What is the spin Hall effect?

The Hall Effect is an old (1879) story, told in first-year undergraduate physics classes for decades. Once students are told about the Lorentz force law, it's easy to make a handwave classical argument that something like the Hall Effect has to exist:  Drive a current in a conductor in the presence of a magnetic induction \(\mathbf{B}\).  Charged particles undergo a \(q \mathbf{v} \times \mathbf{B}\) force that pushes them transverse to their original \(\mathbf{v}\) direction.  In a finite slab of material with current perpendicular to \(\mathbf{B}\), the particles have to pile up at the transverse edge, leading to the development of a (Hall) voltage perpendicular to the direction of current flow and the magnetic induction.  You can measure the Hall voltage readily, and it's used for sensing magnetic fields, as well as figuring out charge carrier densities in materials.

The spin Hall effect, in contrast, is a much newer idea.  It was first proposed by Dyakonov and Perel in 1971 as an extrinsic effect (that is, induced by scattering from impurities in a material), and this was revisited in 1999 by Hirsch and others.  It's also possible to have an intrinsic spin Hall effect (proposed here and here) due just to the electronic structure of a material itself, not involving impurities.

Adapted from here.

So what is the SHE?  In some non-magnetic conductors, in the absence of any external magnetic field, a charge current (say in the \(+x\) direction) results in a build-up of electrons with spin polarized up (down) along the \(z\) direction along the positive (negative) \(y\) edge of the material, as shown in the bottom left drawing of the figure.  Note that there is no net charge imbalance or transverse voltage - just a net spin imbalance. 

The SHE is a result of spin-orbit coupling - it's fundamentally a relativistic effect (!).  While we static observers see only electric fields in the material, the moving charge carriers in their frame of reference see effective magnetic fields, and that affects carrier motion.  In the extrinsic SHE, scattering of carriers from impurities ends up having a systematic spin dependence, so that spin-up carriers are preferentially scattered one way and spin-down carriers are scattered the other.  In the intrinsic SHE, there ends up being a spin-dependent term in the semiclassical velocity that one would get from the band structure, because of spin-orbit effects.  (The anomalous Hall effect, when one observes a Hall voltage correlated with the magnetization of a magnetic conductor, is closely related.  The net charge imbalance shows up because the populations of different spins are not equal in a ferromagnet.)  The result is a spin current density \(\mathbf{J}_{\mathrm{s}}\) that is perpendicular to the charge current density \(\mathbf{J}_{\mathrm{c}}\), and is characterized by a (material-dependent) spin Hall angle, \(\theta_{\mathrm{SH}}\), so that \(J_{\mathrm{s}} = (\hbar/2e)\theta_{\mathrm{SH}}J_{\mathrm{c}}\).

There is also an inverse SHE:  if (appropriately oriented) spin polarized charge carriers are injected into a strong spin-orbit coupled non-magnetic metal (say along \(+x\) as in the bottom right panel of the figure), the result is a transverse (\(y\)-directed) charge current and transverse voltage build-up.  (It's this inverse SHE that is used to detect spin currents in spin Seebeck effect experiments.)

The SHE and ISHE have attracted a lot of interest for technological applications.  Generating a spin current via the SHE and using that to push around the magnetization of some magnetic material is called spin orbit torque, and here is a recent review discussing device ideas.