SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

36 CHAPTER 2. ELECTRONIC LEVELS IN SEMICONDUCTORS

EXAMPLE 2.1Calculate the density of states of electrons in a 3D system and a 2D system at an energy
of 1.0 eV. Assume that the background potential is zero.
The density of states in a 3D system (including the spin of the electron) is given by (Eis the energy in
Joules)

N(E)=

√

2(m 0 )^3 /^2 E^1 /^2

π^2 ^3

=

√

2(0. 91 × 10 −^30 kg)(E^1 /^2 )

π^2 (1. 05 × 10 −^34 J·s)^3

=1. 07 × 1056 E^1 /^2 J−^1 m−^3

ExpressingEin eV and the density of states in the commonly used units of eV−^1 cm−^3 ,weget

N(E)=1. 07 × 1056 ×(1. 6 × 10 −^19 )^3 /^2 (1. 0 × 10 −^6 )E^1 /^2

=6. 8 × 1021 E^1 /^2 eV−^1 cm−^3

AtE=1. 0 eV we get
N(E)=6. 8 × 1021 eV−^1 cm−^3
For a 2D system the density of states is independent of energy and is

N(E)=m^0

π^2

=4. 21 × 1014 eV−^1 cm−^2

2.3.1 Particle in a periodic potential: Bloch theorem

Band theory, which describes the properties of electrons in a periodic potential arising from
the periodic arrangement of atoms in a crystal, is the basis for semiconductor technology.
The Schrodinger equation in the crystal ̈
[
−^2
2 m 0

∇^2 +U(r)

]

ψ(r)=Eψ(r) (2.3.9)

whereU(r)is the background potential seen by the electrons. Due to the crystalline nature of
the material, the potentialU(r)has the same periodicity

U(r)=U(r+R)

We have noted earlier that if the background potential isV 0 , the electronic function in a volume
Vis

ψ(r)=

eik·r
√
V
and the electron momentum and energy are

p = k

E =

^2 k^2

2 m 0

+V 0