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Resonant second harmonic generation in a two-dimensional electron system

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Pith's one-line read A two-dimensional electron gas with Rashba spin-orbit coupling and an in-plane magnetic field produces a second-harmonic current that spikes when the light frequency matches twice the spin-orbit splitting; the peak height grows as the…

desk verdict Solid formalism, but the headline τ^3 resonance at ω=2Δ may be an artifact of the Δ_k→Δ approximation; exact integration likely splits the peak. read the letter →

arxiv 2411.08947 v2 pith:DMITDSBL submitted 2024-11-13 cond-mat.mes-hall cond-mat.mtrl-sci

Maxim Dzero , Jaglul Hasan , Alex Levchenko This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-sci
keywords secondharmonicgenerationRashbaspin-orbitcouplingtwo-dimensionalelectrongasin-planemagneticfieldnonlinearopticalresponsenonreciprocaltransportdisorderscatteringWignerdistributionfunction
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

The paper argues that a simple two-dimensional electron gas with Rashba spin-orbit coupling can act as a resonant frequency doubler when a static magnetic field is applied in the plane of the electrons. Shining light of frequency $\omega$ on the disordered system produces a current at $2\omega$ whose magnitude and direction are set by two independent vector combinations of the magnetic field, the light polarization, and the out-of-plane spin-orbit axis. The central result is that this second-harmonic current is not a smooth background: when $\omega$ approaches twice the spin-orbit splitting energy $\Delta$, the current is resonantly enhanced, growing as the cube of the elastic scattering time $\tau$ and narrowing with width $\tau^{-1}$. This resonance makes the effect potentially observable in clean quantum wells and gives the response a nonreciprocal character, so reversing the magnetic field changes the current. If correct, the work identifies a mechanism for second-harmonic generation that does not rely on Berry curvature or a polar crystal axis, only on Rashba coupling, disorder, and a magnetic field.

What carries the argument

The machinery is the quantum kinetic equation for the Wigner distribution function $W(k,\epsilon;t)$, a $2\times 2$ spin matrix describing electron occupation in phase space. The paper derives this equation from the nonequilibrium Green's function Dyson equations, treats disorder in the self-consistent Born approximation, and solves it order by order in the electric field; the second-order solution $W^{(2)}$ is integrated against the bare velocity to give $j_{2\omega}$. The load-bearing algebraic object is the combination $b_k = \alpha_{\rm so}[k\times \hat{z}]+h$ that enters the spin projections and the denominators $z_{2\omega}^2+4b_k^2$; after angular averaging and expansion to linear order in the magnetic field, the momentum integrals over products of these denominators with Fermi-function differences produce the functions $A$, $C$, $C'$ and the resonant factor $[(1-i\omega\tau)^2+4\zeta^2]^{-3}$. This denominator, together with the two angular-average identities in Eq. (38), fixes the resonance frequency, width, and the two vector structures of the current.

What would settle it

A direct numerical evaluation of the k-integrals in Eqs. (A3), (C1), and (C5) with the full momentum-dependent splitting $\Delta_k=\alpha_{\rm so}k$ kept in the Fermi functions would settle the claim: if the resulting $S(\omega)$ and $Q(\omega)$ fail to peak at $\omega\approx 2\Delta$ with width $\tau^{-1}$ and a $(\omega_{\rm res}-\omega-i\gamma)^{-3}$ line shape, the resonance result does not survive.

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Extended reading notes

Core claim

On its own terms, the paper's claim is that the second-harmonic current density of a weakly disordered Rashba two-dimensional electron gas in an in-plane magnetic field is $$j_{2\omega} = -\frac{2m}{\pi}\left(\frac{e\alpha_{\rm so}}{\omega}\right)^{3}\frac{g\mu_B}{\omega}\left\{S(\omega)(\hat{z}\times H)$E^{2}$ + Q(\omega)([\hat{z}\times H]\cdot E)E\right\},$$ where $S(\omega)=A(\omega)-2C(\omega)+C'(\omega)$ and $Q(\omega)=2(2C(\omega)-C'(\omega))$. The functions $A$, $C$, and $C'$ all carry the same resonant denominator $[(1-i\omega\tau)^2+4\zeta^2]^3$ with $\zeta=\alpha_{\rm so}k_F\tau$, so near $\omega=2\Delta$, with $\Delta=\alpha_{\rm so}k_F$, the current behaves as $j_{2\omega}\propto (\omega_{\rm res}-\omega-i\gamma)^{-3}$, $\gamma=\tau^{-1}$. This is the paper's central discovery: the earlier clean-limit result, which decays as $\omega^{-4}$ and has no resonance, misses a sharp interband resonance that appears once disorder is included, and the resonant peak height scales as $\tau^3$. The same expression leaves room for the topological Berry-dipole contribution to $j_{2\omega}$, and the total current reverses when $H$ is reversed, so the effect is nonreciprocal.

Load-bearing premise

The calculation replaces the momentum-dependent spin-orbit splitting inside the electron occupation functions by its value at the Fermi surface, asserting that the error is small; right at the resonance this replacement fixes the peak's position and width, and the paper does not demonstrate its accuracy there in detail.

Editorial extensions

If this is right

  • At resonance $\omega\approx 2\Delta$, the second-harmonic current scales as $\tau^3$, so cleaner samples give dramatically larger signals and the line shape is a third-order Lorentzian with width $\tau^{-1}$.
  • The resonance position directly measures the spin-orbit splitting $2\Delta=2\alpha_{\rm so}k_F$, and the linewidth measures the elastic scattering rate $\tau^{-1}$.
  • Because the current contains two independent vector terms, $S(\omega)(\hat{z}\times H)E^2$ and $Q(\omega)([\hat{z}\times H]\cdot E)E$, polarization- and field-angle-resolved measurements separate $S$ and $Q$, and reversing $H$ reverses the current.
  • Away from resonance the local term decays as $\omega^{-4}$, so the topological Berry-dipole contribution dominates off-resonance, whereas the resonant local term dominates near $\omega=2\Delta$ in sufficiently clean samples.
  • For oblique incidence, nonlocal terms can overwhelm the local current away from resonance, so normal-incidence geometry is needed to see the resonant local contribution.

Reading between the lines

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

  • Going beyond the paper, the same resonant denominator should also control the rectified (dc) and third-harmonic responses of the Rashba model, so checking for a $\tau^3$-enhanced peak at $\omega=2\Delta$ in those observables would test whether the mechanism is generic.
  • The paper does not state it, but because the resonance frequency is set by $\alpha_{\rm so}k_F$, the peak should shift with electron density, making the resonance electrostatically tunable by gating.
  • The paper does not analyze long-range or small-angle disorder; in that regime momentum and spin relaxation rates differ, so the $\tau^3$ scaling and the linewidth may change.
  • The comparison with the Berry-dipole term implies a cleanliness- and spin-orbit-strength-dependent crossover between the two mechanisms, which the paper leaves implicit and an experiment varying mobility and Rashba coupling could map.
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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

1 major / 4 minor

Summary. The paper derives the second-harmonic current density of a disordered two-dimensional Rashba electron gas in an in-plane magnetic field, using a Keldysh/Wigner-function approach. The main result, Eq. (42), expresses j_{2ω} as a sum of two vector combinations with frequency-dependent coefficients S(ω) and Q(ω). The authors show that, near the spin-orbit splitting frequency ω=2Δ, the response develops a resonant cubic pole, Eq. (4), with width γ=τ^{-1} and peak height scaling as τ^3. They compare this resonant local contribution with the topological Berry-dipole term and with nonlocal terms, and give estimates for the relative magnitudes.

Significance. If the central calculation is correct, the paper provides an analytically tractable microscopic derivation of a disorder-enabled resonant second-harmonic mechanism that is distinct from the Berry-dipole and nonlocal contributions, with explicit clean-limit (Eq. (3)) and Drude (Eq. (28)) consistency checks. The Wigner-function route and the explicit angular and energy integrals in the appendices are valuable and could be used for further calculations. However, the quantitative resonant prediction currently rests on an approximation whose validity in the ballistic regime is questionable, as detailed below.

major comments (1)
  1. [Appendix A, Eqs. (A3)-(A4); Appendix C, Eqs. (C1)-(C2), (C5)-(C6)] The central resonant claim, Eq. (4) and the corresponding curves in Fig. 3, rests on the replacement Δ_k=α_so k → Δ=α_so k_F inside the Heaviside functions in Appendix A (Eqs. (A3)–(A4)) and Appendix C (Eqs. (C1)–(C2), (C5)–(C6)). This replacement is not an O((α_so/v_F)^2) correction in the resonant regime. The exact spectrum E_s(k)=k^2/2m+sα_so k gives Fermi momenta k_F^s≈k_F−s mα_so, so the interband thresholds for the two spin bands are ω_s=2α_so k_F^s=2Δ∓2mα_so^2, i.e., two resonances split by 4mα_so^2=4(α_so/v_F)Δ. For the parameters quoted in Fig. 2 and plotted in Fig. 3 (α_so/v_F=0.1, ζ=8), this splitting equals 0.4Δ=3.2γ, so the two resonances are well resolved. The approximation collapses these two thresholds into one at 2Δ, so the cubic-pole line shape of Eq. (4), the peak position, and the τ^3 peak height at ω=2Δ are not reliable for ζ α_so/v_F≳1. I ask the authors to recompute the k-integrals retaining the momentum dependence of Δ_k in the occupation functions (or at least the first-order shift of k_F^s), and to correct Eq. (4), Fig. 3, and the associated estimates accordingly.
minor comments (4)
  1. [Section IV.B, text after Eq. (33)] The word 'assymptotes' should be 'approaches' or 'asymptotes'.
  2. [Fig. 3 caption] The phrase 'normalized on the pot' appears to be a typo for 'normalized on the plot'.
  3. [Eq. (23)] The quantity z_ω is used before it is defined; please define z_ω=-iω+1/τ at first occurrence.
  4. [Section V, after Eq. (43)] The phrase 'de Broglie wave length' should be 'de Broglie wavelength'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SHG result is derived from the model Hamiltonian via the Keldysh/Wigner kinetic equation, with no fitted parameters and no load-bearing self-citation chain.

full rationale

The derivation chain is self-contained. Starting from the Rashba Hamiltonian and the disorder model, the authors derive the Keldysh/Wigner kinetic equation (Eq. (18)), solve it iteratively to first and second order in the electric field (Eqs. (23) and (29)), and integrate the traced distribution function to obtain the total second-harmonic current (Eq. (42)). No parameter is fitted to the target observable: the scale Delta = alpha_so k_F and the dimensionless parameter zeta = alpha_so k_F tau are defined from the model parameters and the disorder scattering rate, not extracted from the second-harmonic response. The clean-limit comparison, Eq. (3), is attributed to Ref. [35] and is used only as a high-frequency limit check (A(omega) -> 1); the Drude limit check, Eq. (28), is likewise an external consistency check. The self-citations (Refs. [10], [22], [24], [25], [38], [40]) are methodological or contextual, not load-bearing: the paper explicitly declines the spin-Hall method it cites and instead uses the Wigner-distribution approach of Ref. [37]. The resonance at omega = 2Delta follows from the model's spin-orbit band splitting within the explicitly stated approximation Delta_k -> Delta inside the occupation functions (Appendix A, Eqs. (A3)-(A4); Appendix C, Eqs. (C1)-(C2)); whether that approximation is quantitatively reliable very near resonance is an accuracy concern, not a circularity. No equation in the paper reduces to its own input by construction, and no prediction is a renamed fit.

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

No free parameters are fitted; all inputs (α_so, τ, k_F, g, H) are physical constants of the model. The main paper-specific axiom is the Δ_k → Δ replacement in the Fermi function arguments. No new entities are postulated.

assumptions (5)
  • domain assumption Two-dimensional electron gas with Rashba spin-orbit coupling, short-range Gaussian disorder, and in-plane magnetic field described by Hamiltonian (5)
    The entire calculation is performed for this model; the predicted resonance and its line shape are specific to this Hamiltonian.
  • domain assumption Weak disorder / ballistic limit τ^{-1} << min{ω, α_so k_F}, used to neglect the collision integral in Eq. (18) to leading order
    This justifies treating the SCBA self-energy broadening in the Green's functions as the only disorder effect and underlies the τ^3 scaling.
  • domain assumption Weak magnetic field, response computed only to first order in H (gμ_B H << α_so k_F)
    The current is expanded to linear order in H; all results are H-linear.
  • ad hoc to paper Momentum-dependent spin-orbit splitting Δ_k replaced by Δ = α_so k_F inside the Fermi functions
    This approximation is introduced to make the k-integrals in Appendices A and C tractable; the authors claim corrections of order (α_so/v_F)^2 but do not present a detailed error estimate near resonance.
  • domain assumption Local limit / normal incidence: dependence of the Wigner function on r is ignored
    The local response is computed for a spatially uniform field; away from normal incidence, nonlocal terms (Eq. (45)) can dominate.

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Cite this review

Pith. "Pith review of Resonant second harmonic generation in a two-dimensional electron system." pith.science (2026). https://pith.science/paper/DMITDSBL

@misc{pith2026241108947,
  author       = {Pith},
  title        = {Pith review of: Resonant second harmonic generation in a two-dimensional electron system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DMITDSBL}},
  note         = {Machine review of arXiv:2411.08947}
}
read the original abstract

We consider the nonlinear response of a disordered two-dimensional electronic system, lacking inversion symmetry, to an external alternating electric field. The application of an in-plane static magnetic field induces local contributions to the current density that are quadratic in the electric field and linear in the magnetic field. This current oscillates at twice the frequency of the external irradiation and there are two linearly independent vector combinations that contribute to the current density. This particular mechanism coexists with the topological Berry-dipole contribution to the second harmonic of the current density, which can be generated by quantum confinement. Additional nonlocal terms in the current density are possible in the regime away from the normal incidence. The total current exhibits a nonreciprocal character upon reversal of the magnetic field direction. We evaluate the magnitude of this effect by computing its dependence on the strength of spin-orbit coupling and the disorder scattering rate. Importantly, we show that these local second-harmonic contributions can be resonantly excited when the frequency of the external radiation approaches the energy separation between the spin-orbit split bands.

Figures

Figures reproduced from arXiv: 2411.08947 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic of the system under consideration: [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Energy bands of the electron spectrum in the presence [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Normalized frequency dependence of the second har [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Reference graph

Works this paper leans on

42 extracted references · 35 canonical work pages

  1. [1]

    Design of high-energy terahertz sources based on optical rectification,

    J. A. F¨ ul¨ op, L. P´ alfalvi, G. Alm´ asi, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express 18, 12311–12327 (2010)

    work page 2010

  2. [2]

    Electrodynamics of correlated electron materials,

    D. N. Basov, Richard D. Averitt, Dirk van der Marel, Martin Dressel, and Kristjan Haule, “Electrodynamics of correlated electron materials,” Rev. Mod. Phys. 83, 471–541 (2011)

    work page 2011

  3. [3]

    Topo- logical nature of nonlinear optical effects in solids,

    Takahiro Morimoto and Naoto Nagaosa, “Topo- logical nature of nonlinear optical effects in solids,” Science Advances 2, e1501524 (2016), https://www.science.org/doi/pdf/10.1126/sciadv.1501524

    doi:10.1126/sciadv.1501524 2016

  4. [4]

    Nonreciprocal re- sponses from non-centrosymmetric quantum materials,

    Yoshinori Tokura and Naoto Nagaosa, “Nonreciprocal re- sponses from non-centrosymmetric quantum materials,” Nature Communications 9, 3740 (2018)

    work page 2018

  5. [5]

    Nonreciprocal transport and optical phenomena in quantum materials,

    Naoto Nagaosa and Youichi Yanase, “Nonreciprocal transport and optical phenomena in quantum materials,” Annual Review of Condensed Matter Physics 15, 63–83 (2024)

    work page 2024

  6. [6]

    Revealing novel aspects of light-matter coupling in terahertz two-dimensional coherent spectroscopy: the case of the amplitude mode in superconductors

    Kota Katsumi, Jacopo Fiore, Mattia Udina, Ralph Romero III au2, David Barbalas, John Jesudasan, Pratap Raychaudhuri, Goetz Seibold, Lara Benfatto, and N. P. Armitage, “Revealing novel aspects of light-matter coupling in terahertz two-dimensional coherent spec- troscopy: the case of the amplitude mode in supercon- ductors,” (2023), arXiv:2311.16449 [cond-m...

    work page Pith review arXiv 2023

  7. [7]

    Microwave-induced inverse Faraday effect in superconductors,

    A. Hamed Majedi, “Microwave-induced inverse Faraday effect in superconductors,” Phys. Rev. Lett. 127, 187001 (2021)

    work page 2021

  8. [8]

    Inverse Faraday effect in superconduc- tors with a finite gap in the excitation spectrum,

    A. V. Putilov, S. V. Mironov, A. S. Mel’nikov, and A. A. Bespalov, “Inverse Faraday effect in superconduc- tors with a finite gap in the excitation spectrum,” JETP Letters 117, 827–833 (2023)

    work page 2023

Show all 42 references
  1. [9]

    Inverse Faraday effect for superconducting condensates,

    S. V. Mironov, A. S. Mel’nikov, I. D. Tokman, V. Vadi- mov, B. Lounis, and A. I. Buzdin, “Inverse Faraday effect for superconducting condensates,” Phys. Rev. Lett. 126, 137002 (2021)

    2021

  2. [10]

    Inverse Faraday effect in superconduc- tors with potential impurities,

    Maxim Dzero, “Inverse Faraday effect in superconduc- tors with potential impurities,” Phys. Rev. B110, 054506 (2024)

    2024

  3. [11]

    Theory of optical harmonic genera- tion at a metal surface,

    Sudhanshu S. Jha, “Theory of optical harmonic genera- tion at a metal surface,” Phys. Rev. 140, A2020–A2030 (1965)

    1965

  4. [12]

    Paul N Butcher, The elements of nonlinear optics (Cam- bridge ; New York : Cambridge University Press, 1991)

    1991

  5. [13]

    E. L. Ivchenko, Optical spectroscopy of semiconductor nanostructures (Harrow, U.K. : Alpha Science, 2005)

    2005

  6. [14]

    Sturman and Vladimir M

    Boris I. Sturman and Vladimir M. Fridkin, The Photo- voltaic and Photorefractive Effects in Noncentrosymmet- ric Materials (Gordon and Breach Science Publishers, 1992)

    1992

  7. [15]

    The photogalvanic effect in media lacking a center of symmetry,

    V. I. Belinicher and V. M. Sturman, “The photogalvanic effect in media lacking a center of symmetry,” Sov. Phys. - Uspekhi 23, 199 (1980)

    1980

  8. [16]

    Kinetic theory of the displacement photovoltaic effect in piezoelectrics,

    V. I. Belinicher, E. L. Ivchenko, and B. I. Sturman, “Kinetic theory of the displacement photovoltaic effect in piezoelectrics,” Sov. Phys. JETP 56, 359 (1982)

    1982

  9. [17]

    Second-order optical re- sponse in semiconductors,

    J. E. Sipe and A. I. Shkrebtii, “Second-order optical re- sponse in semiconductors,” Phys. Rev. B 61, 5337–5352 (2000)

    2000

  10. [18]

    ∝#𝐳×𝐇𝐄! ζ=2 ζ=4 ζ=6 ζ=8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 ω/Δ Re[j2ω/j2ω res] 1.6 1.8 2.0 2.2 2.4 -0.5 0.0 0.5 ω/Δ Im[j2ω/j2ωres] 𝐣!

    and references therein). For matched frequencies, ω1 = ω2 = ω the E2 term gives rise to second harmonic generation (SHG). In materials with topologically nontrivial band struc- tures, finite Berry curvature contributes to unique pho- tocurrents [19–22], the quantum nonlinear H...

    2024

  11. [19]

    Collision integral for elastic scattering of electrons and phonons,

    B. I. Sturman, “Collision integral for elastic scattering of electrons and phonons,” Phys. Usp. 27, 881–884 (1984)

    1984

  12. [20]

    Semiclassical theory of the photogalvanic effect in non-centrosymmetric systems,

    E. Deyo, L. E. Golub, E. L. Ivchenko, and B. Spi- vak, “Semiclassical theory of the photogalvanic effect in non-centrosymmetric systems,” (2009), arXiv:0904.1917 [cond-mat.mes-hall]

    2009 arXiv

  13. [21]

    Confinement-induced Berry phase and helicity-dependent photocurrents,

    J. E. Moore and J. Orenstein, “Confinement-induced Berry phase and helicity-dependent photocurrents,” Phys. Rev. Lett. 105, 026805 (2010)

    2010

  14. [22]

    Quantized circular photogal- vanic effect in Weyl semimetals,

    Fernando de Juan, Adolfo G. Grushin, Takahiro Mori- moto, and Joel E Moore, “Quantized circular photogal- vanic effect in Weyl semimetals,” Nature Communica- tions 8, 15995 (2017)

    2017

  15. [23]

    Photogalvanic effect in Weyl semimetals,

    E. J. K¨ onig, H.-Y. Xie, D. A. Pesin, and A. Levchenko, “Photogalvanic effect in Weyl semimetals,” Phys. Rev. B 96, 075123 (2017)

    2017

  16. [24]

    Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials,

    Inti Sodemann and Liang Fu, “Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials,” Phys. Rev. Lett. 115, 216806 (2015)

    2015

  17. [25]

    Quantum kinetics of anomalous and nonlinear Hall effects in topological semimetals,

    Elio J. K¨ onig and Alex Levchenko, “Quantum kinetics of anomalous and nonlinear Hall effects in topological semimetals,” Annals of Physics 435, 168492 (2021)

    2021

  18. [26]

    Gyrotropic Hall effect in Berry-curved materials,

    E. J. K¨ onig, M. Dzero, A. Levchenko, and D. A. Pesin, “Gyrotropic Hall effect in Berry-curved materials,” Phys. Rev. B 99, 155404 (2019)

    2019

  19. [27]

    Conversion of spin into directed electric current in quantum wells,

    S. D. Ganichev, E. L. Ivchenko, S. N. Danilov, J. Eroms, W. Wegscheider, D. Weiss, and W. Prettl, “Conversion of spin into directed electric current in quantum wells,” Phys. Rev. Lett. 86, 4358–4361 (2001)

    2001

  20. [28]

    Obser- vation of the orbital circular photogalvanic effect,

    P. Olbrich, S. A. Tarasenko, C. Reitmaier, J. Karch, D. Plohmann, Z. D. Kvon, and S. D. Ganichev, “Obser- vation of the orbital circular photogalvanic effect,” Phys. Rev. B 79, 121302 (2009)

    2009

  21. [29]

    Giant anisotropic nonlinear optical response in transition metal monopnictide Weyl semimetals,

    Liang Wu, S. Patankar, T. Morimoto, N. L. Nair, E. The- walt, A. Little, J. G. Analytis, J. E. Moore, and J. Oren- stein, “Giant anisotropic nonlinear optical response in transition metal monopnictide Weyl semimetals,” Nature Physics 13, 350–355 (2017)

    2017

  22. [30]

    Resonance-enhanced op- tical nonlinearity in the Weyl semimetal TaAs,

    Shreyas Patankar, Liang Wu, Baozhu Lu, Manita Rai, Jason D. Tran, T. Morimoto, Daniel E. Parker, Adolfo G. Grushin, N. L. Nair, J. G. Analytis, J. E. Moore, J. Oren- stein, and D. H. Torchinsky, “Resonance-enhanced op- tical nonlinearity in the Weyl semimetal TaAs,” Phys. Rev....

    2018

  23. [31]

    Second-harmonic generation in GaAs: Experiment versus theoretical predictions of χ(2) xyz ,

    S. Bergfeld and W. Daum, “Second-harmonic generation in GaAs: Experiment versus theoretical predictions of χ(2) xyz ,” Phys. Rev. Lett. 90, 036801 (2003)

    2003

  24. [32]

    Dispersion of the second-order nonlinear sus- ceptibility in ZnTe, ZnSe, and ZnS,

    H. P. Wagner, M. K¨ uhnelt, W. Langbein, and J. M. Hvam, “Dispersion of the second-order nonlinear sus- ceptibility in ZnTe, ZnSe, and ZnS,” Phys. Rev. B 58, 10494–10501 (1998)

    1998

  25. [33]

    Symmetry of bands in wurzite-type crys- tals. 1. symmetry of bands disregarding spin-orbit inter- action,

    E. I. Rashba, “Symmetry of bands in wurzite-type crys- tals. 1. symmetry of bands disregarding spin-orbit inter- action,” Sov. Phys. Solid. State 1, 368 (1959)

    1959

  26. [34]

    Properties of a 2D electron gas with lifted spectral degeneracy,

    Yu. A. Bychkov and E. I. Rashba, “Properties of a 2D electron gas with lifted spectral degeneracy,” Sov. Phys. JETP Lett. 39, 66 (1984)

    1984

  27. [35]

    Theory of the bulk photovoltaic effect in pure crystals,

    Ralph von Baltz and Wolfgang Kraut, “Theory of the bulk photovoltaic effect in pure crystals,” Phys. Rev. B 23, 5590–5596 (1981)

    1981

  28. [36]

    Second-harmonic generation in two- dimensional systems without inversion centers,

    V. M. Edel’shtein, “Second-harmonic generation in two- dimensional systems without inversion centers,” Sov. 11 Phys. - JETP 68, 1446 (1988)

    1988

  29. [37]

    Spin current and polarization in impure two- dimensional electron systems with spin-orbit coupling,

    E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, “Spin current and polarization in impure two- dimensional electron systems with spin-orbit coupling,” Phys. Rev. Lett. 93, 226602 (2004)

    2004

  30. [38]

    Small-angle impurity scattering and the spin Hall conductivity in two-dimensional semiconductor sys- tems,

    A. V. Shytov, E. G. Mishchenko, H.-A. Engel, and B. I. Halperin, “Small-angle impurity scattering and the spin Hall conductivity in two-dimensional semiconductor sys- tems,” Phys. Rev. B 73, 075316 (2006)

    2006

  31. [39]

    Keldysh tech- nique and non-linear σ-model: basic principles and ap- plications,

    Alex Kamenev and Alex Levchenko, “Keldysh tech- nique and non-linear σ-model: basic principles and ap- plications,” Advances in Physics 58, 197–319 (2009), https://doi.org/10.1080/00018730902850504

    2009 doi

  32. [40]

    Alex Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge University Press, 2011)

    2011

  33. [41]

    Spin Hall conduc- tivity of interacting two-dimensional electron systems,

    Maxim Dzero and Alex Levchenko, “Spin Hall conduc- tivity of interacting two-dimensional electron systems,” Phys. Rev. B 106, 245302 (2022)

    2022

  34. [42]

    In fact, for a Berry flux linear in k any linear combination of kx and ky can be brought to the above form by a rotation

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