We describe a new effect in semiconductor spintronics that leads to dissipationless spin-currents in paramagnetic spin-orbit coupled systems. We argue that in a high mobility two-dimensional electron system with substantial Rashba spin-orbit coupling, a spin-current that flows perpendicular to the charge current is intrinsic [1].
In what follows, the basic physics of this effect is illustrated schematically.
In a translationally invariant 2DES, electronic eigenstates have definite momentum and, because of spin-orbit coupling, a momentum dependent effective magnetic field that causes the spins (red arrows) to align perpendicular to the momenta (green arrows), as illustrated in the following figure
The 2D electronic eigenstates in a Rashba spin-orbit coupled system are labeled by momentum (green arrows). For each momentum the two eigenspinors point in the azimuthal direction (red arrows).
In the presence of an electric field, which we take to be in the x̂ direction and indicate by blue arrows, electrons are accelerated and drift through momentum space at the rate ṗ = - e E x̂.
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In an electric field (blue) the Fermi surface (blue circle) is displaced an amount |eEx t0/ℏ| at time t0 (shorter than typical scattering times). While moving in momentum space, electrons experience an effective torque which tilts the spins up for py >0 and down for py<0, creating a spin-current in the y-direction.
The principal result is summarized in the following figure. where the spin-hall conductivity is plotted as a function of carrier density and Rashba coupling strength in the zero quasiparticle spectral broadening limit.
Plot of σsH vs. the two-dimensional electron density n2D and the Rashba coupling constant λ. The color scale is yellow = e/8π and dark red=0. The dashed line indicates the boundary between the universal region (constant σsH) and the linear density dependence region. Most of the current experimental samples are deep in the universal value region. We expect the intrinsic spin-Hall contribution to be present only when the Rashba splitting is larger than the disorder broadening of the quasiparticle energy levels, i.e. when λ pFτ/ℏ2 > 1.
Unlike the universal Hall conductivity value on a 2DES quantum Hall plateau, the universality of the intrinsic spin Hall effect is not robust against disorder and will be reduced whenever the disorder broadening is larger than the spin-orbit coupling splitting.
[1] Phys. Rev. Lett. 92, 126603 (2004), doi:10.1103/PhysRevLett.92.126603.
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