S. Scandolo, ICTP

Lesson 11 (Bloch's theorem)

(If you notice anything wrong with this summary, (spelling, wrong reasoning, factual errors, etc...) please send an email to Peter Papai: ppapai at ictp dot it)

The lesson starts with stating Bloch's theorem. The consequence of the periodic potential is that there must be a basis of energy eigenfunctions of Schroedinger's equation which is also a basis of eigenfunctions of the discrete translation operator. Thus, each of these functions can be expressed as a periodic function (with the periodicity of the potential) multiplied by a general wave function. (This was a tough statement if you don't remember basic quantum mechanics.) The eigenvalues of the translation operator are parametrized by 'k' wavenumber (which can be chosen to be inside the Brillouin zone (BZ)). For each wavenumber there are multiple energy eigenvalues which define energy bands. (This gives us 'n', our second quantum number with the fist being 'k'.)

The professor argues that the energy of each band continuously changes with 'k'. He also speaks about properties of the solution: the energy must be a symmetric and periodic (as the reciprocal lattice) function of 'k'. A consequence: dE/dk must be 0 on the boundary of the BZ.

Since these bands describe an electron system, we can answer the question how many electrons are there in each band at most? The answer is two for each cell, one for each spin. A real life example is given, the Lithium bcc crystal, in which low energy bands follow atomic energy levels. Since the 2s atomic level is only half-filled for Lithium, the ground state of the system fills only the lower half of the corresponding band. This should cause a qualitative difference between this and a Magnesium bcc crystal. The bands are filled completely in the ground state of the latter; thus, the Mg crystal cannot be excited by an arbitrarily small amount of energy. Based on this, Mg should be an insulator: so something is wrong with this picture. Resolution is given by noticing that the energies of bands can overlap and the ground state fills two bands half way instead of one band fully. However, odd number of electrons always corresponds to a metal.