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  • Letter
  • Open Access

Kubo formula for dc conductivity: Generalization to systems with spin-orbit coupling

I. A. Ado1,2,*, M. Titov1, Rembert A. Duine2,3, and Arne Brataas4

  • 1Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands
  • 2Institute for Theoretical Physics, Utrecht University, 3584 CC Utrecht, The Netherlands
  • 3Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
  • 4Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, Høgskoleringen 5, 7491 Trondheim, Norway

  • *iv.a.ado@gmail.com
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Phys. Rev. Research 6, L012057 – Published 11 March, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L012057

Abstract

We revise the Kubo formula for the electric dc conductivity in the presence of spin-orbit coupling (SOC). We discover that each velocity operator that enters this formula differs from ∂H/∂p, where H is the Hamiltonian and p is the canonical momentum. Moreover, we find an additional contribution to the Hall dc conductivity from noncommuting coordinates that is missing in the conventional Kubo-Středa formula. This contribution originates from the “electron-positron” matrix elements of the velocity and position operators. We argue that the widely used Rashba model does in fact provide a finite anomalous Hall dc conductivity in the metallic regime (in the noncrossing approximation) if SOC-corrections to the velocity and position operators are properly taken into account. While we focus on the response of the charge current to the electric field, linear response theories of other SOC-related effects should be modified similarly.

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Phys. Rev. Research 6, L012057– Published 11 March, 2024
  • Received 12 October 2023
  • Revised 31 October 2023
  • Accepted 25 January 2024
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