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Lundstrom ECE 305 S16
ECE-305: Spring 2016
Intro to PN Junctions: II
Professor Mark Lundstrom Electrical and Computer Engineering
Purdue University, West Lafayette, IN USA lundstro@purdue.edu
2/18/15
Pierret, Semiconductor Device Fundamentals (SDF) pp. 195-209
NP junction (equilibrium)
2
N P
p0 ! NA
ρ ! 0 n0 ! ND
ρ ! 0
xp−xn 0
“transition region”
Lundstrom ECE 305 S16
p0 < NAn0 < ND
energy band diagram
3
EF
EC
EV
x
E
Ei
x = xpx = 0x = −xn
qVbi
p0 < NAn0 < ND
Lundstrom ECE 305 S16
electrostatics: V(x)
4
V
x
N P
xp−xn
qVbi = kBT lnNAND
ni2
Lundstrom ECE 305 S16
electrostatics: E (x)
5
E
xN P xp−xn
Lundstrom ECE 305 S16
carrier densities vs. x
6
log10 n x( ), log10 p x( )
xN P xp−xn
p0P = NA
p0N = ni2 ND
n0N = ND
n0 p = ni2 NA
n0N << ND p0P << NA
Lundstrom ECE 305 S16
electrostatics: rho(x)
7
ρ
x
N P
ρ = q p0 x( ) − n0 x( ) + ND+ x( ) − NA
− x( )⎡⎣ ⎤⎦
xp−xn
qND
−qNA
Lundstrom ECE 305 S16
n0N << ND
p0P << NA
NP junction electrostatics
8
How do we calculate rho(x), E(x), and V(x)?
Lundstrom ECE 305 S16
Gauss’s Law
9
+Q n̂
“Gaussian surface”
!D = ε0
!E
!D = KSε0
!E
!D i d
!S"∫ = Q
Lundstrom ECE 305 S16
Gauss’s Law in 1D
10
!D i d
!S"∫ = Q
xx x + dx
ρ x( )C/cm3
D x + dx( )D x( )
n̂n̂
Area = A
−D x( )A+ D x + dx( )A = Q
Q = ρ x( )Adx
D x + dx( )− D x( )dx
= ρ x( )
dDdx
= ρ x( )Lundstrom ECE 305 S16
the Poisson equation
11
dEdx
=ρ x( )KSε0
dDdx
= ρ x( )
∇ i!D = ρ x( )
!D i d
!S"∫ = Q
D = KSε0E
Lundstrom ECE 305 S16
electrostatics: rho(x)
12
ρ
x
N P
ρ = q p0 x( ) − n0 x( ) + ND+ x( ) − NA
− x( )⎡⎣ ⎤⎦
xp−xn
qND
−qNA
“depletion approximation”
Lundstrom ECE 305 S16
n < ND
p < NA
the “depletion approximation”
13
dEdx
=ρ x( )KSε0
ρ
x
N P
−xn
ρ = +qND
xp
ρ = −qNA
qNDxn = qNAxp
NDxn = NAxpLundstrom ECE 305 S16
but first
14
E
x
V = 0V > 0
d
E = - dV
dx
V = − E
x1
x2
∫ dx
E = V
d
dEdx
=ρ x( )KSε0
= 0 →E is constant
Lundstrom ECE 305 S16
electric field between the plates
15
E
x
+ d2
− d2
E = V
d V = − E
−d /2
+d /2
∫ dx
Lundstrom ECE 305 S16
the NP junction
16
dEdx
=ρ x( )KSε0
is not constant!
ρ
x
+qND
ρ = −qNA
xn + xp
P
V = 0
N
V =Vbi > 0
1) Make depletion approximation
2) Solve
3) Find
NP junction electrostatics
17
dEdx
=ρ x( )KSε0
ρ
x
N P
−xn
ρ = +qND
xp
ρ = −qNA
E x( ),V x( ), xn , xp
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