SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

2.10. TAILORING ELECTRONIC PROPERTIES 79

The occupation of the light hole subband can be ignored.
In many electronic devices used for information processing, a quantum well with a “triangu-
lar” shape is produced. The potential for electrons may be written in the form

V(x)=∞ x< 0

= eEx x > ̃ 0 (2.10.11)

The potential energy of the particle is of the form

V(x)=Fx+constant (2.10.12)

whereFis the force on the particle (say, an electron) and has a valueeE. We choose the
constant in the potential energy to be such that atx=0,V(x)=Exas shown in figure 2.34.
The solutions to this problem are the Airy functions

Φ(ξ)=

1

√

π

∫∞

0

cos

(

u^3

3

+uξ

)

du (2.10.13)

with a normalized solution
ψ(ξ)=AΦ(ξ) (2.10.14)

The normalization constant can be shown to have the value

A=

(2m)^1 /^3

π^1 /^2 E^16 ^2 /^3

(2.10.15)

The Airy functions have the following asymptotic behavior:

Φ(ξ) ∼

1

2

(ξ)−^1 /^4 exp

(

−

2 ξ^3 /^2

3

)

,ξ> 0

Φ(ξ) ∼|ξ|−^1 /^4 sin

(

2 |ξ|^3 /^2

3

+

π

4

)

,ξ< 0 (2.10.16)

Note that atx=0the second form is to be used, sinceξ< 0.
The solutions for the energy levels turn out to be:

En=

(

^2

2 m

) 1 / 3 (

3

2

πE

) 2 / 3 (

n−

1

4

) 2 / 3

,n=1, 2 ,... (2.10.17)

As shown in figure 2.35, in electronic devices such as a MOSFET or a MODFET the device
consists of an insulator-semiconductor junction. Electrons are injected at the interface on the
semiconductor side by a controlling electrode (the gate). The free charge causes a bending of
the semiconductor band to produce an approximately triangular quantum well, as shown. The
triangular quantum well is defined by an electric fieldEswhich is related to the areal charge
density by Gauss’s law

Es=

ens

s

(2.10.18)

As a result of the confinement, quantized energy levels are formed in the triangular well.
Approximate positions of these levels can be obtained from the results given above.