Suppose an electron in a crystal(periodic potential) has the Bloch wave function [itex]\psi (k_0,x)[/itex] at t=0. Applying a constant electric field E(at t=0), the electron would still have Bloch-type wave function at any time t with [itex]k(t)=-(1/\hbar) eEt+k_0[/itex]. Now According to this formula for k(t), one can see that k's are now continuous(wheras we had as many as N different ks in the first brillouin zone before exerting the electric field). Does electronic band structure change by applying the electric field or not? If yes, How?
Not in semiconductors and regular metals. The electrons move, but the bands remain in place. The kinetic energy of the carriers increases quadratically with wave vector. The band minimum stays in place. So when an electric field is applied, the kinetic energy of the carrier increases until there is a collision. The collision takes away kinetic energy, so the carrier winds up back at the minimum of the band. The band structure in superconductors does move when an electric field is applied. In a type I superconductor, conduction electrons are bound in Cooper pairs by a phonon field. The dispersion relation of the Cooper pairs is still parabolic, in that the kinetic energy of the pair increases quadratically with wave number. However, the band minimum moves in the presence of an electric field. Because the Cooper pairs are always at the bottom of the band, they always have zero kinetic energy even while they are "moving". The Cooper pairs have no kinetic energy even when their center of masses are moving at top speed. So that collisions don't take away kinetic energy. The electrons in the Cooper pairs just keep accelerating in "group velocity" without absorbing any energy from the electric field. So the superconductor doesn't have a resistivity that is analogous to the resistivity of a normal conductor. There are some ways that a superconductor can "resist" electric current. However, they are very strange from the standpoint of normal electronics. The London equation shows how the penetrance of a magnetic field is effected by the superconductivity property. Superconductors are very weird. Quantum mechanics is needed to understand superconductors. They are harder to explain with classical analogs then regular metals and semiconductors However, the electronic structure in semiconductors and metals are fixed. Superconductors are "the exception that proves the rule."
Electronic band structure changes not ONLY by applying the electric field. Electronic band structure changes even spontaneously without applying any electric field. There are cases of phase transitions to another crystall lattice. There are cases of phase transitions due to Peierls instability (doubling lattice period). There are cases of phase transitions due to Minich instability (pseudogap in insulator/metal transition). And so on. By applying the electric field you get nonperiodic potential, so Bloch theory is nonapplicable. Peierls instability lead to changing period, but Bloch theory is applicable. Minich instability lead to nonperiodic potential from the beginning, so Bloch theory (fermi liquid picture) is nonapplicable.
But according to a lot of solid state text-books, by applying an electric field, under the ideal conditions the band structure would not change and only electron's wave vector will change with time so that the electron would oscillate in the crystal. As mentioned earlier, k(t) will change continuously while in the FBZ k(0)s are quantized so how k(t) will move on k(0)s?
"according to a lot of solid state text-books" there are approximations, which might be true in some cases or might be untrue in other cases. Invention of LED (light emittng diods) and heterojunction systems are the cases which point us which solid state text-books we should not read to get true physics.