ABSTRACT
Beyond the general form indicated above, rigorous analytical solutions are impossible to obtain. Depending on the nature of the crystal, various simplifications can be introduced to obtain the dispersion relation E = f(k). For metals, simplifications result in the nearly free-electron approximation. For a periodic crystalline potential, the perturbation on the propagation of the valence electrons is a small correction. The resulting dispersion relation deviates from the parabolic free electron relation only when the wave vector k approaches the limits of the Brillouin zones where the Bragg conditions are nearly satisfied. When the Bragg conditions are satisfied strictly, the corresponding electronic waves cannot propagate: the amplitude of reflected waves is equal to that of incident waves and results in standing waves. This causes the appearance of a forbidden energy band of width that is proportional to (the Fourier component of) the corresponding potential energy. In contrast, for insulators and semiconductors, the tight-binding approximationis typically used. It starts from the initial electronic states of isolated atoms and considers the effect of condensing them into the solid state via a perturbation, which is responsible for the appearance of a chemical bond between atoms and their neighbors. Using wave functions satisfying the Bloch theorem, this approximation is also known as the linear combination of atomic orbitals (LCAO). It leads to a dispersion relation of the form: E k E e
ik j( ) = - - Â -0 a g r (1) in which E0 represents the initial electron energy, α is the energy of the orbital, and γ is the overlap energy between neighboring atoms (which are separated by a distance ρj and are generally limited to the nearest neighbors j). α and γ are positive so that α determines the cohesion of the crystal even though the width of the energy band will be proportional to γ: the weaker the coupling between neighbors, the narrower the band. The determination of the detailed band structure is a complex task that is beyond the scope of this book. Assuming the validity of expressions similar to Equation (1), for example, the goal here is to study the influence of the crystal potential on the Fermi surface and the density of states in order to understand the electronic and optical properties of some important crystals.
