I have a set of coupled ODEs:
J = mu*e*n(x)*E(x) + mu*K*T0* dn(x)/dx
and
dE(x)/dx = (e/eps)*[N_D(x) - n(x)]
These are the drift diffusion equation and Gauss's law for a unipolar N+ N N+ device. The doping profile N_D(x), the mobility mu, T0 are known. The DC current, J, is also known, as are the boundary values: n(0) = N+, N(L) = N+. I want to self consistently solve the above two equations using finite differences, but I am unsure how to go about doing so.
Thanks in advance!
Have you experience with the explicit Euler method? Ii is only first order accurate but I suggest to use it first to get easily an idea of the solution.
Then, you can use the Heun method or high order RK methods.
Fillipo, thanks for your reply. Can you offer some more specific advice on how to go about solving once the equations have been discretized? Is there perhaps a simple 1d model example of a comparable system?
consider for example the system
df1/dx=G1(f1,f2,x)
df2/dx=G2(f1,f2,x)
that you can integrate from x0 to x0+h:
f1(x0+h)=f1(x0) + Int[x0,x0+h] G1 dx
f2(x0+h)=f2(x0) + Int[x0,x0+h] G2 dx
Now you can introduce the approximation by substituting the exact integral with its numerical discretization.
The first order explicit Euler scheme works as
f1(x0+h)=f1(x0) + h* G1[f1(x0),f2(x0),x0]
f2(x0+h)=f2(x0) + h* G2[f1(x0),f2(x0),x0]
where the RHS of the equations is known.
Repeating these equations at point x0+h you can proceed by integrating over a finite set of node covering the global interval.
This type of integration has to fulfill some numerical stability constraint.
You can read some textbook of numerical analysis with several examples about these problems.
This is the classic reference on how to solve this system of equations:
Analysis and Simulation of Semiconductor Devices by S. Selberherr
To solve the current density equations, you would need to use the Scharfetter Gummel discretization, which is described in the above reference. Simple gradients of the carrier density would be too unstable.
- Badges
- Science topic