Lundstrom ECE 305 S15
ECE-305: Spring 2015
Intro to PN Junctions: II
Professor Mark Lundstrom Electrical and Computer Engineering
Purdue University, West Lafayette, IN USA [email protected]
2/20/15
Pierret, Semiconductor Device Fundamentals (SDF) pp. 197-209
outline
2
1) PN Junctions (qualitative)
2) The Poisson equation
3) PN Junctions (quantitative)
Lundstrom ECE 305 S15
NP junction (equilibrium)
3
N P
p0 ! NA
ρ ! 0 n0 ! ND
ρ ! 0
xp−xn 0
“transition region”
Lundstrom ECE 305 S15
p0 < NAn0 < ND
the built-in potential
4
EC
EVEFP
Ei V = 0
EC
EV
Ei
EFN
qVbiV = Vbi
n0 = nieEFN −Ei( ) kBT p0 = nie
Ei−EFP( ) kBT
n0p0 = NDNA = ni2e EFN −EFP( ) kBT = eqVbi kBT
Vbi =kBTqln NDNA
ni2
⎛⎝⎜
⎞⎠⎟
energy band diagram
5 5
EF
EC
EV
x
E
Ei
x = xpx = 0x = −xn
qVbi
p0 < NAn0 < ND
electrostatics: V(x)
6
V
x
N P
xp−xn
qVbi
electrostatics: E (x)
7
E
xN P
ρ = q p0 x( ) − n0 x( ) + ND+ x( ) − NA
− x( )⎡⎣ ⎤⎦
xp−xn
carrier densities vs. x
8
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
electrostatics: rho(x)
9
ρ
x
N P
ρ = q p0 x( ) − n0 x( ) + ND+ x( ) − NA
− x( )⎡⎣ ⎤⎦
xp−xn
qND
−qNA
NP junction electrostatics
10
How do we calculate rho(x), E(x), and V(x)?
question
11
1) The built-in potential of an NP junction is a little less than what?
a) The thermal voltage, kBT/q . b) 3/kBT/2q. c) 110 V. d) The bandgap of the semiconductor in eV. e) The electron affinity of the semiconductor in eV.
Vbi =kBTqln NDNA
ni2
⎛⎝⎜
⎞⎠⎟
question
12
2) Which of the following is true about the electron density in the transition region (-xn < x < xp) of an NP junction?
a) It is less than ni everywhere. b) It is zero everywhere. c) It is less than the doping density over most of the transition region. d) It varies with space as exp(-x/Ln) . e) It varies with space as cosh(x/Ln) .
“Depletion approximation”
outline
13
1) PN Junctions (qualitative)
2) The Poisson equation and …
3) PN Junctions (quantitative)
Lundstrom ECE 305 S15
the Poisson equation
14
dEdx
=ρ x( )KSε0
dDdx
= ρ x( )
∇ i!D = ρ x( )
D = KSε0E
Gauss’s Law
15
+Q n̂
“Gaussian surface”
!D = ε0
!E
!D = KSε0
!E
!D i d
!S"∫ = Q
Gauss’s Law in 1D
16
!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( )
the Poisson equation
17
dEdx
=ρ x( )KSε0
dDdx
= ρ x( )
∇ i!D = ρ x( )
!D i d
!S"∫ = Q
D = KSε0E
electrostatics: rho(x)
18
ρ
x
N P
ρ = q p0 x( ) − n0 x( ) + ND+ x( ) − NA
− x( )⎡⎣ ⎤⎦
xp−xn
qND
−qNA
“depletion approximation”
the “depletion approximation”
19
dEdx
=ρ x( )KSε0
ρ
x
N P
−xn
ρ = +qND
xp
ρ = −qNA
qNDxn = qNAxp
NDxn = NAxp
but first
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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
electric field between the plates
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E
x
+ d2
− d2
E = V
d V = − E
−d /2
+d /2
∫ dx
the NP junction
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dEdx
=ρ x( )KSε0
ρ
x
N P
−xn
ρ = +qND
xp
ρ = −qNA
the NP junction
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dEdx
=ρ x( )KSε0
is not constant!
ρ
x
N P
+qND
ρ = −qNA
xn + xp
V = 0V =Vbi > 0
outline
24
1) PN Junctions (qualitative)
2) The Poisson equation and …
3) PN Junctions (quantitative)
Lundstrom ECE 305 S15
