Editing Superlattice (section)

== Miniband structure ==

The schematic structure of a periodic superlattice is shown below, where A and B are two semiconductor materials of respective layer thickness ''a'' and ''b'' (period: <math>d=a+b</math>). When ''a'' and ''b'' are not too small compared with the interatomic spacing, an adequate approximation is obtained by replacing these fast varying potentials by an effective potential derived from the band structure of the original bulk semiconductors. It is straightforward to solve 1D Schrödinger equations in each of the individual layers, whose solutions <math> \psi</math> are linear combinations of real or imaginary exponentials.

For a large barrier thickness, tunneling is a weak perturbation with regard to the uncoupled dispersionless states, which are fully confined as well. In this case the dispersion relation <math> E_z(k_z) </math>, periodic over <math>2 \pi /d </math> with over <math> d=a+b </math> by virtue of the Bloch theorem, is fully sinusoidal:

:<math>\ E_z(k_z)=\frac{\Delta}{2}(1-\cos(k_z d))</math>

and the effective mass changes sign for <math> 2\pi /d</math>:

:<math>\ {m^* = \frac{\hbar^2}{\partial^2 E / \partial k^2}}|_{k=0}</math>

In the case of minibands, this sinusoidal character is no longer preserved. Only high up in the miniband (for wavevectors well beyond <math>2 \pi /d</math>) is the top actually 'sensed' and does the effective mass change sign. The shape of the miniband dispersion influences miniband transport profoundly and accurate dispersion relation calculations are required given wide minibands. The condition for observing single miniband transport is the absence of interminiband transfer by any process. The thermal quantum ''k<sub>B</sub>T'' should be much smaller than the energy difference <math> E_2-E_1</math> between the first and second miniband, even in the presence of the applied electric field.