Advanced Solid State Physics

(^001020304050)
1
D(E) E
F
E/EF
(a)
(^0012345)
0.2
0.4
0.6
0.8
1
1.2
E/EF
2 E
D(E)/F
π
(b)
(^0020406080100)
1
2
3
4
5
6
7
8
9
10
E/EF
2 E
D(E)/F
π
(c)
Figure 20: a) Density of states of an electron gas in 3 dimensions; b) Density of states of an electron
gas in 2 dimensions; c) Density of states of an electron gas in 1 dimension

7.1.2 Chemical Potential

The Fermi Energy is of course a theoretical construct, becauseT= 0can’t be reached, so we start to
ask ourselves what happens at non-zero temperatures. In case of non-zero temperatures one has to
take the Fermi-statistic into account. With this in mind, we adapt eqn. (45) for the electron density

n=

∫∞

0

D(E)f(E)dE (46)

wheref(E)is the Fermi-function, so the equation fornreads as:

n=

∫∞

0

D(E)dE

e

E−μ

kBT+ 1

In this equation,μstands for thechemical potential, which can be seen as the change of the charac-
teristic state function per change in the number of particles. More precisely, it is the thermodynamic
conjugate of the particle number. It should also be noted, that forT = 0, the Fermi Energy and the
chemical potential are equal.