Empty lattice approximation

The Empty Lattice Approximation is a theoretical electronic band structure model in which the periodic potential of the crystal lattice is not defined more precisely than "periodic" and it is assumed that the potential is weak. This model mainly serves to illustrate a number of concepts which are fundamental to all electronic band structure phenomena.

Introduction

 

Free electron bands in a FCC crystal structure

 

Free electron bands in a BCC crystal structure

 

Free electron bands in a HCP crystal structure

The potential must be weak because otherwise the electron wouldn't be free, but it is just strong enough to scatter the electrons. How strong must a potential be to be able to scatter an electron? The answer is that it depends on the topology of the system how large topologically defined parameters, like scattering cross sections, depend on the magnitude of the potential and the size of the potential well. One thing is clear for currently known 1, 2 and 3-dimensional spaces: potential wells do always scatter waves no matter how small their potentials are or what their sign is and how limited their sizes are. For a particle in a one-dimensional lattice, like the Kronig-Penney model, it is easy to substitute the values for the potential and the size of the potential well.^[1]

In theory the lattice is infinitely large, so a weak periodic scattering potential will eventually be strong enough to reflect the wave. The scattering process results in the well known Bragg reflections of electrons in the periodic potential of the crystal structure. The division of k-space in Brillouin zones still is the result of this scattering process. The dispersion relation is

${\displaystyle E_{n}({\mathbf {k}})={\frac {\hbar ^{2}({\mathbf {k}}+{\mathbf {G_{n}}})^{2}}{2m}}}$ 

and consists of a increasing number of free electron bands ${\displaystyle E_{n}({\mathbf {k}})}$  when the energy rises. ${\displaystyle {\mathbf {G}}_{n}}$  is the reciprocal lattice vector to which the band ${\displaystyle E_{n}({\mathbf {k}})}$  belongs. Electrons with larger wave vectors outside the first Brillouin zone are mapped back into the first Brillouin zone by a so called Umklapp process.

Nearly free electron model

In the NFE model the Fourier transform, ${\displaystyle U_{\mathbf {G}}}$ , of the lattice potential, ${\displaystyle V({\mathbf {r}})}$ , in the NFE Hamiltonian, can be reduced to an infinitesimal value. When the values of the off-diagonal elements ${\displaystyle U_{\mathbf {G}}}$  between the reciprocal lattice vectors in the Hamiltonian almost go to zero. As a result the magnitude of the band gap ${\displaystyle 2|U_{\mathbf {G}}|}$  collapses and the Empty Lattice Approximation is optained.

Second, third and higher Brillouin zones

"Free electrons" that move through the lattice of a solid with wave vectors ${\displaystyle {\mathbf {k}}}$  far outside the first Brillouin zone are still reflected back into the first Brillouin zone. See the external links section for sites with examples and figures.

References

1. C. Kittel (1953–1976). Introduction to Solid State Physics. Wiley & Sons. ISBN 0-471-49024-5.