2 and optical absorption. The numerical results and dis- cussion for the behavior of desired quantities in terms of the alloy concentration and the scattering-strength pa- rameter are given in Sec. III. The concluding remarks are given in section IV. We consider a multilayer structure where the layers are stacked along the z -direction and labeled by integer number n . The number of layers in this direction is N$_{z}$ , hence 1 ≤ n ≤ N$_{z}$ . The lattice structure of the sys- tem is assumed to be a simple cubic with lattice spacing a and the (001) orientation of the layers is taken to be normal to these layers. The multilayer is a semiconduct- ing alloy of the form A$_{1}$$_{−}$$_{x}$B$_{x}$ , where A atom (such as Si) and B atom (such as Ge) occupy randomly the lat- tice sites with concentration x . We use the single-band tight-binding approximation with nearest-neighbor hop- ping and the on-site delta function-like potential. The Hamiltonian for a single electron (or a Frenkel exciton) in this system is given by CPA with an effective medium Green’s function ¯ G = ( ω − H $_{eff}$) − 1 defined by the following Dyson equation [16] FIG. 1: (Color online) Band structure of the clean multilayer with N = 5 along the high symmetry axes in the 1BZ (see inset). MODEL AND FORMALISM ¯ G$_{n,n}$ ′ ( r , r $^{′}$; ω ) = G 0 n,n ′ ( r , r $^{′}$; ω ) + N z ∑ n ′′ $_{=1}$∑ r ′′ G 0 n,n ′′ ( r , r $^{′′}$; ω ) × Σ$_{n}$ ′′ ( ω ) ¯ G$_{n}$ $^{′′}$,n′ ( r $^{′′}$, r $^{′}$; ω ) , (3) H = ∑ r ,n ∑ r $^{′}$,n′ [ u$_{r}$$_{,n}$δ$_{n,n}$ ′ δ$_{r}$$_{,}$$_{r}$ ′ − t$_{r}$$_{n,}$$_{r}$ $^{′}$n′ ] | r , n 〉〈 r $^{′}$, n′ | , (1) G 0 n,n ′ ( r , r $^{′}$; ω ) = 1 N$_{‖}$ N z ∑ ℓ $_{=1}$∑ k$_{‖}$ G 0 n,n ′ ( ℓ, k$_{‖}$ ; ω ) e i k$_{‖}$ · ( r − r $^{′}$) , (4) where G 0 is the clean system Green’s function and its matrix element is given by where r denotes the positional vector in the x − y plane and the on-site potential u$_{r}$$_{,n}$ is assumed to be E$_{A}$ and E$_{B}$ with probabilities 1 − x and x when the lattice site ( r , n ) is occupied by the A and B atoms, respectively. The arrangement of these atoms is completely random. The hopping integral t$_{r}$$_{n,}$$_{r}$ $^{′}$n′ is equal to t for the nearest neighbor sites and zero otherwise. We assume that the hopping integral only depends on the relative position of the lattice sites, that is, only the diagonal disorder is considered. The introduced randomness is treated by the CPA [7, 8]. In this approximation, one seeks to replace a disordered alloy with an effective periodic medium. This CPA medium is characterized by a local (i.e., momentum- independent), energy-dependent self-energy, Σ( ω ), called the coherent potential. In some disordered low dimen- sional systems such as the quantum wires [15, 16] and the multilayers [17, 18], the self-energy depends on the position of atomic sites, due to the absence of transla- tional symmetry in some directions. Therefore, in this study the self-energy depends on the layer number, i.e., Σ( ω ) ≡ Σ$_{n}$( ω ) because of the quantum confinement along the z -direction. Accordingly, the multilayer Hamiltonian H is replaced by an effective medium Hamiltonian de- fined by here, G 0 n,n ′ ( ℓ, k$_{‖}$ ; ω ) is the mixed Bloch-Wannier repre- sentation of G 0 and is expressed as and is the clean system band structure. In Eq. (4), k$_{‖}$ ≡ ( k$_{x}$, k$_{y}$ ) is a wave vector parallel to the layer and the summation is over all the wave vectors in the first Bril- louin zone (1BZ) of the two-dimensional lattice [7], ℓ is the mode of the subband, N$_{‖}$ is the number of lattice sites in each layer, and η is a positive infinitesimal. G 0 n,n ′ ( ℓ, k$_{‖}$ ; ω ) = 2 ( N z +1) sin( ℓπ N z $_{+1}$n ) sin( ℓπ N z $_{+1}$n $^{′}$) ω + iη − ε$_{ℓ}$ ( k$_{‖}$ ) , (5) ε$_{ℓ}$ ( k$_{‖}$ ) = − 2 t [cos( k$_{x}$a ) + cos( k$_{y}$a ) + cos( ℓπ N$_{z}$ + 1 )] , (6) H eff = ∑ r ,n ∑ r $^{′}$,n′ [Σ$_{n}$( ω ) δ$_{n,n}$ ′ δ$_{r}$$_{,}$$_{r}$ ′ − t$_{r}$$_{n,}$$_{r}$ $^{′}$n′ ] | r , n 〉〈 r $^{′}$, n′ | . (2) Now, to determine the coherent potential in each layer (say n ), we consider an arbitrary chosen site in the effec- tive medium (layer) describe by Σ$_{n}$. Then, we apply the condition that the average scattering of a carrier by the chosen site in the medium is zero. This condition for any The physical properties of the real system can be ob- tained from the configurationally averaged Green’s func- tion 〈 G 〉$_{av}$ = 〈 ( ω − H ) − $^{1}$〉$_{av}$ which is replaced in the