SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

532 APPENDIX C. DENSITY OF STATES

per electron state is(^2 Lπ)^3. Therefore, the number of electron states in the region betweenkand
k+dkis
4 πk^2 dk
8 π^3

V=

k^2 dk

2 π^2

V

Denoting the energy and energy interval corresponding tokanddkasEanddE, we see that
the number of electron states betweenEandE+dEper unit volume is

N(E)dE=

k^2 dk

2 π^2

Using theEversuskrelation for the free electron, we have

k^2 dk=

√

2 m∗^3 /^2 (E−V 0 )^1 /^2 dE

^3

and

N(E)dE=

m∗^3 /^2 (E−V 0 )^1 /^2 dE

√

2 π^2 ^3

The electron can have a spin state/ 2 or−/ 2. Accounting for spin, the density of states
obtained is simply multiplied by 2

N(E)=

√

2 m∗^3 /^2 (E−V 0 )^1 /^2

π^2 ^3

Density of States in Sub-Three-Dimensional Systems
In quantum wells electrons are free to move in a 2-dimensional space. The two-dimensional den-
sity of states is defined as the number of available electronic statesperunitareaperunitenergy
around an energyE. Similar arguments as used in the derivation show that the density of states
for a parabolic band (for energies greater thanV 0 ) is (see figure C.3b)

N(E)=

m∗

π^2

The factor of 2 resulting from spin has been included in this expression. 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 statesperunitlengthperunitenergy around an
energyE. In a 1D system or a “quantum wire” the density of states is (including spin) (see
figure C.3c)

N(E)=

√

2 m∗^1 /^2

π

(E−V 0 )−^1 /^2

Notice that as the dimensionality of the system changes, the energy dependence of the density
of states also changes. As seen in figure C.4, 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. The changes in density of states
with dimensions are exploited in electronic and optoelectronic devices.