# 6. Free Electron Fermi Gas by 396NK3

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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
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
Energy Levels in One Dimension
d 2
2
H n             n n
2m dx 2

Orbital: solution of a 1-e Schrodinger equation

Boundary conditions:               n  0   n  L   0   Particle in a box

 n                     
 n  A sin         x   A sin  2   x
 L                n    
2
n  1, 2,              n      L
n

 n 
2            2

n         
2m  L 
Pauli-exclusion principle: No two electrons can occupy the same quantum state.

Degeneracy: number of orbitals having the same energy.

Fermi energy εF = energy of topmost filled orbital when system is in ground state.

 nF  
2           2
N free electrons:                                            N
F                          nF 
2m  L                          2
Effect of Temperature on the Fermi-Dirac Distribution
1                            1
Fermi-Dirac distribution :                        f                                     
    
e                1                 k BT

Chemical potential μ = μ(T) is determined by                                       N   d g   f     g = density of states

1     
At T = 0:      f     for
0     

→             0   F

1
For all T :         f   
2

f   e
     
For ε >> μ :

(Boltzmann distribution)
3D e-gas
Free Electron Gas in Three Dimensions
 d 2 d 2 d 2 
2
H  r        2  2  2     r 
2m  dx   dy   dz 

Particle in a box (fixed) boundary conditions:

 n  0, y, z   n  L, y, z   n  x,0, z   n  x, L, z   n  x, y,0  n  x, y, L   0

n           ny    nz              
→            n  A sin  x       x  sin    y  sin           z               ni  1, 2,          Standing
 L               L   L                                                  waves
Periodic boundary
conditions:
 k  x, y, z    k  x  L, y, z    k  x, y  L, z    k  x, y, z  L 

 k  A e i k r                          2ni                                        Traveling
→                                               ki                        ni  0,  1,  2,
L                                          waves
2
k2
k 
2m
p k   k          k k             → ψk is a momentum eigenstate with eigenvalue k.
i
k
p k                  v
m
V 4 3
N free electrons:        N  2           kF
8 3
3
1/3
 3 2 N 
kF          
   V 

2/3
2
2
kF    2
 3 2 N 
F                  
2m    2m  V 
1/3
kF            3 2 N 
vF                      
m           m V 
Density of states:
3/2
V       dSk              V 4 k 2  V mk     V  2m 
D    2 3                      3 2                2 2            
8 k  k  k          4   k/m    2 2    2     

V  2m  F 
3/2
V 3
N  2 kF  2         
3     3    2


3 N    
→         D   
2 F   F

3 N
D  F  
2 F
Heat Capacity of the Electron Gas
(Classical) partition theorem: kinetic energy per particle = (3/2) kBT.
3
N free electrons:                  Ce      N kB                ( 2 orders of magnitude too large at room temp)
2

T
Pauli exclusion principle →                    Ce ~ N k B                                   TF ~ 104 K for metal
TF


f   
1                                                            
 d  D   f   
3 N
U                                                                                                         D                       
1


e       1               k BT
2 F   F
Using the Sommerfeld expansion formula                                                                                      free electrons
                             
d 2 n 1 H
 d  H   f     d  H      2  2                                     2n  k T 

21 n                       2n

d  2 n 1
B
                                                   n 1                                                           

 dD             
U     d D       2  k T                         D      O T 4 
2
B             d             
                                                              

 O T 4 
dD
N          d  D       2  k BT 
2


d 

 O T 4 
dD
N           d  D       2  k BT 
2


d 
F
 O T 4 
dD
       d  D         F  D   F     2  kBT 
2


d  F
F
dD
 d D                               F  D  F     2 kBT 
2
N                             →                                                                             0

d  F
dD                                     dD     1
            F    2  k BT 
2

D d      F
D d  2
for 3-D e-gas

 dD             
U    d D       2  k T                          D      O T 4 
2
B             d             
                                                              
F
 dD                 
 d  D          D                             2  k BT           F      D   F    O T 4 
2
F                F    F                              d                
                                                                                        F             
F
2
 d D       2 k T  D    O T                                                             2 
2                     4
B                 F
                                                                                                                6
 U               2                                                              2
CV                         D  F  k T2
CV                N kB
kB T
3-D e-gas
 T  N
B
3                                                               2            F
dD     1        dD       1
               
dD         D d  2        D d    2
    F    2  k BT 
2

D d   F
for 3-D e-gas   for 1-D e-gas

  k BT
Experimental Heat Capacity of Metals
For T <<  and T << TF :             C    T  AT3           el + ph

C
  AT2
T

1/3
   2                              3N     3 N 2m                   3 2 N 
C             D  F  kB T
2     D  F                            kF          
3                                 2 F         2
2 2 kF                    V 

mth     obs 
Deviation from e-gas value is described by mth :          
m   e  gas 
mth     obs    Possible causes:
                                    Heavy fermion:
m   e  gas     e-ph interaction
mth ~ 1000 m
 e-e interaction
UBe3 , CeAl3, CeCu2Si2.
Electrical Conductivity and Ohm’s Law
Free particle in constant E field               dp
dp
 p, H                p ,  q Er                                   qE
Heisenberg picture:      i
dt
      q Ei       dt
dp              1         dk
Lorentz force on free electron:                   F  e  E  v  B  
dt              c         dt

eEt
No collision:        k t   k  0               k t 

 k           ne 2                     Ohm’s      ne 2 1
Collision time  :   j  nq v  n  e                                                               
m               m
E                    law         m     
Experimental Electrical Resistivity of Metals
Dominant mechanisms
high T: e-ph collision.
low T: e-impurity collision.

Matthiessen’s rule:
   ph  imp                    imp indep of T
phonon   impurity
1        1          1
        ph        imp

K
Residual                           0 imp         Sample dependent
resistivity:
 ph T    T   imp       Sample independent

Resistivity ratio:             Troom 
imp
imp ~ 1  ohm-cm per atomic percent of impurity
Consider Cu with resistivity ratio of 1000:
From Table 3, we have         L  295K  1.7 ohm-cm
  295K 
         imp                        1.7 103  ohm-cm
resistivity ratio

imp ~ 1  ohm-cm per atomic percent of impurity

Impurity concentration:       ci  1.7  103  102   = 17 ppm
Very pure Cu sample:           4K  105 300K 

  4K  2 109 s           vF    1.57  108 cm s 1         l  4K   vF  4K  0.3cm

For T >  :         T            See App.J
Umklapp Scattering

Normal:       k  k  q

Umklapp:       k  k  q  G
Large scattering angle ( ~  ) possible

For Fermi sphere completely inside BZ, U-processes are possible only for q > q0

q0 = 0.267 kF for 1e /atom Fermi sphere inside a bcc BZ.

Number of phonon available for U-process  exp(U /T )

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

Equation of motion with relaxation time  :              d 1                 1      
     k  F  q  E  v  B 
 dt                 c      
 d 1        1   
m   v  qE  vB                                 q = –e for electrons
 dt        c   

Let    e   1
ˆ
, e2 , e//  B      be a right-handed orthogonal basis

 d 1               1                                    Steady state:
m    v 1  q  E1  v 2 B 
 dt               c       
q       q
v 1      E 1  c v  2
 d 1               1                                              m        q
m    v 2  q  E2  v 1 B 
 dt               c       
v 2   
q       q
E 2  c v 1
m        q
 d 1
m    v / /  q E/ /                                               q
 dt                                                        v//    E/ /
m
q B
c       = cyclotron frequency
mc
Hall Effect
q
vx       Ex
jy  0 →                m
q      q
E y  c vx  0
m       q
q
vz       Ez  0
m
q
     Ey      c Ex  qB  E x
q          mc

Hall
coefficient:
electrons                                  qB
 Ex
Ey     mc          1
RH        2        
jx B  nq         nqc
Ex B
m
Thermal Conductivity of Metals
1
From Chap 5:         K  Cvl
3
1 2      kB T              2     k T
Fermi gas:          Kel     N kB       vF  vF     N kB B 
3 2        F               3       m

In pure metal, Kel >> Kph for all T.

Wiedemann-Franz Law:
2          kB T
N kB                            2
K
 3             m             kB 
2
       T T
             nq   2
3 q 
m
2
K       2  kB 
Lorenz number:            L                          2.45 108 watt-ohm/deg 2
T  3 q 