126 Solving the Schrödinger equation in periodic solids
The eigenstates of the Hamiltonian can thus be written in the form of a plane wave
times a periodic function. Equivalently, evaluating such a wave function at two
positions, separated by a lattice vectorR, yields a difference of a phase factor eik·R,
according to our previous formulation of Bloch’s theorem.
Electronic structure methods for periodic crystals are usually formulated in recip-
rocal space, by solving an equation like (6.9) in which the basis functions are plane
waves labelled by reciprocal space vectors. In fact, Bloch’s theorem allows us to
solve for the full electronic structure in real space by considering only one cell
of the lattice for eachkand applying boundary conditions to the cell as dictated
by the Bloch condition (6.11). In particular, each facet of the unit cell boundary
has a ‘partner’ facet which is found by translating the facet over a lattice vector
R. The solutions to the Schrödinger equation should on both facets be equal up to
factor exp(ik·R). These boundary conditions determine the solutions inside the
cell completely. We see that we can try to solve the Schrödinger equation either
in reciprocal space or in real space. For nonperiodic systems, real-space methods
enjoy an increasing popularity [ 4 – 7].
6.3 Approximations
The Schrödinger equation for an electron in a crystal can be solved in two limiting
cases: the nearly free electron approximation, in which the potential is considered
to be weak everywhere, and the tight-binding approximation in which it is assumed
that the states are tightly bound to the nuclei. Both methods aim to reduce the
difficulty of the band structure problem and to increase the understanding of band
structures by relating them to those of two different systems which we can easily
describe and understand: free electrons and electrons in single-atom orbitals. The
tight-binding method has led to many computational applications. We shall apply
it to graphene and carbon nanotubes.
6.3.1 The nearly free electron approximationIt is possible to solve Eq. (6.9) if the potential is small, by using perturbative
methods. This is called the nearly free electron (NFE) approximation. You might
consider this to be inappropriate as the Coulomb potential is certainly not small
near the nuclei. Surprisingly, the NFE bands closely resemble those of aluminium,
for example! We shall see later on that the pseudopotential formalism provides an
explanation for this.
The main results of the NFE are that the bands are perturbed by an amount
which is quadratic in the size of the weak potentialVexcept close toBragg planes