Tamtam Proceedings - lamsin

378 Ben Abdallah et al.where B ∓ (E) is the 2 × 2 matrix transforming the canonical basis on to the basis vectors(e ∓ c (E), e ∓ v (E)) and B −1∓ (E) its inverse matrix. Indeed, D ∓ (E) is a diagonal matrix. Itis equal to( )∓kc (E) 0D ∓ (E) =. (8)0 ∓k v (E)The one dimensional Poisson equation is− d2 Vdx 2 (x) = e ɛ r(n D − n), with V (0) = 0 and V (1) = V 1 (9)where ɛ r is the relative dielectric constant and n D is the ionized donor concentration andn is the electron density distribution. The wave function Ψ and electron density n arerelated byn(x) =∫ +∞where g 0 (k) the one dimensional full Fermi-Dirac statistic0g 0 (k)|Ψ(x)| 2 dk (10)g 0 (k) = m∗ k b T−Ec(k) + eµπ 2 log(1 + exp( )). (11)k b TEc(k) is the conduction energy band, T is the temperature, m ∗ is the effective mass andµ is the Fermi potential related to the Fermi energy E F by E F = −eµ.In this case, we use an iteration procedure to obtain a self-consistent solution for equations[1]-[3] and [9]. Starting with a trial potential V (x), the wave function correspondingto each wave vector k can be used to calculate the electron density distribution n(x) using[10]. The computed n(x) and a given donor concentration n D (x) can be used to calculatethe Poisson potential via equation [9]. The new potential V is then obtained. The subsequentiteration will yield the final self-consistent solution for V and n which satisfy acertain error criteria.3. Numerical methodsThe simple way to solve the linear system [1]-[3] is to make a variable change whichleads to an ordinary differential equation. Then, the previous problem can be solvedusing for example fourth order Runge-Kutta method. For the discretization of the Poissonequation, we use the linear Gummel method where equation [9] becomes− d2 V newdx 2 (x) + e2ɛ r k B T |n(x)|V new = e (n D − n +eɛ r k B T |n(x)|V old)), (12)V new (0) = 0 and V new (1) = V 1 . (13)TAMTAM –Tunis– 2005