SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

530 APPENDIX C. DENSITY OF STATES

Periodic boundary conditions are shown in figure C.2b. Even though we focus our attention on a
finite volumeV, the wave can be considered to spread in all space as we regard the entire space
as made up of identical cubes of sidesL.Then

ψ(x, y, z+L)=ψ(x, y, z)

ψ(x, y+L, z)=ψ(x, y, z)

ψ(x+L, y, z)=ψ(x, y, z)

ψ(x–L)

y

x

ENERGY

LEVELS

W

AVEFUNCTIONS

0 L

λ= 32 L

λ= L

λ=^2 L

(a) (b)

L

L

ψ(x) ψ(x+L)

Figure C.1: Two types of boundary conditions. A schematic showing (a) the stationary boundary
conditions; (b) the periodic boundary conditions.

In this case the allowed values ofkare (nare integers—positive and negative)

kx=

2 πnx

L

; ky=

2 πny

L

; kz=

2 πnz

L

IfLis large, the spacing between the allowedk-values is very small. Also it is important to
note that the results one obtains for properties of the particles in alargevolumeareindependent
ofwhetherweusethestationaryorperiodicboundaryconditions. It is useful to discuss the
volumeink-spacethateachelectronicstateoccupies. As can be seen from figure C.2, this
volume is (in three dimensions) (
2 π
L

) 3

=

8 π^3

V

(C.1)