main_06._FreeElectronFermiGas1234567890abcdefghij.ppt

Free electron fermi gas

1. 6. Free Electron Fermi Gas
•Energy Levels in One Dimension
•Effect of Temperature on the Fermi-Dirac Distribution
•Free Electron Gas in Three Dimensions
•Heat Capacity of the Electron Gas
•Electrical Conductivity and Ohm’s Law
•Motion in Magnetic Fields
•Thermal Conductivity of Metals
•Nanostructures

2. Introduction
Free electron model:
Works best for alkali metals (Group I: Li, Na, K, Cs, Rb)
Na: ionic radius ~ .98A, n.n. dist ~ 1.83A.
Successes of classical model:
Ohm’s law.
σ / κ
Failures of classical model:
Heat capacity.
Magnetic susceptibility.
Mean free path.
Quantum model ~ Drude model

3. Energy Levels in One Dimension
2 2
2
2
n n n
d
H
m dx

  

Orbital: solution of a 1-e Schrodinger equation
Boundary conditions: 0 0
n n
L  
sin
n
n
A x
L


 

 
 
2
n
L
n

Particle in a box
2
sin
n
A x


 
  
 
1,2,n
2
2
2
n
n
m L


 

 
 


4. Pauli-exclusion principle: No two electrons can occupy the same quantum state.
Quantum numbers for free electrons: (n, m
s
) ,
s
m 
Degeneracy: number of orbitals having the same energy.
Fermi energy ε
F = energy of topmost filled orbital when system is in ground
state.
N free electrons:
2
2
2
F
F
n
m L


 

 
 

2
F
N
n

5. Effect of Temperature on the Fermi-Dirac Distribution
Fermi-Dirac distribution : 

1
1
f
e
  




1
B
k T

Chemical potential μ = μ(T) is determined by N d g f  
g = density of states
At T = 0:
1
for
0
f
 

 



→ 0
F
 

1
2
fFor all
T :
For ε >> μ :

f e
  

 

(Boltzmann distribution)
3D e-gas

6. Free Electron Gas in Three Dimensions
 
2 2 2 2
2 2 2
2
d d d
H
m dx dy dz
  
 
 
    
 
r r

Particle in a box (fixed) boundary conditions:
      0, , , , ,0, , , , ,0 , , 0y z L y z x z x L z x y x y L          
n n n n n n
sin sin sin
yx z
nn n
A x y z
L L L
 

    
     
    
n
Periodic boundary
conditions:
Standing
waves
    , , , , , , , ,x y z x L y z x y L z x y z L        
k k k k
→
→
i
Ae


k r
k
2
i
i
n
k
L

 0, 1, 2,
i
n   
2 2
2
k
m

k

Traveling
waves
1,2,
i
n

7. i
  
k k
p


k
k → ψ
k
is a momentum eigenstate with eigenvalue k.
p k
m

k
v

N free electrons:
3
3
4
2
8 3
F
V
N k


  
1/3
2
3
F
N
k
V
 
 
 
2 2
2
F
F
k
m


2/3
2 2
3
2
N
m V
 
 
 

F
F
k
v
m


1/3
2
3N
m V
 
 
 


9. Density of states:

3
2
8
dSV
D
 

 




k
k
k k
2
3 2
4
4 /
V k
k m




2 2
V mk



3/2
2 2
2
2
V m


 

 
 
3
2
3
F
V
N k


3/2
2 2
2
3
F
mV 

 

 
 
→

3
2
F
F
N
D



3
2
F F
N
D


 


10. Heat Capacity of the Electron Gas
(Classical) partition theorem: kinetic energy per particle = (3/2) k
B
T.
N free electrons:
3
2
e B
C Nk ( 2 orders of magnitude too large at room temp)
Pauli exclusion principle →~
e B
F
T
C Nk
T
T
F
~ 10
4
K for metal
U d D f   

 



1
1
f
e
  




1
B
k T
  
3
2
F F
N
D

  
 

free electrons
Using the Sommerfeld expansion formula
 

 
2 1
22 1
2 1
1
2 2 2
n
nn
B n
n
d H
d H f d H n k T
d


     

 


   
    
  
2
4
2
B
dD
U d D k T D O T
d


     

 
 
     
 
 

 
2
4
2
B
dD
N d D k T O T
d


  

 
  

11.   
2
4
2
B
dD
U d D k T D O T
d


     

 
 
     
 
 

   
2
4
2
F
F
F F F B F F
dD
d D D k T D O T
d


         

 
 
      
 
 


 
2
4
2
B
dD
N d D k T O T
d


  

 
  
  
2
4
2
F
F
F F B
dD
d D D k T O T
d


     

 
   

→ 
2
2 0
F
F F B
dD
D k T
d

   

  

2
2
F
F B
dD
k T
D d

  


 
2
4
2
F
B F
d D k T D O T

    
 
 


F
N d D

 
 


2
2
3
V F B
N
U
C D k T
T


 
 
 


2
2
6


2
2
B
V B
F
k T
C Nk


 3-D e-gas
1
2
d D
D d 

for 3-D e-gas

12. 
2
2
F
F B
dD
k T
D d

  


B
k T
1
2
d D
D d 

for 3-D e-gas
1
2
d D
D d 

for 1-D e-gas

13. Experimental Heat Capacity of Metals
For T <<  and T << T
F :
3
C T AT el + ph
2C
AT
T


 
th
obsm
m e gas





2
2
3
F B
C D k T

 
3
2
F
F
N
D


2 2
3 2
2
F
N m
k


1/3
2
3
F
N
k
V
 
 
 
Deviation from e-gas value is described by m
th
:

14. 
 
th
obsm
m e gas




Possible causes:
 e-ph interaction
 e-e interaction
Heavy fermion:
m
th
~ 1000 m
UBe
3
, CeAl
3
, CeCu
2
Si
2
.

15. Electrical Conductivity and Ohm’s Law
d
dt

p
FLorentz force on free electron:
1
e
c
 
  
 
 
E v B
d
dt

k

No collision:0
e t
t 
E
k k

tk
Collision time
 :
nqj v 

n e
m
 
 
k
2
ne
m

 E
Ohm’s
law
2
1ne
m



 
Heisenberg picture:  ,
d
i H
dt

p
p  ,q  p E r q iE
d
q
dt

p
E
Free particle in constant E field

16. Experimental Electrical Resistivity of Metals
Dominant mechanisms
high T: e-ph collision.
low T:e-impurity collision.
phonon impurity
1 1 1
ph imp
  
 
ph imp
   
Matthiessen’s rule:
0
imp
 
Sample dependent

ph imp
T T   
Sample independent
Residual
resistivity:
Resistivity ratio:
room
imp
T


imp ~ 1  ohm-cm per atomic percent of impurity
K

imp
indep of T
(collision freq
additive)

17. Consider Cu with resistivity ratio of 1000:
 
3
295
1.7 10 ohm-cm
resistivity ratio
imp
K
 

  
3 2
1.7 10 10
i
c
 
  Impurity concentration: = 17 ppm
Very pure Cu sample:  
5
4 10 300K K 

9
4 2 10K s

  4 4 0.3
F
l K v K cm 
8 1
1.57 10
F
v cm s

 
For T >  : T See App.J
From Table 3, we have 295 1.7 ohm-cm
L
K  


imp
~ 1  ohm-cm per atomic percent of impurity

19. Umklapp Scattering
Normal:  k k q
Umklapp
:
  k k q G
Large scattering angle ( ~  ) possible
Number of phonon available for U-process  exp(
U
/T )
For Fermi sphere completely inside BZ, U-processes are possible only for q > q
0
q
0
= 0.267 k
F
for 1e /atom Fermi sphere inside a bcc BZ.

For K, 
U
= 23K,  = 91K  U-process negligible for T < 2K

20. Motion in Magnetic Fields
1d
dt


 
 
 
 
k FEquation of motion with relaxation time  :
1
q
c
 
  
 
 
E v B
1 1d
m q
dt c
   
      
  
v E v B
// //
1d
m q
dt
 
 
 
 
v E
1 1 2
1 1d
m q B
dt c
  
   
     
  
v E v
be a right-handed orthogonal basis 1 2 //
ˆ
, ,
 
e e e BLet
2 2 1
1 1d
m q B
dt c
  
   
     
  
v E v
Steady state:
1 1 2 c
q q
m q


  
 v E v
2 2 1 c
q q
m q


  
 v E v
// //
q
m

v E
c
q B
mc
 = cyclotron frequency
q = –e for electrons

21. Hall Effect
0
y
j→
0
y c x
q q
E v
m q

 
x x
q
v E
m


0
z z
q
v E
m

 
y c x
q
E E
q

 x
qB
E
mc

Hall
coefficient:
y
H
x
E
R
j B

2
x
x
qB
E
mc
nq
E B
m



1
nqc

electrons

23. Thermal Conductivity of Metals
From Chap 5:
1
3
K C vl
Fermi gas:
2
1
3 2
B
el B F F
F
k T
K Nk v v



   
2
3
B
B
k T
Nk
m


In pure metal, K
el
>> K
ph
for all T.
Wiedemann-Franz Law:
2
2
3
B
B
k T
Nk
K m
nq
m




2
2
3
B
k
T
q
 

 
 
T
Lorenz number:
K
L
T

2
2
3
B
k
q
 

 
 
8 2
2.45 10 watt-ohm/deg

 
for free electrons

24. 8 2
2.45 10 watt-ohm/degL

  for free electrons