SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

Appendix C

DENSITY OF STATES

In semiconductor devices we use the effective mass approximation to describe the properties
of electrons in a crystal. Using the effective mass picture the Schrodinger equation for electrons ̈
can be written as a “free’ electron problem with a background potentialV 0 ,

−^2

2 m∗

(

∂^2

∂x^2

+

∂^2

∂y^2

+

∂^2

∂z^2

)

ψ(r)=(E−V 0 )ψ(r)

A general solution of this equation is

ψ(r)=

1

√

V

exp (±ik·r)

and the corresponding energy is

E=

^2 k^2

2 m

+V 0

where the factor 1 /

√

Vin the wavefunction occurs because we wish to have one particle per
volumeVor ∫

V

d^3 r|ψ(r)|^2 =1

We assume that the volumeVis a cube of sideL.
An important aspect of electronic bands is the density of states which tells us how many
allowed energy levels there are between two energies. To obtain macroscopic properties inde-
pendent of the chosen volumeV, two kinds of boundary conditions are imposed on the wave-
function. In the first one the wavefunction is considered to go to zero at the boundaries of the
volume, as shown in figure C.1a. In this case, the wave solutions are standing waves of the form
sin(kxx)orcos(kxx), etc., andk-values are restricted to positive values:

kx=

π

L

,

2 π

L

,

3 π

L

···

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