SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

32 CHAPTER 2. ELECTRONIC LEVELS IN SEMICONDUCTORS

2 a

V V 0

–W

2

W

2

x = 0

0

x

Figure 2.2: Schematic of a quantum well of width 2 aand infinite barrier height or barrier height
V 0.

so that the wavefunction is separable and of the form

ψ(r)=ψ(x)ψ(y)ψ(z)

We will briefly discuss the problem of the square potential well, and in section 2.10 we will
use the quantum well physics to discuss semiconductor quantum wells of importance in devices.
The simplest form of the quantum well is one where the potential is zero in the well and
infinite outside. The equation to solve then is (the wave function is non-zero only in the well
region)

−

^2

2 m

d^2 ψ

dx^2

=Eψ (2.2.4)

which has the general solutions

ψ(x)=Bcos

nπx

2 a

, nodd

= Asin

nπx

2 a

, neven (2.2.5)

The energy is

E=

π^2 ^2 n^2

8 ma^2

(2.2.6)

Note that the well size is 2 a.
The normalized particle wavefunctions are

ψ(x)=

√

2

W

cos

nπx

W

,nodd

=

√

2

W

sin

nπx

W

,neven (2.2.7)