Lecture #9
OUTLINE
• Continuity equations
• Minority carrier diffusion equations
• Minority carrier diffusion length
• Quasi-Fermi levels
Read: Sections 3.4, 3.5
Derivation of Continuity Equation
• Consider carrier-flux into/out-of an infinitesimal volume:
Area A, volume Adx
JN(x)
JN(x+dx)
dx
1
n
 n 
Adx    J N ( x) A  J N ( x  dx) A 
Adx
q
n
 t 
Spring 2007
EE130 Lecture 9, Slide 2
J N ( x)
J N ( x  dx)  J N ( x) 
dx
x
n 1 J N ( x) n



t q x
n
Continuity
Equations:
Spring 2007
n 1 J N ( x) n


 GL
t q x
n
p
1 J P ( x) p


 GL
t
q x
p
EE130 Lecture 9, Slide 3
Derivation of Minority Carrier Diffusion Equation
• The minority carrier diffusion equations are derived
from the general continuity equations, and are
applicable only for minority carriers.
• Simplifying assumptions:
– The electric field is small, such that
n
n
J N  q n n  qDN
 qDN
in p-type material
x
x
p
p
J P  q p p  qDP
 qDP
in n-type material
x
x
– n0 and p0 are independent of x (uniform doping)
– low-level injection conditions prevail
Spring 2007
EE130 Lecture 9, Slide 4
• Starting with the continuity equation for electrons:
n 1 J N ( x) n


 GL
t q x
n
 n0  n  1  
 n0  n   n

qDN

 GL


t
q x 
x
 n
n
 2 n n
 DN

 GL
2
t
x
n
Spring 2007
EE130 Lecture 9, Slide 5
Carrier Concentration Notation
• The subscript “n” or “p” is used to explicitly
denote n-type or p-type material, e.g.
pn is the hole (minority-carrier) concentration in ntype material
np is the electron (minority-carrier) concentration in
n-type material
• Thus the minority carrier diffusion equations are
n p
t
 DN
 2 n p
x
2

n p
n
 GL
pn
 2 pn pn
 DP

 GL
2
t
x
p
Spring 2007
EE130 Lecture 9, Slide 6
Simplifications (Special Cases)
•
•
n p
pn
Steady state:
0
0
t
t
2
2
 n p
 pn
No diffusion current: DN
 0 DP
0
2
2
x
x
• No R-G:
• No light:
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n p
n
0
p n
p
0
GL  0
EE130 Lecture 9, Slide 7
Example
• Consider the special case:
– constant minority-carrier (hole) injection at x=0
– steady state; no light absorption for x>0 pn (0)  pn 0
 pn pn
0  DP

2
x
p
2
 2 pn
pn
pn

 2
2
x
DP p LP
LP is the hole diffusion length:
Spring 2007
EE130 Lecture 9, Slide 8
LP  DP p
 2 pn pn
 2
The general solution to the equation
2
x
LP
is
pn ( x)  Ae
 x / LP
 Be
x / LP
where A, B are constants determined by boundary conditions:
pn ()  0  B  0
pn (0)  pn 0  A  pn 0
Therefore, the solution is
pn ( x)  pn 0 Ae
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 x / LP
EE130 Lecture 9, Slide 9
Minority Carrier Diffusion Length
• Physically, LP and LN represent the average distance
that minority carriers can diffuse into a sea of majority
carriers before being annihilated.
• Example: ND=1016 cm-3; p = 10-6 s
Spring 2007
EE130 Lecture 9, Slide 10
Quasi-Fermi Levels
• Whenever n = p  0, np  ni2. However, we
would like to preserve and use the relations:
n  ni e
( E F  Ei ) / kT
p  ni e
( Ei  E F ) / kT
• These equations imply np = ni2, however. The
solution is to introduce two quasi-Fermi levels FN
and FP such that
n  ni e ( FN  Ei ) / kT
p  ni e
n
FN  Ei  kT ln  
 ni 
 p
FP  Ei  kT ln  
 ni 
Spring 2007
EE130 Lecture 9, Slide 11
( Ei  FP ) / kT
Example: Quasi-Fermi Levels
Consider a Si sample with ND = 1017 cm-3 and
n = p = 1014 cm-3.
What are p and n ?
What is the np product ?
Spring 2007
EE130 Lecture 9, Slide 12
• Find FN and FP :
n
FN  Ei  kT ln  
 ni 
 p
FP  Ei  kT ln  
 ni 
Spring 2007
EE130 Lecture 9, Slide 13
Summary
• The continuity equations are established based on
conservation of carriers, and therefore are general:
n 1 J N ( x) n


 GL
t q x
n
p
1 J P ( x) p


 GL
t
q x
p
• The minority carrier diffusion equations are
derived from the continuity equations, specifically for
minority carriers under certain conditions (small Efield, low-level injection, uniform doping profile):
n p
t
 DN
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 2 n p
x
2

n p
n
 GL
pn
 2 pn pn
 DP

 GL
2
t
x
p
EE130 Lecture 9, Slide 14
• The minority carrier diffusion length is the average
distance that a minority carrier diffuses before it
recombines with a majority carrier:
LP  DP p
LN  DN n
• The quasi-Fermi levels can be used to describe the
carrier concentrations under non-equilibrium conditions:
n
FN  Ei  kT ln  
 ni 
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 p
FP  Ei  kT ln  
 ni 
EE130 Lecture 9, Slide 15