Methods for Calculating and Measuring the Band Structure

Abstract

In the previous chapter we examined two simple methods based on opposite approaches to calculate the energies and band structure of one-particle electron states. In the nearly-free-electron model the potential created by the lattice of atoms was considered as a perturbation with respect to the kinetic energy of electrons. In the tight-binding method we started with localized atomic states, and treated the propagation of electrons in the lattice as perturbation. Both methods gave a good qualitative picture about how the allowed energies of Bloch electrons form bands. In the first approach, even though free electrons can have arbitrary energies, certain energies are found to be forbidden in the presence of the potential, while in the second approach discrete atomic energy levels are observed to broaden into bands. Although in certain cases-e.g., for simple metals-the two methods lead to even quantitatively correct results, the kinetic energy of electrons and the potential arising from the interactions with atoms and other electrons are equally important in general , and neither of them can be treated as a perturbation with respect to the other. The accurate calculation of the band structure requires the solution of a difficult numerical problem in which every state-including deep core states-must be taken into account in principle, since even those are broadened into bands. Two problems arise in connection with the Schrödinger equation (17.1.3). Firstly, the choice of the one-particle potential is dictated by the specific problem , secondly, when U (r) is given, a suitable and efficient numerical method is needed to compute the energy eigenvalues quickly and accurately. We shall not deal with the choice of the potential here: it will be deferred to Volume 3 devoted to the study of electron-electron interactions. We shall just note that, since the influence of all other electrons must also be lumped into the potential , the solutions for the wavefunction must be consistent with the electron density used for specifying the potential-that is, self-consistent solutions have to be found. Assuming that the potential is known, we shall first briefly present various computational methods. As mentioned before, all these methods eventually