This document summarizes a lecture on PN junctions. It begins by discussing diffusion and how a concentration gradient causes particles to diffuse from an area of high concentration to low concentration. It then explains how bringing a p-type and n-type semiconductor together forms a PN junction. Minority carriers diffuse across the junction, leaving an electric field. The depletion approximation is introduced, where the junction region is assumed to be completely depleted of free carriers, simplifying the analysis of the electric field.
A parallel-polarized uniform plane wave is incident obliquely on a lo.pdfaroraenterprisesmbd
A parallel-polarized uniform plane wave is incident obliquely on a lossless dielectric slab that is
embedded in a free-space medium, as shown in Figure P5-17. Derive expressions for the total
reflection and transmission coefficients in terms of the electrical constitutive parameters,
thickness of the slab, and angle of incidence.
Solution
In this section we look at the power or energy transmitted and reflected at an interface between
two insulators. To do so, we must evaluate the time-averaged power in the incident, reflected,
and transmitted waves which is done by calculating the Poynting vector. The energy current
density toward or away from the interface is then given by the component of the Poynting vector
in the direction normal to the interface. In the second medium, where there is just a single
(refracted) wave, the normal component of S is unambiguously the transmitted power per unit
area. But in the first medium, the total electromagnetic field is the sum of the fields of the
incident and reflected waves. In evaluating E × H, one finds three kinds of terms. There is one
which is the cross-product of the fields in the incident wave, and its normal component gives the
incident power per unit area. A second is the cross-product of the fields in the reflected wave,
giving the reflected power. But there are also two cross-terms involving the electric field of one
of the plane waves and the magnetic field of the other one. It turns out that the time-average of
the normal component of these terms is zero, so that they may be ignored in the present context.
Bearing this in mind, we have the following quantities of interest: The time-averaged incident
power per unit area:
P =< S > ·n = c 8 s ² µ |E0| 2 k · n k
The time-averaged transmitted power per unit area:
P 0 =< S 0 > ·n = c 8 s ² 0 µ0 |E 0 0 | 2k 0 · n k 0 d
The reflection coefficient R and the transmission coefficient T are defined as the ratios of the
reflected and transmitted power to the incident power. We may calculate the reflection and
transmission coefficients for the cases of polarization perpendicular and parallel to the plane of
incidence by using the Fresnel equations. If an incident wave has general polarization so that its
fields are linear combinations of these two special cases, then there is once again the possibility
of cross terms in the power involving an electric field with one type of polarization and a
magnetic field with the other type. Fortunately, these turn out to vanish, so that one may treat the
two polarizations individually. For the case of polarization perpendicular to the plane of
incidence, we use the Fresnel equations (52) and (54) for the reflected and transmitted
amplitudes and have
T = q ² 0 µ0 4n 2 cos2 i cos r (n cos i+(µ/µ0) n02n2 sin2 i) 2 q ² µ cosi
Making use of the relations n = ²µ, n 0 = ² 0µ0 , sin r = (n/n0 )sin i, and cosi = 1 sin2 i,
T = 4n(µ/µ0 ) cosi n02 n2 sin2 i [n cosi + (µ/µ0 ) n02 n2 sin2 i] 2 .
By similar means one can write the .
A parallel-polarized uniform plane wave is incident obliquely on a lo.pdfaroraenterprisesmbd
A parallel-polarized uniform plane wave is incident obliquely on a lossless dielectric slab that is
embedded in a free-space medium, as shown in Figure P5-17. Derive expressions for the total
reflection and transmission coefficients in terms of the electrical constitutive parameters,
thickness of the slab, and angle of incidence.
Solution
In this section we look at the power or energy transmitted and reflected at an interface between
two insulators. To do so, we must evaluate the time-averaged power in the incident, reflected,
and transmitted waves which is done by calculating the Poynting vector. The energy current
density toward or away from the interface is then given by the component of the Poynting vector
in the direction normal to the interface. In the second medium, where there is just a single
(refracted) wave, the normal component of S is unambiguously the transmitted power per unit
area. But in the first medium, the total electromagnetic field is the sum of the fields of the
incident and reflected waves. In evaluating E × H, one finds three kinds of terms. There is one
which is the cross-product of the fields in the incident wave, and its normal component gives the
incident power per unit area. A second is the cross-product of the fields in the reflected wave,
giving the reflected power. But there are also two cross-terms involving the electric field of one
of the plane waves and the magnetic field of the other one. It turns out that the time-average of
the normal component of these terms is zero, so that they may be ignored in the present context.
Bearing this in mind, we have the following quantities of interest: The time-averaged incident
power per unit area:
P =< S > ·n = c 8 s ² µ |E0| 2 k · n k
The time-averaged transmitted power per unit area:
P 0 =< S 0 > ·n = c 8 s ² 0 µ0 |E 0 0 | 2k 0 · n k 0 d
The reflection coefficient R and the transmission coefficient T are defined as the ratios of the
reflected and transmitted power to the incident power. We may calculate the reflection and
transmission coefficients for the cases of polarization perpendicular and parallel to the plane of
incidence by using the Fresnel equations. If an incident wave has general polarization so that its
fields are linear combinations of these two special cases, then there is once again the possibility
of cross terms in the power involving an electric field with one type of polarization and a
magnetic field with the other type. Fortunately, these turn out to vanish, so that one may treat the
two polarizations individually. For the case of polarization perpendicular to the plane of
incidence, we use the Fresnel equations (52) and (54) for the reflected and transmitted
amplitudes and have
T = q ² 0 µ0 4n 2 cos2 i cos r (n cos i+(µ/µ0) n02n2 sin2 i) 2 q ² µ cosi
Making use of the relations n = ²µ, n 0 = ² 0µ0 , sin r = (n/n0 )sin i, and cosi = 1 sin2 i,
T = 4n(µ/µ0 ) cosi n02 n2 sin2 i [n cosi + (µ/µ0 ) n02 n2 sin2 i] 2 .
By similar means one can write the .
MAHARASHTRA STATE BOARD
CLASS XI AND XII
PHYSICS
CHAPTER 8
ELECTROSTATICS
Introduction.
Coulomb's law
Calculating the value of an electric field
Superposition principle
Electric potential
Deriving electric field from potential
Capacitance
Principle of the capacitor
Dielectrics
Polarization, and electric dipole moment
Applications of capacitors.
I am Luther H. I am a Diffusion Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of Illinois, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Diffusion.
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Investigation of Steady-State Carrier Distribution in CNT Porins in Neuronal ...Kyle Poe
In this work, the carrier distribution of a carbon nanotube inserted into the spinal ganglion neuronal membrane is examined. After primary characterization based on previous work, the nanotube is approximated as a one-dimensional system, and the Poisson and Schrödinger equations are solved using an iterative finite-difference scheme. It was found that carriers aggregate near the center of the tube, with a negative carrier density of ⟨ρn⟩ = 7.89 × 10^13 cm−3 and positive carrier density of ⟨ρp⟩ = 3.85 × 10^13 cm−3. In future work, the erratic behavior of convergence will be investigated.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
MAHARASHTRA STATE BOARD
CLASS XI AND XII
PHYSICS
CHAPTER 8
ELECTROSTATICS
Introduction.
Coulomb's law
Calculating the value of an electric field
Superposition principle
Electric potential
Deriving electric field from potential
Capacitance
Principle of the capacitor
Dielectrics
Polarization, and electric dipole moment
Applications of capacitors.
I am Luther H. I am a Diffusion Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of Illinois, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Diffusion.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Diffusion Assignments.
Investigation of Steady-State Carrier Distribution in CNT Porins in Neuronal ...Kyle Poe
In this work, the carrier distribution of a carbon nanotube inserted into the spinal ganglion neuronal membrane is examined. After primary characterization based on previous work, the nanotube is approximated as a one-dimensional system, and the Poisson and Schrödinger equations are solved using an iterative finite-difference scheme. It was found that carriers aggregate near the center of the tube, with a negative carrier density of ⟨ρn⟩ = 7.89 × 10^13 cm−3 and positive carrier density of ⟨ρp⟩ = 3.85 × 10^13 cm−3. In future work, the erratic behavior of convergence will be investigated.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
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• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
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• Compatible with MAFI CCR system.
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• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
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1. Department of EECS University of California, Berkeley
EECS 105 Spring 2005, Lecture 27
Lecture 27:
PN Junctions
Prof. Niknejad
2. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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?
3. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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:
4. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Diffusion Equations
Assume that the mean free path is λ
Find flux of carriers crossing x=0 plane
)
(
n
)
0
(
n
)
(
n
0
th
v
n )
(
2
1
th
v
n )
(
2
1
)
(
)
(
2
1
n
n
v
F th
dx
dn
n
dx
dn
n
v
F th
)
0
(
)
0
(
2
1
dx
dn
v
F th
dx
dn
qv
qF
J th
5. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Einstein Relation
The thermal velocity is given by kT
kT
v
m th
n 2
1
2
*
2
1
c
th
v
Mean Free Time
dx
dn
q
kT
q
dx
dn
qv
J n
th
n
n
q
kT
D
*
*
2
n
c
n
c
c
th
th
m
q
q
kT
m
kT
v
v
6. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
dn
qD
nE
q
J
J
J n
n
diff
drift
2
1 J
J
2
2
1
1 E
E
1
2
2
1
E
E
)
( 1
1
J
)
( 2
2
J
7. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
dn
qD
E
qn
J o
n
n
n
0
0
0
dx
d
n
kT
q
E
n
D
dx
dn
o
o
n
n
o 0
0
0
0
0
0
n
dn
V
n
dn
q
kT
d th
o
8. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
)
(
)
(
0
0
0
0
0
0
x
n
x
n
V
x
x th
i
n
x
n
x
)
(
0
)
( 0
0
0
0
0
0
0
0
n
dn
V
n
dn
q
kT
d th
o
9. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
th
V
x
ie
n
x
n /
)
(
0
0
)
(
th
V
x
ie
n
x
p /
)
(
0
0
)
(
2
/
)
(
/
)
(
2
0
0
0
0
)
(
)
( i
V
x
V
x
i n
e
e
n
x
p
x
n th
th
10. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
10
0
0
0
0
0
0
10
)
(
log
10
ln
mV
26
)
(
)
(
ln
mV
26
)
(
)
(
ln
)
(
x
n
x
n
x
n
x
n
x
n
V
x
i
i
th
10
0
0
10
)
(
log
mV
60
)
(
x
n
x
10
0
0
10
)
(
log
mV
60
)
(
x
p
x
11. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
+
12. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
PN Junction Currents
Consider the PN junction in thermal equilibrium
Again, the currents have to be zero, so we have
dx
dn
qD
E
qn
J o
n
n
n
0
0
0
dx
dn
qD
E
qn o
n
n
0
0
dx
dn
n
q
kT
n
dx
dn
D
E
n
o
n
0
0
0
0
1
dx
dp
p
q
kT
n
dx
dp
D
E
p
o
p
0
0
0
0
1
13. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
PN Junction Fields
n-type
p-type
ND
NA
)
(
0 x
p
a
N
p
0
d
i
N
n
p
2
0
diff
J
0
E
a
i
N
n
n
2
0
Transition Region
diff
J
d
N
n
0
– – + +
0
E
0
p
x
0
n
x
14. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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:
)
(
)
( 0
0
0 a
d N
N
n
p
q
x
)
(
)
( 0
0 a
N
p
q
x
0
)
(
0
x
0
p
Na
0
0
x
xp
15. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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:
)
(
)
( 0
0 d
N
n
q
x
0
0 n
x
x
0
)
(
0
x
0
n
Nd
a
i
N
n
n
2
0
Transition Region
diff
J
d
N
n
0
– – + +
0
E
16. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
“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
/
)
(
/
)
(
0
0
)
(
0
)
(
)
(
0
0
n
V
x
i
d
po
a
V
x
i
x
x
e
n
N
q
x
x
N
e
n
q
x
th
th
s
x
dx
d
dx
dE
)
(
0
2
2
0
17. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
)
(
n
d
po
a
x
x
qN
x
x
qN
x
s
x
dx
dE
)
(
0
0
)
(
'
)
'
(
)
( 0
0
0
0
0
p
x
x
s
x
E
dx
x
x
E
p
Field zero outside
transition region
18. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Depletion Approximation (2)
Since charge density is a constant
If we start from the n-side we get the following
result
)
(
'
)
'
(
)
(
0
0
0 po
s
a
x
x
s
x
x
qN
dx
x
x
E
p
)
(
)
(
)
(
'
)
'
(
)
( 0
0
0
0
0
0
0
x
E
x
x
qN
x
E
dx
x
x
E n
s
d
x
x
s
n
n
)
(
)
( 0
0 x
x
qN
x
E n
s
d
Field zero outside
transition region
19. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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-type
p-type
ND
NA
– – – – –
– – – – –
– – – – –
– – – – –
+ + + + +
+ + + + +
+ + + + +
+ + + + +
Depletion
Region
)
(
)
( 0
0 x
x
qN
x
E n
s
d
)
(
)
(
0 po
s
a
x
x
qN
x
E
20. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
( 0
0
x
E
x
qN
x
qN
x
E p
no
s
d
po
s
a
n
no
d
po
a x
qN
x
qN
21. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
po
s
a
po
p
dx
x
x
qN
x
x
0
'
)
'
(
)
(
)
(
x
x
po
s
a
p
p
x
x
x
qN
x
0
'
2
'
)
(
2
22. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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)
2
0 )
(
2
)
( p
s
a
p
p
o x
x
qN
x
2
0 )
(
2
)
( n
s
d
n
n x
x
qN
x
)
0
(
2
2
)
0
( 2
0
2
0 p
p
s
a
p
n
s
d
n
n x
qN
x
qN
23. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Solve for Depletion Lengths
We have two equations and two unknowns. We are
finally in a position to solve for the depletion
depths
2
0
2
0
2
2
p
s
a
p
n
s
d
n x
qN
x
qN
no
d
po
a x
qN
x
qN
(1)
(2)
d
a
a
d
bi
s
no
N
N
N
qN
x
2
a
d
d
a
bi
s
po
N
N
N
qN
x
2
0
p
n
bi
24. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
0
2
lim
0
a
d
d
d
bi
s
N
n
N
N
N
qN
x
d
a
bi
s
a
d
d
a
bi
s
N
p
qN
N
N
N
qN
x
d
2
2
lim
0
25. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
d
a
bi
s
n
p
d
N
N
q
x
x
X
1
1
2
0
0
0
μ
1
10
1
2
15
0
q
X bi
s
d
cm
V
10
μ
1
V
1 4
pn
E
26. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Have we invented a battery?
Can we harness the PN junction and turn it into a
battery?
Numerical example:
2
ln
ln
ln
i
A
D
th
i
A
i
D
th
p
n
bi
n
N
N
V
n
N
n
N
V
mV
600
10
10
10
log
mV
60
ln
mV
26 20
15
15
2
i
A
D
bi
n
N
N
?
27. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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:
mn
pm
bi
0
p
n
mn
pm
+
−
bi
)
( mn
pm
bi
28. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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:
no
d
po
a x
qN
x
qN
29. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
+
−
D
bi V
D
V 0
D
V
30. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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:
d
a
D
bi
s
D
n
D
p
D
d
N
N
q
V
V
x
V
x
V
X
1
1
)
(
2
)
(
)
(
)
(
bi
D
n
d
a
a
d
D
bi
s
D
n
V
x
N
N
N
qN
V
V
x
1
)
(
2
)
( 0
bi
D
p
d
a
d
a
D
bi
s
D
p
V
x
N
N
N
qN
V
V
x
1
)
(
2
)
( 0
bi
D
d
D
d
V
X
V
X
1
)
( 0
31. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
D
a
D
p
a
D
J
V
qN
V
x
qN
V
Q
1
)
(
)
(
)
(
)
(
)
( D
D
J
D
D
J v
q
V
Q
v
V
Q
32. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Derivation of Small Signal Capacitance
From last lecture we found
Notice that
D
V
D
D
J
D
D
J v
dV
dQ
V
Q
v
V
Q
D
)
(
)
(
R
D V
V
bi
p
a
V
V
j
D
j
j
V
x
qN
dV
d
dV
dQ
V
C
C
1
)
( 0
bi
D
j
bi
D
bi
p
a
j
V
C
V
x
qN
C
1
1
2
0
0
d
a
d
a
bi
s
d
a
d
a
bi
s
bi
a
bi
p
a
j
N
N
N
N
q
N
N
N
qN
qN
x
qN
C
2
2
2
2
0
0
33. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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!
d
a
d
a
bi
s
j
N
N
N
N
q
C
2
0
0
1
0
1
1
2 d
s
d
a
bi
s
s
j
X
N
N
q
C
)
(
)
(
D
d
s
D
j
V
X
V
C
34. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
A Variable Capacitor (Varactor)
Capacitance varies versus bias:
Application: Radio Tuner
0
j
j
C
C
35. Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 27
Part II: Currents in PN Junctions
36. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
D
N A
N
-
-
-
-
-
-
-
-
-
-
-
-
-
+
+
+
+
+
+
+
+
+
+
+
+
+
0
E
bi
q
,
p diff
J
,
p drift
J
,
n diff
J
,
n drift
J
−
−
+
+
−
−
Thermal
Generation
Recombination
Carrier with energy
below barrier height
Minority Carrier Close to Junction
37. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
D
N A
N
-
-
-
-
-
-
-
+
+
+
+
+
+
+
( )
bi R
q V
+
−
38. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
D
N A
N
-
-
-
-
-
-
-
+
+
+
+
+
+
+
( )
bi R
q V
+
−
39. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
1
d
qV
kT
d S
I I e
d
qV
kT
d
s
I
I
1
( )
d d S
I V I
40. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Minority Carriers at Junction Edges
Minority carrier concentration at boundaries of
depletion region increase as barrier lowers …
the function is
)
(
)
(
p
p
n
n
x
x
p
x
x
p (minority) hole conc. on n-side of barrier
(majority) hole conc. on p-side of barrier
kT
Energy
Barrier
e /
)
(
(Boltzmann’s Law)
kT
V
q D
B
e /
)
(
A
n
n
N
x
x
p )
(
41. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
“Law of the Junction”
Minority carrier concentrations at the edges of the
depletion region are given by:
kT
V
q
A
n
n
D
B
e
N
x
x
p /
)
(
)
(
kT
V
q
D
p
p
D
B
e
N
x
x
n /
)
(
)
(
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
42. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
A
qV
kT
n
p e
0 0
( ) 1
A
p
x
qV
L
kT
n n n
p x p p e e
0
n
p
0
p
n
0
A
qV
kT
p
n e
Minority Carrier
Diffusion Length
43. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
A
qV
kT
n
p e
0
n
p
0
p
n
0
A
qV
kT
p
n e
This also happens if the minority carrier
diffusion lengths are much larger than Wn,p
, ,
n p n p
L W
44. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Diode Current Densities
0 1
A
p
qV
p
diff n kT
n n p
p
x x
dn D
J qD q n e
dx W
0 0
( )
( )
A
qV
kT
p p p
p p
dn n e n
x
dx x W
p side n side
-Wp Wn
xn
-xp
0
A
qV
kT
n
p e
0
n
p
0
p
n
0
A
qV
kT
p
n e
0 1
A
n
qV
p
diff n kT
p p n
x x n
D
dp
J qD q p e
dx W
2
1
A
qV
p
diff n kT
i
d n a p
D D
J qn e
N W N W
2
0
i
p
a
n
n
N
45. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Diode Small Signal Model
The I-V relation of a diode can be linearized
( )
1
d d d d
q V v qV qv
kT kT kT
D D S S
I i I e I e e
( )
1 d d
D D D
q V v
I i I
kT
2 3
1
2! 3!
x x x
e x
d
D d d
qv
i g v
kT
46. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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
0
1.4
S
j j
dep
C A C
X
47. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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 d
q V v
kT
n
p e
0
n
p
0
p
n
( )
0
d d
q V v
kT
p
n e
1
2
d
d
qI
C
kT
Time to cross junction
(or minority carrier lifetime)
48. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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.
49. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
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+
annode
cathode
p
50. EECS 105 Fall 2003, Lecture 27 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Diode Circuits
Rectifier (AC to DC conversion)
Average value circuit
Peak detector (AM demodulator)
DC restorer
Voltage doubler / quadrupler /…