7 Electrons

7.1 Free Electron Fermi Gas

The density of statesD(E)of an electron gas isD(E) =D(k)dEdk, as stated previously. The energy of
a free electron regardless of the numbers of dimensions considered is always:

E=~ω=
~^2 k^2
2 m
But the densityD(k)is of course dependent on the number of dimensions:

D(k) 1 −D=

2

π

, D(k) 2 −D=

k

π

, D(k) 3 −D=

k^2

π^2

So computing the derivativedEdk and expressing everything in terms ofEleads us to these equations:

D(E) 1 −D=

1

~π

√

2 m

E

, D(E) 2 −D=

m

~^2 π

, D(E) 3 −D=

(2m)

(^32)
2 ~^3 π^2

√

E

7.1.1 Fermi Energy

We assume a system of many electrons at temperatureT= 0, so our system has to be in the lowest
possible energy configuration. Therefore, we group the possible quantum states according to their
energy and fill them up with electrons, starting with the lowest energy states. When all the particles
have been put in, theFermi energyis the energy of the highest occupied state. The mathematical
definition looks like this

n=

∫EF

0

D(E)dE (45)

wherenis the electron density,D(E)the density of states in terms of the energy andEFis the Fermi
Energy.

Performing the integral for the densities of states in 1, 2, and 3 dimensions and plugging in these
Fermi energies in the formulas for the density of states gives us (Nis the number of electrons andL
is the edge length of the basic cube, see also chapter 3):

1-D:

EF=

~^2 π^2

8 m

(

N

L

) 2

, D(E) =

√

2 m

~^2 π^2 E

=

n

2

√

EFE

[

1

Jm

]

2-D:

EF=

~^2 πN

mL^2

, D(E) =

m

~^2 π

=

n

EF

[

1

Jm^2

]

3-D:

EF=

~^2

2 m

(

3 π^2 N

L^3

) (^23)
, D(E) =
π
2
(
2 m
~^2 π^2
) (^32)

3 n
2 E
(^32)
F

√

E

[

1

Jm^3

]