What is the orbital Hall effect?

In the course of thinking about how best to revise my too-math-infused post about quantum geometry, I realized that writing about the orbital Hall effect lays nice groundwork.  

I've previously written about the spin Hall effect (SHE), in which a charge current \(\mathbf{j}_{\mathrm{c}}\) directed along \(\hat{\mathbf{x}}\) generates a net flow of \(\hat{\mathbf{y}}\)-directed spin angular momentum along the \(\hat{\mathbf{z}}\) direction.  This is a consequence of spin-orbit coupling, and it was first predicted in 1971 with a major revival sparked in 1999.  Electrically generating angular momentum currents has proven very useful, leading to many ideas about magnetic memory devices.  Microscopically, it's not easy to develop an intuition about the SHE, though as a spin-orbit effect, it is expected to be much stronger in heavier metals, since the spin-orbit coupling in atomic orbitals scales like \(Z^{4}\), and electronic bands in solids are built from those orbitals.  

That fact, that the electronic bands originate from atomic orbitals, is something that can get lost in a Bloch wave/nearly-free electron treatment of electronic structure.  In the orbital Hall effect, this idea is paramount.  This was explained clearly in this PRL (arXiv here).  The little \(p\)-orbitals are drawn on top of the \(k_{x}-k_{y}\) plane, to illustrate the idea that the electronic states in \(\mathbf{k}\)-space have different orbital content, depending on \(\mathbf{k}\).   The blue circle represents the "Fermi disk", with \(\mathbf{k}\)-states inside the circle occupied, and \(\mathbf{k}\)-states outside the circle empty.  

Adapted from Fig. 1 here.

When no electric field is applied, the Fermi disk is centered on \(\mathbf{k} = 0\); there is no net current, and there is no net orbital angular momentum once all the filled states are considered.  When an electric field is applied in the \(+x\) direction, though, the Fermi disk is shifted away from the origin in the \(-x\) direction (because of our convention that electrons are negatively charged).  Now adding up the \(z\)-directed orbital angular momentum contained within the Fermi disk, there is net \(+z\) orbital angular momentum carried by states with positive \(k_{y}\), and net \(-z\) orbital angular momentum carried by states with negative \(k_{y}\).  So, for this orbital texture, a charge current \(\mathbf{j}_{\mathrm{c}}\) directed along \(+\hat{\mathbf{x}}\) generates a net flow of \(\hat{\mathbf{z}}\)-directed orbital angular momentum along the \(+\hat{\mathbf{y}}\) direction.  Charge current generates a transverse flow of orbital angular momentum, entirely due to the way atomic orbitals come together to make Bloch states in \(\mathbf{k}\)-space, independent of any spin-orbit physics.  That's why the orbital Hall effect has been inferred experimentally in several materials with weak spin-orbit effects, like chromium and titanium.

These effects can be large, and orbital Hall physics plus some \(\mathbf{L}\cdot\mathbf{S}\) coupling may be responsible for some of the results labeled as spin Hall.  See here for a discussion.  Electrically pumping around angular momentum through orbital and spin Hall effects, and their inverses, is the idea behind a variety of device concepts for memory (e.g. here) and logic.  Fun stuff.