Docstoc

The effective mass v2

Document Sample
The effective mass v2 Powered By Docstoc
					         The effective mass
• Conductivity effective mass – determines
  mobility.
• Density of states effective mass –
  determines NC
• Cyclotron effective mass – can be
  measured directly
 Electron in a periodic potential
• Why does the semiconductor industry use
  single crystal material (when possible) ?
Electrons are not scattered
 by a periodic potential –
 move with a constant
 velocity as in vacuum !
  Electron in vacuum
V ( x)  0

( x)  A exp[ i(kx  t )]


     (k ) 2     p2
          E
      2m         2m
        Dispersion
            (k )

( x)  A exp[ i(kx  t )]
  Electron in vacuum
V ( x)  0

( x)  A exp[ i(kx  t )]


     (k ) 2     p2
          E
      2m         2m
Electron in a periodic potential

    V ( x  a)  V ( x)
     ( x)  u ( x) exp[ i (kx  t )]
    u ( x  a)  u ( x)
           2
    E (k     )  E (k )
            a
 Electron in a 3D periodic potential
                
V (r  R)  V (r )

R  Bravais lattice vector
                      
 (r )  u (r ) exp[ i (k r  t )]
               
u (r  R)  u (r )
                 
E (k  K )  E (k )
 
K  Reciprocal lattice vector
 Expansion of E(kx,ky,kz) near a
minimum value E0= E(kx0,ky0,kz0)


             1         E
                         2
 E (k )  E0                 (ki  ki 0 )(k j  k j 0 )
              2 i , j ki k j
 i , j  x, y , z
 Expansion of E(kx,ky,kz) near a
minimum value E0= E(kx0,ky0,kz0)
              1        2E
 E ( k )  E0                 (ki  ki 0 )(k j  k j 0 )
               2 i , j ki k j
 i , j  x, y , z
 or
                                                            k x  k x0 
                                                                      
 E (k )  E0 k x  k x 0 , k y  k y 0 , k z  k z 0 M  k y  k y 0 
                    2
                                                       1

            2                                              k k 
                                                            z       z0 

 where

         1 2E
 M ij 1  2             inverse effective mass tensor
          ki k j 
                    k0
        1 2E
M ij 1  2             the inverse effective mass tensor
         ki k j 
                   k0
                         is a symmetrical matrix -
                           can be diagonaliz ed
In the coordinate system in which
    the effective mass tensor is
              diagonal
           m1                  
                               
       M         m2           
                            m3 
                               
        1  E
                  E      E               
       v       ˆ
                x       ˆ
                        y                 z
                                          ˆ
            k
           x     k y    k z             
                                            
           m11                    k x  k x 0 
                       1
                                                
                m2               k y  k y 0 
                               1 
          
                             m3  k z  k z 0 
                                                 
Acceleration due to an electric filed (F)
            m11                   k x  k x 0 
                       1
                                                
      v          m2              k y  k y 0 
                               1 
           
                             m3  k z  k z 0 
                                                 

                    
       dv          dk
      a     M 1
         dt         dt

         
        dk     
           qF - (the semiclassi cal model )
        dt

            
      Ma  -qF
Effective mass tensor – valid near
  E(k) minima and maxima only


                 
           Ma  -qF
Constant energy surfaces in crystal
   momentum space cookies
Constant energy surfaces near a
    minimum are ellipsoids

                                                            k x  k x0 
                                                                      
 E (k )  E0 k x  k x 0 , k y  k y 0 , k z  k z 0 M  k y  k y 0 
               2
                                                       1

            2                                              k k 
                                                            z       z0 




         m11            
                 1
                          
 M 1        m2         
        
                      1 
                    m3  
  Constant energy surfaces in Si and
  Ge near a minimum are ellipsoids
            of revolution
                                                           k x  k x0 
                                                                     
E (k )  E0 k x  k x 0 , k y  k y 0 , k z  k z 0 M  k y  k y 0 
           2                                          1

           2                                              k k 
                                                           z       z0 




        ml 1            
                 1
                          
M 1         mt         
       
                      1 
                    mt  
Acceleration of an electron near an
    energy minimum in silicon
F  Fx x  Fy y  FZ z - electric field (main coordinate system)
       ˆ      ˆ      ˆ
-qF  Ma -in each valley
       1 6
-qF   M i a -average
       6 i 1
      1 6
a    M i -1qF
       6 i 1
               1   2
a x  qFx (         )
              3ml 3mt
1      1   2
  *
        
m     3ml 3mt
Electron transport effective mass in
      silicon and germanium



        1      1   2
          *
                
        m     3ml 3mt
Homework competition – find a shorter way
to prove this equation for germanium than
      given in last year’s home exam




         1      1   2
           *
                 
         m     3ml 3mt
         The spherical case -
                         m    
                              
                      M   m  
                             m
                              
• Electrons in GaAs
• Holes in Si, Ge, GaAs
Cyclotron resonance effective mass

   qH

    m
Cyclotron resonance effective mass –to
        be shown in the tutorial

              m1m2 m3
m 
       ˆ        ˆ          ˆ
       H12 m1  H 2 2 m2  H 32 m3
  1 
        2
         ˆ        ˆ          ˆ
         H12 m1  H 2 2 m2  H 32 m3    ˆ
                                        H12   ˆ
                                              H 22   ˆ
                                                     H 32
                                            
 m             m1m2 m3               m2 m3 m1m3 m1m2
Cyclotron resonance effective mass –to
        be shown in the tutorial
Density of states effective mass

             Ec  E f 
  n  Nce
                           kT


                                32
          2 m c kT  *
  Nc  2      2     
          h         
         Density of states
       number of allowed energy states
g(E) 
                dEdV


                 V- Volume
                 E- Energy
      Example: density of states of
            hydrogen gas
g ( E )  n gi ( E  Ei )
           i

Ei - allowed energy level
gi - number of allowed states in energy level i (degeneracy)
n - density of hydrogen atoms    E




                                                       g(E)
Density of states of solids
 E




                    g(E)
     Density of electrons in an energy
                   band

                                            g  E  f  E  dE
 E
                                  n
                                          Band

Ef
                                1.2

                                 1

                                0.8
                         f(E)   0.6

                                0.4

                                0.2

                                 0
                  g(E)                0     0.5    1     1.5   2
                                                  E/Ef
     Density of holes in an energy band

                                             g  E [1  f  E ]dE
 E
                         p
                                        Band

Ef
                                  1.2

                                   1

                                  0.8


                         1-f(E)
                                  0.6

                                  0.4

                                  0.2

                                   0
                  g(E)                  0       0.5    1     1.5   2
                                                      E/Ef
Approximation for the Fermi Dirac
    distribution for E-Ef>3KT


                        1
f FD  E             E  EF 
                                        e    EF  E    kT
                                                                f MB  E 
               1 e                kT
  Density of electrons in an energy
                band      E


                                         Ef

n     g  E  f  E  dE
     Band
                                                     g(E)

n     g  E  exp[( E  E
     Band
                               f   ) / KT )]dE

n  NC exp[( EC  E f ) / KT ]

NC      g  E  exp[( E  E
       Band
                                   C   ) / KT )]dE
Density of states of solids in K
            space

number of allowed energy states      1
                                2
        dk x dk y dk z dV          (2 )3


         V- Volume
   The density of states in an energy interval is
proportional to the volume in K space between two
            constant energy surfaces



                        E2
                1
            2
              (2 ) 3   
                        E1
                          dk x dk y dk z
Constant energy surfaces in crystal
   momentum space cookies
Volume of an ellipsoid

 x2 y 2 z 2
   2
      2  2 1
 a    b   c



            4
 Volume=     abc
             3
Volume of a constant energy (E’)
            ellipsoid
                 2
                       k12 k2 2 k32 
E '  k   Ec ,v                  
                    2  m1 m2 m3 
        k12                k2 2           k3 2
                                                   1
2m1 E ' Ec ,v       2m2 E ' Ec ,v   2m3 E ' Ec ,v
       2                 2                 2




            4
   E ' 
                                  3
              3
                8m1m2 m3 E  Ec,v
            3
Density of states near a conduction band minimum
 or valence band maximum, and the definition of
        the density of states effective mass


                             N elipso
            g c ,v  E                 2m1m2 m3 E  Ec ,v
                                2 3




            g c ,v  E 
                                 1
                                                               
                                                                    32
                                        2 E  Ec ,v m   *

                                2 3                     c ,v




                           Nelipso          m1m2m3 
                 *                       23              13
             m    c ,v
Conduction band density of states
         effective mass
 n     g  E  f  E  dE
      Band

 n  NC exp[( EC  E f ) / KT ]

 NC      g  E  exp[( E  E
        Band
                                  C   ) / KT )]dE

                        32
         2 m c kT 
                *
 Nc  2      2     
         h         
Valence band density of states
       effective mass
p      g  E [1  f  E ]dE
     Band

p  NV exp[( E f  EV ) / KT ]

NV      g  E  exp[( E
       Band
                            V    E ) / KT )]dE

                       32
        2 m kT 
                *
NV  2         2
                 V
                 
           h    
                  Summary
• Conductivity effective        1      1   2
  mass – determines               *
                                        
                                m     3ml 3mt
  mobility.
• Density of states
  effective mass –         m   Nelipso           m1m2m3 
                            *                 23                13
                             c
  determines NC
• Cyclotron effective
  mass – can be             
                           m 
                                  ˆ
                                          m1m2 m3
                                           ˆ          ˆ
                                  H12 m1  H 2 2 m2  H 32 m3
  measured directly
      Elective home exercise
• Derive conductivity and density of states
  effective mass for holes.
                                       k



                                  hh

                          lh




                     E

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:5
posted:1/19/2012
language:English
pages:47