Things you should know when you leave…
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M.J. Gilbert ECE 340 – Lecture 19 10/10/12

Diffusion and Recombination
For the last 2 lectures we’ve been discussing diffusion…
Define the carrier flux for electrons and holes:
n
=
D n dn ( ) dx
p
D p dp
= dx
And the corresponding current densities associated with diffusion…
J
J n diff p diff
= qD n dn dx
= − qD n dp dx
M.J. Gilbert
Carriers move together, currents opposite directions.
ECE 340 – Lecture 19 10/10/12

Diffusion and Drift of Carriers
And what happens when drift and diffusion are occurring simultaneously… e-­‐
E
The total current must be the sum of the electron and hole currents resulting from the drift and diffusion processes h+   n(x)
J n
= q µ n n ( ) ( ) + qD n dn
φ p
(diff) φ p
(drift) dx p(x)
J n
( )
= q µ p p ( ) ( ) x
Drift
Where are the particles and currents flowing?
qD p dp dx
Diffusion
φ n
(diff)
J =
Electrons
Holes
J n
+ J p
φ n
(drift)
J p
(diff) J p
(drift) J n
(diff) J n
(drift)
Dashed Arrows = Particle Flow !
!!
M.J. Gilbert
!
Solid Arrows = Resulting Currents
ECE 340 – Lecture 19
!
10/10/12

Diffusion and Recombination
But from previous lectures, we know more in happening…
We have completely ignored recombination !
!
Type 1: Direct recombination
•
Electron and hole drift into the same vicinity and recombine.
•
They can give off light if the semiconductor has a direct bandgap.
Type 2: R-G Center recombination
•
R-G centers may be impurity atoms or lattice defects.
•
Create states in the band gap.
•
Electrons see a potential well and get trapped losing energy. Holes are attracted to the electron and annihilates it giving off heat to the lattice.
Type 3: Auger recombination
•
Collision between two like carriers.
•
Energy released by recombination is given to the surviving carrier.
•
Surviving electron then loses excess energy through lattice collisions.
M.J. Gilbert ECE 340 – Lecture 19 10/10/12

Diffusion and Recombination
So what does this mean?
•
The hole current density leaving the differential area may be larger or smaller than the current density that enters the area.
•
This is a result of recombination and generation.
•
Net increase in hole concentration per unit time, dp/dt, is difference between hole flux per unit volume entering and leaving, minus the recombination rate.
M.J. Gilbert ECE 340 – Lecture 19 10/10/12

Diffusion and Recombination
How can we explain this?
The net increase in hole concentration per unit time is the difference between the hole flux entering and leaving minus the recombination rate…
∂ p
∂ t x → x + Δ x
1
= q
J
P
− J
P
( x +
Δ x
Δ x )
−
Δ p
τ p
Rate of hole buildup.
Increase in hole concentration in
Δ xA per unit time.
Recombination rate
As Δ x goes to zero, we can write the change in hole concentration as a derivative, just like in diffusion…
These relations form the continuity equations .
∂
∂ p n
∂
∂
( ) t
( ) t
,
, t t
=
=
∂
∂
∂
∂ t n t p
=
=
−
1 q
1 q
∂
∂
J
∂ x
∂
N
J x
P
−
−
τ
Δ
N
Δ p
τ n
P
Holes
Electrons
M.J. Gilbert ECE 340 – Lecture 19 10/10/12

Diffusion and Recombination
Are there any simplifications?
If the current is carried mainly by diffusion (small drift) we can replace the currents in the continuity equation…
J n diff
J p diff
=
= qD
N
− qD
P
∂ n
∂ x
∂ p
∂ x
J n
J n
= q µ n n ( ) ( ) +
= q µ p p qD n
qD p dn dx dp dx
We put this back into the continuity equations …
∂ p
∂ t
∂ n
∂ t
=
∂ p
∂ t
= −
1 q
∂ J
∂ x
P
−
Δ p
τ
P
Diffusion equation for electrons
∂
∂ t n
= D
N
∂
2
∂ x n
2
=
∂ n
∂ t
=
1 q
∂ J
∂ x
N
−
τ
Δ n
N Diffusion equation for holes
∂ p
∂ t
D
P
∂
2
=
∂ x
Useful mathematical equation for many different physical situations…
2 p
M.J. Gilbert ECE 340 – Lecture 19
−
−
Δ
Δ
N
P n p
10/10/12

Steady State Carrier Injection
To this point, we been assuming that the perturbation was removed…
∂ n
∂ t
∂ p
∂ t
= D
N
= D
P
∂
2 n
∂ x 2
∂ n
−
N
∂
2
∂ x 2 p ∂ p
−
P
What happens if we keep the perturbation?
•
The time derivatives disappear d 2 n dx 2
=
Δ n
D
N
τ
N d 2 dx 2 p
=
Δ p
D
P
τ
P
≡
Δ n
L 2
N
≡
Δ p
L 2
P
M.J. Gilbert
Electrons
Holes
Where
L
L
N
P
=
=
D
N
D
P
P
N
Diffusion Length
ECE 340 – Lecture 19 10/10/12

Steady State Carrier Injection
Let’s consider the following situation…
Let’s assume that we are injecting excess holes into a sample of silicon.
•
We do not remove the perturbation, so we maintain a constant excess hole concentration at the injection point, δ p (x = 0) = Δ p.
•
The injected holes then begin to diffuse along the bar recombining with a characteristic lifetime, τ p
.
M.J. Gilbert ECE 340 – Lecture 19 10/10/12

Steady State Carrier Injection
What should we expect?
We expect the excess hole concentration to decay to zero at large distances from the perturbation.
•
This is due to recombination .
Solving the partial differential equation for holes, the solution has the form…
δ p ( )
= x
C
1 e L
P + C
2 e x
−
L
P
The boundary conditions are:
δ p = 0 for large x
δ p = Δ p for x = 0
Get constants from boundary conditions. Which are what?
C1 = 0
C2 = Δ p
Average distance a hole travels before it recombines.
p =
pe x
−
L
P
M.J. Gilbert ECE 340 – Lecture 19 10/10/12

Steady State Carrier Injection
But we really want the average diffusion length…
What is the probability that an injected hole recombines in a particular interval?
We know that the probability that a hole injected at x = 0 survives to x is: p
p
= e
−
L
P x
Ratio of steady state concentrations
We know that the probability that a hole at x recombines in dx is: p ( ) − p ( x + Δ x ) p
=
− ( dp p
/ dx ) dx
= dx
L
P
Multiply the two probabilities…
⎜
⎜
⎝
⎛ e
−
L
P x
⎠
⎞
⎟
⎟
⎛
⎝
⎜⎜ dx
L
P
⎠
⎟⎟
⎞
=
1
L
P x e
−
L
P dx
M.J. Gilbert
Compute expectation value x =
0
∞ x e
−
L
P x
L
P dx
L
P
ECE 340 – Lecture 19
Distance minority carriers diffuse into a sea of majority carriers
10/10/12

Steady State Carrier Injection
The distribution of excess holes causes a current…
We have a diffusion current of holes moving from high concentration to low:
J p diff
= − qD
P
∂ p
∂ x
Plug in what we know about the rate of change of the hole concentration with position:
J p diff
= − qD
P
∂ δ p
∂ x
Begin to reduce the equation…
J p diff
= q
D
P
L
P
Δ pe
−
L
P x
= q
D
P
L
P
δ p
The diffusion current depends on the excess carriers at a point and not on the initial concentration.
M.J. Gilbert ECE 340 – Lecture 19 10/10/12