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ECE-305: Spring 2016
Semiconductor Equations: II
Professor Peter BermelElectrical and Computer Engineering
Purdue University, West Lafayette, IN [email protected]
Pierret, Semiconductor Device Fundamentals (SDF)pp. 104-124
9/19/2016 1
Bermel ECE 305 F16 2
outline
1. Semiconductor equations
2. Equilibrium versus non-equilibrium
3. Minority carrier diffusion equation
9/19/2016
drift- diffusion equation
3
current = drift current + diffusion current
total current = electron current + hole current
D
p
p D
n
n k
BT q
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continuity equation for holes
4
in-flow
out-flow
p t
recombinationgeneration
in-flow - out-flow + G - Rp
t
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Continuity Equations for Electron/Holes
Bermel ECE 305 F16
Continuity Equations for Electron/Holes
1n N N
nJ g r
t q
+ -
1 - + -
P P P
pJ g r
t q
( )J x
( )J x dx+
Ng
n
Nr
p
( )pJ x
( )pJ x dx+
rp
gp
5
Bermel ECE 305 F166
outline
1. Semiconductor equations
2. Equilibrium versus non-equilibrium
3. Minority carrier diffusion equation
✓
equilibrium (no G-R)
7
in-flow
out-flow
p t
9/19/2016 Bermel ECE 305 F16
current and QFL’s
8
p nieEi-Fp kBT
dp
dx nie
Ei-Fp kBT´1
kBT
dEi
dx-
dFp
dx
p
kBT
dEi
dx-
dFp
dx
dEi
dx qE x
n nieFn-Ei kBT
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Direct Band-to-band Recombination
Bermel ECE 305 F16
Photon(light)
GaAs, InP, InSb (3D)
Lasers, LEDs, etc.
In real space … In energy space …
Photon
9
Indirect Recombination (Trap-assisted)
Bermel ECE 305 F16
Phonon (Thermal Energy)
Ge, Si, ….
Transistors, Solar cells, etc.
10
Auger Recombination
Bermel ECE 305 F16
Phonon (Thermal Energy)
InP, GaAs, …
Lasers, etc.
12
3
1 2
4
3
4
11
Bermel ECE 305 F1612
outline
1. Semiconductor equations
2. Equilibrium versus non-equilibrium
3. Minority carrier diffusion equation
✓
✓
Minority Carrier Equation
Bermel ECE 305 F16
0 0 0
D A
D A
D q p n N N
q p p n p N N
+ -
+ -
- + -
® + D - - D + - ®
0
20
2
1 1p pNN N N N
n
p p p p
N p
n
n n nnr g g
t q t q x
n n n n nD g
t t x
+ D D - + ® - +
+ D D D D - +
JJ
t
t
( ~0)
N N N
N
qn qD n
nqD
x
+
®
J E
E
13
Various approximations …
Bermel ECE 305 F16
2
2
p p p
N p
n
n n nD g
t x
D D D - +
t
Time dependence
density gradient
recombination
generation
14
Summary: Equation of State
Bermel ECE 305 F16
0 0 0D A D AD q p n N N q p p n p N N+ - + - - + - ® + D - - D + - ®
2
2
1 1 p n nP P P p P p
n n
p p p pr g g D g
t q q x x
- D D D - + ® - - + ® - - +
JJ
t t
( ~0)P P P P
pqp qD p qD
x
- ® -
J E E
2
2
1 1 p pNN N N N N p
n n
n nn nr g g D g
t q q x x
D D D - + ® - + ® - - +
JJ
t t
( ~0)N N N N
nqn qD n qD
x
+ ®
J E E
15
when is the electric field zero?
17
x
n x ND x
1017 cm-3
1018 cm-3
n x » ND x
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e-band diagram
18
EF
EC x
EV x
Ei x
x
E qE x
dEC
dx
Dp
t¹ Dp
d 2Dp
dx2-Dp
t p
+GL
9/19/2016 Bermel ECE 305 F16
example #1: N-type sample in ll injection
19
Steady-state, uniform generation, no spatial variation
Solve for Δp and for the QFL’s.
1) Simplify the MCDE2) Solve the MCDE3) Deduce Fp from Δp
Dp
t Dp
d2Dp
dx2-Dp
t p
+GL
9/19/2016 Bermel ECE 305 F16
example #1: solution
20
x
Dp x
Dp x GLt p
x L 200 mx 0
Steady-state, uniform generation, no spatial variation9/19/2016 Bermel ECE 305 F16
Example 2A: Transient, No Illumination
Bermel ECE 305 F16
1N N N
nr g
t q
- +
J J + N N Nqn E qD n
(uniform)
0( )
n
n n nG
t t
+ D D - +
Acceptor doped
1 - - +
J p P p
pr g
t q - J p p Pqp E qD p
(uniform)
0( )
t
+ D D - +
p
p p pG
tMajority carrier
0 0 0+ - + - - + - + D- - + - DD A D AD q p n N N q p n N Npn21
Dn
time
Example 2A: Transient, No Illumination
Bermel ECE 305 F16
( )
n
n nG
t
D D - +
t
( , ) ntn x t A Be-D + tAcceptor doped
22
000,
00,
nBnxn
Axn
DDD
¥D
ntt
entxn-
DD 0,
Dn
time
Example 2B: Transient, Uniform Illumination
Bermel ECE 305 F16
1N N N
nr g
t q
- +
J
1
J + N N Nqn E qD n
(uniform)
0( )
n
n n nG
t t
+ D D - +
Acceptor doped
1 - - +
J p P p
pr g
t q - J p p Pqp E qD p
(uniform)
0( )
t
+ D D - +
p
p p pG
tMajority carrier
0 0 0+ - + - - + - + D- - + - DD A D AD q p n N N q p n N Npn23
Example 2B: Transient, Uniform Illumination
Bermel ECE 305 F16
1
( )
n
n nG
t
D D - +
t
( , ) ntn x t A Be-D + t
( , ) 1 ntnn x t G e tt -D -
Acceptor doped
0, ( ,0) 0
, ( , ) n
t n x A B
t n x G A
D -
®¥ D ¥ t
time
24
example #3
25
Solve for Δp and for the QFL’s.
1) Simplify the MCDE2) Solve the MCDE3) Deduce Fp from Δp
Dp
t Dp
d2Dp
dx2-Dp
t p
+GL
Transient, no generation, no spatial variation
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example #3
26
x
Dp x
Dp t 0 GLt p
x L 200 mx 0
transient, no generation, no spatial variation
Dp t Dp t 0 e-t /t p
9/19/2016 Bermel ECE 305 F16
example #4
27
Steady-state, sample long compared to the diffusion length.i.e., a short diffusion length
fixedDp x 0
1) Simplify the MCDE2) Solve the MCDE3) Deduce Fp from Δp
Dp
t Dp
d2Dp
dx2-Dp
t p
+GL
9/19/2016 Bermel ECE 305 F16
example #4
28
x
Dp x
Dp x ®¥ 0
Dp 0
Dp x Dp 0 e-x/Lp
x L 200 mx 0
Lp Dpt p << L
Steady-state, sample long compared to the diffusion length.9/19/2016 Bermel ECE 305 F16
Continuity Equations…
Bermel ECE 305 F16 29
1 - - +
JP P P
pr g
t q
D AD q p n N N+ - - + -
P P Pqp qD p - J E
1N N N
nr g
t q
- +
J
N N Nqn qD n + J E
1
time
Dn
time
Analytic solutions
ntt
entxn-
DD 0,
( , ) 1 ntnn x t G e tt -D -
Dn
Example 5: One sided Minority Diffusion
Bermel ECE 305 F16
1 nN N
n dJr g
t q dx
- +
N N N
dnqn E qD
dx +J
2
20 N
d nD
dx
Steady state, no generation/recombination, acceptor dopedLong diffusion length
30
1
0,' D txn
a 0x’
Metal contact
Example: One sided Minority Diffusion
Bermel ECE 305 F16
, ( ' ) 0 D -x a n x a C Da
'( , ) ( 0 ') 1
D D -
xn x t n x
a
2
20 N
d nD
dx
( , ) 'n x t C DxD +
x’
a
Metal contact
0 ', ( ' 0 ') D x n x C
0x’
31
0,' D txn
Conclusions
1) We will often be using minority carrier diffusion equation to understand the mechanics of carrier transport in electronic devices. Review the problem carefully to see if the assumption of minority carrier transport is satisfied.
2) Divide all complex problems into solvable parts, solve the parts sequentially and then put the partial solutions back by using proper boundary conditions to arrive at the complete solution.
3) Explore analytical solution whenever possible, however numerical solutions are also of great value.
Bermel ECE 305 F16 32
