Empty lattice approximation

The Empty Lattice Approximation is a theoretical electronic band structure model in which the potential is defined not more precisely than periodic and weak. The Empty Lattice Approximation describes a number of properties of energy dispersion relations of non-interacting free electrons that move through a crystal lattice. The energy of the electrons in the "empty lattice" is the same as the energy of free electrons. The model is useful because it clearly illustrates a number of, the sometimes very complex, features of energy dispersion relations in solids 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 in this free electron model must be weak because otherwise the electrons wouldn't be free. The strength of the scattering mainly depends on the geometric 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, the lattice spacing and the size of 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 by the periodic potential of the crystal structure. The periodicity of the dispersion relation 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.

In three-dimensional space the Brillouin zone boundaries are planes. The dispersion relations show conics of the free-electron energy dispersion parabolas for all possible reciprocal lattice vectors.

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

 

FCC Brillouin zone

"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 BCC crystal structure

 

Free electron bands in a HCP crystal structure

Metals crystallize in three kinds of crystal structures: the BCC and FCC cubic crystal structures and the hexagonal close-packed HCP crystal structure.

•  
  Body-centered cubic (I)
•  
  Face-centered cubic (F)
•  
  Hexagonal

 

Free electron bands in a FCC crystal structure

External links

• Brillouin Zone simple lattice diagrams by Thayer Watkins
• Brillouin Zone 3d lattice diagrams by Technion.
• DoITPoMS Teaching and Learning Package- "Brillouin Zones"

References

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