Accurate and fully analytical expressions for quantum energy levels in finite potential wells for nanoelectronic compact modeling

Abstract

In this paper, we present fully analytical closed-form expressions to provide the energy levels of finite quantum wells. The model accounts for unequal electronic effective masses in the well and barrier regions. The analytical expressions were validated by comparing the results provided by our model with those arising from the numerical solution of the eigenenergy transcendental equations. As a result, excellent accuracy is demonstrated within the parameter space investigated.

Appendix

It is worth mentioning that, since $\alpha \left[ i\right]$ is the result of a third degree equation, two additional values of $\alpha \left[ i\right]$ would also be available to develop alternative expressions, if desired:

$$\begin{aligned} \begin{aligned} \alpha \left[ i\right]&=-\frac{H_{1}\left[ i\right] }{24}+\frac{3G_{1} \left[ i\right] }{2H_{1}\left[ i\right] }\\&\quad +\frac{pp\left[ i\right] }{6}+\frac{I\sqrt{3}}{4}\left( \frac{H_{1}\left[ i \right] }{6}+\frac{6G_{1}\left[ i\right] }{H_{1}\left[ i\right] }\right) , \\ \alpha \left[ i\right]&=-\frac{H_{1}\left[ i\right] }{24}+\frac{3G_{1} \left[ i\right] }{2H_{1}\left[ i\right] }\\&\quad +\frac{pp\left[ i\right] }{6}-\frac{I\sqrt{3}}{4}\left( \frac{H_{1}\left[ i \right] }{6}+\frac{6G_{1}\left[ i\right] }{H_{1}\left[ i\right] }\right) . \end{aligned} \end{aligned}$$

In our case, the choice of $\alpha \left[ i\right]$ is dictated by the goal to have the simplest possible final expressions.