Lecture 10
Equations of State, Minority Carrier Diffusion
Equation and Quasi-Fermi Levels
Reading:
(Cont’d) Notes and Anderson2 sections 3.4-3.11 and
chapter 4
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ECE 3080 - Dr. Alan Doolittle
Ways carrier concentrations can be altered
Continuity Equations
Drift Current out
∆n, ∆p due to Thermal
R-G, Direct
recombination
light and other
processes
Drift Current in
Diffusion Current in
∂n ∂n
=
∂t ∂t
∂p ∂p
=
∂t ∂t
Drift
Drift
+
∂n
∂t
∂p
+
∂t
Diffusion
Diffusion
+
∂n
∂t
∂p
+
∂t
Re combination − Generation
Re combination − Generation
Diffusion Current out
∂n
∂t
All other processes
such as light , etc ...
∂p
+
∂t
All other processes
such as light , etc ...
+
There must be spatial and time continuity in the carrier concentrations
Georgia Tech
ECE 3080 - Dr. Alan Doolittle
Ways carrier concentrations can be altered
Continuity Equations
∂n
∂t
∂p
∂t
Drift
Drift
∂n
+
∂t
∂p
+
∂t
Diffusion
Diffusion
1 ⎛ ∂J Nx ∂J Ny ∂J Nz ⎞ 1
⎟⎟ = ∇ ⋅ J N
= ⎜⎜
+
+
q ⎝ ∂x
∂y
∂z ⎠ q
1 ⎛ ∂J Px ∂J Py ∂J Pz ⎞
1
⎟⎟ = − ∇ ⋅ J P
= − ⎜⎜
+
+
q ⎝ ∂x
q
∂y
∂z ⎠
Divergence in the current
∂n 1
∂n
= ∇⋅ JN +
∂t q
∂t
Re combination −Generation
∂p
∂p
1
= − ∇⋅ JP +
∂t
∂t
q
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∂n
+
∂t
Re combination − Generation
All other processes
such as light , etc ...
∂p
+
∂t
All other processes
such as light , etc ...
ECE 3080 - Dr. Alan Doolittle
Continuity Equations: Special Case known as “Minority Carrier
Diffusion Equation”
Simplifying Assumptions:
1) One dimensional case. We will use “x”.
Continuity Equation
Minority Carrier
Diffusion Equation
2) We will only consider minority carriers
3) Electric field is approximately zero in regions subject to
analysis.
4) The minority carrier concentrations IN EQUILIBRIUM are not
a function of position.
5) Low-level injection conditions apply.
6) SRH recombination-generation is the main recombinationgeneration mechanism.
7) The only “other” mechanism is photogeneration.
Georgia Tech
ECE 3080 - Dr. Alan Doolittle
Continuity Equations: Special Case known as “Minority Carrier
Diffusion Equation”
Because of (3) - no electric field …
Since E = 0, ...
0
0
J n = J n | Drift + J n | Diffusion = qµ n nE + qDn ∇n
∂n
J n = J n | Diffusion = qDn
∂x
∂ 2 ( n o + ∆n )
1
1 ∂J N
∂ 2 ( ∆n )
∂ 2n
= Dn
= Dn 2 = Dn
∇⋅ JN =
2
q
q ∂x
∂x
∂x 2
∂x
Georgia Tech
ECE 3080 - Dr. Alan Doolittle
Continuity Equations: Special Case known as “Minority Carrier
Diffusion Equation”
Because of (5) - low level injection …
∂n
∂t
Re combination −Generation
=−
∆n
τn
Because of (7) - no other process …
∂n
∂t
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All other processes
such as light , etc ...
= GL
ECE 3080 - Dr. Alan Doolittle
Continuity Equations: Special Case known as “Minority Carrier
Diffusion Equation”
Finally …
∂n ∂ (no + ∆n) ∂ (∆n)
=
=
∂t
∂t
∂t
∂n ∂n
=
∂t ∂t
Drift
∂n
+
∂t
Diffusion
∂ (∆n p )
∂n
+
∂t
∂ 2 (∆n p )
2
−
All other processes
such as light , etc ...
(∆n p )
+ G L Minority
∂t
τn
∂x
Carrier
Diffusion
or
Equations
2
∂ (∆p n )
∂ (∆p n ) (∆p n )
= DP
−
+ GL
2
∂t
τp
∂x
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= DN
Re combination −Generation
∂n
+
∂t
ECE 3080 - Dr. Alan Doolittle
Continuity Equations: Special Case known as “Minority Carrier
Diffusion Equation” - Further simplifications
Steady State …
∂ (∆n p )
∂t
→ 0 and
∂ (∆p n )
→0
∂t
No minority carrier diffusion gradient …
∂ 2 (∆n p )
∂ 2 ( ∆p n )
DN
→0
→ 0 and D P
2
2
∂x
∂x
No SRH recombination-generation …
∆n p
τn
= 0 and
∆p n
τp
=0
No Light …
GL → 0
Georgia Tech
ECE 3080 - Dr. Alan Doolittle
Solutions to the Minority Carrier Diffusion Equation
Consider a semi-infinite p-type silicon sample with NA=1015 cm-3
constantly illuminated by light absorbed in a very thin region of the
material creating a steady state excess of 1013 cm-3 minority carriers.
What is the minority carrier distribution in the region x>0?
Light
x
Light absorbed
in a thin skin
∂ ( ∆n p )
∂t
0
= DN
DN
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∂ ( ∆n p )
2
∂x
∂ 2 ( ∆n p )
∂x
2
2
=
−
( ∆n p )
τn
0
+ GL
( ∆n p )
τn
ECE 3080 - Dr. Alan Doolittle
Solutions to the Minority Carrier Diffusion Equation
General Solution …
∆n p ( x) = Ae ( − x / LN ) + Be ( + x / LN )
where LN ≡ Dnτ n
LN is the “diffusion length” the average distance a minority carrier
can move before recombining with a majority carrier.
∆n p ( x = ∞) = 0 = A(0) + Be (+ ∞ / LN )
⇒B=0
∆n p ( x) = 1013 e (− x / LN )
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cm −3
∆np(x)
Boundary Condition …
∆n p ( x = 0) = 1013 cm −3 = A + B
x
ECE 3080 - Dr. Alan Doolittle
Solutions to the Minority Carrier Diffusion Equation
Consider a p-type silicon sample with NA=1015 cm-3 and minority carrier
lifetime τ=1 uS constantly illuminated by light absorbed uniformly
throughout the material creating an excess 1013 cm-3 minority carriers per
second. The light has been on for a very long time. At time t=0, the light
is shut off. What is the minority carrier distribution in for t<0?
Light absorbed uniformly
x
0
0
2
∂ ( ∆n p )
∂ ( ∆n p ) ( ∆n p )
−
+ GL
= DN
2
τn
∂t
∂x
∆n p (all x, t < 0) = GLτ n = 10 7 cm −3
Georgia Tech
ECE 3080 - Dr. Alan Doolittle
Solutions to the Minority Carrier Diffusion Equation
In the previous example: What is the minority carrier distribution in for t>0?
∂t
= DN
∂x
2
−
( ∆n p )
[
τn
]
∆n p (t ) = ∆n p (t = 0) e
∆n p (t ) = 10 e
7
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(
−t
0
+ GL
⎛− t ⎞
⎜
τ n ⎟⎠
⎝
1e −6
)
∆np(all x)
∂ ( ∆n p )
0
∂ 2 ( ∆n p )
t
ECE 3080 - Dr. Alan Doolittle
Quasi - Fermi Levels
Equilibrium:
no = ni e
(
)
⎛ E f − Ei
⎞
⎟
⎜
kT
⎝
⎠
Non-equilibrium: ⎛ ( FN − Ei )
⎜
n = ni e ⎝
⎞
kT ⎟⎠
(
and
and
N-type Low level injection
Ec
Ef, FN
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FN
Ef
Ei
FP
Ev
p = ni e
⎛ ( Ei − FP ) ⎞
⎜
kT ⎟⎠
⎝
N-type High level injection
Ec
Ei
po = ni e
)
⎛ Ei − E f
⎞
⎜
⎟
kT
⎝
⎠
Ev
FP
ECE 3080 - Dr. Alan Doolittle
Quasi - Fermi Levels
p = ni e
⎛ ( Ei − FP ) ⎞
⎜
kT ⎟⎠
⎝
ni ⎛⎜⎝ ( Ei − FP ) kT ⎞⎟⎠
(∇Ei − ∇FP )
∇p =
e
kT
⎛ qp ⎞
⎛ p ⎞
∇p = ⎜ ⎟ E − ⎜ ⎟∇FP
⎝ kT ⎠
⎝ kT ⎠
J p = qµ p pE − qD p ∇p
qD p ⎞
⎛
⎛ qD p ⎞
⎟⎟ pE + ⎜⎜
⎟⎟ p∇FP
J p = q⎜⎜ µ p −
kT ⎠
⎝
⎝ kT ⎠
qD P
but... µ p =
leading to...
kT
J p = µ p p∇FP and similarly J N = µ n n∇FN
If there is a change in quasi-fermi levels, current flows!
Georgia Tech
ECE 3080 - Dr. Alan Doolittle