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.

 Quantum numbers for free electrons: (n, ms )         ms , 

  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
                                (collision freq additive)
            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 
                                                                     for free electrons
L  2.45 108 watt-ohm/deg 2   for free electrons

								
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