A2.14
(i) A simple model of a crystal consists of an infinite linear array of sites equally spaced with separation . The probability amplitude for an electron to be at the -th site is . The Schrödinger equation for the is
where is real and positive. Show that the allowed energies of the electron must lie in a band , and that the dispersion relation for written in terms of a certain parameter is given by
What is the physical interpretation of and ?
(ii) Explain briefly the idea of group velocity and show that it is given by
for an electron of momentum and energy .
An electron of charge confined to one dimension moves in a periodic potential under the influence of an electric field . Show that the equation of motion for the electron is
where is the group velocity of the electron at time . Explain why
can be interpreted as an effective mass.
Show briefly how the absence from a band of an electron of charge and effective mass can be interpreted as the presence of a 'hole' carrier of charge and effective mass .
In the model of Part (i) show that
(a) for an electron behaves like a free particle of mass ;
(b) for a hole behaves like a free particle of mass .