An efficient one-dimensional Poisson solver: We wish to solve the Poisson equation in a heterostructure. The heterostructure is divided into series of atomic layers $i$ of width $a_i$ and centered at the position $x_i$. The dielectric constant $\epsilon_i$ is uniform in each atomic layer $i$.
(a) We first assume that the charge distribution $\rho(x)$ and dielectric constant $\epsilon_i$ are uniform in each atomic layer $i$ :
$$
\rho(x)=\sum_i \rho_i\left[\mathrm{u}\left(x-x_i+\frac{a_i}{2}\right)-\mathrm{u}\left(x-x_i-\frac{a_i}{2}\right)\right],
$$
Fig. 2.9 can't copy. Charge distribution $\rho(x)$ for the step approximation.
where $\rho_i$ is the value of $\rho(x)$ in the interval $i$ (see Figure 2.9). We assume initially that the charge distribution $\rho_i$ is known.
Verify that the following relations are obtained if we integrate the Poisson equation exactly from the site 1 to the site $n$ :
$$
\begin{aligned}
& \epsilon_n F\left(X_n^{+}\right)=\epsilon_1 F\left(X_1^{+}\right)+\sum_{j=1}^{n-1} \rho_j a_j, \\
& V\left(X_n\right)=V\left(X_1\right)-\sum_{j=1}^{n-1} F\left(X_j^{+}\right) a_j-\frac{1}{2} \sum_{j=1}^{n-1} \frac{\rho_j a_j^2}{\epsilon_j}
\end{aligned}
$$
with $X_i=x_i-a_i / 2$.
(b) Assume now that the charge distribution in each atomic layer is given by an impulse function (see Figure 2.10):
$$
\rho(x)=\sum_i \rho_i a_i \delta\left(x-x_i\right)
$$
Compare the field and potential distribution. Do the expressions given in (a) still hold? Assume that $\rho_i$ is known and is the same as in (a).
(c) The charge distribution $\rho(x, V)$ is now assumed to depend on the voltage $V(x)$ as well as position $x$. The charge distribution is assumed to be $\rho_i=\rho\left(x_i, V\left(x_i\right)\right)$ for both the uniform charge distribution (a) and the impulse distribution (b). Verify that a close-form solution can easily be obtained with method (b) but not with method (a).
Note: This method of integrating a non-linear Poisson equation using the initial conditions $F\left(X_i^{+}\right)$and $V\left(X_i\right)$ gives results quite similar to the Runge-Kutta algorithm. This new algorithm, however, is much simpler and therefore faster than the Runge-Kutta algorithm. Note that the solution obtained is exact. However, the uniform charge distribution (a) is easier to justify than the impulse distribution (b).
Fig. 2.10 can't copy. Charge distribution $\rho(x)$ for the impulse approximation.