Lecture 4
Electrons and Holes in Semiconductors
In this lecture you will learn:
• Generation-recombination in semiconductors in more detail
• The basic set of equations governing the behavior of electrons and holes
in semiconductors
• Shockley Equations
• Quasi-neutrality in conductive materials
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Majority and Minority Carriers
In N-doped Semiconductors:
Electrons are the majority carriers
Holes are the minority carriers
In P-doped Semiconductors:
Holes are the majority carriers
Electrons are the minority carriers
Golden Rule of Thumb:
When trying to understand semiconductor devices, always
first see what the minority carriers are doing
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
1
Generation and Recombination in Semiconductors
Generation process:
+
Rate = G
Recombination process:
We can write the following
equations for the carrier
densities:
n t 
GR
t
p t 
GR
t
These equations tell how the
electron and hole densities
change in time as a result of
recombination and
generation processes
+
Rate = R
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Generation and Recombination in Thermal Equilibrium
• From the first lecture, in thermal equilibrium:
The recombination rate  Ro  k no po
equals the generation rate  Go
i.e. Go  Ro
• Then in thermal equilibrium:
no t 
 Go  Ro  0
t
po t 
 Go  Ro  0
t
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
2
Light Absorption in Semiconductors
Generation of electrons and holes by photons in semiconductors:

Photon
  E g
A perfect silicon crystal
lattice
Negatively charged
free electron
Positively charged
hole
+
A silicon crystal lattice with one broken bond
(one electron and one hole)
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Generation and Recombination Out of Thermal Equilibrium
1) Consider a P-doped slab of Silicon:
 no  po 
Electron-hole recombination rate in thermal equilibrium  Ro  k no po
equals the generation rate  Go  k ni2
2) Now turn light on at time t = 0:
light
• Light breaks the Si-Si covalent bonds and generates excess electron-hole pairs
• The net generation rate now becomes: G  Go  GL
extra part
3) Mathematical model of the above situation:
n  no  n' t 
p  po  p' t 
• n’(t ) and p’(t) are the excess electron and hole densities
• It must be that: n' t   p' t 
• We also assume that: n' t , p' t   po
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
3
Generation and Recombination Out of Thermal Equilibrium
• We can use the equations:
nt 
GR
t
n  no  n' t 
pt 
GR
t
p  po  p' t 
• Generation rate:
G  Go  GL
• Recombination rate:
Assumptions: n' , p'  po
Rknp
 k no  n'  po  p'
1
 k no  n' po
n
 k no po  k n' po
 Ro 
n'
 n is the lifetime of the
minority carriers (i.e. electrons)
 k po
The excess recombination rate is proportional
to the excess MINORITY carrier density
n
• The equation for excess minority carriers (i.e. electrons) becomes:
n' t 
n' t 
 GL 
n
t
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Generation and Recombination Out of Thermal Equilibrium
n' t 
n' t 
 GL 
t
n
• Solution with the boundary condition, n' t  0   0 , is:
n’(t )

 


n' t   GL  n  1  e  n 




t
light turned-on at t = 0
GL  n
t 0
t
• Excess hole density is, of course :
p' t   n' t 
• As t  
the excess electron and hole densities reach a steady state value
n' t     GL  n
p' t     GL  n
and
nt     no  GL  n
pt     po  GL  n
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
4
Generation and Recombination Out of Thermal Equilibrium
Now suppose that light had been turned-on for a very very long time and it was
turned-off at time t = 0
At time t = 0 : n'  GL  n
n  no  GL  n
and
p'  GL  n
n’(t )
p  po  GL  n
light turned-off at t = 0
??
t 0
t
• Since n  no , and p  po , the carrier densities are not equal to their thermal
equilibrium values. Thermal equilibrium must get restored since the light has been
turned-off
Question: How does thermal equilibrium gets restored??
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Generation and Recombination Out of Thermal Equilibrium
• We can use the equations:
nt 
GR
t
pt 
GR
t
n  no  n' t 
p  po  p' t 
• Generation rate:
G  Go
• Recombination rate:
Assumptions: n' , p'  po
Rknp
 k no  n'  po  p'
 k no  n' po
 k no po  k n' po
 Ro 
The excess recombination rate is proportional
to the excess MINORITY carrier density
n'
n
• The equation for excess minority carriers (i.e. electrons) becomes:
n' t 
n' t 

n
t
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
5
Generation and Recombination Out of Thermal Equilibrium
n' t 
n' t 

n
t
• Solution is:

t
n' t   n' t  0  e  n
n’(t )

Excess electron density decays exponentially
to zero from its initial value
light turned-off at t = 0
The excess carrier densities
decay with time and thermal
equilibrium values for carrier
densities are restored
t 0
t

t
• The excess hole density will also decay in the same way: p' t   p' t  0  e  n
• As t  
the electron and hole densities reach their equilibrium values:
n' t     0
and
p' t     0
n t     no
p t     po
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Generation and Recombination in Doped Semiconductors
Whenever you have to find an expression for R use the following recipe:
• If it is a p-doped semiconductor:
R  Ro 
n'  x , t 
n
 n is the minority carrier lifetime
• If it is a n-doped semiconductor:
R  Ro 
p'  x , t 
p
 p is the minority carrier lifetime
The excess recombination rate (i.e. R - Ro ) is always proportional to the excess
MINORITY carrier density
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
6
Electron and Hole Current Density Equations
From last lecture…………
J n  x   q n x  n E  x   q Dn
d n x 
dx
1
J p  x   q p  x   p E  x   q Dp
d p x 
dx
2
These are two of Shockley’s equations !
Shockley, Bardeen, and Brattain
from Bell Labs were awarded the
Nobel Prize for inventing the
semiconductor transistor
William
Shockley
John
Bardeen
Walter
Brattain
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Electron and Hole Current Continuity Equations
• You have already seen the equations:
n  x , t 
GR
t
p  x , t 
GR
t
These equations tell how the electron and hole densities change in time as a
result of recombination and generation processes.
• Carrier densities can also change in time if the current densities change in
space !!!
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
7
Electron and Hole Current Continuity Equations
Consider the infinitesimal strip between x and x+x
x
Jp x 
x
x  x
J p  x  x 
The difference in hole fluxes at x and x+x must result in piling up of holes
in the infinitesimal strip …….
Note that q p  x , t  is the hole
 p  x , t  x
J p  x , t   J p  x  x , t   q
charge density


J p  x  x , t   J p  x , t 
J p  x , t 
x
x
q
 p x , t 
t
t
 p x , t 
q
t
 p x , t 
1 J p  x , t 

t
q
x
Now add recombination and generation to the above equation:
 p x , t 
1 J p  x , t 
GR
t
q
x
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Electron and Hole Current Continuity Equations - III
One can do the same for electrons as well……
x
Jn  x 
x
x  x
J n  x  x 
 n x , t 
1 J n  x , t 
GR
t
q
x
So now we have two new equations,
 p x , t 
1 J p  x , t 
GR
t
q
x
3 
 n x , t 
1 J n  x , t 
GR
t
q
x
4 
These are two more of Shockley’s equations !
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
8
Gauss’s Law and Electrostatics
The net charge density in a semiconductor is,
  x , t   q   Nd  x   Na  x   p x , t   n x , t  
Gauss’s Law in differential form:
 E x, t   x , t 

s
x
 E  x , t  q  Nd  x   N a  x   p  x , t   n  x , t  

x
s
5 
This is the fifth and the last of the Shockley’s equations !
 o  8.85  10 12 Farads/m
For Silicon:  s  11.7 o
 8.85  10 14 Farads/cm
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
The Five Shockley Equations
J n  x , t   q n x , t  n E  x , t   q Dn
d n x , t 
dx
1
J p  x , t   q p x , t   p E  x , t   q Dp
d p x , t 
dx
2
 p x , t 
1 J p  x , t 
GR
t
q
x
3 
 n x , t 
1 J n  x , t 
GR
t
q
x
4 
 E  x , t  q  Nd  x   N a  x   p  x , t   n  x , t  

x
s
5 
Using these equations one can understand the behavior of semiconductor
microelectronic devices !!
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
9
Quasi-Neutrality
Materials with large conductivities are “quasi-neutral”
“Quasi-neutrality” implies that there cannot be large charge densities or electric fields
inside a conductive material
Lets see why this is true…..and how deviations from quasi-neutrality disappear…….
Consider an infinite and conductive N-doped semiconductor with a net charge
density at time t=0:
N-doped


s
Charge density
The charge density will generate electric fields (by Gauss’ law):
N-doped


E
s
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Quasi-Neutrality
The electric field will generate electrical currents:
J n  x , t   q n  x , t   n E  x , t   E  x , t 
N-doped

E

s
Jn
The electrical currents will pile electrons on top of the charge density and neutralize it
and then there is no charge density left in the medium…………
N-doped

s
This whole process takes a time of the order of the dielectric relaxation time  d :
d 
s
~ 10 15  10 13 Seconds

ECE 315 – Spring 2005 – Farhan Rana – Cornell University
10
Appendix: Restoration of Quasi-Neutrality
N-doped

E

s
Jn
From Gauss’ law:

 
 r , t 
.E r , t  
s
Current equation:
 
 
J r , t   E r , t 
Use the continuity equation for charge:

 
 


 r , t 
 r , t 
 .J r , t   .E r , t   
s
t


s
 r , t   r , t 



0
 d 
d

t

Solution:


 r , t    r , t  0 e

t
d
Charge density in a
conductive medium
disappears on a time
scale of d
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
11