Tunneling of Bloch electrons through vacuum barrier

Tunneling of Bloch electrons through a vacuum barrier introduces new physical effects in comparison with the textbook case of free (plane wave) electrons. For the latter, the exponential decay rate in the vacuum is minimal for electrons with the parallel component of momentum ${\bf k}_\parallel=0$, and the prefactor is defined by the electron momentum component in the normal to the surface direction. However, the decay rate of Bloch electrons may be minimal at an arbitrary ${\bf k}_\parallel$ (``hot spots''), and the prefactor is determined by the electron's group velocity, rather than by its quasimomentum.