Kinetics of spin relaxation in quantum wires and channels: Boundary spin echo and formation of a persistent spin helix

In this paper we use a spin kinetic equation to study spin-polarization dynamics in one-dimensional (1D) wires and 2D channels. The spin kinetic equation is valid in both diffusive and ballistic spin transport regimes and therefore is more general than the usual spin drift-diffusion equations. In particular, we demonstrate that in infinite 1D wires with Rashba spin-orbit interaction the exponential spin-relaxation decay can be modulated by an oscillating function. In the case of spin relaxation in finite length 1D wires, it is shown that an initially homogeneous spin polarization spontaneously transforms into a persistent spin helix. We find that a propagating spin-polarization profile reflects from a system boundary and returns back to its initial position similarly to the reflectance of sound waves from an obstacle. The Green's function of the spin kinetic equation is derived for both finite and infinite 1D systems. Moreover, we demonstrate explicitly that the spin relaxation in specifically oriented 2D channels with Rashba and Dresselhaus spin-orbit interactions of equal strength occurs similarly to that in 1D wires of finite length. Finally, a simple transformation mapping 1D spin kinetic equation into the Klein-Gordon equation with an imaginary mass is found thus establishing an interesting connection between semiconductor spintronics and relativistic quantum mechanics.