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# Theory of spin transport in an $n$-typed GaAs quantum well

M.W. Wu and M.Q. Weng
Dept. of Physics, Univ. of Science & Technology of China, CN

Keywords: spintronics, spin transport, dephasing, decoherence

Abstract:
We perform a many-body investigation of the spin diffusion/transport in an $n$-doped GaAs QW based on a set of many-body kinetic transport equations. These equations are derived for the spatial inhomogeneous system by using nonequilibrium Green function method with generalized Kadanoff-Baym Ansatz in a two-spin-band model\cite{wu1,wu2,wu3}. Our theory takes all the inhomogeneous broadening effect, the spin diffusion due to the spacial inhomogeneity and the spin dephasing/decoherence into account and gets the results self-consistently. We reexamine the wildly adopted quasi independent electron model (QIEM)\cite{schmidt,spintronics,flatte_prl_2000,zutic_0106085,zutic} and pointed out an important many-body spin decoherence effect which is missing in the single electron model. The new decoherence effect is based on interference effect of electrons/spins with different momentum ${\bf k}$ along the diffusion direction, which is referred as inhomogeneous broadening effect'' in our paper. We have shown that this inhomogeneous broadening effect can cause the spin decoherence alone even without the scattering and that the resulting decoherence can be more important than the dephasing effect due to DP term together with the scattering part. Our study shows the inadequacy of the QIEM. Therefore, it is important to use the full many-body theory to study the spin transport. We further study the spin diffusion/transport from the full many-body theory with the DP terms (The spin dephasing mechanism for $n$-typed GaAs QW at high temperature is the DP mechanism.) and the scattering included. By numerically solving the kinetic Bloch equations, together with the Poisson equation, we are able to investigate the spin diffusion in the steady state under the constant spin injection. We have shown the spin diffusion in the absence/presence of an applied electric field along the diffusion direction as well as with/without impurities. By applying an electric field along the diffusion direction, one gets much longer spin diffusion length as the electrons are driven by the electric field and get a net drift velocity. Also in the presence of the electric field, the spin diffusion length is reduced if one introduces impurities into the sample. However, when there is no applied electric field, the spin diffusion length is {\em enlarged} by adding impurities into the sample. This is contrary to what is predicted by the QIEM. The reason of this qualitative difference is also discussed. We also study the effects of the magnetic field in the Voigt configuration and the applied electric fields along the QW growth direction to the spin diffusion. In the present of the magnetic field, the spin polarization exhibits oscillation along the direction of diffusion and the decay due to the interference is much more effective than that of the dephasing and therefore the spin diffusion length is greatly reduced. We also investigate the spin diffusion at different temperatures. We find that as the temperature increases, the interference effect reduces as the electron distribution near $k_x=0$, which is main contributor to the inhomogeneous broadening, becomes smaller. As a result, the spin diffusion length increases with the temperature. We show that the applied electric field along the growth direction makes the Rashba term more pronounced and hence both the decoherence and the dephasing get enlarged. Consequently the diffusion length is reduced. We have also demonstrated the time evolution of the diffusion of a spin package. The spin signals near the center of the package always decay with time due to the diffusion as well as the dephasing. Whereas the spin signals away from the center first increase then drop. For positions beyond the regime of the initial spin package, the spin polarization can be opposite to the initial one due to the spin flipping by the relatively large local effective magnetic field originated from the DP term together with the spin coherence $\rho_{\sigma-\sigma}$, with the later coming from both the diffusion and the spin precession. We also predict the spin oscillations with time at some positions. These features cannot be obtained from the QIEM.

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