34 CHAPTER 2. ELECTRONIC LEVELS IN SEMICONDUCTORS
Density of states for a three-dimensional system
We will now discuss the extremely important concept of density of states. The concept of
density of states is extremely powerful, and important physical properties in materials, such as
optical absorption, transport, etc., are intimately dependent upon this concept. 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.
Accounting for spin, the density of states can be shown to be (see Appendix C)
N(E)=
√
2 m^30 /^2 (E−V 0 )^1 /^2
π^2 ^3(2.3.5)
In figure 2.4a we show the form of the three-dimensional density of states.
Density of states in sub-three-dimensional systems
The use of heterostructures has allowed one to make sub-three-dimensional-systems. In these
systems the electron can be confined in two-dimensions (forming a quantum well) or in one-
dimensional (quantum wire) and zero-dimensional (quantum dot) space. The two-dimensional
density of states is defined as the number of available electronic statesperunitareaperunitenergy
around an energyE. It can be shown that the density of states for a parabolic band (for energies
greater thanV 0 ) is (see figure 2.3b)
N(E)=
m 0
π^2(2.3.6)
Finally, we can consider a one-dimensional system often called a “quantum wire.” The one-
dimensional density of states is defined as the number of available electronic statesperunit
lengthperunitenergy around an energyE. In a 1D system or a “quantum wire” the density of
states is (including spin) (see figure 2.3c)
N(E)=
√
2 m^10 /^2
π(E−V 0 )−^1 /^2 (2.3.7)
Notice that as the dimensionality of the system changes, the energy dependence of the density
of states also changes. As shown in figure 2.3, for a three-dimensional system we have(E−
V 0 )^1 /^2 dependence, for a two-dimensional system we have no energy dependence, and for a
one-dimensional system we have(E−V 0 )−^1 /^2 dependence.
We will see later in the next section that when a particle is in a periodic potential, its wave-
function is quite similar to the free particle wavefunction. Also, the particle responds to external
forces as if it is a free particleexceptthatitsenergy-momentumrelationismodifiedbythe
presenceoftheperiodicpotential. In some cases it is possible to describe the particle energy by
the relation
E=^2 k^2
2 m∗+Eedge (2.3.8)wherem∗is called the effective mass in the material andEedgeis the bandedge energy. The ef-
fective mass in general summarizes the appropriate way to modify the free electron mass based