, while the measured crystals are Cr1.33Te2 according to EDXS and lattice parameters (Sec. III.A). The text calls the supercell "a close approximation" but provides no calculation at the actual composition and no test of how the Berry-curvature cancellation depends on δ. The paper itself cites Fujisawa et al. [37] to state that Ωz and σxy^int can switch sign as δ varies and cross through zero. A δ difference of 0.08 is therefore potentially material. The authors should either compute σxy^int for a Cr1.33Te2 supercell or show explicitly that the cancellation is robust over the measured δ range and over intercalant arrangements.
[Sec. III.E, Eq. (5)] The conclusion that skew scattering dominates the AHE rests on a three-parameter fit to ρAHE_xy versus ρxx, but the paper reports only the best-fit values α′ = -0.042, α′′ = 0.037, β = 4.41 S/cm. No uncertainties, correlation matrix, or goodness-of-fit statistic are given, and alternative scaling forms (e.g., a single exponent q, or different separation of residual and phonon terms) are not discussed. The authors should provide error bars and a measure of fit quality, and demonstrate that the βρxx^2 term is negligible within the uncertainties.
[Sec. IV, Eq. (7) and Fig. 7(d)] The central theoretical claim is that the intrinsic AHC is "effectively zero," but the manuscript gives no numerical value of σxy^int at EF and no quantitative comparison with the experimental σAHE_xy shown in Fig. 6(b). Because Fig. 7(d) shows sign flips near EF, the exact position of EF is critical. The authors should state the computed AHC at EF, including its sensitivity to k-mesh, smearing, and Hubbard U, and compare it with the experimental magnitude to make the cancellation claim quantitative.
[Sec. II.B, Fig. 7] The Hubbard U is chosen in the range 0.5-0.8 eV to match magnetic moments, but no U dependence of the Berry curvature or AHC is reported. Since the gaps at the avoided crossings, and hence the Berry curvature and σxy^int, are expected to depend on U and on the magnetic order, the near-cancellation could be specific to the chosen interaction strength. A short U-dependence scan of σxy^int would test this directly.

minor comments (5)

[Sec. V] In the conclusion, "δS" should be ΔSM, matching the notation used in Sec. III.B.
[Sec. II.B] "V ASP" should be "VASP."
[Sec. IV, Fig. 7(a)] The Fig. 7(a) caption and the main text disagree on the colors used for the bands with and without SOC; please align them.
[Sec. III.E] The phrase "the inset shows the enlarge view" should be "the inset shows an enlarged view."
[Sec. I] "alternative staking" should be "alternative stacking."

Circularity Check

1 steps flagged · score 2.0 of 10

Minor circularity: the magnetic-moment 'agreement' recycles the fitted Hubbard U, but the central Berry-curvature cancellation is an independent first-principles result.

fitted input called prediction [Sec. II.B (Methods, Theoretical Calculations) and Sec. IV (Ab initio Calculation)]
"Matching the magnetic moments requires an effective Coulomb interaction (U) in the range of 0.5-0.8 eV. ... The overall magnetic moment of Cr1.25Te2 is calculated to be 3.08 μB per formula unit, which is very similar to the experimentally observed value."

The Hubbard U is tuned until the computed Cr moment matches the experimental saturation moment, so the statement that the calculated moment is 'very similar' to experiment reports the fitted constraint rather than an independent first-principles prediction. This is a minor circular step: the central AHC claim is not forced because the Berry-curvature cancellation is computed from the resulting band structure via Eqs. (6) and (7), and is not fitted to the measured anomalous Hall signal.

full rationale

The paper's central claim is that the intrinsic AHC in Cr1+δTe2 is suppressed by cancellation of positive (Γ-M) and negative (Γ-K) Berry-curvature contributions. This is a first-principles result obtained from DFT+U, Wannier interpolation, and the Kubo formula (Eqs. 6 and 7), not a fit to the measured AHE. The experimental attribution to skew scattering is a model interpretation of the ρ_AHE vs ρxx scaling fit (Eq. 5), which is underdetermined but not circular. The use of Cr10Te16 (=Cr1.25Te2) as a proxy for the measured Cr1.33Te2, together with the paper's own citation of δ-dependent Berry-curvature sign changes [37], is a validity concern about the theoretical explanation, but it is not circularity: the computation is not defined in terms of the experimental outcome. Self-citations ([6], [32], [34], [35]) appear only in background context and are not load-bearing. The only concrete reduction-by-construction is the magnetic-moment agreement, where U is matched to the experimental moment and the result is then reported as agreement; this is minor and does not propagate to the central AHC conclusion.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the DFT supercell approximation, the GGA+U choice, and the assumed collinear magnetic order, plus the scaling-law decomposition used to attribute the measured AHE to skew scattering. No new physical entities are introduced. The fitted Hubbard U and the three scaling coefficients are the main fitted inputs; other fits (Curie-Weiss, magnetocaloric, thermal expansion) support secondary characterization.

free parameters (5)

Hubbard U (GGA+U) = 0.5-0.8 eV
Tuned so computed magnetic moment matches experimental 3.08 µB/f.u.; affects band structure and Berry curvature, so it influences the central cancellation claim.
AHE scaling coefficients α', α'', β (Eq. 5) = -0.042, 0.037, 4.41 S/cm
Fitted to the measured ρ_AHE vs ρ_xx; the small β is the basis for concluding skew scattering dominates. No error bars are reported.
MCE Landau fit A, B, H0 (Eq. 2) = H∥c: A=1.79(9), B=-0.10(3); H⊥c: A=-0.11(5), B=0.23(3); H0 not stated
Fitted to field dependence of maximum magnetic entropy change; not central to AHE claim but a fitted input to the magnetocaloric analysis.
Thermal expansion fit V0, κ, θ_D (Eq. 3) = V0=80.5(1) Å³, κ=0.01(1), θ_D=310 K
Fitted to unit-cell volume vs T; used to support magneto-structural coupling but not central to AHE.
Curie-Weiss parameters θ_C, C (μ_eff) = θ_C=196 K, μ_eff=4.96 μB/Cr
Fitted to χ^{-1}(T) between 240-315 K at 0.1 T; used to characterize exchange, not central to AHE.

assumptions (6)

domain assumption GGA+U with U=0.5-0.8 eV approximates the exchange-correlation and correctly captures Cr 3d correlations
Used in all DFT band structure and Berry curvature results; validity is assumed, not benchmarked beyond matching magnetic moment.
ad hoc to paper Cr10Te16 (Cr1.25Te2) supercell is a sufficient proxy for experimental Cr1.33Te2
Explicitly stated in Section IV. Since reference [37] shows Berry curvature can change sign with Cr content δ, this is load-bearing for the cancellation claim.
domain assumption Magnetic order is collinear along the easy c-axis in the DFT calculation
Used when including SOC; if the actual magnetic structure has canting (suggested by M_s < 3 μB/Cr and staircase jumps), Berry curvature results could change.
domain assumption The scaling law Eq. 5 uniquely separates skew scattering (linear) from side-jump/intrinsic (quadratic)
Used in Section III E to conclude skew scattering dominates; the decomposition is model-dependent and does not include error analysis.
domain assumption Kubo formula with Wannier-interpolated tight-binding Hamiltonian and 8×8×8 k-grid gives converged AHC
Section IV; no convergence test shown, and AHC typically needs dense k-meshes.
standard math Inversion symmetry P and PT symmetry arguments for K-point degeneracy (Ref [74])
Symmetry analysis in Section IV supports the unoccupied K node; standard group theory.

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Pith. "Pith review of Suppression of Intrinsic Hall Effect through Competing Berry Curvature in Cr$_{1+\delta}$Te$_2$." pith.science (2026). https://pith.science/paper/ZFJFGFST

@misc{pith2026241114045,
  author       = {Pith},
  title        = {Pith review of: Suppression of Intrinsic Hall Effect through Competing Berry Curvature in Cr$_1+\delta$Te$_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZFJFGFST}},
  note         = {Machine review of arXiv:2411.14045}
}

abstract

We conducted a comprehensive analysis of the magnetic and electronic transport properties of the layered chalcogenide Cr$_{1+\delta}$Te$_2$ in its single crystalline form. This material exhibits a ferromagnetic transition at a critical temperature of $T_C = 191$ K, characterized by significant thermal hysteresis in the magnetization data below this temperature. Measurements of isothermal magnetization, magnetocaloric effect, and magnetoresistance indicate that the system exhibits strong magnetocrystalline anisotropy, with the $c$-axis serving as the easy axis of magnetization. The Cr$_{1+\delta}$Te$_2$ compound shows pronounced anomalous Hall effect (AHE); however, existing experimental and theoretical data do not provide a clear understanding of the nature and origin of this phenomenon. Our experimental findings suggest that the skew scattering mechanism primarily accounts for the observed AHE. In contrast, our theoretical study reveals the presence of gapped nodal points accompanied by non-zero Berry Curvature, which are expected to contribute towards intrinsic AHE. A detailed analysis of the electronic band structure, obtained through density functional theory calculations, reveals that the Berry Curvature at different nodal points exhibit both positive and negative signs. These opposing contributions largely cancel each other out, thereby significantly diminishing the intrinsic contribution to the AHE.

Figures

Figures reproduced from arXiv: 2411.14045 by the authors.

**Figure 2.** FIG. 2. Temperature ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Temperature dependence of magnetic entropy [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: (b) shows the schematic of the measurement configuration for both directions. For µ0H ∥ c, unsaturated negative MR has been observed for all the T, with a maximum value of ∼ 17 % near TC [ [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 4.** Figure 4: FIG. 4. (a) PXRD data along with Rietveld refinement at 10 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 6.** Figure 6: FIG. 6. (a) Hall resistivity [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. (a) The band structure of Cr [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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