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Department of EECS University of California, Berkeley
EECS 105 Spring 2005, Lecture 27
Lecture 27: PN Junctions
Prof. Niknejad
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Diffusion
Diffusion occurs when there exists a concentration gradient
In the figure below, imagine that we fill the left chamber with a gas at temperate T
If we suddenly remove the divider, what happens? The gas will fill the entire volume of the new
chamber. How does this occur?
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Diffusion (cont)
The net motion of gas molecules to the right chamber was due to the concentration gradient
If each particle moves on average left or right then eventually half will be in the right chamber
If the molecules were charged (or electrons), then there would be a net current flow
The diffusion current flows from high concentration to low concentration:
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Diffusion Equations
Assume that the mean free path is λ Find flux of carriers crossing x=0 plane
)(n)0(n
)( n
0
thvn )(2
1 thvn )(2
1
)()(2
1 nnvF th
dx
dnn
dx
dnnvF th )0()0(
2
1
dx
dnvF th
dx
dnqvqFJ th
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Einstein Relation
The thermal velocity is given by kT
kTvm thn 212*
21
cthv Mean Free Time
dx
dn
q
kTq
dx
dnqvJ nth
nn q
kTD
**2
n
c
n
ccthth m
q
q
kT
mkTvv
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Total Current and Boundary Conditions
When both drift and diffusion are present, the total current is given by the sum:
In resistors, the carrier is approximately uniform and the second term is nearly zero
For currents flowing uniformly through an interface (no charge accumulation), the field is discontinous
dx
dnqDnEqJJJ nndiffdrift
21 JJ
2211 EE
1
2
2
1
E
E
)( 11 J
)( 22 J
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Carrier Concentration and Potential
In thermal equilibrium, there are no external fields and we thus expect the electron and hole current densities to be zero:
dx
dnqDEqnJ o
nnn 000
dx
dn
kT
qEn
Ddx
dnoo
n
no 00
0
0
00 n
dnV
n
dn
q
kTd th
o
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Carrier Concentration and Potential (2)
We have an equation relating the potential to the carrier concentration
If we integrate the above equation we have
We define the potential reference to be intrinsic Si:
)(
)(ln)()(
00
0000 xn
xnVxx th
inxnx )(0)( 0000
0
0
00 n
dnV
n
dn
q
kTd th
o
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Carrier Concentration Versus Potential
The carrier concentration is thus a function of potential
Check that for zero potential, we have intrinsic carrier concentration (reference).
If we do a similar calculation for holes, we arrive at a similar equation
Note that the law of mass action is upheld
thVxienxn /)(
00)(
thVxienxp /)(
00)(
2/)(/)(200
00)()( iVxVx
i neenxpxn thth
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
The Doping Changes Potential
Due to the log nature of the potential, the potential changes linearly for exponential increase in doping:
Quick calculation aid: For a p-type concentration of 1016 cm-3, the potential is -360 mV
N-type materials have a positive potential with respect to intrinsic Si
100
0
0
0
00 10
)(log10lnmV26
)(
)(lnmV26
)(
)(ln)(
xn
xn
xn
xn
xnVx
iith
100
0 10
)(logmV60)(
xnx
100
0 10
)(logmV60)(
xpx
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
n-type
p-type
ND
NA
PN Junctions: Overview The most important device is a junction
between a p-type region and an n-type region When the junction is first formed, due to the
concentration gradient, mobile charges transfer near junction
Electrons leave n-type region and holes leave p-type region
These mobile carriers become minority carriers in new region (can’t penetrate far due to recombination)
Due to charge transfer, a voltage difference occurs between regions
This creates a field at the junction that causes drift currents to oppose the diffusion current
In thermal equilibrium, drift current and diffusion must balance
− − − − − −
+ + + + +
+ + + + ++ + + + +
− − − − − −− − − − − −
−V+
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
PN Junction Currents
Consider the PN junction in thermal equilibrium Again, the currents have to be zero, so we have
dx
dnqDEqnJ o
nnn 000
dx
dnqDEqn o
nn 00
dx
dn
nq
kT
ndxdn
DE
n
on
0
000
1
dx
dp
pq
kT
ndxdp
DE
p
op
0
000
1
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
PN Junction Fields
n-typep-type
NDNA
)(0 xpaNp 0
d
i
N
np
2
0 diffJ
0E
a
i
N
nn
2
0
Transition Region
diffJ
dNn 0
– – + +
0E
0px 0nx
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Total Charge in Transition Region
To solve for the electric fields, we need to write down the charge density in the transition region:
In the p-side of the junction, there are very few electrons and only acceptors:
Since the hole concentration is decreasing on the p-side, the net charge is negative:
)()( 000 ad NNnpqx
)()( 00 aNpqx
0)(0 x0pNa
00 xxp
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Charge on N-Side
Analogous to the p-side, the charge on the n-side is given by:
The net charge here is positive since:
)()( 00 dNnqx 00 nxx
0)(0 x0nNd
a
i
N
nn
2
0
Transition Region
diffJ
dNn 0
– – + +
0E
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
“Exact” Solution for Fields
Given the above approximations, we now have an expression for the charge density
We also have the following result from electrostatics
Notice that the potential appears on both sides of the equation… difficult problem to solve
A much simpler way to solve the problem…
0/)(
/)(
00)(
0)()(
0
0
nVx
id
poaVx
i
xxenNq
xxNenqx
th
th
s
x
dx
d
dx
dE
)(0
2
20
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Depletion Approximation
Let’s assume that the transition region is completely depleted of free carriers (only immobile dopants exist)
Then the charge density is given by
The solution for electric field is now easy
0
0 0
0)(
nd
poa
xxqN
xxqNx
s
x
dx
dE
)(00
)(')'(
)( 000
00
p
x
xs
xEdxx
xEp
Field zero outsidetransition region
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Depletion Approximation (2)
Since charge density is a constant
If we start from the n-side we get the following result
)(')'(
)(0
00 po
s
ax
xs
xxqN
dxx
xEp
)()()(')'(
)( 0000
00
0
xExxqN
xEdxx
xE ns
dx
xs
n
n
)()( 00 xxqN
xE ns
d
Field zero outsidetransition region
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Plot of Fields In Depletion Region
E-Field zero outside of depletion region Note the asymmetrical depletion widths Which region has higher doping? Slope of E-Field larger in n-region. Why? Peak E-Field at junction. Why continuous?
n-typep-type
NDNA
– – – – –– – – – –– – – – –– – – – –
+ + + + +
+ + + + +
+ + + + +
+ + + + +
DepletionRegion
)()( 00 xxqN
xE ns
d
)()(0 pos
a xxqN
xE
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Continuity of E-Field Across Junction
Recall that E-Field diverges on charge. For a sheet charge at the interface, the E-field could be discontinuous
In our case, the depletion region is only populated by a background density of fixed charges so the E-Field is continuous
What does this imply?
Total fixed charge in n-region equals fixed charge in p-region! Somewhat obvious result.
)0()0(00
xExqN
xqN
xE pno
s
dpo
s
an
nodpoa xqNxqN
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Potential Across Junction
From our earlier calculation we know that the potential in the n-region is higher than p-region
The potential has to smoothly transition form high to low in crossing the junction
Physically, the potential difference is due to the charge transfer that occurs due to the concentration gradient
Let’s integrate the field to get the potential:
x
x pos
apo
p
dxxxqN
xx0
')'()()(
x
x
pos
ap
p
xxxqN
x
0
'2
')(
2
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Potential Across Junction
We arrive at potential on p-side (parabolic)
Do integral on n-side
Potential must be continuous at interface (field finite at interface)
20 )(
2)( p
s
ap
po xx
qNx
20 )(
2)( n
s
dnn xx
qNx
)0(22
)0( 20
20 pp
s
apn
s
dnn x
qNx
qN
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Solve for Depletion Lengths
We have two equations and two unknowns. We are finally in a position to solve for the depletion depths
20
20 22 p
s
apn
s
dn x
qNx
qN
nodpoa xqNxqN
(1)
(2)
da
a
d
bisno NN
N
qNx
2
ad
d
a
bispo NN
N
qNx
2
0 pnbi
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Sanity Check
Does the above equation make sense? Let’s say we dope one side very highly. Then
physically we expect the depletion region width for the heavily doped side to approach zero:
Entire depletion width dropped across p-region
02
lim0
ad
d
d
bis
Nn NN
N
qNx
d
a
bis
ad
d
a
bis
Np qNNN
N
qNx
d
22lim0
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Total Depletion Width
The sum of the depletion widths is the “space charge region”
This region is essentially depleted of all mobile charge
Due to high electric field, carriers move across region at velocity saturated speed
da
bisnpd NNq
xxX112
000
μ110
12150
qX bis
d
cm
V10
μ1
V1 4pnE
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Have we invented a battery?
Can we harness the PN junction and turn it into a battery?
Numerical example:
2lnlnln
i
ADth
i
A
i
Dthpnbi n
NNV
n
N
n
NV
mV60010
1010logmV60lnmV26
20
1515
2
i
ADbi n
NN
?
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Contact Potential
The contact between a PN junction creates a potential difference
Likewise, the contact between two dissimilar metals creates a potential difference (proportional to the difference between the work functions)
When a metal semiconductor junction is formed, a contact potential forms as well
If we short a PN junction, the sum of the voltages around the loop must be zero:
mnpmbi 0
pnmn
pm
+
−
bi )( mnpmbi
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
PN Junction Capacitor
Under thermal equilibrium, the PN junction does not draw any (much) current
But notice that a PN junction stores charge in the space charge region (transition region)
Since the device is storing charge, it’s acting like a capacitor
Positive charge is stored in the n-region, and negative charge is in the p-region:
nodpoa xqNxqN
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Reverse Biased PN Junction
What happens if we “reverse-bias” the PN junction?
Since no current is flowing, the entire reverse biased potential is dropped across the transition region
To accommodate the extra potential, the charge in these regions must increase
If no current is flowing, the only way for the charge to increase is to grow (shrink) the depletion regions
+
−
Dbi V DV 0DV
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Voltage Dependence of Depletion Width
Can redo the math but in the end we realize that the equations are the same except we replace the built-in potential with the effective reverse bias:
da
DbisDnDpDd NNq
VVxVxVX
11)(2)()()(
bi
Dn
da
a
d
DbisDn
Vx
NN
N
qN
VVx
1
)(2)( 0
bi
Dp
da
d
a
DbisDp
Vx
NN
N
qN
VVx
1
)(2)( 0
bi
DdDd
VXVX
1)( 0
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Charge Versus Bias
As we increase the reverse bias, the depletion region grows to accommodate more charge
Charge is not a linear function of voltage This is a non-linear capacitor We can define a small signal capacitance for small
signals by breaking up the charge into two terms
bi
DaDpaDJ
VqNVxqNVQ
1)()(
)()()( DDJDDJ vqVQvVQ
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Derivation of Small Signal Capacitance
From last lecture we found
Notice that
DV
DDJDDJ v
dV
dQVQvVQ
D
)()(
RD VVbipa
VV
jDjj
VxqN
dV
d
dV
dQVCC
1)( 0
bi
D
j
bi
Dbi
paj
V
C
V
xqNC
112
00
da
da
bi
s
da
d
a
bis
bi
a
bi
paj NN
NNq
NN
N
qN
qNxqNC
2
2
220
0
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Physical Interpretation of Depletion Cap
Notice that the expression on the right-hand-side is just the depletion width in thermal equilibrium
This looks like a parallel plate capacitor!
da
da
bi
sj NN
NNqC
20
0
1
0
11
2 d
s
dabissj XNN
qC
)()(
Dd
sDj VX
VC
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
A Variable Capacitor (Varactor)
Capacitance varies versus bias:
Application: Radio Tuner
0j
j
C
C
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27
Part II: Currents in PN Junctions
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Diode under Thermal Equilibrium
Diffusion small since few carriers have enough energy to penetrate barrier Drift current is small since minority carriers are few and far between: Only
minority carriers generated within a diffusion length can contribute current Important Point: Minority drift current independent of barrier! Diffusion current strong (exponential) function of barrier
p-type n-type
DN AN
-
-
-
-
-
-
-
-
-
-
-
-
-
+++++++++++++
0E
biq
,p diffJ
,p driftJ
,n diffJ
,n driftJ
−
−
+
+
−
−
ThermalGeneration
Recombination Carrier with energybelow barrier height
Minority Carrier Close to Junction
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Reverse Bias
Reverse Bias causes an increases barrier to diffusion
Diffusion current is reduced exponentially
Drift current does not change Net result: Small reverse current
p-type n-type
DN AN
-
-
-
-
-
-
-
+++++++
( )bi Rq V
+−
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Forward Bias
Forward bias causes an exponential increase in the number of carriers with sufficient energy to penetrate barrier
Diffusion current increases exponentially
Drift current does not change Net result: Large forward current
p-type n-type
DN AN
-
-
-
-
-
-
-
+++++++
( )bi Rq V
+−
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Diode I-V Curve
Diode IV relation is an exponential function
This exponential is due to the Boltzmann distribution of carriers versus
energy
For reverse bias the current saturations to the drift current due to minority
carriers
1dqV
kTd SI I e
dqV
kT
d
s
I
I
1
( )d d SI V I
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Minority Carriers at Junction Edges
Minority carrier concentration at boundaries of depletion region increase as barrier lowers … the function is
)(
)(
pp
nn
xxp
xxp (minority) hole conc. on n-side of barrier
(majority) hole conc. on p-side of barrier
kTEnergyBarriere /)(
(Boltzmann’s Law)
kTVq DBe /)(
A
nn
N
xxp )(
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
“Law of the Junction”
Minority carrier concentrations at the edges of thedepletion region are given by:
kTVqAnn
DBeNxxp /)()(
kTVqDpp
DBeNxxn /)()(
Note 1: NA and ND are the majority carrier concentrations on the other side of the junction
Note 2: we can reduce these equations further by substituting
VD = 0 V (thermal equilibrium)Note 3: assumption that pn << ND and np << NA
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Minority Carrier Concentration
The minority carrier concentration in the bulk region for forward bias is a decaying exponential due to recombination
p side n side
-Wp Wn xn -xp
0
AqV
kTnp e
0 0( ) 1A
p
xqVLkT
n n np x p p e e
0np0pn
0
AqV
kTpn e
Minority CarrierDiffusion Length
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Steady-State Concentrations
Assume that none of the diffusing holes and electrons recombine get straight lines …
p side n side
-Wp Wn xn -xp
0
AqV
kTnp e
0np0pn
0
AqV
kTpn e
This also happens if the minority carrier diffusion lengths are much larger than Wn,p
, ,n p n pL W
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Diode Current Densities
0 1A
p
qVpdiff n kT
n n ppx x
dn DJ qD q n e
dx W
0 0( )( )
AqV
kTp p p
p p
dn n e nx
dx x W
p side n side
-Wp Wn xn -xp
0
AqV
kTnp e
0np
0pn
0
AqV
kTpn e
0 1A
n
qVpdiff n kT
p p nx x n
DdpJ qD q p e
dx W
2 1AqV
pdiff n kTi
d n a p
D DJ qn e
N W N W
2
0i
pa
nn
N
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Diode Small Signal Model
The I-V relation of a diode can be linearized
( )
1d d d dq V v qV qv
kT kT kTD D S SI i I e I e e
( )1 d d
D D D
q V vI i I
kT
2 3
12! 3!
x x xe x
dD d d
qvi g v
kT
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Diode Capacitance
We have already seen that a reverse biased diode acts like a capacitor since the depletion region grows and shrinks in response to the applied field. the capacitance in forward bias is given by
But another charge storage mechanism comes into play in forward bias
Minority carriers injected into p and n regions “stay” in each region for a while
On average additional charge is stored in diode
01.4Sj j
dep
C A CX
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Charge Storage
Increasing forward bias increases minority charge density By charge neutrality, the source voltage must supply equal
and opposite charge A detailed analysis yields:
p side n side
-Wp Wn xn -xp
( )
0
d dq V v
kTnp e
0np0pn
( )
0
d dq V v
kTpn e
1
2d
d
qIC
kT
Time to cross junction(or minority carrier lifetime)
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Forward Bias Equivalent Circuit
Equivalent circuit has three non-linear elements: forward conductance, junction cap, and diffusion cap.
Diff cap and conductance proportional to DC current flowing through diode.
Junction cap proportional to junction voltage.
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Fabrication of IC Diodes
Start with p-type substrate Create n-well to house diode p and n+ diffusion regions are the cathode and annode N-well must be reverse biased from substrate Parasitic resistance due to well resistance
p-type
p+
n-well
p-type
n+
annodecathode
p
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Diode Circuits
Rectifier (AC to DC conversion) Average value circuit Peak detector (AM demodulator) DC restorer Voltage doubler / quadrupler /…