SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

2.3. ELECTRONS IN CRYSTALLINE SOLIDS 33

If the potential barrier is not infinite, we cannot assume that the wavefunction goes to zero at
the boundaries of the well. Let us define two parameters.

α =

√

2 mE

^2

β =

√

2 m(V 0 −E)

^2

(2.2.8)

The conditions for the allowed energy levels are given by the transcendental equations

αW

2

tan

αW

2

=

βW

2

(2.2.9)

and
αW
2

cot

αW

2

=−

βW

2

(2.2.10)

An important outcome of these solutions is that as in the H-atom case, only some energies are
allowed for the electron. This result is of importance in electronic devices as will be discussed
in section 2.10.

2.3 ELECTRONS IN CRYSTALLINE SOLIDS

The devices discussed in this text are made from crystalline materials. It is, therefore, impor-
tant to understand the electronic properties of these materials. Let us first examine the simpler
problem of electrons in free space. It turns out that electrons in crystals can be considered to
behave as if they are in free space except they have a different “effective properties”. In the
free space problem the background potential energy is uniform in space. The time-independent
equation for the background potential in a solid equal toV 0 is

−^2

2 m

(

∂^2

∂x^2

+

∂^2

∂y^2

+

∂^2

∂z^2

)

ψ(r)=(E−V 0 )ψ(r) (2.3.1)

A general solution of this equation is

ψ(r)=

√^1

V

e±ik·r (2.3.2)

and the corresponding energy is

E=

^2 k^2

2 m

+V 0 (2.3.3)

where the factor√^1 Vin the wavefunction occurs because we wish to have one particle per volume
Vor ∫

V

d^3 r|ψ(r)|^2 =1 (2.3.4)

We assume that the volumeVis a cube of sideL. Note that if we assign the momentum of the
electron askthe energy-momentum relation of free electrons is the same as that in classical
physics. Later we will see that in crystalline material one can use a similar relationship except
the mass of the electron is modified by an effective mass.