Thermodynamics and Statistical
Mechanics
Fermi-Dirac Statistics
Thermo & Stat Mech Spring 2006 Class 22
1
Free Electrons in Metals
A model based on Fermi-Dirac statistics
explains why metals do not have a much larger
heat capacity than insulators. At one time it was
expected that the “free electron gas” would
provide a large addition to the lattice heat
capacity, but it does not.
Thermo & Stat Mech - Spring 2006
Class 22
2
Fermi Function
fj 
Nj
gj
1

 j 
e
1
f ( ) 
kT
Number per state
 
e
kT
1
1
Thermo & Stat Mech - Spring 2006
Class 22
3
Properties of Fermi Function
1
f ( ) 
 
e
kT
1
For    , e
 
kT
For    , e
For    , e
1
f ( ) 
 
kT
0
f ( )  1

f ( )  0
 
kT
1
2
Thermo & Stat Mech - Spring 2006
Class 22
4
Properties of Fermi Function
1
f ( ) 
Consider as T  0
 
e
kT
1
For    , e
For    , e
 
kT
 
kT
 e   0
f ( )  1

f ( )  0
e 
Thermo & Stat Mech - Spring 2006
Class 22
5
Properties of Fermi Function
1.2
 = 0.1 eV
1
0.8
FD(T1)
0.6
FD(T2)
0.4
T1 = 1K
T2 = 200K
T3 = 400K
FD(T3)
0.2
0
0
0.05
0.1
0.15
0.2
0.25
-0.2
E(eV)
Thermo & Stat Mech - Spring 2006
Class 22
6
Free Electron Gas
Density of States for Particles (Lec 17)
V  2m 


g ( )d  
2  2 
4   
3/ 2
 d
  2 for electrons
V  2m 


g ( )d 
2  2 
2   
3/ 2
Thermo & Stat Mech - Spring 2006
Class 22
 d
7
Energy of Electrons
1
dU   f ( ) g ( )d  
e
V  2m 
dU 
2 
2 
2   
3/ 2

e
3/ 2
(   )
kT
1
V  2m 
2 
2 
2   
3/ 2
 d
d
(   )
kT
1
Thermo & Stat Mech - Spring 2006
Class 22
8
Fermi Energy
F Energy of highest energy state occupied at T = 0 K
N 
F
0
V  2m 
g ( )d 
2  2 
2   
V  2m 
N
2  2 
2   
3/ 2
  2 N
F 
 3

2m 
V
2
3/ 2

F
0

1/ 2
d
2 3/ 2
F
3
2/3
V  2m 
2  2 
2   
Thermo & Stat Mech - Spring 2006
Class 22
3/ 2
3N
 3/ 2
2 F
9
Electrons’ Energy at T = 0 K
U0  
F
0
3 N
 g ( )d 
3/ 2
2 F

F
0

3/ 2
d
3 N 2 5/ 2
U0 
F
3/ 2
2 F 5
3
U 0  N F
5
Thermo & Stat Mech - Spring 2006
Class 22
10
Electrons’ Energy at T > 0 K
 kT 
3
U  N F  N  kT
5
 F 
3
N
2
U  N F  (kT )
5
F
Thermo & Stat Mech - Spring 2006
Class 22
11
Heat Capacity of Electrons
3
N
2
U  N F 
(kT )
5
F
 U
CV  
 T
 kT 

N
2
 
2k T  2 Nk  
V  F
 F 
 kT 
Exact result : CV 
Nk  
2
 F 

2
Thermo & Stat Mech - Spring 2006
Class 22
12
Heat Capacity of Metals
At normal temperatures, the heat capacity due
to the electrons is much less than the heat
capacity of the lattice, since kT << F.
At low temperatures, the lattice heat capacity
gets very small, and the contribution of the
electrons becomes significant.
Thermo & Stat Mech - Spring 2006
Class 22
13
At Low Temperatures
12 Nk  T 


C Lattice 
5
 D 
4
Cmetal  AT  BT
Cmetal
2
 A  BT
T
3
Celectrons
 Nk
A
2 F
2
3
 kT 

Nk  
2
 F 

2
12 Nk
B
3
5 D
Thermo & Stat Mech - Spring 2006
Class 22
2
4
14
Better Value for Electron’s Energy
To calculate the energy of the electrons
correctly, it is necessary to carry out the
integral below. In order to do that, a value
is needed for the chemical potential, .

U    f ( ) g ( )d
0
Thermo & Stat Mech - Spring 2006
Class 22
15
Approximate Method
Consider t he integral :

I   f ( ) H ( )d
0
dh( )
where H ( ) 
, and h(0)  0
d

Then, h( )   H ( ' )d '
0
Thermo & Stat Mech - Spring 2006
Class 22
16
Approximate Method
I 
dh( )
f ( )
d
d

0
df ( )
I   f ( )h( )   h( )
d
0
d
f ( )  0
h(0)  0, so

0
I  

0

df ( )
h( )
d
d
Thermo & Stat Mech - Spring 2006
Class 22
17
df
Graph of 
d
Fermi-Dirac
160
140
120
-df/d 
100
80
60
40
20
0
0
0.05
0.1
0.15
0.2
0.25
E (eV)
Thermo & Stat Mech - Spring 2006
Class 22
18
Expansion
h( )  h(  )  (   )h' (  )  (   ) h" (  )  ...
2
1
2


0
0
I  h(  )  f ' ( )d  h' (  )  (   ) f ' ( )d

 h" (  )  (   ) f ' ( )d  ...
1
2
I   h(  ) 


2
0

f ' ( )d  h' (  )  (   ) f ' ( )d


 12 h" (  )  (   ) 2 f ' ( )d  ...

Thermo & Stat Mech - Spring 2006
Class 22
19
Evaluate each term
 h(  ) 


f ' ( )d  h(  ) f ()  f ()

 h(  )   H ( )d
0

 h' (  )  (   ) f ' ( )d  0 an odd function

Thermo & Stat Mech - Spring 2006
Class 22
20
Evaluate each term
Let :
 
kT
 x, then
2 x
x
e dx
2
2
1
1
 2 h" (  )  (   ) f ' ( )d  2 h" (  )( kT ) 

  (e x  1) 2




2
6
2
6
(kT ) 2 h" (  )
(kT ) 2 H ' (  )
Thermo & Stat Mech - Spring 2006
Class 22
21
Apply to Chemical Potential


0
0
I   f ( ) H ( )d   H ( )d 


0
0
N   f ( ) g ( )d   g ( )d 
where
2
6
2
6
(kT ) 2 H ' (  )
(kT ) 2 g ' (  )
3N 1 2
g ( )  3 2 
2 F
Thermo & Stat Mech - Spring 2006
Class 22
22
Chemical Potential


0
0
N   f ( ) g ( )d   g ( )d 
However,
so
and

F
0
F

2
6
(kT ) 2 g ' (  )
F
N   g ( )d
0

g ( )d   g ( )d 
0
g ( )d 
2
6
2
6
(kT ) 2 g ' (  )
(kT ) 2 g ' (  )
3N 1 2
where g ( )  3 2 
2 F
Thermo & Stat Mech - Spring 2006
Class 22
23
Chemical Potential
3N 1 2
g ( )  3 2 
2 F
F

g ( )d  g (  )( F   ) 
3N 1
g ' ( )  3 2 1 2
2 F 2
2
6
(kT ) 2 g ' (  )
1
3N
g ' ( )  3 2
2 F 2 1 2
1
2
3N 1 2
2 3N
(kT )
 ( F   ) 
32
2 F3 2 2 1 2
6
2 F
Thermo & Stat Mech - Spring 2006
Class 22
24
Finally, the Chemical Potential!
3N 1 2
2
1
2 3N
 ( F   ) 
(kT )
32
2 F
6
2 F3 2 2 1 2

 (kT )
( F   ) 
(kT )

12
 12  F
2
2
1
2
2
2
2

 (kT )
  kT  
  
  F 
  F 1 
12  F
 12   F  
2
2
Thermo & Stat Mech - Spring 2006
Class 22
25
Electron Energy


0
0
U    f ( ) g ( )d  
3N  

 g ( ) 
2  F
U 

0



32
2
d
 g ( )d  (kT )
[ g ( )]
6
d
2
d
9N 1 2
[ g ( )] 
d
4  F3 2
2
12
9
N

 g ( )d  (kT ) 2
6
4  F3 2
Thermo & Stat Mech - Spring 2006
Class 22
26
Approximate again!
U 

0


0

9N 
 g ( )d  (kT )
32
6
4 F
2
12
2
F

 g ( )d    g ( )d    g ( )d
0
F
3
 N F  (    F ) F g ( F )
5
Thermo & Stat Mech - Spring 2006
Class 22
27
Still more!


0
 g ( )d  U 0  (    F ) F g ( F )
 2 (kT ) 2
( F   ) 
12  F
3N 1 2 3N
 F g ( F )   F 3 2  F 
2
2 F


0
 g ( )d  U 0 
2
8
N
(kT ) 2
F
Thermo & Stat Mech - Spring 2006
Class 22
28
And more!
2

d
U    g ( )d 
(kT )
[ g ( )]
0
6
d

 2 (kT ) 2
0  g ( )d  U 0  8 N  F
2
d
9N 1 2 9N
[ g ( )] 

32
d
4 F
4 F
U  U0 

2
8
N
(kT )
F
2

2
9N

(kT )
6
4 F
Thermo & Stat Mech - Spring 2006
Class 22
2
29
Finally!!
U  U0 
U  U0 
2
8
2
4
N
N
(kT ) 2
F
2
9N

(kT )
6
4 F
2
(kT ) 2
F
 kT 

 U 
CV  
Nk  
 
2
 T V
 F 
2
Thermo & Stat Mech - Spring 2006
Class 22
30