Download - Qualitative Derivation
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Dr.-Ing. Erwin SitompulPresident University
Lecture 8Semiconductor Device Physics
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Majoritycarriers
Majoritycarriers
Qualitative DerivationChapter 6 pn Junction Diodes: I-V Characteristics
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Current Flow in a pn Junction DiodeChapter 6 pn Junction Diodes: I-V Characteristics
When a forward bias (VA > 0) is applied, the potential barrier to diffusion across the junction is reduced.
Minority carriers are “injected” into the quasi-neutral regions Δnp > 0, Δpn > 0.
Minority carriers diffuse in the quasi-neutral regions, recombining with majority carriers.
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Ideal Diode: AssumptionsChapter 6 pn Junction Diodes: I-V Characteristics
Steady-state conditions.Non-degenerately doped step junction.One-dimensional diode.Low-level injection conditions prevail in the quasi-neutral
regions.No processes other than drift, diffusion, and thermal R–G take
place inside the diode.
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N n N n N( )( ) dn d nx q n qD q n qD
dx dx
J E E
P p P p P( )( ) dp d px q p qD q p qD
dx dx
J E E
Current Flow in a pn Junction DiodeChapter 6 pn Junction Diodes: I-V Characteristics
Current density J = JN(x) + JP(x)
JN(x) and JP(x) may vary with position, but J is constant throughout the diode.
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n-sidep-side
p0 p A
2i
p0 pA
( )
( )
p x N
nn xN
n0 n D
2i
n0 nD
( )
( )
n x N
np xN
p p A( )p x N n n D( )n x N
Carrier Concentrations at –xp, xn
Chapter 6 pn Junction Diodes: I-V Characteristics
Consider the equilibrium carrier concentrations at VA = 0:
If low-level injection conditions prevail in the quasi-neutral regions when VA 0, then:
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i P
N i
( )i
( )i
E F kT
F E kTp n en n e
A2i qV kTnp n e
N ii P
N P
( )( )2i
( )2i
F E kTE F kT
F F kT
np n e e
n e
p nfor x x x
“Law of the Junction”Chapter 6 pn Junction Diodes: I-V Characteristics
The voltage VA applied to a pn junction falls mostly across the depletion region (assuming that low-level injection conditions prevail in the quasi-neutral regions).
Two quasi-Fermi levels is drawn in the depletion region:
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p p A( )p x N
n-sidep-side
n n D( )n x N
A
2i
p pA
( ) ( 1)qV kTnn x eN
A
2i
n nD
( ) ( 1)qV kTnp x eN
Excess Carrier Concentrations at –xp, xn
Chapter 6 pn Junction Diodes: I-V Characteristics
A
A
2i
p pA
p0
( )
qV kT
qV kT
n en xN
n e
A
A
2i
n nD
n0
( )
qV kT
qV kT
n ep xN
p e
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A 0.6 0.02586 12 3p p p0( ) 100 1.192 10 cmqV kTn x n e e
12 12 3p p p p p0( ) ( ) 1.192 10 100 1.192 10 cmn x n x n
Example: Carrier InjectionChapter 6 pn Junction Diodes: I-V Characteristics
A pn junction has NA=1018 cm–3 and ND=1016 cm–3. The applied voltage is 0.6 V.
a) What are the minority carrier concentrations at the depletion-region edges?
b) What are the excess minority carrier concentrations?
A 4 0.6 0.02586 14 3n n n0( ) 10 1.192 10 cmqV kTp x p e e
14 4 14 3n n n n n0( ) ( ) 1.192 10 10 1.192 10 cmp x p x p
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An n n0( ) ( 1)qV kTp x p e
2n n
P 2p
0 , 0d p pD xdx
P Pn 1 2( ) x L x Lp x A e A e
0x 0 x
P P pL D for 0x
Excess Carrier DistributionChapter 6 pn Junction Diodes: I-V Characteristics
From the minority carrier diffusion equation,
We have the following boundary conditions:
n ( ) 0p
For simplicity, we develop a new coordinate system:
Then, the solution is given by:
• LP : hole minority carrier diffusion length
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An n0( 0) ( 1)qV kTp x p e
A Pn n0( ) ( 1) , 0qV kT x Lp x p e e x
P P'/ '/n 1 2( ) x L x Lp x A e A e
NAp p0( ) ( 1) , 0x LqV kTn x n e e x
A /1 n0 ( 1)qV kTA p e 2 0A
Excess Carrier DistributionChapter 6 pn Junction Diodes: I-V Characteristics
New boundary conditions
n ( ) 0p x
From the x’ → ∞,
From the x’ → 0,
Therefore
Similarly,
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A Pn PP P n0
P
( )( ) ( 1)qV kT x Ld p x Dx qD q p e edx L
J
NAp NN N p0
N
( )( ) ( 1) x LqV kTd n x Dx qD q n e e
dx L
J
n-side
p-side
p nN P N P0 0x x x x x x
J J J J J
A2 N Pi
N A P D
( 1)qV kTD Dqn eL N L N
J
pn Diode I–V CharacteristicChapter 6 pn Junction Diodes: I-V Characteristics
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A0
2 N P0 i
N A P D
( 1)qV kTI I eD DI Aqn
L N L N
I A J
pn Diode I–V CharacteristicChapter 6 pn Junction Diodes: I-V Characteristics
A2 N Pi
N A P D
( 1)qV kTD DAqn eL N L N
• Shockley Equation,for ideal diode
• I0 can be viewed as the drift current due to minority carriers generated within the diffusion lengths of the depletion region
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I0 can vary by orders of magnitude, depending on the semiconductor material, due to ni
2 factor. In an asymmetrically doped pn junction, the term associated
with the more heavily doped side is negligible. If the p side is much more heavily doped,
If the n side is much more heavily doped,
2 NP0 i
P D N A
DDI AqnL N L N
2 P0 i
P D
DI AqnL N
2 N0 i
N A
DI AqnL N
Diode Saturation Current I0
Chapter 6 pn Junction Diodes: I-V Characteristics
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N P J J J
N p n N p
P p n P n
( ) ( )
( ) ( )
x x x x
x x x x
J J
J J
Diode Carrier CurrentsChapter 6 pn Junction Diodes: I-V Characteristics
• Total current density is constant inside the diode
• Negligible thermal R-G throughout depletion region dJN/dx = dJP/dx = 0
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n0p
p0p
p0n
n0n
A2i qV kTnp n e
p A
n D
p Nn N
Excess minoritycarriers
Excess minoritycarriersA P
n n0( ) ( 1)qV kT x Lp x p e e NAp p0( ) ( 1) x LqV kTn x n e e
Carrier Concentration: Forward BiasChapter 6 pn Junction Diodes: I-V Characteristics
Law of the Junction
Low level injection conditions
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Carrier Concentration: Reverse BiasChapter 6 pn Junction Diodes: I-V Characteristics
Deficit of minority carriers near the depletion region.Depletion region acts like a “sink”, draining carriers from the
adjacent quasineutral regions
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Forward-bias current Reverse-bias current
“Breakdown”
“Slope over” No saturation
Smaller slope
Deviations from the Ideal I-V BehaviorChapter 6 pn Junction Diodes: I-V Characteristics
Si pn-junction Diode, 300 K.
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Dominant breakdown mechanism is tunneling
BR 0.75B
1 VN
Empirical Observations of VBR
Chapter 6 pn Junction Diodes: I-V Characteristics
• VBR : breakdown voltage
VBR decreases with increasing N,
VBR decreases with decreasing EG.
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2S CR A D
BR biA D2
N NV Vq N N
E
A DCR bi BR
S A D
2 N Nq V VN N
E• At breakdown, VA=–VBR
Breakdown Voltage, VBR
Chapter 6 pn Junction Diodes: I-V Characteristics
If the reverse bias voltage (–VA) is so large that the peak electric field exceeds a critical value ECR, then the junction will “break down” and large reverse current will flow
Thus, the reverse bias at which breakdown occurs is
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2 A DCR BR
S A D
2 N Nq VN N
ESmall E-field:
High E-field:
Low energy, causing lattice vibration and localized heating
High energy, enabling impact ionization which causing avalanche, at doping level N < 1018 cm–3
Breakdown Mechanism: Avalanching Chapter 6 pn Junction Diodes: I-V Characteristics
• ECR : critical electric field in the depletion region
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Breakdown Mechanism: Zener Process Chapter 6 pn Junction Diodes: I-V Characteristics
Zener process is the tunneling mechanism in a reverse-biased diode.Energy barrier is higher than the
kinetic energy of the particleThe particle energy remains
constant during the tunneling process
Barrier must be thin dominant breakdown mechanism when both junction sides are heavily doped
Typically, Zener process dominates when VBR < 4.5V in Si at 300 K and N > 1018 cm–3.
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2i
thermal p 1 n 1R-G( ) ( )
np nnt n n p p
n
p
R-GthermalR G
x
x
nI qA dxt
T i( )1 i
E E kTn n e
Effect of R–G in Depletion RegionChapter 6 pn Junction Diodes: I-V Characteristics
R–G in the depletion region contributes an additional component of diode current IR–G.
The net generation rate is given by
i T( )1 i E E kTp n e
• ET: trap-state energy level
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For reverse bias, with the carrier concentrations n and p being negligible,
n
p
2i
R-Gp 1 n 1( ) ( )
x
x
np nI qA dxn n p p
iR-G
02qAnWI
1 10 p n
i i
1 2
n pn n
where
Effect of R–G in Depletion RegionChapter 6 pn Junction Diodes: I-V Characteristics
Continuing,
• Reverse biases with VA< – few kT/q
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A 2R-G i
qV kTI qAnWe
Effect of R–G in Depletion RegionChapter 6 pn Junction Diodes: I-V Characteristics
n
p
2i
R-Gp 1 n 1( ) ( )
x
x
np nI qA dxn n p p
Continuing,
For forward bias, the carrier concentrations n and p cannot be neglected,
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A2 N PDIFF i
N A P D
( 1)qV kTD DI Aqn eL N L N
A
A
iR-G
0 n p 2bi A
0
( 1)2
12
qV kT
qV kT
qAnW eIV V e
kT q
DIFF R GI I I
Diffusion, ideal diode
Effect of R–G in Depletion RegionChapter 6 pn Junction Diodes: I-V Characteristics
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J A SV V IR
Voltage drop, significant for high I
A S
J
( )0
0 A bi,
q V IR kT
qV kTI I e
I e V V
RS can be determined experimentally
SV IR
Sslope=R
Effect of Series ResistanceChapter 6 pn Junction Diodes: I-V Characteristics
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n n0n n (for a p+n junction)
(for a pn+ junction)p p0p p
Effect of High-Level InjectionChapter 6 pn Junction Diodes: I-V Characteristics
As VA increases and about to reach Vbi, the side of the junction which is more lightly doped will eventually reach high-level injection:
This means that the minority carrier concentration approaches the majority doping concentration.
Then, the majority carrier concentration must increase to maintain the neutrality.
This majority-carrier diffusion current reduces the diode current from the ideal.
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Perturbation of both minority and majority carrier
High-Level Injection EffectChapter 6 pn Junction Diodes: I-V Characteristics
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Forward-bias current Reverse-bias current
Deviations from ideal I-V
Due to avalanching and Zener process
Due to high-level injection and series resistance in
quasineutral regions
Due to thermal generation in depletion region
Due to thermal recombination in depletion region
SummaryChapter 6 pn Junction Diodes: I-V Characteristics
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This time no homework. Prepare well for Quiz 2.
Chapter 5 pn Junction Electrostatics
Homework