Pith. sign in

REVIEW 2 major objections 3 minor 54 references

On warped topological-insulator surfaces, in-plane pure spin currents appear at second order in the applied electric field, with directions fixed by crystal symmetry.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-04 21:06 UTC pith:CWUQ5YWR

load-bearing objection The intrinsic second-order spin current in Bi2Te3 surface states is a real result, but the extrinsic collinear spin current vanishes by time-reversal symmetry, so the paper's main new claim is unsupported. the 2 major comments →

arxiv 2608.01782 v1 pith:CWUQ5YWR submitted 2026-08-03 cond-mat.mes-hall

Non-linear spin current in the surface states of topological insulators

Srijan Chatterjee , Tarun Kanti Ghosh This is my paper
classification cond-mat.mes-hall
keywords conserved spin currentnonlinear spin Hall effectspin-weighted Berry curvatureBerry connection polarizabilityBerry curvature dipolehexagonal warpingtopological insulator surface statestime-reversal symmetry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that on the warped surface of a three-dimensional topological insulator such as Bi2Te3, pure spin currents that are absent in the linear response appear at second order in an applied electric field. Working within the conserved-spin-current formalism, it derives linear and nonlinear spin Hall conductivities for the Dirac surface Hamiltonian with hexagonal warping, whose k^3 warping term gives a finite Berry curvature without breaking time-reversal symmetry. The second-order response decomposes into intrinsic and extrinsic channels: the intrinsic channel is governed by the Berry connection polarizability dipole and a spin-dependent Berry-curvature term, while the extrinsic channel comes from the Fermi-surface dipole of the spin-weighted Berry curvature. The C3v anisotropy of the dispersion acts as a selection rule, leaving a field-polarized x-spin current in the intrinsic channel and a collinearly polarized y-spin current in the extrinsic channel. A sympathetic reader would care because these nonlinear in-plane spin currents could enable pure spin injection and field-free magnetization switching without an external magnetic field.

Core claim

On the paper's own terms, the central discovery is that the second-order response of the conserved spin current is not empty for time-reversal-invariant hexagonal-warped surface states. The linear spin Hall conductivity is purely out-of-plane; the in-plane spin components vanish not because of time-reversal symmetry alone, but because the C3 anisotropy makes their momentum integrals cancel. At order E^2, the same anisotropy selects nonzero components: the intrinsic conductivity Γ^x is polarized along the applied field, and the extrinsic Fermi-surface conductivity Γ^y is collinear with the current. Both are expressed through band-geometric quantities—spin-weighted Berry curvature, Berry conne

What carries the argument

The central object is the spin-weighted Berry curvature, Ξ^l_ab = Ω^ab S^l, the product of the band Berry curvature and the spin polarization. At linear order it produces the out-of-plane spin Hall conductivity; at second order its Fermi-surface dipole generates the extrinsic in-plane spin current. The intrinsic second-order channel is carried by the Berry connection polarizability (BCP) dipole and by a Berry-curvature/spin-matrix-element term that arise from first-order corrections to the Bloch states. The enabling ingredient is the k^3 hexagonal warping term in the surface Hamiltonian, which makes the Berry curvature finite and anisotropic while leaving time-reversal symmetry intact.

Load-bearing premise

The central calculation depends on the spin-weighted Berry curvature having a nonzero Fermi-surface dipole; if symmetry forces that dipole to vanish, the extrinsic second-order spin current disappears and only the intrinsic terms remain.

What would settle it

Compute the Fermi-surface integral in Eq. (35) at a single Fermi energy with the paper's own Ξ^l(k). The extrinsic second-order conductivity survives only if this integral is nonzero; a second-harmonic spin-current measurement on Bi2Te3 can then check whether the predicted collinear component actually appears.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • Linear response on these surfaces carries only out-of-plane spin polarization; in-plane spin currents are a second-order phenomenon, so they appear at twice the driving frequency in ac experiments.
  • The intrinsic second-order spin conductivity is predicted to be on the order of 0.4 e µm/V, large enough to produce measurable edge spin accumulation or spin-orbit torque.
  • The extrinsic contribution is a Fermi-surface effect and is much smaller than the intrinsic one, so it should be distinguishable by its relaxation-time scaling.
  • The C3v symmetry acts as a selection switch: among the three spin components, only the components even under the mirror symmetries survive the momentum integration.
  • A collinear spin current, with spins parallel to the travel direction, emerges from the extrinsic channel and could apply field-free torques to an adjacent magnet.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extension: strain or a tilted surface that lowers the C3v symmetry to C_s should unlock additional spin-conductivity tensor components, since the selection rules derived here are purely symmetry-driven.
  • Extension: the same spin-weighted-Berry-curvature machinery could be applied to other hexagonal-warped Dirac materials, such as Bi2Se3 or Sb2Te3, to predict the sign and magnitude of their nonlinear spin currents without new formalism.
  • Extension: a first-principles band-structure calculation of Bi2Te3's surface states could test whether the low-energy C3v model captures the dominant contributions or whether higher-order warping terms change the surviving components.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. This paper studies linear and second-order pure spin currents in the hexagonal-warping surface states of Bi2Te3 within the conserved-spin-current formalism. Starting from the Fu model, the authors derive the k-resolved Berry curvature, BCP, and spin-weighted Berry curvature, and combine first-order perturbation theory with a Boltzmann relaxation-time description. They report (i) a purely out-of-plane linear spin Hall conductivity, (ii) a second-order intrinsic in-plane spin current polarized along the applied electric field, arising from BCP and Berry-curvature terms, and (iii) a second-order extrinsic in-plane collinear spin current attributed to a spin-weighted Berry curvature dipole. The paper also discusses experimental probes and potential applications of the collinear spin current.

Significance. If correct, the paper would identify a new extrinsic nonlinear spin response in topological insulator surfaces. The linear out-of-plane result and the intrinsic second-order field-polarized response are indeed consistent with the symmetries of the model, and the derivations in Eqs. (32)-(34) and Eqs. (23)-(24) have the expected parity properties. The work usefully shows that the conserved-spin-current formalism can be applied to the warped TI surface without introducing a mass term. However, the extrinsic second-order response is the paper's main new predictive claim, and it vanishes identically under time reversal. Since the abstract, Sec. IV, and Fig. 4(b) all rely on this extrinsic result, the central claim of the paper is unsupported by the paper's own equations.

major comments (2)
  1. [III.B, Eq. (35) and Appendix B] The extrinsic SOSC vanishes identically. From Eq. (26), d(-k) = -d(k), so the dispersion is even, ε(-k)=ε(k); the spin expectation is odd, S^l(-k)=-S^l(k); and Eq. (28) gives Ω^xy(-k)=-Ω^xy(k). Therefore Ξ^l_yx(-k)=Ω^xy(-k)S^l(-k)=Ξ^l_yx(k), i.e. the spin-weighted Berry curvature is even under k→-k. Since f^(0)(k) is even, ∂f/∂k_x is odd, so the integrand in Eq. (35) is an odd function of k. The integral over the inversion-symmetric Brillouin zone is therefore zero for every spin component l and for both bands. The same conclusion follows from the Fermi-surface form in Eq. (B2): k_F and the denominator are even under φ→φ+π, while ∂ε/∂k_x is odd, so the angular integral vanishes. Thus Γ^{l,ext}=0; the nonzero Γ^{y,ext} in Fig. 4(b) cannot be reproduced from the paper's own formulas.
  2. [Abstract and Sec. IV] The abstract's claim that the second-order response exhibits an extrinsic in-plane spin current 'induced by ... Berry curvature dipole (BCD)', and Sec. IV's conclusion that 'The extrinsic SOSC ... leads to a collinearly polarized spin current', rest entirely on Γ^{l,ext} in Eq. (35). Since that quantity is zero by time-reversal symmetry, these statements are unsupported. This is not a matter of numerical magnitude or of the relaxation-time approximation; it is a symmetry property of the model. A revision that wishes to retain an extrinsic response would need to break time-reversal symmetry (e.g. by magnetic doping) or change the definition of the distribution function, neither of which is present in the manuscript.
minor comments (3)
  1. [Sec. IV] The terminology 'spin-dependent BCD' is misleading: the quantity in Eq. (35) is the product of Berry curvature and spin expectation, and its dipole is not a standard Berry curvature dipole. The vanishing of this dipole is a direct consequence of time-reversal symmetry, and the text should reflect that.
  2. [Fig. 4(b)] The figure shows a nonzero extrinsic y-polarized conductivity, but the analytic integral in Eq. (B2) is odd under φ→φ+π and hence integrates to zero. The figure should be re-examined; if it was obtained numerically, the integration grid or the inclusion of the Jacobian may have broken the symmetry.
  3. [Sec. II.A] The relaxation-time approximation in Eq. (13) is applied to the surface states without modeling surface-bulk coupling. This is a secondary issue for the intrinsic response, but it would need discussion if the extrinsic response were retained.

Circularity Check

0 steps flagged

No significant circularity: the paper computes model-dependent response coefficients from a published Hamiltonian and external transport formalism without fitting the target observable or importing a load-bearing self-citation.

full rationale

The derivation chain is self-contained in the relevant sense. The model Hamiltonian (Eq. 25) is taken from the established topological-insulator surface literature (Fu, Ref. [25]; Zhang et al., Refs. [21,22]), and the conserved-spin-current response formalism is taken from external references (Refs. [14,15,17]). The linear and second-order spin conductivities are obtained by direct perturbative expansion of Bloch states (Eqs. 2-11) and by solving the Boltzmann equation in the relaxation-time approximation (Eqs. 13-15); no parameter is fitted to the quantity being predicted. The spin-weighted Berry curvature Xi^l = Omega^ab S^l is introduced by definition, not as an input that already encodes the final conductivities. The intrinsic second-order terms are derived from the BCP and Berry curvature, and the extrinsic term (Eq. 35) is an integral over the same defined quantity with the Fermi-function gradient; even if that integral has a symmetry-based problem, that is a correctness or mathematical-validity concern, not circularity. The self-citations present (Refs. [18,27,45]) are used as background or as comparisons of magnitude, not as the load-bearing justification for the main results. The paper does not rename a known result as a new one, does not invoke a uniqueness theorem from the authors' own prior work, and does not smuggle an ansatz in via self-citation. Therefore no circular step is exhibited.

Axiom & Free-Parameter Ledger

1 free parameters · 6 axioms · 1 invented entities

No new particles, forces, or symmetries are postulated. The only new object is the spin-BCD tensor, whose associated current is symmetry-forbidden. The calculation relies on the standard conserved-spin formalism, the RTA Boltzmann equation, and the literature model Hamiltonian.

free parameters (1)
  • Relaxation time tau = 1 ps (assumed)
    Chosen for the extrinsic SOSC plot (Fig. 4b). It enters only the extrinsic channel, which is identically zero, so it does not affect any surviving prediction.
axioms (6)
  • domain assumption Conserved spin current definition j^l_a = e Omega^ab S^l E_b (Eq. 1) from Refs 14 and 17
    The entire calculation operates within the proper/conserved spin current formalism; the results are only as meaningful as this definition of spin current.
  • domain assumption Relaxation-time approximation with a single scalar tau (Eq. 13)
    The Boltzmann equation is solved iteratively with a constant tau; disorder details, skew scattering and side jumps are omitted.
  • domain assumption The Fu model Hamiltonian (Eq. 25) with v0 = 2.55 eV Angstrom and lambda = 250 eV Angstrom^3 from Ref 25, bulk and surface-bulk coupling neglected
    The 2D surface Dirac model with hexagonal warping is the standard low-energy description of the Bi2Te3 (111) surface; bulk bands and surface-bulk coupling are outside the model.
  • standard math First-order perturbative correction to Bloch states (Eq. 2) and truncation of the response at second order in E
    Standard perturbation expansion underlying the BCP and Berry-curvature response; assumes weak fields and no interband coherence beyond first order.
  • domain assumption Low-temperature limit with d f / d epsilon = - delta(epsilon - mu) for the extrinsic term (Appendix B)
    The extrinsic angular integral is evaluated at T = 0; finite temperature would smear the Fermi surface but cannot make a time-reversal-odd integral nonzero.
  • standard math Two-level BCP metric relation G^ab_+/- = +/- hbar^2 g^ab_+/- / (4 d) (Eq. 21)
    Standard relation between Berry connection polarizability and quantum metric for two-band models, taken from Ref 42.
invented entities (1)
  • Spin-weighted Berry curvature dipole (spin-BCD) no independent evidence
    purpose: Introduced as the driver of the extrinsic second-order spin current (Eq. 35, Appendix B)
    This is a response tensor, not a physical entity. It has no observable handle outside the paper, and the current it is claimed to produce vanishes identically because the spin-weighted curvature Omega S is time-reversal even while d f / d k is time-reversal odd.

pith-pipeline@v1.3.0-daily-deepseek · 11377 in / 60718 out tokens · 667232 ms · 2026-08-04T21:06:23.110666+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Non-linear spin current in the surface states of topological insulators." pith.science (2026). https://pith.science/paper/CWUQ5YWR

@misc{pith2026260801782,
  author       = {Pith},
  title        = {Pith review of: Non-linear spin current in the surface states of topological insulators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWUQ5YWR}},
  note         = {Machine review of arXiv:2608.01782}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Spin Hall effect is one of the primary sources of pure spin current in spintronic devices, observed in spin-orbit coupled materials. In this work, we investigate linear and non-linear pure spin currents in the hexagonally warped surface states of topological insulators like $\rm Bi_2Te_3$, using the formalism of conserved spin transport. The hexagonal warping effect enables the application of this formalism by providing a well-defined Berry curvature without breaking the time-reversal symmetry. The absence of in-plane spin currents in the linear response motivates the extension of the formalism to the non-linear regime, by combining the perturbation of Bloch states with the Boltzmann transport equation. The second-order response exhibits both intrinsic and extrinsic in-plane spin currents induced by the spin-weighted Berry curvature, Berry connection polarizability (BCP) dipole and Berry curvature dipole (BCD) respectively. The band anisotropy due to the $C_{3v}$ symmetry of the material ($\rm Bi_2Te_3$) propagates through the spin textures and the band geometric quantities, resulting in the selection of certain components of the spin conductivity tensors.

Figures

Figures reproduced from arXiv: 2608.01782 by Srijan Chatterjee, Tarun Kanti Ghosh.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Top panel [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (a) [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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

54 extracted references · 36 canonical work pages

  1. [1]

    Herek x andk y are in units ofk 0 and range from−1 to 1. as, Gyx ± =− 2v2 0 16ε3 0 ˜d3 9 2 ˜k4 sin(4ϕ) + 1 2 ˜d2 n ˜k2 sin(2ϕ) −6 ˜k6 sinϕcos(3ϕ)−9 ˜k10 sin(4ϕ) cos2(3ϕ) o , (29) Gxx ± = v2 0 16ε3 0 ˜d3 1 + 9˜k4 cos2(2ϕ) − 1 ˜d2 n ˜kcosϕ+ 3 ˜k5 cos(2ϕ) cos(3ϕ) o2 ,(30) where ˜d= q ˜k2 + ˜k6 cos2(3ϕ). They have been plotted in Fig. (2(b),(c)).G yx shows a ...

    work page

  2. [2]

    Similar to the second-order spin conductivity in 2D hole gas[45], we find that the intrinsic response is much larger in magnitude than the extrinsic one

    Converting to real units, Γ x,int y turns out to be close to 0.4e µm/V, which is of the same order as the intrinsic second-order spin conductivity, reported in the 2D Rashba-Dresselhaus electron gas. Similar to the second-order spin conductivity in 2D hole gas[45], we find that the intrinsic response is much larger in magnitude than the extrinsic one. The...

    work page

  3. [3]

    ˇZuti´ c, J

    I. ˇZuti´ c, J. Fabian, and S. Das Sarma, Spintronics: Fun- damentals and applications, Rev. Mod. Phys.76, 323 (2004)

    2004

  4. [4]

    Bandyopadhyay and M

    S. Bandyopadhyay and M. Cahay,Introduction to Spin- tronics (2nd ed.)(CRC Press, 2015)

    work page 2015

  5. [5]

    Fabian, A

    J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. ˇZuti´ c, Semiconductor spintronics, Acta Physica Slo- vaca. Reviews and Tutorials57, 10.2478/v10155-010- 0086-8 (2007)

    work page doi:10.2478/v10155-010- 2007

  6. [6]

    Sinova, D

    J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung- wirth, and A. H. MacDonald, Universal intrinsic spin Hall effect, Phys. Rev. Lett.92, 126603 (2004)

    2004

  7. [7]

    Murakami, N

    S. Murakami, N. Nagaosa, and S.-C. Zhang, Dissipation- less quantum spin current at room temperature, Science 301, 1348 (2003)

    2003

  8. [8]

    Sinova, S

    J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects, Rev. Mod. Phys.87, 1213 (2015)

    2015

  9. [9]

    S. G. Tan, M. B. A. Jalil, C. S. Ho, Z. Siu, and S. Mu- rakami, Gauge physics of spin Hall effect, Sci Rep5, 18409 (2015)

    work page 2015

  10. [10]

    Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Observation of the spin Hall ef- fect in semiconductors, Science306, 1910 (2004), https://www.science.org/doi/pdf/10.1126/science.1105514

    doi:10.1126/science.1105514 1910

  11. [11]

    Schliemann and D

    J. Schliemann and D. Loss, Anisotropic transport in a two-dimensional electron gas in the presence of spin-orbit coupling, Phys. Rev. B68, 165311 (2003)

    2003

  12. [12]

    Schliemann and D

    J. Schliemann and D. Loss, Spin-Hall transport of heavy holes in III-V semiconductor quantum wells, Phys. Rev. B71, 085308 (2005)

    work page 2005

  13. [13]

    B. A. Bernevig and S.-C. Zhang, Intrinsic spin Hall effect in the two-dimensional hole gas, Phys. Rev. Lett.95, 016801 (2005)

    work page 2005

  14. [14]

    Y. A. Bychkov and E. I. Rashba, Oscillatory effects and the magnetic susceptibility of carriers in inversion layers, Journal of Physics C: Solid State Physics17, 6039 (1984)

    work page 1984

  15. [15]

    Dresselhaus, Spin-orbit coupling effects in zinc blende structures, Phys

    G. Dresselhaus, Spin-orbit coupling effects in zinc blende structures, Phys. Rev.100, 580 (1955)

    1955

  16. [16]

    Xiao and Q

    C. Xiao and Q. Niu, Conserved current of nonconserved quantities, Phys. Rev. B104, L241411 (2021)

    2021

  17. [17]

    J. Shi, P. Zhang, D. Xiao, and Q. Niu, Proper definition of spin current in spin-orbit coupled systems, Phys. Rev. Lett.96, 076604 (2006)

    2006

  18. [18]

    Xiang, H

    L. Xiang, H. Jin, and J. Wang, Spin transport revealed by spin quantum geometry, Phys. Rev. Lett.135, 146303 (2025)

    2025

  19. [19]

    H. Liu, J. H. Cullen, and D. Culcer, Topological nature of the proper spin current and the spin-Hall torque, Phys. Rev. B108, 195434 (2023)

    work page 2023

  20. [20]

    Kapri, B

    P. Kapri, B. Dey, and T. K. Ghosh, Role of Berry curva- ture in the generation of spin currents in Rashba systems, Phys. Rev. B103, 165401 (2021)

    work page 2021

  21. [21]

    Fujita, M

    T. Fujita, M. B. A. Jalil, and S. G. Tan, Unified descrip- tion of intrinsic spin-Hall effect mechanisms, New Journal of Physics12, 013016 (2010)

    work page 2010

  22. [22]

    Chang, Effect of in-plane magnetic field on the spin Hall effect in a Rashba-Dresselhaus system, Phys

    M.-C. Chang, Effect of in-plane magnetic field on the spin Hall effect in a Rashba-Dresselhaus system, Phys. Rev. B71, 085315 (2005)

    work page 2005

  23. [23]

    Zhang, C.-X

    H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.- C. Zhang, Topological insulators in Bi 2Se3,Bi 2Te3 and Sb2Te3 with a single Dirac cone on the surface, Nature Phys5, 438 (2009)

    work page 2009

  24. [24]

    Liu, X.-L

    C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.- C. Zhang, Model Hamiltonian for topological insulators, Phys. Rev. B82, 045122 (2010)

    2010

  25. [25]

    Saha and A

    S. Saha and A. Narayan, Nonlinear Hall effect in Rashba systems with hexagonal warping, Journal of Physics: 8 Condensed Matter35, 485301 (2023)

    work page 2023

  26. [26]

    T. Nag, S. K. Das, C. Zeng, and S. Nandy, Third-order Hall effect in the surface states of a topological insulator, Phys. Rev. B107, 245141 (2023)

    work page 2023

  27. [27]

    Fu, Hexagonal warping effects in the surface states of the topological insulator Bi 2Te3, Phys

    L. Fu, Hexagonal warping effects in the surface states of the topological insulator Bi 2Te3, Phys. Rev. Lett.103, 266801 (2009)

    work page 2009

  28. [28]

    Wong and F

    A. Wong and F. Mireles, Spin Hall and longitudinal con- ductivity of a conserved spin current in two dimensional heavy-hole gases, Phys. Rev. B81, 085304 (2010)

    work page 2010

  29. [29]

    Pal and T

    O. Pal and T. K. Ghosh, Polarization and third-order Hall effect in III-V semiconductor heterojunctions, Phys. Rev. B109, 035202 (2024)

    work page 2024

  30. [30]

    C. M. Wang and F. J. Yu, Effects of hexagonal warping on surface transport in topological insulators, Phys. Rev. B84, 155440 (2011)

    work page 2011

  31. [31]

    R. S. Akzyanov and A. L. Rakhmanov, Surface charge conductivity of a topological insulator in a magnetic field: The effect of hexagonal warping, Phys. Rev. B97, 075421 (2018)

    work page 2018

  32. [32]

    Li and J

    Z. Li and J. P. Carbotte, Hexagonal warping on opti- cal conductivity of surface states in topological insulator Bi2Te3, Phys. Rev. B87, 155416 (2013)

    work page 2013

  33. [33]

    Y. Imai, T. Yamaguchi, A. Yamakage, and H. Kohno, Spintronic properties of topological surface Dirac elec- trons with hexagonal warping, Phys. Rev. B103, 054402 (2021)

    work page 2021

  34. [34]

    A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, Spin-transfer torque generated by a topological insulator, Nature511, 449 (2014)

    2014

  35. [35]

    R. S. Akzyanov and A. L. Rakhmanov, Bulk and surface spin conductivity in topological insulators with hexago- nal warping, Phys. Rev. B99, 045436 (2019)

    work page 2019

  36. [36]

    J. H. Cullen and D. Culcer, Spin-Hall effect due to the bulk states of topological insulators: Extrinsic contribu- tion to the proper spin current, Phys. Rev. B108, 245418 (2023)

    work page 2023

  37. [37]

    Asgharpour, C

    A. Asgharpour, C. Gorini, S. Essert, K. Richter, and i. d. I. m. c. Adagideli, Gate-controlled spin extraction from topological insulator surfaces, Phys. Rev. B102, 035401 (2020)

    work page 2020

  38. [38]

    Yar, Jasra, and K

    A. Yar, Jasra, and K. Sabeeh, Nonlinear Hall effect in topological insulator Bi 2Te3 with hexagonal warping, Journal of Applied Physics131, 184401 (2022)

    work page 2022

  39. [39]

    H. Wang, H. Liu, X. Feng, J. Cao, W. Wu, S. Lai, W. Gao, C. Xiao, and S. A. Yang, Intrinsic nonlinear spin Hall effect and manipulation of perpendicular mag- netization, Phys. Rev. Lett.134, 056301 (2025)

    work page 2025

  40. [40]

    H. Pan, Z. Liu, D. Hou, Y. Gao, and Q. Niu, Thermody- namic ground and Berry-phase origin of the planar spin Hall effect, Phys. Rev. Res.6, L012034 (2024)

    work page 2024

  41. [41]

    W. Ding, X. Ma, X. Cheng, and X. Zhai, Intrinsic spin Hall effect in two-dimensional Dirac-Rashba systems, Phys. Rev. B112, 085403 (2025)

    work page 2025

  42. [42]

    Kaplan, T

    D. Kaplan, T. Holder, and B. Yan, Unification of nonlin- ear anomalous Hall effect and nonreciprocal magnetore- sistance in metals by the quantum geometry, Phys. Rev. Lett.132, 026301 (2024)

    work page 2024

  43. [43]

    N. W. Ashcroft and N. D. Mermin,Solid State Physics (Holt-Saunders, 1976)

    1976

  44. [44]

    Liu, X.-B

    T. Liu, X.-B. Qiang, H.-Z. Lu, and X. C. Xie, Quantum geometry in condensed matter, National Science Review 12, nwae334 (2024)

    2024

  45. [45]

    Shen, Three-dimensional topological insulators, in Topological Insulators: Dirac Equation in Condensed Matter(Springer Singapore, Singapore, 2017) pp

    S.-Q. Shen, Three-dimensional topological insulators, in Topological Insulators: Dirac Equation in Condensed Matter(Springer Singapore, Singapore, 2017) pp. 125– 152

    work page 2017

  46. [46]

    Y. Liu, G. Shi, D. Kumar, T. Kim, S. Shi, D. Yang, J. Zhang, C. Zhang, F. Wang, S. Yang, Y. Pu, P. Yu, K. Cai, and H. Yang, Field-free switching of perpen- dicular magnetization at room temperature using out- of-plane spins from TaIrTe 4, Nature Electronics6, 732 (2023)

    work page 2023

  47. [47]

    Chatterjee and T

    S. Chatterjee and T. K. Ghosh, Non-linear in-plane spin current in spin–orbit coupled 2D hole gases, Journal of Applied Physics140, 044301 (2026)

    work page 2026

  48. [48]

    Wunderlich, B

    J. Wunderlich, B. Kaestner, J. Sinova, and T. Jung- wirth, Experimental observation of the spin-Hall effect in a two-dimensional spin-orbit coupled semiconductor system, Phys. Rev. Lett.94, 047204 (2005)

    2005

  49. [49]

    Stamm, C

    C. Stamm, C. Murer, M. Berritta, J. Feng, M. Gabureac, P. M. Oppeneer, and P. Gambardella, Magneto-optical detection of the spin Hall effect in Pt and W thin films, Phys. Rev. Lett.119, 087203 (2017)

    2017

  50. [50]

    Saitoh, M

    E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Conver- sion of spin current into charge current at room temper- ature: Inverse spin-Hall effect, Applied Physics Letters 88, 182509 (2006)

    2006

  51. [51]

    E. A. Karashtin and A. A. Fraerman, Spin current and second harmonic generation in non-collinear magnetic systems: the hydrodynamic model, Journal of Physics: Condensed Matter30, 165801 (2018)

    work page 2018

  52. [52]

    L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Spin-torque ferromagnetic resonance induced by the spin Hall effect, Phys. Rev. Lett.106, 036601 (2011)

    work page 2011

  53. [53]

    Liu, C.-F

    L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Spin-torque switching with the giant spin Hall effect of tantalum, Science336, 555 (2012), https://www.science.org/doi/pdf/10.1126/science.1218197

    doi:10.1126/science.1218197 2012

  54. [54]

    Garello, C

    K. Garello, C. O. Avci, I. M. Miron, M. Baumgartner, A. Ghosh, S. Auffret, O. Boulle, G. Gaudin, and P. Gam- bardella, Ultrafast magnetization switching by spin-orbit torques, Applied Physics Letters105, 212402 (2014)

    work page 2014