The valley degree of freedom of an electron

Under the periodic potential of solid, the movement of an electron obeys the Bloch theorem. In addition to the charge and real spin degree of freedom, Bloch electrons in solids are endowed with valley degree of freedom representing the local energy extrema of the Bloch energy bands. Here we will review the intriguing electronic properties of valley degree of freedom of solid materials ranging from conventional bulk semiconductors to two-dimensional atomic crystals such as graphene, silicene, and transition metal dichalcogenides. The attention is paid to how to break the valley degeneracy via different ways including strain, electric field, optic field, etc. Conventional semiconductors usually have multiple valley degeneracy, which have to be lifted by quantum confinement or magnetic field. This can alleviate the valley degeneracy problem, but lead to simultaneously more complex many-body problems due to the remnant valley interaction in the bulk semiconductor. Two-dimensional materials provide a viable way to cope with the valley degeneracy problem. The inequivalent valley points in it are in analogy with real spin as long as the inversion symmetry is broken. In the presence of electric field, the nonvanishing Berry curvature drives the anomalous transverse velocity, leading to valley Hall effect. The valley degree of freedom can be coupled with other degree of freedom, such as real spin, layer, etc, resulting in rich physics uncovered to date. The effective utilization of valley degree of freedom as information carrier can make novel optoelectronic devices, and cultivate next generation electronics-valleytronics.