Roninkoi / Difidi

Diode finite differences

Difidi - diode finite differences

Simulates an MOS (metal-oxide semiconductor) diode with a voltage applied through it. Solves the 1D Poisson equation

$$ \phi''(z) = -\frac{\rho(z)}{\epsilon(z)} = f(z) $$

using the finite difference method and computes Fermi-Dirac integrals

$$ n(z) = \frac{\sqrt{2} m_e^{3/2}}{\pi^2 \hbar^3} \int_{E_c(z)}^\infty \frac{\sqrt{E - E_c(z)}}{1 + \exp(E / k_B T)} \ \text{d} E, $$

$$ p(z) = \frac{\sqrt{2} m_e^{3/2}}{\pi^2 \hbar^3} \int_{-\infty}^{E_v(z)} \frac{\sqrt{E_v(z) - E}}{1 + \exp(E / k_B T)} \ \text{d} E, $$

to determine the charge density distribution $\rho(z) = e (-N_A - n(z) + p(z))$. A solution satisfying all equations is found by an iterative algorithm using damping.

Compilation and running

Compilation:

make

Compilation requires the LAPACK and GSL libraries.

Usage:

./difidi diode.in diode.out > out.log

The file diode.in contains the simulation parameters. The file diode.out contains the solution in the format:

<z position> <rho(z)> <phi(z)> <n(z)> <p(z)> <Ev(z)> <Ec(z)>

Plotting

Plotting:

./plot.py [diode.out]

This produces plots for potential, charge distribution, charge carrier distributions and the band diagram.

Solver residual can be monitored in real time using:

gnuplot residual.gp