APPENDIX C. DENSITY OF STATES 531
2 π
L
kxky
2 π
L
Figure C.2:k-Space volume of each electronic state. The separation between the various allowed
components of thek-vector is^2 Lπ.
IfΩis a volume ofk-space, the number of electronic states in this volume is
ΩV
8 π^3
It is easy to verify that stationary and periodic boundary conditions lead to the same density
of states value as long as the volume is large.
Density of States for a Three-Dimensional System
Important physical properties in materials such as optical absorption, transport, etc., are inti-
mately dependent upon how many allowed states there are. Density of states is the number of
available electronic statesperunitvolumeperunitenergy around an energyE. If we denote the
density of states byN(E), the number of states in a unit volume in an energy intervaldEaround
an energyEisN(E)dE. To calculate the density of states, we need to know the dimensionality
of the system and the energy versuskrelation that the particles obey. We will choose the particle
of interest to be the electron, since in most applied problems we are dealing with electrons. Of
course, the results derived can be applied to other particles as well. For the free electron case we
have the parabolic relation
E=^2 k^2
2 m∗+V 0
The energiesEandE+dEare represented by surfaces of spheres with radiikandk+dk,as
shown in figure C.3. In a three-dimensional system, thek-space volume between vectorkand
k+dkis (see figure C.3a) 4 πk^2 dk. We have shown in equation C.1 that thek-space volume