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PN Junction
PN Junction Electrostatics
Built-in potential, depletion approximation
Current-Voltage characteristics
Ideal I-V characteristics
Deviation from the ideal
Breakdown, high-level injection
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PN Junction Basics
Physical view of a PN junction
Step junction
metal
n+ p
oxide
n+ p
Cross-section
ND-NA
x
Step junction
idealization
Actual profile
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PN Junction Electrostatics
Qualitative Analysis: Thermal Equil ibrium
Energy Band Diagram at Thermal
Equilibrium
Fermi level should be aligned Vacuum level (and hence EC, EV,
and Ei) should be continuous
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PN Junction Electrostatics
Qualitative Analysis: Thermal Equil ibrium
Electrostatic Variables at Thermal Equilibrium
The V vs x should have the same functional
form of “upside down” of EC (EV, or Ei)
The E vs x can be drawn from the derivation
of V over x (E=-dV/dx)
The ρ vs x can be drawn from the slope of E
)()( iFnFpibi E E E E qV
)ln()()(
2
i
A DiFnFpi
bi
n
N N
q
kT
q
E E E E V
“Built-In Potential”
Since , we have
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PN Junction Electrostatics
Qualitative Analysis: Thermal Equil ibrium
Depletion region -- how charge
redistributed?
Electrons diffusion from n-region to p-
region; while holes diffusion from p-region to
n-region Unbalanced dopants are left and form built-
in field which directs from n-region to p-
region
Equilibrium condition is met when the
electrons and holes diffusions are balancedby their drifts due to built-in field.
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Depletion approximation
Bulk p and n regions assumed as
charge neutral
Transition region assumed to be
depleted of mobile carriers (depletionregion)
Depletion region can be treated as
insulator to withstand voltage, but ischarged due to unbalanced dopants
Quite accurate
PN Junction Electrostatics
Qualitative Analysis: Thermal Equil ibrium
Depletion
approximation
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PN Junction Electrostatics
Quantitative Analysis: Thermal Equil ibrium
Charge density
In n region near the joint, as electrons
are depleted (depletionapproximation), the region should be
positive charged and equal to qND
Similarly, the p-region near the joint
should be negative charged andequal to –qN A
So, we have charge density:
)(0
)0(
)0()(0
n
n D
p A
p
x x
x xqN
x xqN x x
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PN Junction Electrostatics
Quantitative Analysis: Thermal Equil ibrium
Electric Field0
sdx
dE
By integration, we have
)()(0
x xqN
x E p
S
A
)()(0
x xqN x E n
S
D
for –x p x 0
for 0 x xn
0 E for x>xn or x -x p
A note: at x=0, the E-fields equal fromabove two equations, thus
n D p A x N x N
(can also obtained from charge neutrality)
n D p A xqN xqN
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PN Junction Electrostatics
Quantitative Analysis: Thermal Equil ibrium
Electric Potential
By integration, we have
A note: From electrical potential continuity atx=0, one obtains
dx
dV E
With boundary conditions
nbi
p
x xV
x x
V ,
,0
2
0
)(2
)( x xqN
xV p
S
A
2
0
)(2
)( x xqN
V xV n
S
Dbi
for –x p x 0
for 0 x xn
2
0
2
0 22 n
S
D
p
S
A
bi xqN
xqN
V
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PN Junction Electrostatics
Quantitative Analysis: Thermal Equil ibrium
Depletion width
Recall: from continues of E-field and E-
potential at x=0, we have
xn and xp can be calculated as below
2
0
2
0 22 n
S
D
p
S
A
bi xqN
xqN
V
n D p A x N x N
2/1
0
)(
2
bi D A D
AS
n
V N N N
N
q x
2/1
0
)(
2
bi
D A A
DS p V
N N N
N
q x
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PN Junction Electrostatics
Quantitative Analysis: Thermal Equil ibrium
Depletion width
The total depletion width, W=xn+xp, can be
given as below
2/1
0 )(2
bi
D A
D AS
pn V N N
N N
q x xW
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PN Junction Electrostatics
Quantitative Analysis: V A Applied
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PN Junction Electrostatics
Quantitative Analysis: V A Applied
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PN Junction Electrostatics
Quantitative Analysis: V A Applied
Using (Vbi-V A) to replace Vbi
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PN Junction Electrostatics
Quantitative Analysis: V A Applied
Depletion capacitance
W
AC S 0
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PN Junction Electrostatics
Example: Asymmetric Junction
Very often we have N+/P or P+/N junction
For example:ni/pi ~ 1010cm-3
ND+/N A+ ~ 1018cm-3
ND/N A ~ 1015cm-3
For N+/P, only x p has to be considered
N A x p=
2SiV bi
q1( )
1/2
xn~ 0
x p
N D
N A