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Scaling Behavior of Magnetoresistance and Hall Resistivity in Altermagnet CrSb

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper argues that CrSb's large magnetoresistance and nonlinear Hall resistivity come from ordinary multi-band Lorentz-force transport, not from its altermagnetic order.

desk verdict Solid magneto-transport study of CrSb with a first Hall-scaling observation in an altermagnet, but the load-bearing simulation uses a nonmagnetic band structure and the scaling collapse is fitting-driven. read the letter →

arxiv 2412.12263 v1 pith:KWMDM3ZG submitted 2024-12-16 cond-mat.mtrl-sci cond-mat.str-elcond-mat.supr-con

Xin Peng , Yuzhi Wang , Shengnan Zhang , Yi Zhou , Yuran Sun , Yahui Su , Chunxiang Wu , Tingyu Zhou
show 9 more authors
Le Liu Hangdong Wang Jinhu Yang Bin Chen Zhong Fang Jianhua Du Zhiwei Jiao Quansheng Wu Minghu Fang
This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.str-elcond-mat.supr-con PACS 72.15.Gd75.50.Ee71.18.+y
keywords altermagnetCrSbmagnetoresistanceHallresistivityextendedKohler'srulemulti-bandtransportLorentzforcescalinglaw
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 sets out to show that the large magnetoresistance and nonlinear Hall resistivity measured in the altermagnet CrSb -- a magnetic material with zero net magnetization but spin-split electronic bands -- are ordinary Lorentz-force magneto-transport, not signatures of the material's altermagnetic order. The authors grew CrSb single crystals, confirmed the antiferromagnetic transition at $T_N = 712$ K, and measured longitudinal and Hall resistivity in fields up to 7 T between 6 K and 150 K. They find that both transport channels obey scaling laws -- an extended Kohler's rule with a temperature-dependent carrier-density factor $n_T$ for the magnetoresistance and a parallel scaling relation for the Hall resistivity -- and that numerical simulation from the calculated band structure reproduces the data. The central conclusion is that the nonlinear Hall response comes from a three-band (two electron, one hole) Fermi surface rather than from an anomalous Hall effect (a magnetization-related Hall voltage), so altermagnetic spin splitting does not dominate these transport coefficients.

What carries the argument

The argument is carried by three connected tools. The extended Kohler's rule $MR = \alpha(H/n_T\rho_0)^m$, with a temperature-dependent carrier-density factor $n_T$, absorbs the change of carrier density with temperature and collapses the longitudinal magnetoresistance data onto one curve. The Hall-resistivity scaling law $\rho_{yx}/(n\rho_0/m) = h(B/(n\rho_0/m))$, derived under the condition that all charge carriers share the same temperature-dependent relaxation time, collapses the transverse data using the same $n_T$ values. A three-band model with two electron pockets and one hole pocket, together with Boltzmann-transport simulation of the calculated band structure, reproduces the measurements and identifies the nonlinear Hall response as a multi-band effect rather than an anomalous Hall effect; the magnetic point group $\overline{6}'/m'mm'$ of CrSb is what rules out the anomalous Hall interpretation.

What would settle it

Compute the magneto-transport from a DFT band structure that explicitly includes CrSb's antiferromagnetic order and spin splitting, using the same relaxation-time model; if those curves match the measured magnetoresistance and Hall resistivity better than the nonmagnetic simulation does, the paper's conclusion that transport is dominated by the ordinary, nonmagnetic electronic structure fails.

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Extended reading notes

Core claim

The central claim is that in CrSb, a bulk g-wave altermagnet with Néel temperature $T_N = 712$ K and predicted spin splitting up to 1.2 eV, the measured magneto-transport is governed by the Lorentz force acting on ordinary carriers of the intrinsic band structure. Longitudinal magnetoresistance reaches 52.6% at 6 K and 7 T without saturating, and the data collapse under the extended Kohler's rule $MR = \alpha(H/n_T\rho_0)^m$ with $m = 1.42$ and $n_T$ varying from 0.8 at 6 K to 1.72 at 150 K. The Hall resistivity is nonlinear below 150 K, dips and changes sign below 60 K, yet a scaling plot of $\rho_{yx}/(n_T\rho_0)$ against $H/(n_T\rho_0)$ collapses all temperatures onto one curve with the same $n_T$ values used for the longitudinal magnetoresistance. The authors attribute the nonlinearity to a three-band model with two electron pockets and one hole pocket, and they argue from the magnetic point group $\overline{6}'/m'mm'$ that an anomalous Hall effect is not allowed. Numerical Boltzmann-transport simulation using the band structure without antiferromagnetic order almost reproduces the measured curves, which the paper takes as evidence that the transport is intrinsic and Lorentz-force dominated.

Load-bearing premise

The numerical simulations that anchor the Lorentz-force conclusion use the band structure without antiferromagnetic order and assume a shared or constant relaxation time; if the altermagnetic spin splitting or band-dependent scattering changes the Fermi surface enough, the match with experiment would not prove that the transport is ordinary.

Editorial extensions

If this is right

  • Nonlinear Hall resistivity in CrSb should not be taken as evidence of an altermagnetic anomalous Hall effect; the paper attributes it to multi-band transport.
  • The same temperature-dependent $n_T$ collapses both longitudinal magnetoresistance and Hall resistivity, so the two scaling analyses are tied to one carrier-density or relaxation-time scale.
  • Magneto-transport computed from the nonmagnetic band structure predicts the measured curves, so the predicted 1.2 eV spin splitting does not produce a detectable transport anomaly in this field and temperature range.
  • The extended Kohler scaling framework, previously applied to semimetals, now applies to an altermagnet, and Hall-resistivity scaling is reported in an altermagnetic material for the first time.

Reading between the lines

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

  • A sharper test of the Lorentz-force claim than the paper's simulation would be to repeat the calculation with the antiferromagnetic order and spin splitting included; the paper's match with experiment does not by itself identify which Fermi surface is active.
  • If the same analysis were applied to MnTe, where an anomalous Hall effect has been reported, subtracting the multi-band background would show whether a genuine altermagnetic Hall response remains; the paper does not make that comparison.
  • The success of the scaling suggests that other altermagnets could be analyzed the same way, but the extracted $n_T$ values are model-dependent and should be cross-checked against quantum-oscillation or thermoelectric measurements of the carrier density.
  • One consequence the authors leave implicit: if CrSb's transport is purely Lorentz-force in all field geometries, spintronic readout schemes for this altermagnet would have to rely on spin currents or other responses rather than on an anomalous Hall signal.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript reports the growth of high-quality CrSb single crystals and a combined experimental and computational study of magnetotransport in this altermagnet. The authors measure magnetization, longitudinal resistivity, magnetoresistance (MR), and Hall resistivity as functions of temperature and magnetic field, and they perform DFT band-structure, Fermi-surface, and Boltzmann-transport simulations. They confirm an antiferromagnetic transition at TN = 712 K, observe a non-saturating positive MR (52.6% at 6 K, 7 T), and find a nonlinear Hall resistivity at low temperatures. The experimental MR and Hall data are claimed to collapse onto universal scaling curves when a temperature-dependent factor nT is introduced, extending Kohler's rule. The authors conclude that the magnetotransport is dominated by the Lorentz force and originates from the intrinsic electronic structure, and that the nonlinear Hall effect is due to a multi-band structure rather than an anomalous Hall effect. The latter conclusion is supported by a magnetic-point-group symmetry argument and by a three-band model fit.

Significance. If the conclusions hold, the paper provides a useful counterpoint to recent altermagnet transport studies: it shows that ordinary multi-band magnetotransport can reproduce the scaling behavior in CrSb without invoking an anomalous Hall effect, and it reports the first scaling behavior of Hall resistivity in an altermagnet. The experimental data are of good quality (RRR ≈ 9, well-characterized crystals, high-temperature susceptibility), and the symmetry-based exclusion of an AHE is a solid contribution. The numerical simulations are based on standard, reproducible tools (VASP, Wannier90, WannierTools). However, the central interpretation rests on two pillars that are not fully demonstrated: (i) the Boltzmann simulations deliberately use the band structure without AFM order, although CrSb is predicted to have a spin splitting up to 1.2 eV near the Fermi level; and (ii) the scaling collapses are produced by freely adjusting nT at each temperature with no reported fit residuals or error bars. These issues currently limit the strength of the Lorentz-force/multi-band claim.

major comments (3)
  1. [Fig. 3(b) and the paragraph beginning 'Figure 3(b) displays...'] The numerical simulations that are used to conclude that magnetotransport 'originates from the intrinsic electronic structure and is dominated by the Lorentz force' are, by the authors' own statement, based on 'the band structures without considering AFM order.' Since CrSb is predicted to have a spin splitting of up to 1.2 eV near the Fermi level, the nonmagnetic Fermi surface may differ substantially from the altermagnetic one. Without a quantitative comparison of the nonmagnetic and AFM band structures/Fermi surfaces, or a calculation using the AFM spin configuration, the agreement between experiment and this nonmagnetic simulation does not establish the Lorentz-force origin. This is a load-bearing step for the paper's main conclusion and needs to be either fixed by calculation or explicitly justified as an approximation with supporting evidence.
  2. [Figs. 3(d) and 4(d) with insets] The scaling collapse is achieved by choosing a temperature-dependent constant nT for each temperature, with nT = 1 defined at 40 K, and then plotting MR and Hall data in the rescaled variables. No error bars, residuals, or goodness-of-fit statistics are reported for the collapsed curves, and no independent determination of nT is provided. Since any smooth family of curves can often be made to collapse by tuning one parameter per curve, the demonstration would be much stronger if the nT values were independently constrained—for example, from Hall coefficient measurements or from the carrier densities extracted from the three-band model—and if the collapse quality were quantified.
  3. [Eq. (1) and the Hall scaling analysis] Equation (1) is presented as a scaling law where n and m are the carrier density and effective mass, and the condition is that all charge carriers share the same temperature-dependent relaxation time. However, in the analysis the authors replace n and m by a single temperature-dependent nT and plot ρyx/(nT ρ0) against H/(nT ρ0), effectively absorbing the effective mass into nT without comment. The physical interpretation of nT is therefore ambiguous: it could represent carrier density, mobility, or a combination. The authors should state clearly whether Eq. (1) is being used in its full form or in a reduced, empirical form, and should discuss whether the fitted nT values are consistent with the stated condition of a common relaxation time.
minor comments (4)
  1. [Fig. 3(b) caption and text] The phrase 'band structures without considering AFM order' appears in both the main text and the caption; this is an unusual choice for a paper whose central subject is an altermagnet, and it should be flagged and explained more prominently in the text.
  2. [The paragraph discussing Fig. 4(a)] The three-band model used to fit ρyx(H) (2 electron pockets and 1 hole pocket) is mentioned, but no fit curves, parameters, or residuals are shown in the main text or referenced in the Supplemental Material. Since this fit is part of the evidence for the multi-band interpretation, a brief summary of its quality and parameters would be appropriate.
  3. [Experimental methods] The direction of the current relative to the crystal axes is not specified for the MR and Hall measurements. The band structure and Fermi surface are anisotropic, so the orientation of the current and field relative to the crystallographic axes is important for comparing experiment with simulation.
  4. [The concluding paragraph] The sentence 'the scaling law can very well describe the experimental data ... consistent very well with the numerical simulation results' is stronger than the evidence presented, because the simulation agreement is described elsewhere only as 'almost reproduce the main characteristics.' The wording should be adjusted to match the actual quantitative comparison.

Circularity Check

3 steps flagged · score 6.0 of 10

The scaling-law claim is enforced by per-temperature nT fits, and the Hall scaling form is imported from the authors' own prior work; the Lorentz-force conclusion retains independent simulation support.

  1. fitted input called prediction [Fig. 3(d) and preceding paragraph (extended Kohler analysis)]
    "We set nT = 1 at 40 K, and let all the MR data at various temperatures collapse onto the same line with that of 40 K by adjusting nT values. [...] This analysis indicates that all the MR data can be well described by the extended Kohler's scale rule, MR = α(H/nT ρ0)m, with m = 1.42"

    nT is a free temperature-dependent parameter introduced into the scaling form, and the collapse is produced by construction ('by adjusting nT values'). The statement that the data 'can be well described' by the extended Kohler rule is therefore a restatement of the fitting procedure, not an independent confirmation of a scaling law. The extracted nT(T) and m are outputs of an enforced collapse, so the advertised MR scaling behavior reduces to the fit.

  2. fitted input called prediction [Fig. 4(d) and following paragraph (Hall scaling)]
    "We chose a temperature-dependent constant nT for each temperature with setting nT = 1.0 at 40 K, and let all the ρyx/(nT ρ0) data below 150 K collapse onto a single line, as shown in Fig. 4(d), indicating that the Hall resistivity can be described by the scaling law expressed in Eq. (1)."

    The Hall scaling collapse is achieved by the same procedure: one free nT per temperature is chosen so that the curves lie on one line. Hence the conclusion 'the Hall resistivity can be described by the scaling law' is equivalent to the fitting operation. The later observation that the independently chosen nT values are close to the MR ones is a cross-check, but it does not turn the enforced collapse into a parameter-free prediction; the scaling behavior itself is fitted input.

1 more flagged steps
  1. self citation load bearing [Near Eq. (1) in the Hall-scaling section]
    "As discussed by us in Ref. [46], the Hall resistivity may also follow a scaling law under certain conditions, especially when all charge carriers share the same temperature-dependent relaxation time, expressed as: ρyx nρ0/m = h( B nρ0/m ) (1)"

    The specific Hall scaling form used to claim a unified description of longitudinal and transverse resistivity is taken from the authors' own previous work (Ref. [46], by S. Zhang, Q. Wu, Z. Fang et al., overlapping with the present authorship). This paper provides no derivation or external validation of Eq. (1); the load-bearing functional form is therefore justified by a self-citation. The empirical collapse could stand independently, but the 'unified framework' interpretation leans on this circular support.

full rationale

The paper's most novel advertised result is the scaling behavior of MR and Hall resistivity. In both channels the scaling is demonstrated by introducing a temperature-dependent nT and then 'letting' the data collapse: the collapse is a fitting criterion, not a prediction. This is the strongest circularity in the paper. The Hall scaling law's analytic form is imported from the authors' own Ref. [46], making the self-citation load-bearing for the unified-framework interpretation. I do not count the Boltzmann-transport simulation as circular: reproducing MR and Hall from the calculated (nonmagnetic) band structure is an independent consistency test of the Lorentz-force/multi-band picture, even though the omission of AFM order is a substantive correctness risk, not a circularity. The MPG-based exclusion of AHE uses external references and is independent. However, because the title and abstract foreground the scaling behavior, and that behavior is constructed by fitted nT values, the net circularity is partial but real: score 6.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central scaling claims rest on per-temperature fitted parameters nT and a Kohler exponent m, plus domain assumptions about the validity of the constant relaxation time approximation and the nonmagnetic band structure. No invented entities are introduced.

free parameters (2)
  • n_T (temperature-dependent rescaling factor) = 0.8 at 6 K to 1.72 at 150 K, normalized to 1 at 40 K
    Adjusted at each temperature to force the collapse of both MR and Hall data onto single scaling curves (Figs. 3d and 4d). Without this factor the simple Kohler's rule does not collapse the data.
  • m (Kohler exponent) = 1.42
    Fitted to the collapsed MR data in the extended Kohler's rule MR = alpha (H/(n_T rho_0))^m. Determines the power-law field dependence of the magnetoresistance.
assumptions (3)
  • domain assumption Constant relaxation time approximation for Boltzmann transport is valid for CrSb.
    The numerical simulations and the scaling law Eq. (1) assume that all charge carriers share the same temperature-dependent relaxation time. If scattering differs strongly between bands, the scaling argument and the simulation-to-experiment match would break down.
  • domain assumption The nonmagnetic PBE band structure accurately represents the transport-relevant Fermi surface of antiferromagnetic CrSb.
    The simulations are computed from band structures without considering AFM order (Fig. 3(b) text). If the altermagnetic spin splitting or the AFM order modifies the Fermi surface, the agreement between simulation and experiment would not establish a Lorentz-force origin.
  • domain assumption The magnetic point group 6'/m'mm' of CrSb forbids a spontaneous anomalous Hall effect.
    The exclusion of AHE relies on the magnetic point group analysis from Refs. 48 and 50. If the actual magnetic structure differs from the assumed one, the nonlinear Hall could still contain an anomalous component.

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Cite this review

Pith. "Pith review of Scaling Behavior of Magnetoresistance and Hall Resistivity in Altermagnet CrSb." pith.science (2026). https://pith.science/paper/KWMDM3ZG

@misc{pith2026241212263,
  author       = {Pith},
  title        = {Pith review of: Scaling Behavior of Magnetoresistance and Hall Resistivity in Altermagnet CrSb},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KWMDM3ZG}},
  note         = {Machine review of arXiv:2412.12263}
}
abstract

The discovery of altermagnet (AM) marks a significant advancement in magnetic materials, combining characteristics of both ferromagnetism and antiferromagnetism. In this Letter, we focus on CrSb, which has been verified to be an AM and to exhibit substantial spin splitting near the Fermi level. After successfully growing high-quality CrSb single crystals, we performed comprehensive magnetization, magnetoresistance (MR), and Hall resistivity measurements, along with the electronic structure, and Fermi surface (FS) calculations, as well as the magneto-transport property numerical simulations. An antiferromagnetic transition occurring at $T_{N}$ = 712 K was reconfirmed. It was found that both experimental MR and Hall resistivity are consistent with the numerical simulation results, and exhibit obvious scaling behavior. The nonlinear Hall resistivity is due to its multi-band structure, rather than an anomalous Hall effect (AHE). Especially, the scaling behavior in Hall resistivity is first observed within an AM material. These findings demonstrate that the magneto-transport properties in CrSb originate from the intrinsic electronic structure and are dominated by the Lorentz force.

Figures

Figures reproduced from arXiv: 2412.12263 by the authors.

Figure 2
Figure 2. FIG. 2. (Color online) (a) Temperature dependence of suscep [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1. (Color online) (a) Powder XRD pattern with the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) The field dependence of MR measured at various [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

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Pith tools

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