Empty lattice approximation

The Empty Lattice Approximation is a theoretical electronic band structure model in which the periodic potential of the crystal lattice is defined not more precisely than "periodic" and it is assumed that the potential is weak. The Empty Lattice Approximation is a description of the properties of electrons in an electron gas of non-interacting free electrons that move through a weak periodic potential of a crystal structure. The energy of the electrons in the "empty lattice" is the same as the energy of free electrons. The model mainly serves to illustrate a number of concepts which are fundamental to all electronic band structures.

Scattering and periodicity

 

Free electron bands in a one dimensional lattice

The periodic potential of the lattice must be weak because otherwise the electrons wouldn't be free. The strength of the scattering mainly depends on the topology of the system. 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, what their signs are or 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 periodicity and the division of k-space in Brillouin zones is the result of this scattering process. The periodic energy 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.

The 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.

The crystal structures of metals

 

Free electron bands in a FCC crystal structure

 

Free electron bands in a BCC crystal structure

 

Free electron bands in a HCP crystal structure

References

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