In solid-state physics, the nearly free electron model (or NFE model and quasi-free electron model) is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid.
The model is closely related to the more conceptual empty lattice approximation.
The model enables understanding and calculation of the electronic band structures, especially of metals.
This model is an immediate improvement of the free electron model, in which the metal was considered as a non-interacting electron gas and the ions were neglected completely.
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
is a lattice translation vector.)
Because it is a nearly free electron approximation we can assume that
denotes the volume of states of fixed radius
[clarification needed] A solution of this form can be plugged into the Schrödinger equation, resulting in the central equation:
are the Fourier coefficients of the wave function
and the screened potential energy
are the reciprocal lattice vectors, and the discrete values of
are determined by the boundary conditions of the lattice under consideration.
Before doing the perturbation analysis, let us first consider the base case to which the perturbation is applied.
, and therefore all the Fourier coefficients of the potential are also zero.
In this case the central equation reduces to the form
is a non-degenerate energy level, then the second case occurs for only one value of
, the Fourier expansion coefficient
In this case, the standard free electron gas result is retrieved:
is a degenerate energy level, there will be a set of lattice vectors
Non-degenerate and degenerate perturbation theory, respectively, can be applied in these two cases to solve for the Fourier coefficients
of the wavefunction (correct to first order in
An important result of this derivation is that there is no first-order shift in the energy
Particularly, at the Brillouin zone boundary (or, equivalently, at any point on a Bragg plane), one finds a twofold energy degeneracy that results in a shift in energy given by:[clarification needed]
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.
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 severe approximation, 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: