pith. sign in

arxiv: 2605.30881 · v1 · pith:IZMOHUXMnew · submitted 2026-05-29 · ❄️ cond-mat.mes-hall

Spin Hall effect in two-dimensional materials with inverted bands and Mexican-hat dispersion

Bagun S. Shchamkhalova , Vladimir A. Sablikov This is my paper

Pith reviewed 2026-06-28 21:34 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords spin Hall effectMexican-hat dispersiontwo-dimensional topological insulatorsskew scatteringquantum metricband inversionFermi energy dependence
0
0 comments X

The pith

Extrinsic spin-Hall current from skew scattering exceeds intrinsic Berry curvature contribution in Mexican-hat dispersion materials and can change sign with Fermi energy.

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

The paper investigates the spin Hall effect in two-dimensional topological insulators that have inverted bands leading to a Mexican-hat dispersion and ring-shaped Fermi surfaces. It shows that second-order scattering processes enable a spin-dependent skew scattering whose strength is set by inter-contour electron transitions and the quantum metric. This extrinsic spin-Hall current can greatly exceed the intrinsic contribution arising from Berry curvature. The dependence on Fermi energy is non-monotonic: the current reaches a maximum, then falls, and may reverse sign because of the competing energy dependencies of transition probabilities and carrier velocities on the two contours.

Core claim

In materials with Mexican-hat dispersion, the extrinsic spin-Hall current generated by second-order skew scattering significantly exceeds the intrinsic spin-Hall current from Berry curvature; moreover, the extrinsic current reaches a maximum at intermediate Fermi energy, decreases thereafter, and can change sign, due to the interplay of inter- and intra-contour transition probabilities and differing electron velocities.

What carries the argument

Spin-dependent skew scattering arising from second-order scattering processes, determined by the quantum metric and transitions between different isoenergetic contours in the Mexican-hat band structure.

Load-bearing premise

That the extrinsic spin-Hall current is controlled by second-order scattering processes whose probabilities are set solely by the quantum metric and inter-contour transitions, without higher-order effects or disorder details altering the sign change.

What would settle it

A plot of spin-Hall conductivity versus Fermi energy in a Mexican-hat dispersion sample that lacks a maximum followed by decrease and sign change would contradict the prediction; observation of this sequence would confirm it.

Figures

Figures reproduced from arXiv: 2605.30881 by Bagun S. Shchamkhalova, Vladimir A. Sablikov.

Figure 1
Figure 1. Figure 1: FIG. 1. MHD and ring-shaped Fermi surface for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: presents the dependencies of the extrinsic SHC for two systems with parameters 𝑎 = 0.2, 𝛿 = −0.1 and 𝑎 = 0.25, 𝛿 = −0.2 on the 𝐸𝐹 for different temperatures. These dependencies are similar for both systems. At the low 𝐸𝐹 the extrinsic SHC increases with the 𝐸𝐹 . We attribute this to an increase of the carrier number. But with further increase of the 𝐸𝐹 the SHC reaches maximum and then decreases and even ch… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The additions to the distribution function for two [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Intrinsic SHC dependencies on [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: a) shows the dependencies of the extrinsic SHC on the 𝐸𝐹 for systems with different hybridization parameters 𝑎 and the same parameter 𝛿 = −0.1 and temperature 𝑇 = 0.01. It is clearly seen that the value of the extrinsic SHC decreases with the increase of 𝑎. For 𝛿 = −0.1 and 𝑎 < 0.25 the SHC changes the sign at 𝐸𝐹 > 0.5. At 𝑎 > 0.3 the SHC is negative for all values of 𝐸𝐹 < 1. The dependencies of the extrin… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Extrinsic (curves 1-3) and intrinsic (curves 4-6) SHC depen [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

We study the spin Hall effect in two-dimensional topological insulators with "Mexican hat" dispersion and a ring-shaped Fermi surface which are formed due to the band inversion. Electron transitions between different isoenergetic contours and the quantum metric of band states play an important role in the transport properties of such materials, since they largely determine the spatial distribution of the electron charges screening the impurity potential and the scattering probability [Phys.B, 719, 417942 (2025)]. Here we study a spin-dependent skew scattering, which is enabled by the second-order scattering processes, and show that the extrinsic spin-Hall current (SHC) can significantly exceed the intrinsic SHC arising from the Berry curvature. Furthermore, due to Mexican-hat dispersion, the SHC exhibits a very unusual dependence on the Fermi energy ($E_F$). The extrinsic SHC reaches a maximum at some $E_F$, then decreases with increasing $E_F$ and can even change a sign. This complicated behavior reflects an interplay of energy dependencies of such important factors as probabilities of inter- and intra-contour transitions, as well as different electron velocities in two contours.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines the spin Hall effect in 2D topological insulators featuring Mexican-hat dispersion and a ring-shaped Fermi surface arising from band inversion. It focuses on extrinsic spin-Hall conductivity (SHC) generated by spin-dependent skew scattering via second-order processes, where inter-contour electron transitions and the quantum metric determine screening and scattering rates (referencing the authors' prior Phys. B 719, 417942 (2025) work). The central claims are that this extrinsic SHC can substantially exceed the intrinsic contribution from Berry curvature, and that the total SHC displays a non-monotonic Fermi-energy dependence—reaching a maximum, then declining and potentially changing sign—due to the competing energy dependencies of inter- versus intra-contour scattering probabilities and the distinct velocities on the two contours.

Significance. If the perturbative treatment and resulting E_F dependence hold, the work demonstrates how Mexican-hat band structures can make extrinsic mechanisms dominant and produce testable, non-standard energy scaling in the spin Hall effect. This provides a concrete example of quantum-metric-enabled skew scattering controlling transport beyond conventional semiclassical expectations, with potential implications for doping-tuned spin currents in inverted-band 2D materials.

major comments (2)
  1. [Method / Results (scattering probabilities section)] The central claim of non-monotonic E_F dependence (including possible sign change) and extrinsic dominance rests on scattering probabilities and velocities derived in the referenced prior Phys. B paper. The manuscript should reproduce the key expressions for the inter- and intra-contour transition rates (or at minimum the resulting energy-dependent factors entering the skew-scattering term) to allow independent verification of how they produce the reported maximum and sign reversal.
  2. [Results (SHC vs E_F)] The abstract and introduction assert that extrinsic SHC 'can significantly exceed' the intrinsic Berry-curvature contribution, yet no explicit numerical comparison, ratio plot, or table of SHC values versus E_F is referenced. A figure or table quantifying the ratio (extrinsic/intrinsic) across the relevant E_F range is needed to substantiate the 'significantly exceed' statement.
minor comments (2)
  1. [Introduction / Model] Notation for the two isoenergetic contours (inner/outer ring) and the quantum-metric tensor components should be defined explicitly at first use, with a brief reminder of how they enter the second-order scattering amplitude.
  2. [Model] The manuscript would benefit from a short schematic (or reference to one) illustrating the Mexican-hat dispersion, the ring Fermi surface, and the inter-contour transitions that enable the skew scattering.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive suggestions. We address each major comment below and will revise the manuscript accordingly to improve clarity and verifiability.

read point-by-point responses

  1. Referee: [Method / Results (scattering probabilities section)] The central claim of non-monotonic E_F dependence (including possible sign change) and extrinsic dominance rests on scattering probabilities and velocities derived in the referenced prior Phys. B paper. The manuscript should reproduce the key expressions for the inter- and intra-contour transition rates (or at minimum the resulting energy-dependent factors entering the skew-scattering term) to allow independent verification of how they produce the reported maximum and sign reversal.

    Authors: We agree that reproducing the key expressions will facilitate independent verification. In the revised manuscript we will add a dedicated subsection (or appendix) that explicitly states the inter- and intra-contour transition rates and the resulting energy-dependent factors that enter the skew-scattering spin Hall conductivity. These expressions are taken directly from our prior Phys. B 719, 417942 (2025) work; we will highlight the competing E_F scalings that produce the maximum and possible sign change without requiring the reader to consult the earlier paper. revision: yes

  2. Referee: [Results (SHC vs E_F)] The abstract and introduction assert that extrinsic SHC 'can significantly exceed' the intrinsic Berry-curvature contribution, yet no explicit numerical comparison, ratio plot, or table of SHC values versus E_F is referenced. A figure or table quantifying the ratio (extrinsic/intrinsic) across the relevant E_F range is needed to substantiate the 'significantly exceed' statement.

    Authors: We accept the point. The revised manuscript will include a new figure (or inset panel) that plots the ratio of extrinsic to intrinsic SHC versus Fermi energy over the full range of interest, together with a brief table of representative values. This will make the statement that the extrinsic contribution can significantly exceed the intrinsic one quantitatively explicit and directly testable against the non-monotonic behavior already shown in the existing figures. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation relies on a self-citation only for supporting details on screening and scattering probabilities in the Mexican-hat system, but this serves as independent prior input rather than causing the central extrinsic SHC result or its E_F dependence to reduce by construction to the paper's own fitted parameters or definitions. The claimed unusual energy dependence arises from applying standard second-order skew scattering to inter/intra-contour transitions and velocity differences in the given band structure, without any quoted equations showing self-definitional equivalence, fitted inputs renamed as predictions, or load-bearing uniqueness imported solely via overlapping-author citations. The approach remains self-contained within semiclassical transport theory applied to the inverted-band dispersion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on abstract: relies on standard band-inversion producing Mexican-hat dispersion, quantum metric affecting screening, and second-order perturbation for skew scattering. No explicit free parameters or invented entities listed.

axioms (2)
  • domain assumption Band inversion produces Mexican-hat dispersion with ring-shaped Fermi surface having two isoenergetic contours
    Stated in abstract as the origin of the dispersion and contours
  • domain assumption Quantum metric of band states largely determines spatial distribution of screening charges and scattering probability
    Abstract cites prior work but invokes it as key for transport

pith-pipeline@v0.9.1-grok · 5735 in / 1466 out tokens · 21658 ms · 2026-06-28T21:34:46.464955+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 32 canonical work pages

  1. [1]

    and Wunderlich, J

    author author J. Sinova , author S. O. \ Valenzuela , author J. Wunderlich , author C. H. \ Back , \ and\ author T. Jungwirth ,\ 10.1103/RevModPhys.87.1213 journal journal Rev. Mod. Phys. \ volume 87 ,\ pages 1213 ( year 2015 ) NoStop

    work page doi:10.1103/revmodphys.87.1213 2015

  2. [2]

    author author N. A. \ Sinitsyn ,\ 10.1088/0953-8984/20/02/023201 journal journal Journal of Physics: Condensed Matter \ volume 20 ,\ pages 023201 ( year 2007 ) NoStop

    work page doi:10.1088/0953-8984/20/02/023201 2007

  3. [3]

    \ Chang , author C.-X

    author author C.-Z. \ Chang , author C.-X. \ Liu , \ and\ author A. H. \ MacDonald ,\ 10.1103/RevModPhys.95.011002 journal journal Rev. Mod. Phys. \ volume 95 ,\ pages 011002 ( year 2023 ) NoStop

    work page doi:10.1103/revmodphys.95.011002 2023

  4. [4]

    Niu , author H

    author author C. Niu , author H. Wang , author N. Mao , author B. Huang , author Y. Mokrousov , \ and\ author Y. Dai ,\ 10.1103/PhysRevLett.124.066401 journal journal Phys. Rev. Lett. \ volume 124 ,\ pages 066401 ( year 2020 ) NoStop

    work page doi:10.1103/physrevlett.124.066401 2020

  5. [5]

    Wang , author H

    author author H. Wang , author H. Liu , author X. Feng , author J. Cao , author W. Wu , author S. Lai , author W. Gao , author C. Xiao , \ and\ author S. A. \ Yang ,\ 10.1103/PhysRevLett.134.056301 journal journal Phys. Rev. Lett. \ volume 134 ,\ pages 056301 ( year 2025 ) NoStop

    work page doi:10.1103/physrevlett.134.056301 2025

  6. [6]

    Hayami , author M

    author author S. Hayami , author M. Yatsushiro , \ and\ author H. Kusunose ,\ 10.1103/PhysRevB.106.024405 journal journal Phys. Rev. B \ volume 106 ,\ pages 024405 ( year 2022 ) NoStop

    work page doi:10.1103/physrevb.106.024405 2022

  7. [7]

    Manchon,et al., Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems.Rev

    author author A. Manchon , author J. Z Z elezn\'y , author I. M. \ Miron , author T. Jungwirth , author J. Sinova , author A. Thiaville , author K. Garello , \ and\ author P. Gambardella ,\ 10.1103/RevModPhys.91.035004 journal journal Rev. Mod. Phys. \ volume 91 ,\ pages 035004 ( year 2019 ) NoStop

    work page doi:10.1103/revmodphys.91.035004 2019

  8. [8]

    Bai , author B

    author author Y. Bai , author B. Yuan , author Z. Chen , author Y. Dai , author B. Huang , author X. Wang , \ and\ author C. Niu ,\ 10.1103/588c-z8cy journal journal Phys. Rev. Lett. \ volume 136 ,\ pages 046602 ( year 2026 ) NoStop

    work page doi:10.1103/588c-z8cy 2026

  9. [9]

    author author B. A. \ Bernevig \ and\ author S.-C. \ Zhang ,\ 10.1103/PhysRevLett.96.106802 journal journal Phys. Rev. Lett. \ volume 96 ,\ pages 106802 ( year 2006 ) NoStop

    work page doi:10.1103/physrevlett.96.106802 2006

  10. [10]

    Jiabin , author Q

    author author Y. Jiabin , author Q. R. \ Bernevig B. Andrei , author T. P. \ Rossi Enrico , \ and\ author Y. Bohm-Jung ,\ 10.1038/s41535-025-00801-3 journal journal Quantum Materials \ volume 10 ,\ pages 1 ( year 2025 ) NoStop

    work page doi:10.1038/s41535-025-00801-3 2025

  11. [11]

    author author J. E. \ Hirsch ,\ 10.1103/PhysRevLett.83.1834 journal journal Phys. Rev. Lett. \ volume 83 ,\ pages 1834 ( year 1999 ) NoStop

    work page doi:10.1103/physrevlett.83.1834 1999

  12. [12]

    author author M. A. \ R , author M. P. J. \ Lee J. S. , Richardella A.and Grab J. L. , author V. A. \ Fischer M. H. , \ and\ author R. D. C. \ Manchon A. , Kim E.-A.and Samarth N. ,\ 10.1038/nature13534 journal journal Nature \ volume 511 ,\ pages 449 ( year 2014 ) NoStop

    work page doi:10.1038/nature13534 2014

  13. [13]

    author author S. S. \ Krishtopenko ,\ 10.1038/s41598-021-00577-z journal journal Scientific Reports \ volume 11 ,\ pages 21060 ( year 2021 ) NoStop

    work page doi:10.1038/s41598-021-00577-z 2021

  14. [14]

    Stauber , author N

    author author T. Stauber , author N. M. R. \ Peres , author F. Guinea , \ and\ author A. H. \ Castro Neto ,\ 10.1103/PhysRevB.75.115425 journal journal Phys. Rev. B \ volume 75 ,\ pages 115425 ( year 2007 ) NoStop

    work page doi:10.1103/physrevb.75.115425 2007

  15. [15]

    Wickramaratne , author F

    author author D. Wickramaratne , author F. Zahid , \ and\ author R. K. \ Lake ,\ 10.1063/1.4928559 journal journal Journal of Applied Physics \ volume 118 ,\ pages 075101 ( year 2015 ) ,\ http://arxiv.org/abs/https://pubs.aip.org/aip/jap/article-pdf/doi/10.1063/1.4928559/14743158/075101\_1\_online.pdf https://pubs.aip.org/aip/jap/article-pdf/doi/10.1063/1...

    work page doi:10.1063/1.4928559 2015

  16. [16]

    Kremer , author A

    author author G. Kremer , author A. Mahmoudi , author M. Bouaziz , author M. Rahimi , author F. Bertran , author J.-F. \ Dayen , author M. L. D. \ Rocca , author M. Pala , author A. Naitabdi , author J. Chaste , author F. Oehler , \ and\ author A. Ouerghi ,\ https://arxiv.org/abs/2509.06488 title Mexican hat-like valence band dispersion and quantum confin...

    arXiv 2025

  17. [17]

    Jiang , author S

    author author Y. Jiang , author S. Thapa , author G. D. \ Sanders , author C. J. \ Stanton , author Q. Zhang , author J. Kono , author W. K. \ Lou , author K. Chang , author S. D. \ Hawkins , author J. F. \ Klem , author W. Pan , author D. Smirnov , \ and\ author Z. Jiang ,\ 10.1103/PhysRevB.95.045116 journal journal Phys. Rev. B \ volume 95 ,\ pages 0451...

    work page doi:10.1103/physrevb.95.045116 2017

  18. [18]

    Du , author T

    author author L. Du , author T. Li , author W. Lou , author X. Wu , author X. Liu , author Z. Han , author C. Zhang , author G. Sullivan , author A. Ikhlassi , author K. Chang , \ and\ author R.-R. \ Du ,\ 10.1103/PhysRevLett.119.056803 journal journal Phys. Rev. Lett. \ volume 119 ,\ pages 056803 ( year 2017 ) NoStop

    work page doi:10.1103/physrevlett.119.056803 2017

  19. [19]

    Meyer , author J

    author author M. Meyer , author J. Baumbach , author S. Krishtopenko , author A. Wolf , author M. Emmerling , author S. Schmid , author M. Kamp , author B. Jouault , author J.-B. \ Rodriguez , author E. Tournie , author T. Müller , author R. Thomale , author G. Bastard , author F. Teppe , author F. Hartmann , \ and\ author S. Höfling ,\ 10.1126/sciadv.adz...

    work page doi:10.1126/sciadv.adz2408 2025

  20. [20]

    \ Hou , author B.-J

    author author Y.-N. \ Hou , author B.-J. \ Wang , author C.-D. \ Jin , author H. Zhang , author J.-L. \ Wang , author P.-L. \ Gong , author R.-Q. \ Lian , author X.-Q. \ Shi , \ and\ author R.-N. \ Wang ,\ 10.1063/5.0237686 journal journal Journal of Applied Physics \ volume 137 ,\ pages 064302 ( year 2025 ) ,\ http://arxiv.org/abs/https://pubs.aip.org/ai...

    work page doi:10.1063/5.0237686 2025

  21. [21]

    Jiang , author B

    author author W. Jiang , author B. Li , author X. Wang , author G. Chen , author T. Chen , author Y. Xiang , author W. Xie , author Y. Dai , author X. Zhu , author H. Yang , author J. Sun , \ and\ author H.-H. \ Wen ,\ 10.1103/PhysRevB.101.121115 journal journal Phys. Rev. B \ volume 101 ,\ pages 121115 ( year 2020 ) NoStop

    work page doi:10.1103/physrevb.101.121115 2020

  22. [22]

    Rukelj , author Željana Bonačić Lošić , author I

    author author Z. Rukelj , author Željana Bonačić Lošić , author I. Kupčić , \ and\ author A. Akrap ,\ https://doi.org/10.1016/j.physb.2024.416590 journal journal Physica B: Condensed Matter \ volume 695 ,\ pages 416590 ( year 2024 ) NoStop

    work page doi:10.1016/j.physb.2024.416590 2024

  23. [23]

    Zhang , author P

    author author W. Zhang , author P. Jia , author W. kai Lou , author X. Wang , author S. Su , author K. Chang , \ and\ author R.-R. \ Du ,\ 10.1088/0256-307X/43/1/010704 journal journal Chin. Phys. Lett. \ volume 43 ,\ pages 010704 ( year 2026 ) NoStop

    work page doi:10.1088/0256-307x/43/1/010704 2026

  24. [24]

    author author V. A. \ Sablikov ,\ https://doi.org/10.1016/j.physe.2025.116213 journal journal Physica E: Low-dimensional Systems and Nanostructures \ volume 170 ,\ pages 116213 ( year 2025 ) NoStop

    work page doi:10.1016/j.physe.2025.116213 2025

  25. [25]

    Das , author D

    author author P. Das , author D. Wickramaratne , author B. Debnath , author G. Yin , \ and\ author R. K. \ Lake ,\ 10.1103/PhysRevB.99.085409 journal journal Phys. Rev. B \ volume 99 ,\ pages 085409 ( year 2019 ) NoStop

    work page doi:10.1103/physrevb.99.085409 2019

  26. [26]

    author author B. S. \ Shchamkhalova \ and\ author V. A. \ Sablikov ,\ https://doi.org/10.1016/j.physb.2025.417942 journal journal Physica B: Condensed Matter \ volume 719 ,\ pages 417942 ( year 2025 ) NoStop

    work page doi:10.1016/j.physb.2025.417942 2025

  27. [27]

    author author V. A. \ Sablikov \ and\ author A. A. \ Sukhanov ,\ https://doi.org/10.1016/j.physe.2022.115492 journal journal Physica E: Low-dim.Systems and Nanostructures \ volume 145 ,\ pages 115492 ( year 2023 a ) NoStop

    work page doi:10.1016/j.physe.2022.115492 2022

  28. [28]

    author author V. A. \ Sablikov \ and\ author A. A. \ Sukhanov ,\ https://doi.org/10.1016/j.physleta.2023.129006 journal journal Physics Letters A \ volume 481 ,\ pages 129006 ( year 2023 b ) NoStop

    work page doi:10.1016/j.physleta.2023.129006 2023

  29. [29]

    author author B. A. \ Bernevig , author T. L. \ Hughes , \ and\ author S.-C. \ Zhang ,\ 10.1126/science.1133734 journal journal Science \ volume 314 ,\ pages 1757 ( year 2006 ) NoStop

    work page doi:10.1126/science.1133734 2006

  30. [30]

    K\" o nig , author H

    author author M. K\" o nig , author H. Buhmann , author L. W. Molenkamp , author T. Hughes , author C.-X. \ Liu , author X.-L. \ Qi , \ and\ author S.-C. \ Zhang ,\ 10.1143/JPSJ.77.031007 journal journal J. Phys. Society of Japan \ volume 77 ,\ pages 031007 ( year 2008 ) ,\ http://arxiv.org/abs/https://doi.org/10.1143/JPSJ.77.031007 https://doi.org/10.114...

    work page doi:10.1143/jpsj.77.031007 2008

  31. [31]

    Schliemann \ and\ author D

    author author J. Schliemann \ and\ author D. Loss ,\ 10.1103/PhysRevB.68.165311 journal journal Phys. Rev. B \ volume 68 ,\ pages 165311 ( year 2003 ) NoStop

    work page doi:10.1103/physrevb.68.165311 2003

  32. [32]

    author author D. Culcer ,\ https://doi.org/10.1016/j.physe.2011.11.003 journal journal Physica E: Low-dimensional Systems and Nanostructures \ volume 44 ,\ pages 860 ( year 2012 ) ,\ note sI:Topological Insulators NoStop

    work page doi:10.1016/j.physe.2011.11.003 2011

  33. [33]

    Bandyopadhyay , author N

    author author A. Bandyopadhyay , author N. B. \ Joseph , \ and\ author A. Narayan ,\ https://doi.org/10.1016/j.mtelec.2024.100101 journal journal Materials Today Electronics \ volume 8 ,\ pages 100101 ( year 2024 ) NoStop

    work page doi:10.1016/j.mtelec.2024.100101 2024