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I am trying to solve the $\textbf{1-D}$ Poisson equation for a semiconductor structure at equilibrium (There is no external bias applied).

$\textbf{Background}$

\begin{equation} \frac{d^2V}{dx^2} = -\frac{\rho(V)}{\epsilon}\\ \rho(V) = q(N_D-N_A+p(V)-n(V)) \end{equation}

When there are two different semiconductors placed in contact, there will be a redistribution of charge, causing a dependence of charge on potential (the bands will bend).

$\textbf{Solving with the Newton-Rhapson method}$

I am trying to solve the above non-linear equation using the newton-rhapson method. Taking the central difference, the second derivative becomes: \begin{equation} \frac{d^2V}{dx^2}|_i = \frac{(V_{i+1}-2V_i+V_{i-1})}{dx^2} \end{equation}

The issue with solving this now is that this derivative cannot be computed for the two end points of the structure. What are the boundary conditions to be applied on the edges?

Another issue I'm facing is when computing the charge densities by including a non-parabolic coefficient $\alpha$ (for non-parabolic bands), I have to compute these integrals in every iteration - \begin{equation} n = \int^{\infty}_{Ec}F_{1/2}(E)g_n(E)dE \hspace{10pt}p= \int^{E_v}_{-\infty}F_{1/2}(E)g_p(E)dE \end{equation} where $F_{1/2}$ is the 1/2nd order Fermi-distribution and $g(E)$ is the density of states. I have to compute the integrals with $E_c\text{ and }E_v$, but those are unknown. How do I initiate and update their values? How are each of these energies ($E_c,E_v,E_f$) related to each other and the potential of the device?

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    $\begingroup$ "What are the boundary conditions to be applied on the edges?" -- that is a modeling question, not a mathematical one. What do you know about the voltage at the end points of your domain? $\endgroup$ Mar 19, 2023 at 17:41
  • $\begingroup$ Hi. The voltages are unknown as we are not applying any bias across the device. $\endgroup$
    – Photon
    Mar 19, 2023 at 20:10
  • $\begingroup$ So then you don't have Dirichlet boundary conditions. Are there currents across the boundary? $\endgroup$ Mar 19, 2023 at 23:15
  • $\begingroup$ In addition to the point about boundary conditions, there is another modeling concern to address to make the problem tractable: a closure relation relating $E$ and $V$. This again is not really a mathematical problem (although there are mathemtical approaches to getting around unknown closure relations) and is something that will need to be derived or assumed from the physical properties of the system. $\endgroup$
    – whpowell96
    Mar 20, 2023 at 1:21
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    $\begingroup$ This answer and links therein might offer some help scicomp.stackexchange.com/a/30840/17869 $\endgroup$
    – uhoh
    Mar 20, 2023 at 13:29

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