# The effective mass v2

Document Sample

```					         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
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
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