Many semiconductor devices are designed to confine electrons within a layer only a few nanometers thick. If a potential difference is applied across the layer, the electrons act very much as if they are trapped within a microscopic capacitor.

FIGURE 40.25a shows two capacitor plates separated by distance $L$. The left plate is positive, so the electric field points to the right with strength $E=\text{Δ}{V}_0\text{/}L$. Because of its negative charge, an electron launched from the left plate is slowed by a retarding force. The electron makes it across to the right plate if it starts with sufficient kinetic energy; otherwise, it reaches a turning point and then is pushed back toward the positive plate.

FIGURE 40.25 An electron in a capacitor.

This classical analysis is a valid model of a macroscopic capacitor. But if $L$ becomes sufficiently small, comparable to the de Broglie wavelength of an electron, then the wave-like properties of the electron cannot be ignored. We need a quantum-mechanical model.

Let’s establish a coordinate system with $x=0$ at the left plate and $x=L$ at the right plate. We define the electric potential to be zero at the positive plate. The potential decreases in the direction of the field, so the potential inside the capacitor (see Section 25.5) is

The electron, with charge $q=-e$, has potential energy

This potential energy increases linearly for $0<x<L$. If we assume that the capacitor plates act like the walls of a rigid box, then $U\left(x\right)\to \infty$ at $x=0$ and $x=L$.

FIGURE 40.25b shows the electron’s potential-energy function. It is the particle-in-a-rigid-box potential with a sloping “floor” due to the electric field. The figure also shows the total energy line $E$ of an electron in the capacitor. The energy is purely kinetic at $x=0$, where $K=E$, but it is converted to potential energy as the electron moves to the right. The right turning point occurs where the energy line $E$ crosses the potential-energy curve $U\left(x\right)$. If the electron is a classical particle, it must reverse direction at this point.

It is physically impossible for the electron to be outside the capacitor, so the wave function must be zero for $x<0$ and $x>L$. The continuity of $\psi$ requires the same boundary conditions as for a particle in a rigid box: $\psi =0$ at $x=0$ and at $x=L$. The wave functions inside the capacitor are too complicated to find by guessing, so we have solved the Schrödinger equation numerically and will present the results graphically.

FIGURE 40.26 shows the wave functions and probability densities for the first five quantum states of an electron confined in a 5.0-nm-thick layer that has a 0.80 V potential difference across it. Each allowed energy is represented as a horizontal line, with the numerical values shown on the right. They range from $E_1=0.23\text{eV}$ up to $E_5=0.81\text{ eV}$. An electron must have one of the allowed energies shown in the figure. An electron cannot have $E=0.30\text{eV}$ in this capacitor because no de Broglie wave with that energy can match the necessary boundary conditions.

FIGURE 40.26 Energy levels, wave functions, and probability densities for an electron in a 5.0-nm-wide capacitor with a 0.80 V potential difference.

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We can make some observations about the Schrödinger equation solutions:

1. The energies $E_n$ become more closely spaced as $n$ increases, at least to $n=5$. This contrasts with the particle in a box, for which $E_n$ became more widely spaced.

2. The spacing between the nodes of a wave function is not constant but increases toward the right. This is because an electron on the right side of the capacitor has less kinetic energy and thus a slower speed and a larger de Broglie wavelength.

3. The height of the probability density ${\left|\psi \right|}^2$ increases toward the right. That is, we are more likely to find the electron on the right side of the capacitor than on the left. But this also makes sense if, classically, the electron is moving more slowly when on the right side and thus spending more time there than on the left side.

4. The electron penetrates beyond the classical turning point into the classically forbidden region.