Nearly-free Electron Model - Introduction

Introduction

Free electrons are traveling plane waves. Generally the time independent part of their wave function is expressed as

These plane wave solutions have an energy of

The expression of the plane wave as a complex exponential function can also be written as the sum of two periodic functions which are mutually shifted a quarter of a period.

In this light the wave function of a free electron can be viewed as an aggregate of two plane waves. Sine and cosine functions can also be expressed as sums or differences of plane waves moving in opposite directions

Assume that there is only one kind of atom present in the lattice and that the atoms are located at the origin of the unit cells of the lattice. The potential of the atoms is attractive and limited to a relatively small part of the volume of the unit cell of the lattice. In the remainder of the cell the potential is zero.

The Hamiltonian is expressed as

in which is the kinetic and is the potential energy. From this expression the energy expectation value, or the statistical average, of the energy of the electron can be calculated with

E = \langle H \rangle = \int_{\Omega_r}\psi_{\bold{k}}^*(\bold{r})\psi_{\bold{k}}(\bold{r}) d\bold{r}

If we assume that the electron still has a free electron plane wave wave function the energy of the electron is:

E_k = \frac{1}{\Omega_r}\int_{\Omega_r} e^{-i\bold{k}\cdot\bold{r}} \left e^{i\bold{k}\cdot\bold{r}}d\bold{r}

Let's assume further that at an arbitrary -point in the Brillouin zone we can integrate the over a single lattice cell, then for an arbitrary -point the energy becomes

This means that at an arbitrary point the energy is lowered by the lowered average of the potential in the unit cell due to the presence of the attractive potential of the atom. If the potential is very small we get the Empty Lattice Approximation. This isn't a very sensational result and it doesn't say anything about what happens when we get close to the Brillouin zone boundary. We will look at those regions in -space now.

Let's assume that we look at the problem from the origin, at position . If only the cosine part is present and the sine part is moved to . If we let the length of the wave vector grow, then the central maximum of the cosine part stays at . The first maximum and minimum of the sine part are at . They come nearer as grows. Let's assume that is close to the Brillouin zone boundary for the analysis in the next part of this introduction.

The atomic positions coincide with the maximum of the -component of the wave function. The interaction of the -component of the wave function with the potential will be different than the interaction of the -component of the wave function with the potential because their phases are shifted. The charge density is proportional to the absolute square of the wave function. For the -component it is

and for the -component it is

For values of close to the Brillouin zone boundary, the length of the two waves and the period of the two different charge density distributions almost coincide with the periodic potential of the lattice. As a result the charge densities of the two components have a different energy because the maximum of the charge density of the -component coincides with the attractive potential of the atoms while the maximum of the charge density of the -component lies in the regions with a higher electrostatic potential between the atoms.

As a result the aggregate will be split in high and low energy components when the kinetic energy increases and the wave vector approaches the length of the reciprocal lattice vectors. The potentials of the atomic cores can be decomposed into Fourier components to meet the requirements of a description in terms of reciprocal space parameters.

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