Tamtam Proceedings - lamsin

The two-band Schrödinger model 377The outline of this paper is as follows: section 2 is devoted to the introduction of thedifferent equations to solve and the description of the iterative procedure for obtaining theself-consistent two-band Schrödinger and the Poisson solution. In section 3, the numericalmethods are presented. Like in (Ref.3) and (Ref.4) , we deal with an explicit Runge Kuttaof order 4 with adaptatif mesh applied to the two-band Schrödinger equation and theGummel method to solve the Poisson equation. In section 4, the numerical results for anInAs/AlSb/GaSb/AlSb/InAs type structure in the stationary case is presented.2. The different equations to solveIn this section, we use two types of equations in order to solve the coupled twobandSchrödinger-Poisson model. The first one is a one-dimensional non-scaled two-bandSchrödinger equation.⎛d 2⎜ − 2⎝ 2m dx 2 − eV c(x) iP d( ) ( )dx ⎟ ψ1 ψ1iP d − 2 d 2⎠dx 2m dx 2 − E ψ 2 = Eψ 2 . (1)g − eV c (x)Then, we derive in the same spirit of (Ref.5, 6 and 7) the associated Fourier boundaryconditions. Indeed, we getdΨdx (0) − iK −(E)Ψ(0) = i(k c (E)I 2 − K − (E))⃗e + c (E), (2)dΨdx (1) − iK +(E + eV 1 )Ψ(1) = 0, (3)where ψ 1,2 the components of the wave function Ψ according to the two bands, E is theenergy, V is the potential energy, the reduced Planck constant and m is the electronmass. The terms ⃗e ± c (E), k c,v (E) are given respectively by⃗e c ± (E) =⃗e v ± (k) =k c,v (E) = + √√( √ (δ 2 + P 2 k 2 + δ)2 √ δ 2 + P 2 k 2⎞√( √ (δ 2 + P 2 k 2 ∓P k+ δ)2 √ δ 2 + P 2 k 2 1(δ + E + P 22 ) ± √(δ + P 2)1∓P kδ+ √ , (4)δ 2 +P 2 k 2δ+ √ δ 2 +P 2 k 2), (5)2 )2 + EP 2 (6)and K ∓ (.) represent a 2 × 2 matrix wave with ′′ − ′′ for waves at x < 0 and ′′ + ′′ for wavesat x > 1.It is defined by the following expressionK ∓ (E) = B ∓ (E)D ∓ (E)B −1∓ (E), (7)TAMTAM –Tunis– 2005