Lecture 6
OUTLINE
• Semiconductor Fundamentals (cont’d)
– Continuity equations
– Minority carrier diffusion equations
– Minority carrier diffusion length
– Quasi-Fermi levels
– Poisson’s Equation
Reading: Pierret 3.4-3.5, 5.1.2; Hu 4.7, 4.1.3
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 
EE130/230A Fall 2013
Lecture 6, 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:
EE130/230A Fall 2013
n 1 J n ( x) n


 GL
t q x
n
p
1 J p ( x) p


 GL
t
q x
p
Lecture 6, 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:
1. 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  qD p
 qD p
in n-type material
x
x
2. n0 and p0 are independent of x (i.e. uniform doping)
3. low-level injection conditions prevail
EE130/230A Fall 2013
Lecture 6, 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
EE130/230A Fall 2013
Lecture 6, 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 n-type mat’l
np is the electron (minority-carrier) concentration in n-type mat’l
• Thus the minority carrier diffusion equations are
n p
t
 n p
2
 Dn
x
2

n p
n
 GL
pn
 pn pn
 Dp

 GL
2
t
x
p
2
EE130/230A Fall 2013
Lecture 6, Slide 6
Simplifications (Special Cases)
• Steady state:
• No diffusion current:
n p
pn
0
0
t
t
 2 n p
 2 pn
Dn
 0 Dp
0
2
2
x
x
• No R-G:
n p
• No light:
GL  0
EE130/230A Fall 2013
Lecture 6, Slide 7
n
0
p n
p
0
Example
• Consider an n-type Si sample illuminated at one end:
– constant minority-carrier injection at x = 0 pn (0)  pn 0
– steady state; no light absorption for x > 0
Lp is the hole diffusion length: L p  D p p
EE130/230A Fall 2013
Lecture 6, Slide 8
 pn pn
 2
2
x
Lp
2
The general solution to the equation
is
pn ( x)  Ae
 x / Lp
 Be
x / Lp
where A, B are constants determined by boundary conditions:
pn ()  0
pn (0)  pn 0
Therefore, the solution is
pn ( x)  pn0e
EE130/230A Fall 2013
 x / Lp
Lecture 6, 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
EE130/230A Fall 2013
Lecture 6, Slide 10
Summary: Continuity Equations
• The continuity equations are established based on
conservation of carriers, and therefore hold generally:
n 1 J n ( x) n


 GL
t q x
n
p
1 J n ( 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 E-field, low-level injection,
uniform doping profile):
n p
t
 DN
EE130/230A Fall 2013
 2 n p
x
2

n p
n
 GL
pn
 2 pn pn
 DP

 GL
2
t
x
p
Lecture 6, Slide 11
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
EE130/230A Fall 2013
p  ni e
Lecture 6, Slide 12
( 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 ?
EE130/230A Fall 2013
Lecture 6, Slide 13
• Find FN and FP :
n
FN  Ei  kT ln  
 ni 
 p
FP  Ei  kT ln  
 ni 
EE130/230A Fall 2013
Lecture 6, Slide 14
Poisson’s Equation
area A
Gauss’ Law:
 s ( x  x) A   s ( x) A  xA
 ( x  x)   ( x) 
x
E(x+x)
x
s : permittivity (F/cm)
 : charge density (C/cm3)

s

d


dx
s
EE130/230A Fall 2013
E(x)
Lecture 6, Slide 15
Charge Density in a Semiconductor
• Assuming the dopants are completely ionized:
 = q (p – n + ND – NA)
EE130/230A Fall 2013
Lecture 6, Slide 16