PHYSICAL REVIEW B VOLUME 35, NUMBER 17 15 JUNE 1987-I Folding effects in GaAs-AlAs superlattices L. Brey and C. Tejedor Departamento de Fisica del Estado Solido, Uniuersidad Autonoma, CantobIanco, 28049 Madrid, Spain (Received 7 July 1986) Folding effects are important in the electronic structure of GaAs-AlAs superlattices because of the indirect-gap character of the bulk AlAs. Conduction-band states at the center of the Brillouin zone of superlattices grown along the [001] direction originate from both I states of bulk GaAs and X states of bulk AlAs. The only way to properly describe such effects is to use tight-binding Hamil- tonians. We introduce a simple perturbative approach to compute the eigenvalues, eigenstates, and optical transition probabilities in superlattices. The method is applied to several GaAs-A1As super- lattices. Our results show that transition probabilities between X-like conduction-band states in A1As and I -like valence-band states in GaAs are very small because of spatial separation. These transitions are reinforced when a spike of A1As is introduced in the center of the GaAs region. Our results indicate that in this case folding effects must be optically observable. Finally, we apply our method to study sawtooth superlattices made of Gal „Al„As with x varying in the superlattice period. In this case, folding effects increase the asymmetry between conduction- and valence-band states which must be reflected in a large transient photovoltaic effect. I. INTRODUCTION The most widely used superlattices (SL) are those of GaAs-Ga& A As because of their relatively simple growth by means of molecular beam epitaxy (MBE). For many years it has been very common to restrict the whole range of Ga& Al As ternary compounds to cases with x &0.43 in order to work with direct-gap semiconductors so that the electronic structure of the SL can be analyzed by means of simple models such as the Kronig-Penney' or the more sophisticated envelope-function-matching (EFM) approximation. For x &0.43 the indirect band gap of the ternary compound precludes the use of simple matching procedures and simultaneously opens the possi- bility of new phenomena both in the transport and in the optical properties of the SL. Moreover, the case of x = 1 presents the advantage from the MBE-growth point of view of being the SL of the best quality with very good interfaces. ' There are many other SL of interest with components having indirect gaps as Si, Ge, A1Sb, or GaP. Therefore it is very clear that there is a need for as simple a theoretical method as possible, one which is also capable of giving a detailed description of the band structure. In particular the ability to describe the band structure far away from the center of the Brillouin zone is of great in- terest to the analysis of experiments involving E&-type transitions. " Tight-binding methods' are the evident candidate to fulfill the requisites mentioned above. The main practical problem of this approach appears when one is interested in SL with large periods because then very big matrices must be used. The most exact way to solve this problem is the one introduced by Schulman and Chang' where an eigenvalue equation is used to get the complex band struc- ture of the system. That method requires such lengthy calculations that it has rarely been used. The aim of this paper is to present a simple perturba- tive approach to the solution of tight-binding Hamiltoni- ans. This approach was previously introduced to analyze E& transitions in GaSb-A1Sb SL. ' Here we will pay spe- cial attention to the analysis of effects of indirect-gap semiconductors forming the SL. This will bring us to a careful analysis of the precision of the method. More- over, we will compute matrix elements of the momentum operator in order to get information on the transition probabilities involved in different spectroscopical tech- niques. In Sec. II we sketch the method to compute energy bands, wave functions, and optical-transition probabilities in SL. Section III is devoted to the analysis of several types of GaAs-A1As SL. A study on sawtooth Ga& Al As SL is presented in Sec. IV. Finally, Sec. V contains a brief summary of our results and conclusions. II. METHOD One of our main purposes is the analysis of indirect gap effects on the band structure of a SL. Therefore we will use as a typical example the case of a (GaAs) -(AlAs)„ SL where m and n are, respectively, the number of layers of each semiconductor in one period of the SL. Moreover, in order to work with a model that is as simple as possible we use a tight-binding Hamiltonian with an sp s * basis and interactions only up to first neighbors. ' However, our method can be applied to more general cases. Since the spin-orbit interaction must be included, our Hamil- tonian will be represented as having 20(m +n) functions. All the Hamiltonian matrix elements can be fitted to bulk band structures of the two semiconductors, so that the only thing to be specified in the model is the relative posi- tion of the band structures of the two semiconductors. This implies a relative shift between the diagonal terms of the two semiconductors which is usually represented by the valence-band offset AE„. In principle, this magnitude 35 9112 1987 The American Physical Society