Side jumps in the spin Hall effect: Construction of the Boltzmann collision integral

Dimitrie Culcer, E. M. Hankiewicz, Giovanni Vignale, and R. Winkler
Phys. Rev. B 81, 125332 – Published 29 March 2010

Abstract

We present a systematic derivation of the side-jump contribution to the spin Hall current in systems without band-structure spin-orbit interactions, focusing on the construction of the collision integral for the Boltzmann equation. Starting from the quantum Liouville equation for the density operator we derive an equation describing the dynamics of the density matrix in the first Born approximation and to first order in the driving electric field. Elastic scattering requires conservation of the total energy, including the spin-orbit interaction energy with the electric field: this results in a first correction to the customary collision integral found in the Born approximation. A second correction is due to the change in the carrier position during collisions. It stems from the part of the density-matrix off-diagonal in wave vector. The two corrections to the collision integral add up and are responsible for the total side-jump contribution to the spin Hall current. The spin-orbit-induced correction to the velocity operator also contains terms diagonal and off-diagonal in momentum space, which together involve the total force acting on the system. This force is explicitly shown to vanish (on the average) in the steady state: thus the total contribution to the spin Hall current due to the additional terms in the velocity operator is zero.

  • Received 19 October 2009

DOI:https://doi.org/10.1103/PhysRevB.81.125332

©2010 American Physical Society

Authors & Affiliations

Dimitrie Culcer1, E. M. Hankiewicz2, Giovanni Vignale3, and R. Winkler4,5

  • 1Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
  • 2Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany
  • 3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
  • 4Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
  • 5Northern Illinois University, De Kalb, Illinois 60115, USA

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Issue

Vol. 81, Iss. 12 — 15 March 2010

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