the current density associated with an electron in a superlattice, with different or equal effective masses between wells and barriers. Also, we evaluate ...
Electronic properties of superlattices D. AITELHABTI, P. VASILOPOULOS, AND J. F. CURRIE Genie physique, ~ c o l ePolyrechniqlte, C.P. Box 6079, S~tccltrsaleA, Monrre'al, (Quebec), Canada H3C 3A7 Received July 10, 1989
Can. J. Phys. Downloaded from www.nrcresearchpress.com by 199.201.121.12 on 06/05/13 For personal use only.
Using the transfer-matrix method, we evaluate the exact normalized wave function analytically, the band structure, and the current density associated with an electron in a superlattice, with different or equal effective masses between wells and barriers. Also, we evaluate numerically the dispersion relation, the bandwidth, and the current density (in the tight-binding limit) for both equal and different effective masses between wells and barriers. En utilisant la mCthode de matrice de transfert, nous Cvaluons la fonction d'onde normalisCe exacte analytiquement, la structure de bandes et la densit6 de courant associCes avec un Clectron dans un superrCseau, avec des masses effectives d~ffirenresou e'gales entre puits et barrikres. Nous Cvaluons aussi numkriquement la relation de dispersion, la largeur de bande et la densitt de courant (dans la limite des liaisons fortes) dans les cas de masses effectives Cgales ou diffkrentes entre puits et barrikres. [Traduit par la revue] Can. J. Phys. 68, 268 (1990)
1. Introduction In the last years, especially with the advent of molecularbeam epitaxy, semiconductor superlattices have been studied extensively because they are of both technological and fundamental interest. Surprisingly, however, we have not yet seen the complete normalized eigenfunction of an electron in a superlattice, be that with equal or different effective masses between wells and barriers. The wave function of ref. 1 is given only for the wells, when the effective masses of the wells, rnk, and of the barriers, mt, are equal, and is not normalized.' The wave function of ref.- 2 applies for rnk # rn; near the band edges only, and it is not normalized. Various other studies of superlattices use the wave functions of the isolated quantum well with infinite (3) or finite energy barriers (4), simple variational or exact numerical wave functions (5), and envelope wave functions (6). It is evident that it is of interest to have the exact normalized eigenfunction analytically both when rnk = m: and when mk # rn:, regardless of its complexity. In this paper, we generalize the transfer-matrix technique of ref. 1 to take into account the (effective) mass difference between the wells and bamers. We obtain the band structure, the normalized wave function, and the current density. In Sect. 2 we present briefly the transfer-matrix formalism. Subsequently, we use it to study various electronic properties in Sect. 3. We finish with some concluding remarks in the last section. 2. Transfer-matrix formalism Figure 1 shows the periodic square potential of a superlattice with well thickness 2b, banier thickness 2a, and barrier height Vo; in the wells, the potential is zero. Assuming for the moment that only the barrier centered at the origin exists, we can write the solution of Schrijdinger's equation as W(Z)= ~ [l]
'
,
~ i + k ~: ~ - ; k '
~ ( z= ) Ce-P'
+ DeP',
z
