SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

30 CHAPTER 2. ELECTRONIC LEVELS IN SEMICONDUCTORS

The symbolsn, , mare the three quantum numbers describing the solution. The three quantum
numbers have the following allowed values:

principle number,n : Takes values 1, 2 , 3 ,...

angular momentum number, : Takes values 0, 1 , 2 ,...n− 1

magnetic number,m : Takes values−,− +1,...

The principle quantum number specifies the energy of the allowed electronic levels. The
energy eigenvalues are given by

En=−

μe^4

2(4π 0 )^2 ^2 n^2

(2.2.1)

The spectrum is shown schematically in figure 2.1. Due to the much larger mass of the nucleus
as compared with the mass of the electron, the reduced massμis essentially the same as the
electron massm 0. The ground state of the hydrogen atom is given by

ψ 100 =

1

√

πa^30

e−r/a^0 (2.2.2)

The parametera 0 appearing in the functions is called theBohrradiusand is given by

a 0 =

4 π 0 ^2

m 0 e^2

=0. 53 A ̊ (2.2.3)

It roughly represents the spread of the ground state.
As noted earlier the dopant problem is addressed by using the potential of the H-atom

2.2.2 Electrons in a quantum well

As noted in the previous chapter, using semiconductor heterostructures it is possible to fab-
ricate quantum well systems. These systems are used for high-performance devices, such as
transistors, lasers and modulators. The quantum well problem can also be used to understand
how defects create trap levels.
A quantum well potential profile is shown in figure 2.2. The well (i.e., region where potential
energy is lower) is described by a well sizeW =2aas shown and a barrier heightV 0 .In
general the potential could be confining in one dimension with uniform potential in the other
two directions (quantum well), or it could be confining in two dimensions (quantum wire) or
in all three dimensions (quantum dot). As discussed later in this chapter such quantum wells
are formed in semiconductor structures and we can use the results discussed in this section to
understand these problems.
We assume that the potential has a form

V(r)=V(x)+V(y)+V(z)