Lecture 27 Electron Gas

The model, the partition function and the Fermi function Thermodynamic functions at T = 0 Thermodynamic functions at T > 0

Number of free electrons and number of statesThe model assumes ideal gas of non-interacting electrons. The only constrain is that no two electrons can occupy the same quantum state, since electrons are fermions

Number of states per unit volume between energy = 0 to energy = kT for an ideal gas

This number is about 4 orders of magnitude larger than number of electrons is a typical metal. The reason is two fold. (i) The electron density is very high (~ electron per 10Å3) and (ii) electron mass is small, thus G(kT) is small

Consequently a classical model of an ideal gas is not operational

€

G(kT)

V= (2)

π

6

8mkT

h2

⎛

⎝ ⎜

⎞

⎠ ⎟3 / 2

Partition Function It is convenient to use grand canonical ensemble

The total number of electrons

is the sum of number of electrons in single electron quantum states (each quantum state has either zero or one electron)

Since the electrons are not interacting

€

Ξ= e−βE i

i

∑ eγN i

€

N i = Nkik

∑

€

E i = εkNkik

∑

Partition Function - II In term of single energy levels

Which is the same as

For example with just 2 single electron states the top formula gives

which is the same as the bottom formula

€

Ξ= e−(βε k −γNk )

k

∏i

∑

€

Ξ= e−(βε k −γ )η

η = 0

max Nk

∑k

∏

€

Ξ=e−(0)e−(0) + e−(0)e−(βε 2 −γ ) + e−(βε1 −γ )e−(0) + e−(βε1 −γ )e−(βε 2 −γ ) =

e−(0) + e−(βε1 −γ )( )x e

−(0) + e−(βε 2 −γ )( )

Partition Function - III For Nkmax =1 (fermions)

Thus the partition function

And the logarithm of the partition function

€

e−(βε k −γ )η

η = 0

max Nk

∑ =1+ e−(βε k −γ )

€

Ξ= 1+ e−(βε k −γ )( )

k

∏

€

lnΞ = 1+ e−(βε k −γ )( )

k

∑

Average occupation number For Nkmax =1 (fermions)

Which implies that the average number of electrons in a single electron state is

This is called the Fermi function

€

N =∂ lnΞ

∂γ=

1

1+ e(βε k −γ )k

∑ =1

1+ eε k −μ

kT

⎛

⎝ ⎜

⎞

⎠ ⎟k

∑

€

Nk =1

e(ε k −μ ) / kT +1

Fermi function

T = 0 T > 0

€

Nk =1

e(ε k −μ ) / kT +1

Nk

k

Nk

k

kT

Energy Density of states = 2 times density of state of an ideal gas

Energy

where is the zero temperature Fermi level

€

g(ε) = 2π

4

8m

h2

⎛

⎝ ⎜

⎞

⎠ ⎟3 / 2

Vε1/ 3

€

E = g(ε)ε0

μ 0

∫ dε =8π

5

2m

h2

⎛

⎝ ⎜

⎞

⎠ ⎟3 / 2

Vμ05 / 2

€

μ0

Fermi level (T=0)Number of electrons

From which

€

N = g(ε)0

μ 0

∫ dε =8π

3

2m

h2

⎛

⎝ ⎜

⎞

⎠ ⎟3 / 2

Vμ03 / 2

€

μ0 =h2

2m

3

8π

⎛

⎝ ⎜

⎞

⎠ ⎟2 / 3N

V

⎛

⎝ ⎜

⎞

⎠ ⎟2 / 3

T > 0Number of electrons Energy

Integrating by parts

and expanding

around = μ gives

€

N = g(ε)1

e(ε −μ ) / kT +10

∞

∫ dε

€

E = g(ε)ε1

e(ε −μ ) / kT +10

∞

∫ dε

€

N = G(ε)∂

1

e(ε −μ ) / kT +1

⎛

⎝ ⎜

⎞

⎠ ⎟

∂ε0

∞

∫ dε

€

∂ 1

e(ε −μ ) / kT +1

⎛

⎝ ⎜

⎞

⎠ ⎟

∂ε €

μ =μ0 1−π 2

12

kT

μ0

⎛

⎝ ⎜

⎞

⎠ ⎟

2

+ ... ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

E = E0 1+5π 2

12

kT

μ0

⎛

⎝ ⎜

⎞

⎠ ⎟

2

+ .... ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Energy Heat Capacity

Nk

k

kT

€

E = E0 1+5π 2

12

kT

μ0

⎛

⎝ ⎜

⎞

⎠ ⎟

2

+ .... ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

The energy ~ T2 can be also seen in the following: Number of excited electrons ~ kT, and the energy of each excited electrons ~ kT, thus the total energy ~ kT x kT ~ (kT)2

€

CV =∂E

∂T

⎛

⎝ ⎜

⎞

⎠ ⎟V ,N

=π 2

2NkkT

μ0