Question

How does the energy band diagram look with an electric field applied?

Answer 1

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."