Electrons in finite superlattices: the birth of crystal momentum
  Properties of electrons in superlattices (SLs) of a finite length are described using standing waves resulting from the fixed boundary conditions (FBCs) at both ends. These electron properties are compared with those predicted by the standard treatments using running waves (Bloch states) resulting from the cyclic boundary conditions (CBCs). It is shown that, while the total number of eigenenergies in a miniband is the same according to both treatments, the number of different energies is twice higher according to the FBCs. It is also shown that the wave vector values corresponding to the eigenenergies are spaced twice as densely for the FBCs as for the CBCs. The reason is that a running wave is characterized by a single value of wave vector k, while a standing wave in a finite SL is characterized by a pair of wavevectors +/- q. Using numerical solutions of the Schroedinger equation for an electron in an increasing number N of periodic quantum wells (beginning with N = 2) we investigate the "birth" of an energy miniband and of a Brillouin zone according to the two approaches. Using the Fourier transforms of the computed wave functions for a few quantum wells we follow the "birth" of electron's momentum. It turns out that the latter can be discerned already for a system of two wells. We show that the number of higher values of the wave vector q involved in an eigenenergy state is twice higher for a standing wave with FBCs than for a corresponding Bloch state. Experiments using photons and phonons are proposed to observe the described properties of electrons in finite superlattices. 
P. Pfeffer and W. Zawadzki