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Microscopic theory of the inverse spin galvanic effect in anisotropic Rashba models

Alessandro Veneri1,2, Francesco Quintavalle2,4, Thierry Valet3, and Roberto Raimondi2

  • 1Dipartimento di Ingegneria dell'Informazione, Elettronica e Telecomunicazioni, Università La Sapienza, Via Eudossiana, 18, 00184 Roma, Italy
  • 2Dipartimento di Matematica e Fisica, Università Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
  • 3MPhysX OÜ, Harju maakond, Tallinn, Lasnamäe linnaosa, Sepapaja tn 6, 15551, Estonia
  • 4School of Science and Technology, Physics Division, Università di Camerino, Via Madonna delle Carceri, 62032 Camerino (MC), Italy

Phys. Rev. B 113, 125432 – Published 27 March, 2026

DOI: https://doi.org/10.1103/b675-4wsd

Abstract

The Rashba spin-orbit coupling (SOC) is a well-known mechanism for the spin-charge interconversion via the inverse and direct spin galvanic effects. The lack of a full inversion symmetry allows the coupling of the charge current and spin density. In this paper we investigate this phenomenon when the in-plane rotational symmetry is lowered to the C2v and C3v symmetry groups, whereby the electron spectrum becomes anisotropic. We find that in the C2v case, depending on the ratio between the Rashba SOC strengths along the principal axes, the nonequilibrium spin density deviates notably from the 90∘ rotation, with respect to the applied electric field, which is familiar in the isotropic case. In the C3v case, when a warping cubic in momentum term is present, whereas the standard 90∘ rotation of the spin density remains, the spin-charge interconversion depends on the intensity of the warping itself. The microscopic theory takes into account disorder including vertex corrections, via both the diagrammatic implementation of the Kubo formula and the quantum kinetic theory. We show that vertex corrections are crucial to capture the details of the inverse spin galvanic effect in contrast to previous treatments based on the constant broadening approximation.

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References (102)

  1. S. Datta and B. Das, Electronic analog of the electro-optic modulator, Appl. Phys. Lett. 56, 665 (1990).
  2. J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Gate control of spin-orbit interaction in an inverted In0.53Ga0.47as/In0.52Al0.48as heterostructure, Phys. Rev. Lett. 78, 1335 (1997).
  3. R. H. Silsbee, Spin–orbit induced coupling of charge current and spin polarization, J. Phys.: Condens. Matter 16, R179 (2004).
  4. A. Hirohata and K. Takanashi, Future perspectives for spintronic devices, J. Phys. D 47, 193001 (2014).
  5. S. D. Ganichev, M. Trushin, and J. Schliemann, Spin polarization by current, in Handbook of Spin Transport and Magnetism, edited by E. Y. Tsymbal and I. Zutic (Chapman and Hall, Boca Raton, FL, 2016), Chap. 24.
  6. A. Soumyanarayanan, N. Reyren, A. Fert, and C. Panagopoulos, Emergent phenomena induced by spin–orbit coupling at surfaces and interfaces, Nature (London) 539, 509 (2016).
  7. G. Dresselhaus, Spin-orbit coupling effects in zinc blende structures, Phys. Rev. 100, 580 (1955).
  8. E. I. Rashba, Properties of semiconductors with an extremum loop. 1. cyclotron and combinational resonance in a magnetic field perpendicular to the plane of the loop, Fiz. Tverd. Tela (Leningrad) 2, 1224 (1960) [Sov. Phys. Solid State 2, 1109 (1960)].
  9. E. L. Ivchenko and G. E. Pikus, New photogalvanic effect in gyrotropic crystals, Sov. J. Exp. Theor. Phys. Lett. 27, 604 (1978).
  10. L. E. Vorob'ev, E. L. Ivchenko, G. E. Pikus, I. I. Farbshteľn, V. A. Shalygin, and A. V. Shturbin, Optical activity in tellurium induced by a current, J. Exp. Theor. Phys. Lett. 29, 441 (1979).
  11. C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005).
  12. M. I. D'yakonov, Spin Physics in Semiconductor (Springer, Berlin, 2008).
  13. D. Kochan, S. Irmer, and J. Fabian, Model spin-orbit coupling Hamiltonians for graphene systems, Phys. Rev. B 95, 165415 (2017).
  14. T. P. Cysne, A. Ferreira, and T. G. Rappoport, Crystal-field effects in graphene with interface-induced spin-orbit coupling, Phys. Rev. B 98, 045407 (2018).
  15. A. H. Castro Neto and F. Guinea, Impurity-induced spin-orbit coupling in graphene, Phys. Rev. Lett. 103, 026804 (2009).
  16. C. Weeks, J. Hu, J. Alicea, M. Franz, and R. Wu, Engineering a robust quantum spin Hall state in graphene via Adatom deposition, Phys. Rev. X 1, 021001 (2011).
  17. J. Balakrishnan, G. K. W. Koon, A. Avsar, Y. Ho, J. H. Lee, M. Jaiswal, S.-J. Baeck, J.-H. Ahn, A. Ferreira, M. A. Cazalilla, et al., Giant spin Hall effect in graphene grown by chemical vapour deposition, Nat. Commun. 5, 4748 (2014).
  18. I. Žutić, J. Fabian, and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys. 76, 323 (2004).
  19. M. I. D'yakonov and V. I. Perel', Possibility of orienting electron spins with current, JETP Lett. 13, 467 (1971).
  20. J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Universal intrinsic spin Hall effect, Phys. Rev. Lett. 92, 126603 (2004).
  21. S. Murakami, N. Nagaosa, and S.-C. Zhang, Dissipationless quantum spin current at room temperature, Science 301, 1348 (2003).
  22. J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87, 1213 (2015).
  23. D. T. S. Perkins, A. Veneri, and A. Ferreira, Spin Hall effect: Symmetry breaking, twisting, and giant disorder renormalization, Phys. Rev. B 109, L241404 (2024).
  24. A. G. Aronov and Y. B. Lyanda-Geller, Nuclear electric resonance and orientation of carrier spins by an electric field, JETP Lett. 50, 431 (1989) [Pis’ma Zh. Eksp. Teor. Fiz. 50, 398 (1989)].
  25. E. L. Ivchenko, Y. B. Lyanda-Geller, and G. E. Pikus, Current of thermalized spin-oriented photocarriers, Sov. Phys. JETP 71, 550 (1991) [Zh. Eksp. Teor. Fiz. 98, 989 (1990)].
  26. S. D. Ganichev, E. L. Ivchenko, V. V. Belkov, S. A. Tarasenko, M. Sollinger, D. Weiss, W. Wegscheider, and W. Prettl, Spin-galvanic effect, Nature (London) 417, 153 (2002).
  27. K. Shen, G. Vignale, and R. Raimondi, Microscopic theory of the inverse Edelstein effect, Phys. Rev. Lett. 112, 096601 (2014).
  28. C. Gorini, A. Maleki Sheikhabadi, K. Shen, I. V. Tokatly, G. Vignale, and R. Raimondi, Theory of current-induced spin polarization in an electron gas, Phys. Rev. B 95, 205424 (2017).
  29. M. Offidani, M. Milletarì, R. Raimondi, and A. Ferreira, Optimal charge-to-spin conversion in graphene on transition-metal dichalcogenides, Phys. Rev. Lett. 119, 196801 (2017).
  30. A. Veneri, D. T. S. Perkins, and A. Ferreira, Nonperturbative approach to interfacial spin-orbit torques induced by the Rashba effect, Phys. Rev. B 106, 235419 (2022).
  31. H.-A. Engel, B. I. Halperin, and E. I. Rashba, Theory of spin Hall conductivity in n-doped GaAs, Phys. Rev. Lett. 95, 166605 (2005).
  32. W.-K. Tse and S. Das Sarma, Spin Hall effect in doped semiconductor structures, Phys. Rev. Lett. 96, 056601 (2006).
  33. E. M. Hankiewicz and G. Vignale, Phase diagram of the spin Hall effect, Phys. Rev. Lett. 100, 026602 (2008).
  34. R. Raimondi and P. Schwab, Tuning the spin Hall effect in a two-dimensional electron gas, Europhys. Lett. 87, 37008 (2009).
  35. R. Raimondi, P. Schwab, C. Gorini, and G. Vignale, Spin-orbit interaction in a two-dimensional electron gas: A SU(2) formulation, Ann. Phys. 524, 201100253 (2012).
  36. M. Milletarì and A. Ferreira, Quantum diagrammatic theory of the extrinsic spin Hall effect in graphene, Phys. Rev. B 94, 134202 (2016).
  37. C. L. Kane and E. J. Mele, Z2 topological order and the quantum spin Hall effect, Phys. Rev. Lett. 95, 146802 (2005).
  38. C. Huang, Y. D. Chong, and M. A. Cazalilla, Direct coupling between charge current and spin polarization by extrinsic mechanisms in graphene, Phys. Rev. B 94, 085414 (2016).
  39. Y. A. Bychkov and É. I. Rashba, Properties of a 2d electron gas with lifted spectral degeneracy, JETP Lett. 39, 78 (1984).
  40. Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Current-induced spin polarization in strained semiconductors, Phys. Rev. Lett. 93, 176601 (2004).
  41. V. Sih, R. C. Myers, Y. K. Kato, W. H. Lau, A. C. Gossard, and D. D. Awschalom, Spatial imaging of the spin Hall effect and current-induced polarization in two-dimensional electron gases, Nat. Phys. 1, 31 (2005).
  42. J. C. R. Sánchez, L. Vila, G. Desfonds, S. Gambarelli, J. P. Attané, J. M. De Teresa, C. Magén, and A. Fert, Spin-to-charge conversion using Rashba coupling at the interface between non-magnetic materials, Nat. Commun. 4, 2944 (2013).
  43. E. Lesne, S. O. Yu Fu, J. C. Rojas-Sánchez, D. C. Vaz, H. Naganuma, G. Sicoli, J.-P. Attané, M. Jamet, E. Jacquet, J.-M. George, A. Barthélémy, H. Jaffrès, A. Fert, M. Bibes, and L. Vila, Highly efficient and tunable spin-to-charge conversion through Rashba coupling at oxide interfaces, Nat. Mater. 15, 1261 (2016).
  44. Y. Shiomi, K. Nomura, Y. Kajiwara, K. Eto, M. Novak, K. Segawa, Y. Ando, and E. Saitoh, Spin-Electricity Conversion Induced by Spin Injection into Topological Insulators, Phys. Rev. Lett. 113, 196601 (2014).
  45. 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, Nature (London) 511, 449 (2014).
  46. A. A. Burkov, A. S. Núñez, and A. H. MacDonald, Theory of spin-charge-coupled transport in a two-dimensional electron gas with Rashba spin-orbit interactions, Phys. Rev. B 70, 155308 (2004).
  47. C. Gorini, P. Schwab, R. Raimondi, and A. L. Shelankov, Non-abelian gauge fields in the gradient expansion: Generalized Boltzmann and Eilenberger equations, Phys. Rev. B 82, 195316 (2010).
  48. C. Gorini, R. Raimondi, and P. Schwab, Onsager relations in a two-dimensional electron gas with spin-orbit coupling, Phys. Rev. Lett. 109, 246604 (2012).
  49. A. A. Burkov and D. G. Hawthorn, Spin and charge transport on the surface of a topological insulator, Phys. Rev. Lett. 105, 066802 (2010).
  50. A. Ferreira, Theory of spin–charge-coupled transport in proximitized graphene: An SO(5) algebraic approach, J. Phys.: Mater. 4, 045006 (2021).
  51. A. Veneri, D. T. S. Perkins, C. G. Péterfalvi, and A. Ferreira, Twist angle controlled collinear Edelstein effect in van der Waals heterostructures, Phys. Rev. B 106, L081406 (2022).
  52. E. Simon, A. Szilva, B. Ujfalussy, B. Lazarovits, G. Zarand, and L. Szunyogh, Anisotropic Rashba splitting of surface states from the admixture of bulk states: Relativistic ab initio calculations and k·p perturbation theory, Phys. Rev. B 81, 235438 (2010).
  53. S. Chakraborty and S. Raj, Tunable nonlinear anisotropic Rashba splitting in monolayer transition metal dichalcogenide MoS2(1−x)Se2x alloys, Phys. Rev. B 108, 165402 (2023).
  54. B. Geldiyev, M. Ünzelmann, P. Eck, T. Kißlinger, J. Schusser, T. Figgemeier, P. Kagerer, N. Tezak, M. Krivenkov, A. Varykhalov, A. Fedorov, L. Nicolaï, J. Minár, K. Miyamoto, T. Okuda, K. Shimada, D. Di Sante, G. Sangiovanni, L. Hammer, M. A. Schneider, et al., Strongly anisotropic spin and orbital rashba effect at a tellurium–noble metal interface, Phys. Rev. B 108, L121107 (2023).
  55. D. V. Gruznev, L. V. Bondarenko, A. Y. Tupchaya, V. G. Kotlyar, O. A. Utas, A. N. Mihalyuk, S. V. Eremeev, A. V. Zotov, and A. A. Saranin, Two-dimensional metallic (Tl,Au)/Si(100)c(2×2): A Rashba-type system with C2v symmetry, Phys. Rev. B 98, 125428 (2018).
  56. L. Fu, Hexagonal warping effects in the surface states of the topological insulator Bi2Te3, Phys. Rev. Lett. 103, 266801 (2009).
  57. E. Frantzeskakis and M. Grioni, Anisotropy effects on Rashba and topological insulator spin-polarized surface states: A unified phenomenological description, Phys. Rev. B 84, 155453 (2011).
  58. K. Miyamoto, H. Miyahara, K. Kuroda, T. Maegawa, A. Kimura, and T. Okuda, Peculiar Rashba spin texture induced by C3v symmetry on the Bi(111) surface revisited, Phys. Rev. B 97, 085433 (2018).
  59. J. Ibañez-Azpiroz, A. Bergara, E. Y. Sherman, and A. Eiguren, Spin-flip transitions and departure from the Rashba model in the Au(111) surface, Phys. Rev. B 88, 125404 (2013).
  60. E. E. Krasovskii, Microscopic origin of the relativistic splitting of surface states, Phys. Rev. B 90, 115434 (2014).
  61. A. Johansson, J. Henk, and I. Mertig, Theoretical aspects of the Edelstein effect for anisotropic two-dimensional electron gas and topological insulators, Phys. Rev. B 93, 195440 (2016).
  62. I. Miatka, M. Barbieri, and R. Raimondi, Nonlinear inverse spin galvanic effect in anisotropic disorder-free systems, Eur. Phys. J. D 73, 107 (2019).
  63. S. Leiva-Montecinos, J. Henk, I. Mertig, and A. Johansson, Spin and orbital Edelstein effect in a bilayer system with Rashba interaction, Phys. Rev. Res. 5, 043294 (2023).
  64. I. Gaiardoni, M. Trama, A. Maiellaro, C. Guarcello, F. Romeo, and R. Citro, Edelstein effect in isotropic and anisotropic Rashba models, Condens. Matter 10, 15 (2025).
  65. G. Grosso and G. Parravicini, Solid State Physics (Academic Press, New York, 2000).
  66. J. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids, International Series of Monographs on Physics (Oxford University Press, Oxford, 2001).
  67. 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).
  68. A. Khaetskii, Intrinsic spin current for an arbitrary Hamiltonian and scattering potential, Phys. Rev. B 73, 115323 (2006).
  69. C. Xiao and Q. Niu, Semiclassical theory of spin-orbit torques in disordered multiband electron systems, Phys. Rev. B 96, 045428 (2017).
  70. A. Sekine, D. Culcer, and A. H. MacDonald, Quantum kinetic theory of the chiral anomaly, Phys. Rev. B 96, 235134 (2017).
  71. R. Kubo, A general expression for the conductivity tensor, Can. J. Phys. 34, 1274 (1956).
  72. R. Kubo, Statistical-mechanical theory of irreversible processes. I. general theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12, 570 (1957).
  73. A. Bastin, C. Lewiner, O. Betbeder-matibet, and P. Nozieres, Quantum oscillations of the Hall effect of a fermion gas with random impurity scattering, J. Phys. Chem. Solids 32, 1811 (1971).
  74. P. Středa and L. Smrčka, Galvanomagnetic effects in alloys in quantizing magnetic fields, Phys. Status Solidi B 70, 537 (1975).
  75. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  76. I. A. Ado, I. A. Dmitriev, P. M. Ostrovsky, and M. Titov, Anomalous Hall effect with massive Dirac fermions, Europhys. Lett. 111, 37004 (2015).
  77. I. A. Ado, O. A. Tretiakov, and M. Titov, Microscopic theory of spin-orbit torques in two dimensions, Phys. Rev. B 95, 094401 (2017).
  78. N. A. Sinitsyn, A. H. MacDonald, T. Jungwirth, V. K. Dugaev, and J. Sinova, Anomalous Hall effect in a two-dimensional Dirac band: The link between the Kubo-streda formula and the semiclassical Boltzmann equation approach, Phys. Rev. B 75, 045315 (2007).
  79. T. Valet and R. Raimondi, Quantum kinetic theory of the linear response for weakly disordered multiband systems, Phys. Rev. B 111, L041118 (2025).
  80. D. Go, H.-W. Lee, P. M. Oppeneer, S. Blügel, and Y. Mokrousov, First-principles calculation of orbital Hall effect by Wannier interpolation: Role of orbital dependence of the anomalous position, Phys. Rev. B 109, 174435 (2024).
  81. D. García Ovalle, A. Pezo, and A. Manchon, Spin-orbit torque for field-free switching in C3v crystals, Phys. Rev. B 107, 094422 (2023).
  82. R. Raimondi and P. Schwab, Spin-Hall effect in a disordered two-dimensional electron system, Phys. Rev. B 71, 033311 (2005).
  83. J.-i. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Suppression of the persistent spin Hall current by defect scattering, Phys. Rev. B 70, 041303 (2004).
  84. O. V. Dimitrova, Spin-Hall conductivity in a two-dimensional Rashba electron gas, Phys. Rev. B 71, 245327 (2005).
  85. O. Chalaev and D. Loss, Spin-Hall conductivity due to Rashba spin-orbit interaction in disordered systems, Phys. Rev. B 71, 245318 (2005).
  86. V. Bonbien and A. Manchon, Symmetrized decomposition of the Kubo-Bastin formula, Phys. Rev. B 102, 085113 (2020).
  87. L. Smrcka and P. Streda, Transport coefficients in strong magnetic fields, J. Phys. C 10, 2153 (1977).
  88. A. Crépieux and P. Bruno, Theory of the anomalous Hall effect from the Kubo formula and the Dirac equation, Phys. Rev. B 64, 014416 (2001).
  89. C. Di Castro and R. Raimondi, Statistical Mechanics and Applications in Condensed Matter (Cambridge University Press, Cambridge, 2015).
  90. A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge University Press, Cambridge, 2011).
  91. R. Raimondi, M. Leadbeater, P. Schwab, E. Caroti, and C. Castellani, Spin-orbit induced anisotropy in the magnetoconductance of two-dimensional metals, Phys. Rev. B 64, 235110 (2001).
  92. R. Raimondi and P. Schwab, Magnetoconductance of a two-dimensional metal in the presence of spin-orbit coupling, Eur. Phys. J. B 25, 483 (2002).
  93. M. Offidani, R. Raimondi, and A. Ferreira, Microscopic linear response theory of spin relaxation and relativistic transport phenomena in graphene, Condens. Matter 3, 18 (2018).
  94. T. Valet and R. Raimondi, Semiclassical kinetic theory for systems with non-trivial quantum geometry and the expectation value of physical quantities, Europhys. Lett. 143, 26004 (2023).
  95. L. V. Keldysh, Diagram Technique for Nonequilibrium Processes, Sov. Phys. JETP 20, 1018 (1965).
  96. A. Larkin and Y. Ovchinnikov, Nonlinear conductivity of superconductors in the mixed state, Sov. Phys. JETP 41, 960 (1975).
  97. T. Valet, H. Jaffres, V. Cros, and R. Raimondi, Quantum kinetic anatomy of electron angular momenta edge accumulation, Phys. Rev. Lett. 135, 256301 (2025).
  98. R. Raimondi and T. Valet, Quantum kinetic theory of the spin Hall effect for disordered graphene with Rashba spin–orbit coupling, Condens. Matter 10, 4 (2025).
  99. A. Veneri, P. Burghignoli, and D. Comite, Microwave imaging with circular arrays: Comparative analysis between OAM and MIMO, IEEE Trans. Antennas Propag. 73, 3015 (2025).
  100. J. Rammer, Quantum Transport Theory (CRC Press, Boca Raton, FL, 2018).
  101. M. Milletarì, M. Offidani, A. Ferreira, and R. Raimondi, Covariant conservation laws and the spin Hall effect in Dirac-Rashba systems, Phys. Rev. Lett. 119, 246801 (2017).
  102. B. M. Norman, C. J. Trowbridge, D. D. Awschalom, and V. Sih, Current-induced spin polarization in anisotropic spin-orbit fields, Phys. Rev. Lett. 112, 056601 (2014).

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