Chap. 4 Excess carriers in semiconductors
Objectives
1. Understand how photons interact with direct and indirect band gap semiconductors
2. Understand generation-recombination of excess carriers, possibly through trap sites
3. Introduce quasi-Fermi levels in non-equilibrium
4. Calculate diffusion currents from carrier concentration gradients and diffusivity
5. Use the continuity equation to study time dependence of carrier concentration
4.1 Optical absorption
dI ( x)
 I ( x )
dx
I ( x)  I 0 e x

I t  I 0 e l
4.2 Luminescence
4.2.1 Photoluminescence
4.2.2 Electroluminescence
4.3 Carrier lifetime and photoconductivity
4.3.1 Direct recombination of electrons and holes
dn (t )
  r ni2   r n(t )p(t )
dt
dn(t )
  r ni2   r [n 0  n(t )][p0  p(t )]
dt
  r [(n 0  p0 )n(t )  n 2(t )]
If p-type
dn(t )
  r p0n(t )
dt
n(t )  ne r p t  ne t / n
0
4.3.2 Indirect recombination; Trapping
Photoconductive decay
4.3.3 Steady state carrier generation; Quasi-Fermi levels
g (T )   r ni2   r n0 p0
g (T )  g op   r np   r (n0  n)( p0  p )
for steady state recombinat ion and no trapping, n  p
g (T )  g op   r n0 p0   r [(n0  p0 )n  n 2 ]
g op   r (n0  p0 )n 
n  p  g op n
n
n
quasi - Fermi levels
n0  ni e
( E F  Ei ) / kT
p0  ni e ( Ei  EF ) / kT
n  ni e ( Fn  Ei ) / kT
p  ni e
( Ei  F p ) / kT
Quasi-Fermi levels are steady state analogues of the
equilibrium Fermi level.
Fn=Fp=EF at equilibrium.
4.3.4 Photoconductive devices
Light detector
Exposure meter
Optical sensitivity by examining steady state excess carrier concentrations by g op
n   n g op and p   p g op
  qg op ( n  n   p  p )
 n of InSb  105 cm 2 / V  s
Time response of photoconductive cell: recombination time, carrier trapping, carrier drift
Proper choice of material and geometry
Response time vs. sensitivity
4.4 Diffusion of carriers
4.4.1 Diffusion processes
Natural result of random motion of individual molecules
Net diffusion continues as long as gradients exist.
1
1

n1l A  n2l A
2
2
l
n ( x0 )  n1  n2 
2t
n( x)  n( x  x)
n1  n2 
l
x
l2
n( x)  n( x  x)
l 2 dn( x)
n ( x)  lim

2t x0
x
2t dx
l2
: electron diffusion coefficien t, D n
2t
dn( x)
n ( x)   Dn
dx
dp ( x)
 p ( x)   D p
dx
dn( x)
dn( x)
J n (diff .)  (q ) Dn
  qDn
dx
dx
dp ( x)
dp ( x)
J p (diff .)  ( q ) D p
 qD p
dx
dx
4.4.2 Diffusion and drift of carriers; Built-in fields
dn( x)
dx
dp( x)
J p ( x)  q p p( x) ( x)  qD p
dx
J ( x)  J n ( x)  J p ( x)
J n ( x)  q n n( x) ( x)  qDn
Minority carriers can contribute to current.
V ( x) 
E ( x)
q
 ( x)  
dV ( x)
d  E  1 dEi
  i 
dx
dx   q  q dx
at equilibriu m,
 ( x) 
Dp
 ( x) 
D p 1  dEi dEF 



 p kT  dx
dx 
D

D


p
1 dp ( x)
p ( x) dx
kT
: Einstein relation
q
 0.026eV
See table 4-1
4.4.3 Diffusion and recombination; The continuity equation
p
t

x  x  x
1 J p ( x)  J p ( x  x) p

q
x
p
p ( x, t ) p
1 J p p


 , as x approaches zero.
t
t
q x  p
Continuity equation for holes
n 1 J n n


t
q x  n
Diffusion equation for electrons,
n
 2n n
 Dn

t
x 2  n
p
 2p p
 Dp

t
x 2  p
4.4.4 Steady state carrier injection; Diffusion length
d 2n
n
n


dx 2
Dn n L2n
Lp: average distance a hole diffuses before recombination
d p
p
p


dx 2
D p p L2p
2
p( x)  C1e
x / Lp
p( x)  pe
 C2 e
 x / Lp
 x / Lp
4.4.5 The Haynes-Shockley experiment
Drift and diffusion of minority carriers
L
vd 
td
p 
vd

time dependent diffusion eq.
p ( x, t )
 2p
 Dp
t
x 2
 P   x 2 / 4 D t
p
e
p ( x, t )  
 2 D p t 
 P   x 2 / 4 D t
p
e
p( x, t )  
 2 D p t 
x  tvd  t
L
td
4.4.6 Gradients in the Quasi-Fermi levels
dn( x)
dx
n( x)  dFn dEi 




kT  dx
dx 
J n ( x)  q n n( x) ( x)  qDn

dn( x) d

ni e ( Fn  Ei ) / kT
dx
dx

 dF dE 
J n ( x)  q n n( x) ( x)   n n( x) n  i 
dx 
 dx
dF
J n ( x )   n n( x ) n
dx
a modified Ohm' s law,
d ( Fn / q )
d ( Fn / q )
  n ( x)
dx
dx
d ( Fp / q )
d ( Fp / q )
J p ( x)  q p p ( x)
  p ( x)
dx
dx
J n ( x)  q n n( x)
A lack of current implies constant quasi-Fermi levels.
Same as electrochemical potential