The nearly free electron model is a modification of the free electron gas model which includes a weak periodic perturbation meant to model the interaction between the conduction electrons and the ions in a crystalline solid. This model, like the free electron model, does not take into account electron-electron interactions; that is, the independent electron approximation is still in effect.

As shown by Bloch's theorem, introducing a periodic potential into the Schrödinger equation results in a wave function of the form

$\backslash psi\_\{\backslash bold\{k\}\}(\backslash bold\{r\})\; =\; u\_\{\backslash bold\{k\}\}(\backslash bold\{r\})\; e^\{i\backslash bold\{k\}\backslash cdot\backslash bold\{r\}\}$

where the function u has the same periodicity as the lattice:

$u\_\{\backslash bold\{k\}\}(\backslash bold\{r\})\; =\; u\_\{\backslash bold\{k\}\}(\backslash bold\{r\}+\backslash bold\{T\})$

(T here is a lattice translation vector.)

A solution of this form can be plugged into the Schrödinger equation, resulting in the central equation:

$(\backslash lambda\_\{\backslash bold\{k\}\}\; -\; \backslash epsilon)C\_\{\backslash bold\{k\}\}\; +\; \backslash sum\_\{\backslash bold\{G\}\}\; U\_\{\backslash bold\{G\}\}\; C\_\{\backslash bold\{k\}-\backslash bold\{G\}\}=0$

where

$\backslash lambda\_\{\backslash bold\{k\}\}\; =\; \backslash frac\{\backslash hbar^2\; k^2\}\{2m\}$

and C_k and U_G are the Fourier coefficients of the wavefunction ψ(r) and the potential energy U(r), respectively:

$U(\backslash bold\{r\})\; =\; \backslash sum\_\{\backslash bold\{G\}\}\; U\_\{\backslash bold\{G\}\}\; e^\{i\backslash bold\{G\}\backslash cdot\backslash bold\{r\}\}$

$\backslash psi(\backslash bold\{r\})\; =\; \backslash sum\_\{\backslash bold\{k\}\}\; C\_\{\backslash bold\{k\}\}\; e^\{i\backslash bold\{k\}\backslash cdot\backslash bold\{r\}\}$

In any perturbation analysis, one must consider the base case to which the perturbation is applied. Here, the base case is with U(x) = 0, and therefore all the Fourier coefficients are also zero. In this case the central equation reduces to the form

$(\backslash lambda\_\{\backslash bold\{k\}\}\; -\; \backslash epsilon)C\_\{\backslash bold\{k\}\}\; =\; 0$

This identity means that for each k, one of the two following cases must hold:

1. $C\_\{\backslash bold\{k\}\}\; =\; 0$, or
2. $\backslash lambda\_\{\backslash bold\{k\}\}\; =\; \backslash epsilon$

If the values of $\backslash lambda\_k$ are non-degenerate, then the second case occurs for only one value of k, while for the rest, the Fourier expansion coefficient $C\_k$ must be zero. In this non-degenerate case, the standard free electron gas result is retrieved:

$\backslash psi\_k\; \backslash propto\; e^\{i\backslash bold\{k\}\backslash cdot\backslash bold\{r\}\}$

In the degenerate caes, however, there will be a set of lattice vectors k₁, ..., k_m with λ₁ = ... = λ_m. When the energy $\backslash epsilon$ is equal to this value of λ, there will be m independent plane wave solutions of which any linear combination is also a solution:

$\backslash psi\; \backslash propto\; \backslash sum\_\{i=1\}^\{m\}\; A\_i\; e^\{i\backslash bold\{k\}\_i\backslash cdot\backslash bold\{r\}\}$

Non-degenerate and degenerate perturbation theory can be applied in these two cases to solve for the Fourier coefficients C_k of the wavefunction (correct to first order in U) and the energy eigenvalue (correct to second order in U). An important result of this derivation is that there is no first-order shift in the energy ε in the case of no degeneracy, while there is in the case of near-degeneracy, implying that the latter case is more important in this analysis. Particularly, at the Brillouin zone boundary (or, equivalently, at any point on a Bragg plane), one finds a two-fold energy degeneracy that results in a shift in energy given by:

$\backslash epsilon\; =\; \backslash lambda\_\{\backslash bold\{k\}\}\; \backslash pm\; |U\_\{\backslash bold\{k\}\}|$

This energy gap between Brillouin zones is known as the band gap, with a magnitude of $2|U\_K|$.

Results

Introducing this weak perturbation has significant effects on the solution to the Schrödinger equation, most significantly resulting in a band gap between wave vectors in different Brillouin zones.

Justifications

In this model, the assumption is made that the interaction between the conduction electrons and the ion cores can be modeled through the use of a "weak" perturbing potential. This may seem like a weighty assumption, for the Coulomb attraction between these two particles of opposite charge can be quite significant at short distances. It can be partially justified, however, by noting two important properties of the quantum mechanical system:

1. The force between the ions and the electrons is greatest is at very small distances. However, the conduction electrons are not "allowed" to get this close to the ion cores due to the Pauli exclusion principle: the orbitals closest to the ion core are already occupied by the core electrons. Therefore, the conduction electrons never get close enough to the ion cores to feel their full force.
2. Furthermore, the core electrons shield the ion charge magnitude "seen" by the conduction electrons. The resuilt is an effective nuclear charge experienced by the conduction electrons which is significantly reduced from the actual nuclear charge.

References

• Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).
• Charles Kittel, Introduction to Solid State Physics (Wiley: New York, 1996).

Condensed matter physics