C.2 The Finite Potential Well

   To show that the infinite well approximation is sufficient in our case, the energy levels of a finite potential well, Fig. C.2, are calculated and compared to (C.10).

  

Figure C.2: Finite potential well with width d and height E₀. On the right side a comparison of the first ten energy levels for an infinite potential well of width 10 nm and a finite well with the same width and a height of 0.34 eV is shown. The first levels differ only slightly.

$\parbox{20cm}{ \includegraphics{finit_well.eps} \resizebox{8cm}{!}{\includegraphics{el_finit_well.eps}}}$

The Ansatz for the three regions is

The boundary conditions yield the following equations:

Solving for A₂ gives
$$\begin{gather}A_2\left[\left(1+\frac{\text{i}k_2}{k_1}\right)^2e^{\text{i}k_2d}- \left(1-\frac{\text{i}k_2}{k_1}\right)^2e^{-\text{i}k_2d}\right]=0 \end{gather}$$
Since the trivial solution A₂=0 is not of interest, the second term has to be zero. This is the condition fulfilled only for special energies. The transcendent condition can be written as
$$\begin{gather}\frac{2\sqrt{E(E_0-E)}}{2E-E_0}=\tan\left(\frac{d\sqrt{2m^*E}}{\hbar}\right), \end{gather}$$
which is only numerically solvable. A quantitative comparison is given for $d\sqrt{2m^*E_0}/\hbar=30$, which corresponds to a well with a width of 10 nm and a height of 0.34 eV, in the right side of Fig. C.2. The first energy levels differ only slightly.