Wannier—Stark Effect in Transport

Since the first prediction by G.H. Wannier, the energy spectrum of a crystalline solid in an electric field, which is called a Wannier—Stark state or a Stark ladder, has attracted much interest. In 1960, Wannier studied electronic states in the presence of a uniform electric field theoretically and found that eigenstates are localized along the direction of the electric field and have quantized energy levels 2.8.1$$ {E_n} = neFd, $$ where n is an integer, F the electric field, and d the lattice period along the electric field [1]. The localized Stark ladder states are associated with Bloch oscillation. In an electric field F, temporal motion of an electronic states is described by $$ \dot k = eF/\hbar $$. At the edge of the Brillouin zone, electrons are Bragg reflected by the periodic crystal potential and their motion becomes periodic in space, as well as in the k space.