The empty lattice approximation is a theoretical
electronic band structure
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or ' ...
model in which the potential is ''periodic'' and ''weak'' (close to constant). One may also consider an empty irregular lattice, in which the potential is not even periodic.
[Physics Lecture Notes. P.Dirac, Feynman,R.,1968. Internet, Amazon,25.03.2014.] 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.
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Scattering and periodicity
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 geometry and 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
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy (kinetic energy in the case of a gravitational potential well) because it is cap ...
. For 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 possible to calculate the band structure analytically by substituting the values for the potential, the lattice spacing and the size of potential well.
[
] For two and three-dimensional problems it is more difficult to calculate a band structure based on a similar model with a few parameters accurately. Nevertheless, the properties of the band structure can easily be approximated in most regions by
perturbation methods
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
.
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
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns ...
. This is the origin of the periodicity of the dispersion relation and the division of
k-space in Brillouin zones. The periodic energy dispersion relation is expressed
as:
:
The
are the
reciprocal lattice
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
vectors to which the bands
belong.
The figure on the right shows the dispersion relation for three periods in reciprocal space of a one-dimensional lattice with lattice cells of length ''a''.
The energy bands and the density of states
In a one-dimensional lattice the number of reciprocal lattice vectors
that determine the bands in an energy interval is limited to two when the energy rises. In two and three dimensional lattices the number of reciprocal lattice vectors that determine the free electron bands
increases more rapidly when the length of the wave vector increases and the energy rises. This is because the number of reciprocal lattice vectors
that lie in an interval