NANOSCALE DEVICES: APPLICATIONS AND MODELING - Bio-Inspired and Nanoscale Integrated Computing - page 40

4

3

2

1

d

Figure 2.3. A potential well where electrons are confined in space due to energy
barriers on the sides. Electrons in such a ''box'' can only take on discrete energy
levels as shown. The horizontal axis is location and vertical axis is energy of the
electron.

As will be discussed in more detail later in this chapter, the fundamental
equation that governs the behavior of quantum mechanical particles, such as
electrons, is the Schro ¨ dinger equation. The energy levels that an electron is
allowed to have in this one-dimensional potential well can be easily obtained by an
analytical solution of the Schro ¨ dinger equation. The result is

2 p 2 N 2
2md 2

E N ¼ _

; N ¼ 1 ; 2 ; ...

is Planck's constant and d is the width of the well. In other words,
the electron in the well cannot have just any energy, but must take one of the
discrete values given by the above formula. In general, such ''quantization'' of
energy levels also happens in the 3D case. This idea is at the heart of some
of the quantum devices that we will discuss in this section. Now consider a region
in space where a potential well is connected to two metal electrodes through
barriers with finite heights and widths on the sides, such as in Figure 2.4. This is an
RTD. An electron can enter this region from outside, leave the region by
overcoming the barrier heights (by, for instance, acquiring thermal energy and
going to higher energy levels), or move through the barriers by a process called
quantum mechanical tunneling. What is interesting is that in the transport
characteristics of this device, the effect of these discrete energy levels becomes
completely visible.
Now imagine a voltage bias is applied to the structure that leads to a relative
shift in the chemical potentials of the two contact electrodes, such as in Figure 2.5.
An electron will be able to tunnel through the device from one side to the other
only if the biases on the two sides are such that there is an energy level in the well
in the range where electrons exist on the left and empty states exist on the right,
i.e., when there is a level lower than m 1 but higher than m 2. (Remember that in the
electrodes all energy states up to the chemical potentials are filled with electrons.)

Here _