Figure 1

Schematic of the energy spectrum and electronic transitions in a magnetic field. Here $\lambda =\pm 1$ numerates spectrum branches, $k_{+}$ and $k_{-}$ are Fermi momenta for the upper and lower branches, respectively, $n$ are Landau quantum numbers for these branches, $n_{\pm }$ are the lowest unoccupied states at both branches, and $\mu$ is the Fermi energy. Shadowed strips indicate regions of interbranch transitions between the occupied (−) states and unoccupied (+) states. The arrows $n\to n\pm 1$ show allowed interbranch transitions in the “bulk” of the Fermi sea, while the arrows $n_{+}\to n_{+}-1$ and $n_{-}\to n_{-}-1$ indicate two allowed intrabranch transitions at the Fermi edge. Cancellation of the inter- and intrabranch contributions results in the zero bulk spin Hall conductivity ${\mathcal{J}}_{zxy}$.Reuse & Permissions