Intrinsic spin current for an arbitrary Hamiltonian and scattering potential

Abstract

We have described electron spin dynamics in the presence of the spin-orbit interaction and disorder using the spin-density matrix method. Exact solution is obtained for an arbitrary 2D spin-orbit Hamiltonian and arbitrary smoothness of the disorder potential. Spin current depends explicitely on the disorder properties, namely the smoothness of the disorder potential, even in the ballistic limit when broadening by scattering is much smaller than the spin-orbit related splitting of the energy spectrum. In this sense universal intrinsic spin current does not exist. 73.50.Bk Spin-orbit coupling brings about a number of interesting effects, one of which is generation of a spin flux in the plane perpendicular to the charge current direction. This phenomenon occurs in the paramagnetic system and has been very well known for quite a long time, see Ref. 1 . It results from the fact that in the presense of spin-orbit coupling the scattering by impurities has an asymmetric character 2 . Extrinsic contribution exists only beyond the Born approximation in the scattering amplitude and leads to an accumulation of the spin density near the sample surface 1 .It has been recently claimed 3,4 that an analogous phenomenon can exist even without scattering by impurities, i.e. in the ballistic regime, the corresponding contribution being called intrinsic. Later several papers appeared where the effect of scattering by impurities was taken into account for the Rashba model 5,6,7,8,9,10 , besides, the extrinsic contribution was described 11,12 to explain the recent experiment done by Awschalom's group 13 , see also experiment 14 .Here we study the dependence of the intrinsic contribution to the spin current on the form of the Hamiltonian and properties of the disorder potential using the spin-density matrix method 15 . We have considered an arbitrary Hamiltonian, for example, the generalized 2D Rashba Hamiltonian with an arbitrary momentum dependence of the spin-orbit term or the 3D Luttinger Hamiltonian, and the arbitrary smoothness of the disorder potential. In the case of the generalized 2D Rashba model and the arbitrary smoothness of the disorder potential an exact solution for spin current in the ballistic limit ∆τ ≫ 1 is obtained, where ∆ is the spin splitting of the electron spectrum and τ is the transport scattering time. Spin current depends explicitely on the disorder properties, namely, the smoothness of the disorder potential, even in the ballistic limit. In this sense universal intrinsic spin current does not exist. The problem of 2D holes with the p 3 spin-orbit term has been numerically considered recently by several groups 16,17 . Analytically it was studied in Ref.18 for the δ-scattering, where the incorrect answer was obtained due to the wrong calculation method. The case of the arbitrary smoothness of the disorder potential has been analytically considered very recently in Refs. 19,20,21 , some results were also reported in Ref. 22 . Below we will compare the results obtained here with those from Ref.20. I also ...

Spin currents and spin superfluidity

“…But this conclusion is valid only for Rashba spin-orbit coupling linear in the electron wave vector k, and the intrinsic spin Hall effect for Rashba coupling nonlinear in k, or in the Luttinger model for spin-orbit coupled systems [152] is not ruled out. The intrinsic effect in the medium with nonlinear in k Rashba coupling follows from the theory using the quantum Boltzmann equation for the spin density matrix [149,153]. In contrast to the extrinsic spin Hall effect, the intrinsic spin Hall effect is not possible without off-diagonal matrix elements since the z spin current is proportional to them.…”

Unitarity of scattering and edge spin accumulation

“…Therefore, the distribution functions for the plus and minus modes, f ++ ( k + ) and f −− ( k − ), are determined by the electric field and by scattering off the impurities in the bulk of a system. 16 The wave vectors k + and k − , shown in Fig. 1, correspond to a given energy and a given wave vector along the boundary.…”

Quasiclassical approach to the spin Hall effect in the two-dimensional electron gas

“…16,19 In the case of a more general spin-orbit field, a finite spin Hall effect has been reported even in the presence of disorder. 11,18,[20][21][22][23] With the conventional definition of the spin current, given by the anticommutator of the velocity operator and the Pauli matrices, the spin current is not conserved. Hence it is not automatically guaranteed that a current in the bulk induces a polarization at the edges of the sample.…”

“…In this way the spin Hall induced spin accumulation in systems with linear dimensions of several tens of the Fermi wavelength have been studied. 24,25 Macroscopic systems are conveniently described in terms of semiclassical kinetic equations, 11,13,14,[21][22][23]26 and in particular in terms of diffusion equations describing the coupled dynamics of spin and charge degrees of freedom. In clean systems, however, the spin relaxation length can become comparable to the elastic mean free path.…”