Hara, Bahramy, and Murakami (Phys. Rev. B 102, 184404 (2020)) derived the orbital magnetization induced by an electric current through interatomic (itinerant) electron motion in systems without inversion symmetry, estimated a measurable magnitude in the polar metal SnP (about 0.63 G equivalent field at E=104E=10^4 V/m and τ=10−12\tau=10^{-12} s), and explicitly left open the completion of the quantitative picture: inclusion of the intraatomic orbital-magnetization contribution needed for the total magnetization in real materials such as SnP, and combination with ab initio electronic-structure calculation, which they identified as a promising direction for future research. Later literature (2021-2025) has made the orbital Edelstein effect an established subfield with ab initio estimates for several noncentrosymmetric systems and strong indirect experimental evidence, but the total (intraatomic plus interatomic) current-induced orbital magnetization of bulk polar metals such as SnP remains uncomputed from realistic first-principles electronic structure. The problem is to complete this quantitative ab initio account.

Background

Orbital magnetization (OM) is the magnetization carried by electron orbital motion rather than spin. In a crystal with time-reversal symmetry, the equilibrium OM vanishes, and computing OM in extended crystals is subtle because the position operator is ill-defined for delocalized Bloch states. The modern theory of orbital magnetization (MTOM) solves this using localized Wannier functions (spatially localized orbitals built from the Bloch eigenstates). In the Wannier picture the OM decomposes into two physically distinct mechanisms: an intraatomic contribution, the angular momentum inherited by itinerant electrons from their parent atomic orbitals (the part kept in the atom-centered approximation), and an interatomic (itinerant) contribution, the circulating current that flows between different atomic sites, which is absent from any atom-centered treatment.

The Edelstein effect is the induction of a homogeneous nonequilibrium magnetization by a dc electric current in systems with broken inversion symmetry. Its familiar spin form arises in Rashba-split bands (bands split linearly in crystal momentum by spin-orbit coupling, SOC, the coupling of electron spin to its orbital motion) and requires SOC. The orbital analog — the orbital Edelstein effect (OEE), also called current-induced orbital magnetization or a kinetic magnetoelectric effect — needs no SOC: it was introduced for chiral (helical) crystals by Yoda, Yokoyama, and Murakami, who drew an analogy between the current-driven circulating motion in a helical crystal and the field of a classical solenoid.

Hara, Bahramy, and Murakami (2020) extended the theory to the broad class of systems without inversion symmetry. For polar metals (metals whose crystal structure lacks inversion symmetry and has a distinguished polar axis), symmetry dictates that a current perpendicular to the polar axis induces an OM perpendicular to both the current and the polar axis, and perturbation theory shows the microscopic origin: the electric field drives closed current loops through the crystal, like a nanoscale solenoid. Treating the nonequilibrium distribution in the Boltzmann constant-relaxation-time approximation (each carrier relaxes with a single assumed time τ\tau), and using a tight-binding model fitted to the relativistic band structure of the polar phase of SnP, they estimated an induced interatomic OM of about 0.63 G (expressed as an equivalent magnetic field) for a field E=104E=10^4 V/m and τ=10−12\tau=10^{-12} s — in principle experimentally measurable. They also built a method for the in-plane OM of two-dimensional systems by virtually stacking the system into an artificial three-dimensional crystal, and predicted a current-induced OM at the surfaces of and interfaces between insulators.

Crucially, the paper treated only the interatomic contribution. In real materials with orbitally active atoms (angular momentum l≥1l\geq 1, e.g., pp or dd orbitals, as in SnP), the intraatomic contribution coexists and can dominate, so the total current-induced OM requires both. The SnP estimate was obtained within a Boltzmann treatment at zero temperature with a constant relaxation time. In their conclusion the authors noted that current-induced OM is expected in a wide range of materials, that in real materials such as SnP the intraatomic contribution should also be considered, and that combination with ab initio calculation can be a promising direction for future research.

The subsequent rise of orbitronics (the use of the orbital degree of freedom for information processing) has made this quantitative question timely: first-principles calculations now exist for oxide two-dimensional electron gases, ferroelectric GeTe, chiral materials, and magnetic interfaces, and indirect experiments (inverse and reciprocal charge-orbital conversion, nonlinear response, magnetotransport) have confirmed that the orbital channel can dominate over the spin channel. What has still not been achieved is the completion left open by the source paper: a material-resolved first-principles prediction of the total current-induced OM — intraatomic plus interatomic — particularly for bulk polar metals such as SnP in its polar phase.

Problem Statement

Complete the quantitative ab initio account of current-induced orbital magnetization (the orbital Edelstein effect) in systems without inversion symmetry, as left open by Hara, Bahramy, and Murakami (2020). The scientific objective is a predictive, material-resolved theory of the magnitude of the current-induced orbital magnetization: compute, from realistic first-principles electronic structure (for example density-functional theory combined with Wannier tight-binding models and the modern theory of orbital magnetization), the total current-induced orbital magnetization of real noncentrosymmetric materials, combining the interatomic (itinerant) contribution with the intraatomic (on-site atomic orbital angular-momentum) contribution, with both terms reported separately and summed, and using material-dependent electronic-structure parameters rather than a generic model. A complete answer provides such a first-principles prediction of the total current-induced orbital magnetization for SnP in its polar phase, the source paper's own example — for which the existing estimate of about 0.63 G equivalent field at E=104E=10^4 V/m and τ=10−12\tau=10^{-12} s covers only the interatomic contribution within a constant-relaxation-time, zero-temperature Boltzmann treatment — with the relaxation model, temperature, and material parameters underlying the prediction explicitly stated, and with any comparison to the source estimate identifying which contribution and which relaxation assumptions are being compared, thereby replacing the model-level, single-contribution picture with a first-principles total account. Predictions for additional polar metals or other noncentrosymmetric materials are welcome as extensions but are not required for completion.

References

  1. D. Hara, M. S. Bahramy, and S. Murakami, 'Current-induced orbital magnetization in systems without inversion symmetry', Phys. Rev. B 102, 184404 (2020). DOI: 10.1103/PhysRevB.102.184404; arXiv:2010.12159. https://doi.org/10.1103/PhysRevB.102.184404
  2. T. Yoda, T. Yokoyama, and S. Murakami, 'Current-induced orbital and spin magnetizations in crystals with helical structure', Sci. Rep. 5, 12024 (2015). DOI: 10.1038/srep12024. https://doi.org/10.1038/srep12024
  3. T. Yoda, T. Yokoyama, and S. Murakami, 'Orbital Edelstein Effect as a Condensed-Matter Analog of Solenoids', Nano Lett. 18, 916-920 (2018). DOI: 10.1021/acs.nanolett.7b04300; arXiv:1706.07702. https://doi.org/10.1021/acs.nanolett.7b04300
  4. L. Salemi, M. Berritta, A. K. Nandy, and P. M. Oppeneer, 'Orbitally dominated Rashba-Edelstein effect in noncentrosymmetric antiferromagnets', Nat. Commun. 10, 5381 (2019). DOI: 10.1038/s41467-019-13367-z. https://doi.org/10.1038/s41467-019-13367-z
  5. A. Johansson, B. Göbel, J. Henk, M. Bibes, and I. Mertig, 'Spin and orbital Edelstein effects in a two-dimensional electron gas: Theory and application to SrTiO3 interfaces', Phys. Rev. Research 3, 013275 (2021). DOI: 10.1103/PhysRevResearch.3.013275; arXiv:2006.14958. https://doi.org/10.1103/PhysRevResearch.3.013275
  6. K. Osumi, T. Zhang, and S. Murakami, 'Kinetic magnetoelectric effect in topological insulators', Commun. Phys. 4, 211 (2021). DOI: 10.1038/s42005-021-00702-4. https://doi.org/10.1038/s42005-021-00702-4
  7. L. Chirolli, M. T. Mercaldo, C. Guarcello, F. Giazotto, and M. Cuoco, 'Colossal Orbital Edelstein Effect in Noncentrosymmetric Superconductors', Phys. Rev. Lett. 128, 217703 (2022). DOI: 10.1103/PhysRevLett.128.217703; arXiv:2107.07476. https://doi.org/10.1103/PhysRevLett.128.217703
  8. S. Varotto, A. Johansson, B. Göbel, L. M. Vicente-Arche, S. Mallik, J. Bréhin, R. Salazar, F. Bertran, P. Le Fèvre, N. Bergeal, J. Rault, I. Mertig, and M. Bibes, 'Direct visualization of Rashba-split bands and spin/orbital-charge interconversion at KTaO3 interfaces', Nat. Commun. 13, 6165 (2022). DOI: 10.1038/s41467-022-33621-1; arXiv:2207.08698. https://doi.org/10.1038/s41467-022-33621-1
  9. A. El Hamdi, J.-Y. Chauleau, M. Boselli, C. Thibault, C. Gorini, A. Smogunov, C. Barreteau, S. Gariglio, J.-M. Triscone, and M. Viret, 'Observation of the orbital inverse Rashba-Edelstein effect', Nat. Phys. 19, 1303-1308 (2023). DOI: 10.1038/s41567-023-02121-4. https://doi.org/10.1038/s41567-023-02121-4
  10. S. Leiva-Montecinos, J. Henk, I. Mertig, and A. Johansson, 'Spin and orbital Edelstein effect in a bilayer system with Rashba interaction', Phys. Rev. Research 5, 043294 (2023). DOI: 10.1103/PhysRevResearch.5.043294; arXiv:2307.02872. https://doi.org/10.1103/PhysRevResearch.5.043294
  11. K. Tenzin, A. Roy, F. T. Cerasoli, A. Jayaraj, M. Buongiorno Nardelli, and J. Sławińska, 'Collinear Rashba-Edelstein effect in nonmagnetic chiral materials', Phys. Rev. B 108, 245203 (2023). DOI: 10.1103/PhysRevB.108.245203; arXiv:2304.05287. https://doi.org/10.1103/PhysRevB.108.245203
  12. R. Burgos Atencia, A. Agarwal, and D. Culcer, 'Orbital angular momentum of Bloch electrons: equilibrium formulation, magneto-electric phenomena, and the orbital Hall effect', Adv. Phys. X 9, 2371972 (2024). DOI: 10.1080/23746149.2024.2371972; arXiv:2403.07055. https://doi.org/10.1080/23746149.2024.2371972
  13. D. Jo, D. Go, G.-M. Choi, and H.-W. Lee, 'Spintronics meets orbitronics: emergence of orbital angular momentum in solids', npj Spintronics 2, 19 (2024). DOI: 10.1038/s44306-024-00023-6. https://doi.org/10.1038/s44306-024-00023-6
  14. A. Johansson, 'Theory of spin and orbital Edelstein effects', J. Phys.: Condens. Matter 36, 423002 (2024). DOI: 10.1088/1361-648X/ad5e2b. https://doi.org/10.1088/1361-648X/ad5e2b
  15. X.-G. Ye, P.-F. Zhu, W.-Z. Xu, T.-Y. Zhao, and Z.-M. Liao, 'Nonlinear spin and orbital Edelstein effect in WTe2', Phys. Rev. B 110, L201407 (2024). DOI: 10.1103/PhysRevB.110.L201407; arXiv:2412.02938. https://doi.org/10.1103/PhysRevB.110.L201407
  16. S. A. Nikolaev, M. Chshiev, F. Ibrahim, S. Krishnia, N. Sebe, J.-M. George, V. Cros, H. Jaffrès, and A. Fert, 'Large chiral orbital texture and orbital Edelstein effect in Co/Al heterostructure', Nano Lett. 24, 13465-13472 (2024). DOI: 10.1021/acs.nanolett.4c01607; arXiv:2410.09957. https://doi.org/10.1021/acs.nanolett.4c01607
  17. J. M. Lee, M. J. Park, and H.-W. Lee, 'Orbital Edelstein effect of electronic itinerant orbital motion at edges', Phys. Rev. B 110, 134436 (2024). DOI: 10.1103/PhysRevB.110.134436; arXiv:2411.00353. https://doi.org/10.1103/PhysRevB.110.134436
  18. I. A. Ado, M. Titov, R. A. Duine, and A. Brataas, 'Orbital Edelstein effect from the gradient of a scalar potential', arXiv:2407.00516 (2024). https://arxiv.org/abs/2407.00516
  19. J. Voss, I. A. Ado, and M. Titov, 'Orbital magnetization from interface reflections in a conductor with charge current', Phys. Rev. B 111, L121402 (2025). DOI: 10.1103/PhysRevB.111.L121402; arXiv:2405.11979. https://doi.org/10.1103/PhysRevB.111.L121402
  20. S. Leiva-Montecinos, L. Vojáček, J. Li, M. Chshiev, L. Vila, I. Mertig, and A. Johansson, 'Current-induced spin and orbital polarization in the ferroelectric Rashba semiconductor GeTe', Phys. Rev. Research 7, 043329 (2025). DOI: 10.1103/py8f-zk2d; arXiv:2505.21340. https://doi.org/10.1103/py8f-zk2d
  21. W. Gao, L. Liao, H. Isshiki, N. Budai, J. Kim, H.-W. Lee, K.-J. Lee, D. Go, Y. Mokrousov, S. Miwa, and Y. Otani, 'Nonlocal electrical detection of reciprocal orbital Edelstein effect', Nat. Commun. 16, 6380 (2025). DOI: 10.1038/s41467-025-61602-7; arXiv:2502.11040. https://doi.org/10.1038/s41467-025-61602-7
  22. Z. Hua, C. Niu, S. Joy, P. Tan, G. Shi, H. Liu, J. Guo, D. Graf, P. D. Ye, C. Lewandowski, and P. Xiong, 'Tunable interplay of orbital and spin magnetization in trigonal tellurium', arXiv:2507.14292 (2025). https://arxiv.org/abs/2507.14292