# [Getdp] Semiconductors and Poisson equation

Thomas Dufaux Thomas.Dufaux at rwth-aachen.de
Wed Feb 22 12:48:05 CET 2012

```Hello,

I am trying to calculate the charge carrier distribution in a silicon PN-junction. Therefore I have to solve the following coupled equations:

Div eps Grad v = -rho = n – p + Na - Nd
Div mu_e*n Grad psi_n = 0
Div mu_h*p Grad psi_p = 0

In getdp I used the Newton Raphson method, as n[] & p[] are not linear:

Galerkin { [ epsr * {Grad v} , {Grad v} ]; In Vol; Jacobian Vol; Integration Int; }

Galerkin { JacNL[  dp[{psip}-Ev-{v}]*Dof{v} , {v} ]; In Vol; Jacobian Vol; Integration Int; }
Galerkin { JacNL[ -dn[{psin}-Ec-{v}]*Dof{v} , {v} ]; In Vol; Jacobian Vol; Integration Int; }

Galerkin { [  p[{psip}-Ev-{v}] , {v} ]; In Vol; Jacobian Vol; Integration Int; }
Galerkin { [ -n[{psin}-Ec-{v}] , {v} ]; In Vol; Jacobian Vol; Integration Int; }

Galerkin { [  N_D[] - N_A[] , {v} ]; In Vol; Jacobian Vol; Integration Int; }

Galerkin { [ mu_e*n[{psin}-Ec-{v}] * Dof{Grad psin} , {Grad psin} ]; In Vol; Jacobian Vol; Integration Int; }
Galerkin { [ mu_h*p[{psip}-Ev-{v}] * Dof{Grad psip} , {Grad psip} ]; In Vol; Jacobian Vol; Integration Int; }

When I try to solve these equations the first iterations look good, but then the potential v becomes very large and the solution does not converge any more or gives no reasonable results.

To me it looks like there is a problem in the Poisson equation. A possible explanation could be wrong parameters (mu_e, epsr …) or signs, but I have checked them carefully and when I put in values manually the results are correct.

As I am not so familiar with getdp, could it be that I made an error in the formulation of the Galerkin terms? Did I place the Dof correct? Could it be that the exponential behaviour of the n[{v}] and p[{v}] terms make it difficult to solve the problem, as small variations in the potential v can lead to significant changes in the carrier density? Are there any further possibilities to enhance the convergence? I previously already tried the Picard method, but the results were worse.