Editing Superlattice (section)

== Bloch states ==

For an ideal superlattice a complete set of [[eigenstates]] states can be constructed by products of plane waves <math> e^{ i \mathbf{k} \cdot \mathbf{r} }/ 2\pi </math> and a ''z''-dependent function <math>f_k (z)</math> which satisfies the eigenvalue equation

:<math> \left( E_c(z) - \frac{\partial }{\partial z} \frac{\hbar^2}{2 m_c (z)}  \frac{\partial }{\partial z} + \frac {\hbar^2 \mathbf{k} ^2}{2m_c (z)} \right) f_k (z) = E f_k (z) </math>.

As <math> E_c (z) </math> and <math> m_c(z) </math> are periodic functions with the superlattice period ''d'', the eigenstates are [[Bloch state]] <math> f_k (z)= \phi _{q, \mathbf{k}}(z)</math> with energy <math>E^\nu (q, \mathbf{k})</math>. Within first-order [[perturbation theory]] in '''k'''<sup>2</sup>, one obtains the energy

:<math> E^ \nu (q, \mathbf{k}) \approx E^ \nu(q, \mathbf{0}) +  \langle \phi _{q, \mathbf{k}} \mid \frac{\hbar^2 \mathbf{k}^2}{2m_c (z)} \mid \phi _{q, \mathbf{k}} \rangle </math>.

Now, <math> \phi _{q, \mathbf{0}} (z) </math> will exhibit a larger probability in the well, so that it seems reasonable to replace the second term by

:<math> E_k = \frac{\hbar^2 \mathbf{k}^2}{2m_w} </math>

where <math>m_w</math> is the effective mass of the quantum well.