Electrons in finite superlattices: the birth of crystal momentum

Institute of Physics, Polish Academy of SciencesAl.Lotnikow 32/46, 02–668 Warsaw, Poland(Dated: November 7, 2008)Properties of electrons in superlattices (SLs) of a finite length are described using standing wavesresulting from the fixed boundary conditions (FBCs) at both ends. These electron properties arecompared with those predicted by the standard treatments using running waves (Bloch states) re-sulting from the cyclic boundary conditions (CBCs). It is shown that, while the total numberof eigenenergies in a miniband is the same according to both treatments, the number of differentenergies is twice higher according to the FBCs. It is also shown that the wave vector values corre-sponding to the eigenenergies are spaced twice as densely for the FBCs as for the CBCs. The reasonis that a running wave is characterized by a single value of wave vector k, while a standing wave in afinite SL is characterized by a pair of wavevectors ±q. Using numerical solutions of the Schroedingerequation 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 twoapproaches. Using the Fourier transforms of the computed wave functions for a few quantum wellswe follow the ”birth” of electron’s momentum. It turns out that the latter can be discerned alreadyfor a system of two wells. We show that the number of higher values of the wave vector q involved inan eigenenergy state is twice higher for a standing wave with FBCs than for a corresponding Blochstate. Experiments using photons and phonons are proposed to observe the described properties ofelectrons in finite superlattices.