EF202-2-The PN Junction-P.pdf

Electronics
Dr R. M. Howard
4: The PN Junction
Notes:
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(Copyright - Dr R. M. Howard 2010)
1
Background Results - One Dimensional Case
a) The relationships between electrostatic potential and the electrostatic
electric field - here EXx is the component of the electric
field in the x direction:
x
d
EXx = – x x = –
dx
EXd –
b) The potential difference between two points x2 and x1 (potential
at point x2 with respect to point x1 ):
21 = x2 – x1 =
x2 – EXd x1 c) The relationship between charge density and the electric field:
3
1.0 Overview
Physics of Semiconductors
PN Junction
Bipolar Junction Transistor (BJT)
Field Effect Transistor (FET)
Electronic Circuits
References: See books listed in unit outline.
d
EXx dx
x
x
= ---------- EXx =
----------- d
–
2
4

2.0 The PN Junction
In most electronic devices, from transistors to lasers, there are pn
junctions. An understanding of the pn junction is fundamental to
understanding the operation and characteristics of most electronic
devices.
A pn junction is formed when a n type semiconductor material is
bought into close contact with a p type semiconductor material. In
practice the two doped regions are made on the same piece of silicon.
npo =
anode
N A
N A
N D
2
ni ------
NA p type n type
cathode
Impurity Density
junction
N D
metal contact
Carrier Concentrations Prior to Diffusion
anode
cathode
circuit symbol
np ppo = NA linear-log graph
nno = ND p type n type
junction
pno =
2
ni -------
ND x
Notation: the first subscript denotes the semiconductor type. The
second subscript of o denotes the equilibrium concentration prior
to the two types being connected.
5
7
To understand the characteristics of a pn junction it is useful to
consider the case of bringing together a piece of n type semiconductor
and a piece of p type semiconductor where both are at equilibrium,
both are at the same temperature, and the temperature is
such that the assumption of full ionization is valid.
It is assumed that there is no external connections to either the
anode or the cathode.
The n type semiconductor and the p type semiconductor are
assumed to have the ideal impurity doping levels as illustrated
above. Such an impurity profile results in an ‘abrupt junction’.
Full ionization of the doping atoms yields the following carrier
concentrations just prior to the p type semiconductor making contact
with the n type semiconductor. After contact there will be significant
diffusion of carriers.
The physical situation is illustrated below for the case just prior to
diffusion occurring:
vacant electron site (hole)
free electron
anode
-
acceptor atom
+ -
+
donor atom
+
p type
- +
-
n type
cathode
Note: equilibrium and full ionization implies, on average, one free
electron (an electron in the conduction band) being ‘associated’
with each donor atom. The free electron will move around the
crystal lattice and, with high probability, the donor atom will be
positively charged.
Similarly, equilibrium and full ionization implies, on average, one
vacant electron state (hole) being ‘associated’ with each acceptor
atom. A hole will move away from an acceptor atom and, with high
probability, the acceptor atom will be a fixed negative charge (with
4 covalent bonds).
6
8

At the moment of contact of the p and n semiconductor materials
there is a discontinuity in the hole and electron densities at the
junction. This discontinuity leads, after contact of the two materials,
to an initial very high diffusion flow of carriers. After contact a
smooth, and high, gradient for the electron and hole densities
exists in the vicinity of the junction. These large gradients implies
diffusion: holes from the p region diffuse to the n region and electrons
from the n region diffuse to the p region.
conduction band electrons vacant electron states (holes)
anode
npo =
-
p type
-
+
n type
The consequence of this diffusion is as follows:
N A
N D
2
ni ------
NA +
cathode
valence band electrons
np linear-log graph
pp nn 2
p n
n n
i
p
pno = -------
ND p type n type
x
0
Further, the diffusion results in a net negative charge, due to the
ionized acceptor atoms, close to the junction and on the p side.
Similarly, there is a net positive charge, due to the ionized donor
atoms, close to the junction and on the n side:
9
11
1. Near the junction the free conduction band electrons in the n
type side diffuse to the p type side leaving behind positively
charged donor atoms.
2. Valence band electrons in the n type side diffuse to the vacant
electron states in the p type material resulting in negatively
charged acceptor atom sites close to the junction. This is equivalent
to the diffusion of holes from the p type side to the n type side.
This diffusion results in a change in the concentration profile as
shown below:
anode
- - + +
- - +
+
p type
-
- +
+
n type
E ˜
cathode
The separated charge creates an electric field as illustrated below.
Electrons and holes will move in response to this electric field to
create electron and hole drift currents. The electric field is such
that the electron and hole drift currents oppose the electron and
hole diffusion currents:
10
12

ionized acceptor atoms ionized donor atoms
anode
npo =
2
ni ------
NA p type
- - - +++
n type
- - - +++
- - - +++
Carrier Concentrations in Equilibrium
N A
N D
E ˜
cathode
diffusion force on holes
E field force on holes
diffusion force on electrons
E field force on electrons
np depletion region
linear-log graph
p
nn p
2
p n
n n
i
p
pno = -------
ND p type n type
x
– Lp
0 Ln The region between – Lp and Ln is called the ‘depletion region’.
The name arises because, in equilibrium, this region is ‘depleted’
of carriers (holes and electrons); depletion relative to the levels
that existed prior to contact. To see this clearly consider the concentration
of carriers and impurity atoms in equilibrium:
13
15
Equilibrium
Equilibrium is reached when the electric field is such that the drift
current equals that of the diffusion current and the total net current
flow across the junction is zero, i.e.
Jndrift + Jndiffusion = 0
Jhdrift + Jhdiffusion = 0
Here J is the current density and a scalar quantity is assumed.
In equilibrium the following distribution of carriers holds and distinct
lengths: LP in the p region and LN in the n region can be
defined where the concentrations of holes and electrons are close to
the values that existed prior to the two semiconductor types coming
into contact.
Concentrations in Equilibrium (100% Ionization)
N A
N D
n po
np n p
p p
p type n type
acceptor concentration
donor concentration
nn p n
net charge density
p no
positive charge
negative charge
linear-log graph
x
14
16

2.1 Charge Density and Depletion Approximation
In the p region the charge density, assuming ionization of the
acceptor atoms, is
x = qp px– NA – npx For reference: an isolated p type material in equilibrium has a
charge density given by
= qp A + p – N
i A – ni where pA is the hole density due to the ionization of the acceptor
atoms and the subscript i denotes intrinsic concentrations.
In the n region the charge density, assuming ionization of the
donor atoms, is
x = q– nnx+ ND + pnx For reference: an isolated n type material in equilibrium has a
charge density given by
– Lp
qN D
x – qNA charge density
negative charge
positive charge
Effectively, the charge densities are being approximated as follows:
L n
–
qNA – Lp x 0
x = qp px– NA – npx =
0
x– Lp qND
0 xLn x = q– nnx+ ND + pnx =
0 x Ln x
17
19
= q– nD–
n + N
i D + pi where nD is the electron density due to the ionization of the donor
atoms and the subscript i denotes intrinsic concentrations.
For a pn junction in equilibrium it is the case, close to the junction,
that
x – qNA p side
x qND n side
Definition: Depletion Approximation
The assumption which is usually used when analysing the pn junction
is that the depletion region is defined in a binary manner such
that the net charge density is not a smooth curve either side of the
junction but exhibits a step character as illustrated below:
Conservation of charge requires that the total positive charge to
the left of the junction must equal the total negative charge to the
right of the junction, i.e.
qAN A L p
= qANDLn NALp = NDLn Here A is the cross-sectional area of the pn junction.
The depletion approximation implies the following physical situation:
anode
p type
- - - +++
n type
- - - +++
- - - +++
ionized acceptor atoms
depletion region
cathode
ionized donor atoms
charge neutrality
18
20

2.2 Electric Field and Potential Variation Across PN Junction
For the one dimensional case, and with EXx defined as the component
of the electric field in the x direction, the relation
EXx x
= ----------- d
–
implies that the electric field across the pn junction has the form
shown below:
– Lp
– Lp
j
EXx x As the charge in the depletion region to the left of the junction
equals the charge in the depletion region to the right of the junction,
i.e. NALp = NDLn , it follows that the electrostatic potential
across the junction can be written as
L n
x
– qNAL
p
--------------------
L n
x
2
qNALp ----------------
2
+
qNALpL n
-----------------------
2
21
23
– Lp
– Lp
qN D
x – qNA EXx The relationship between electrostatic potential and the electrostatic
electric field, i.e.
EXx yields the form for the change in potential as illustrated below:
L n
L n
x
– qNAL
p
--------------------
x
d
= – x x = – EXd dx
–
2 2
qNALp qNDLn q 2 2
j = ---------------- + ----------------- = ----- N 2 2 2 ALp + NDLn
This equation for the junction potential requires knowledge of the
depletion region widths Lp and Ln . An alternative expression for
the junction potential, which does not require knowledge of Ln and Lp , will be derived below.
Notation: The junction potential j is called the built in potential.
Depletion Region Width
The above defined result for the junction potential is useful in
establishing an expression for the depletion region width
Wd =
LN + LP : See Exercise 2.
x
22
24

2.3 Implication of Junction Potential
Consider the definition of potential:
A potential change of 1 volt implies a change in energy per unit
Coulomb of charge of one Joule.
Hence, a potential change of j volts across the depletion region
implies that a positive charge of q Coulomb has qj Joules of
energy more in the n region than in the p region (assuming a position
away from the depletion region). Likewise an electron has qj Joules of energy more in the p region than the n region (assuming
points away from the depletion region). The following diagram
illustrates the change in energy for an electron in a pn junction:
p type
– Lp
electron energy
qj L n
n type
First: energy band diagram for conduction band:
Electron Energy
drift
E
-
C
Note:
p type n type
– Lp
diffusion
-
Jndrift + Jndiffusion = 0
0
E
˜
---
L n
-----
zero reference potential
x
energy barrier for
electrons
EC x
q j
25
27
2.4 Energy Band Diagrams for Holes and Electrons in PN Junction
In equilibrium the drift current of electrons from the p region to
the n region is equal to the diffusion current of electrons from the
n region to the p region. Similarly, in equilibrium the drift current
of holes from the n region to the p region is equal to the diffusion
current of holes from the p region to the n region.
The first component of the drift current consists of electrons in the
conduction band of the p type material moving across the junction
to the conduction band in the n type material. The second component
of the drift current consists of holes in the valence band in the
n type material moving across the junction to the valence band in
the p type material (i.e. electrons moving from vacant quantum
states in the p type material moving to vacant quantum states in
the n type material).
Consistent with the electron energy being qj Joules higher in the
p region than the n region (away from the depletion region) the following
energy band diagrams are appropriate (Neudeck, ch 3):
If this was not the case there would be accumulation, at some
point, of electrons. This would violate the assumption of equilibrium.
Second: energy band diagram for valence band:
Electron energy
E V
hole
energy
Again:
+++++
+++
+
diffusion
p type n type
– Lp
0
+
drift
L n
E ˜
energy barrier
for holes
E V
x
q j
26
28

Jhdrift + Jhdiffusion = 0
If this was not the case there would be accumulation, at some
point, of holes which would violate the assumption of equilibrium.
Notes:
The change in the conduction and valence band energy levels in
the depletion region is called band bending.
Electrons in the conduction band (assumed to have an energy close
to EC ) of the n region, and away from the depletion region edge,
cannot move to the p region unless they have additional energy
equal to that of the energy barrier, i.e. qj Joules. Similarly, holes
in the valence band (assumed to have energy close to EV ) of the p
region, and away from the depletion region, cannot move to the n
region unless they have additional energy equal to that of the
energy barrier, i.e. qj Joules.
29
31
2.5 Fermi Level in PN Junction
Recall: By definition, the Fermi level is the energy level where the
probability of occupancy of a quantum state equals 0.5. Thus,
under equilibrium conditions, and in a material without external
energy input, that the Fermi level is the same every where in the
material. If this was not the case then carriers would redistribute
themselves, through drift and diffusion mechanisms, such that the
probability of occupancy at a given energy level is the same
throughout the material. In equilibrium, the following energy
band diagram holds consistent with a constant Fermi level:
Energy
EC Ei EF EV 2.6 Junction Potential
p type n type
– Lp
0
energy barrier = q j
potential barrier for electrons
EC E i
EV potential barrier for holes
x
Ln The energy barrier qj can be written in terms of two components:
the contribution due to the p region and the contribution due to
the n region as illustrated below. The basis for doing this is that
the intrinsic Fermi level Ei is equidistance from the conduction
and valence bands throughout the pn junction.
Energy
E C
Ei EF EV p type n type
– Lp
0
L n
energy change in p region
qj EC energy change in n region
Ei EV x
30
32

Hence:
qj = Ei– Ef + Ef– Ei p region
n region
This demarcation allows the junction potential to be determined as
follows: First, away from the junction region the following relationships
apply
These relationships imply
Hence
pNA nie Ei E =
– f
kT
p region
n ND nie Ef E =
– ikT
n region
NA Ei – Ef = kTln------
p region
n
i
ND Ef – Ei = kTln-------
n region
n
i
2.7 External Potential Across PN Junction in Equilibrium
For a pn junction which is open circuited, the potential between
the anode and cathode terminals is zero. Q: how can this be reconciled
with the non-zero junction potential j .
A: A build up of charge on the metal-semiconductor contacts leads
to an electric field, and hence potential variation that counters the
junction potential.
anode
+
Zero -
anode
+
Zero
p type n type
-
cathode cathode
I +
D = 0
-
+
-
+
x -
– Lp
Ln j
x
33
35
and
qj = Ei– Ef + Ef– Ei p region
n region
NA kTln------
kT
ND -------
NAN D
= + ln = kTln
---------------
n
i n
2
i n
i
kT NAN D
------ j = ln---------------
q 2
n
i
Typical values for NA and for ND result in the junction potential
being in the range of 0.6 to 0.7 volts for Silicon.
This relationship is used later to determine the diode equation.
3.0 PN Junction with Applied External Voltage
3.1 Qualitative Analysis
The applied pn junction voltage is denote as VD and is defined consistent
with the following diagram. The diode current ID is defined
as being positive flowing from the p type semiconductor to the n
type semiconductor.
anode
+
VD -
+
p type n type
cathode
anode
ID cathode
Basis: The p and the n regions are at least moderately doped such
that they act as low resistance (high conductivity) materials. This
means that the electric field in the p and the n
regions away from
the depletion regions is at a low level and that the applied potential
appears mainly across the depletion region.
V D
-
34
36

Assumption: the electric field in the regions away from the depletion
region is close to zero. The applied external potential appears,
in full, across the depletion region.
Consider the illustrative diagram below.
anode
+ VD -
p type
- - - +++
n type
- - - +++
- - - +++
Edue to V
VD 0 holes D
˜ electrons
electrons Edue to VD holes
VD 0
˜
cathode
Edue to depletion charge
˜
forward bias
reverse bias
A positive potential, Vd 0,
will oppose the built in potential j and the depletion region will contract. A negative potential, Vd 0,
3.2 Forward Bias
With forward bias the applied voltage opposes the built in voltage
j . The applied electric field creates a force such that majority carriers
move to both sides of the junction. These carriers counteract
some of the immobile acceptor and donor ions and the charge
stored in the depletion regions decreases.
Energy
E C
VD 0
zero bias level
drift
-
p type n type
– Lp
0
diffusion---
L n
-----
E C
qV D
qj– VD x
q j
37
39
will enhance the built in potential j and the depletion region will
expand.
anode
anode
+ VD -
p type
- - - +++
- - - +++
- - - +++
n type
E ˜ decreases
+ VD -
p type
- - - +++
n type
- - - +++
- - - +++
E ˜ increases
VD 0
cathode
VD 0
cathode
Note: The depletion region widths Lp and Ln decrease from the
zero bias case.
The decrease in the potential across the depletion region results in
the number of electrons in the electron rich region n region, which
have enough energy to cross to the p
region, being significantly
increased. The electron diffusion current is significantly enhanced
above the level for the zero bias case. A similar energy diagram
holds for holes in the valence band and the hole diffusion current is
significantly enhanced above the level of the zero bias case. The
overall result is a significant positive current.
The energy band diagrams correspondingly change as illustrated
below:
38
40

Electron Energy
E C
Ei EF EV p type n type
qV D
– Lp
VD 0
0
energy barrier = qj– VD potential barrier for electrons
EC EF Ei EV potential barrier for holes
x
Ln Note: the applied potential of VD volts results in the Fermi level
changing by a corresponding amount across the depletion region.
This follows because an applied potential of V volts results in a
change in the potential energy of an electron by – qV volts. Hence,
the level where the probability of a state being occupied by an electron
with probability 0.5 changes accordingly.
Energy
E C
drift
--
zero bias case
– Lp
p type n type
0
E
˜
diffusion
-
L n
-----
qj– VD EC x
Note: The depletion region widths Lp and Ln increase from the
zero bias case.
The increase in the potential across the depletion region results in
the number of electrons in the electron rich region n region that
have enough energy to cross to the p region being reduced. The
electron diffusion current is reduced below the zero bias case.
---
– qVD
q j
41
43
3.3 Reverse Bias
VD 0
For the reverse bias case the applied voltage enhances the built in
voltage j . The applied electric field creates a force such that
majority carriers move away from both sides of the junction. This
results in the depletion region being increased in width and with
greater charge storage.
For the reverse bias case the electron potential in the p region, relative
to the n region, is higher than the zero bias case, by qVD. The
following diagram then is relevant:
A similar energy diagram holds for holes in the valence band. The
number of holes in the hole rich p region that have enough energy
to cross to the n region is reduced below the zero bias case.
The enhanced electric field across the depletion region will, however,
force minority carriers that reach the depletion region edge
(electrons in the p region and holes in the n
region) across the
junction. Overall drift current dominates and a net small and negative
current flows across the junction. The current is at a low level
as it is dominated by minority carriers and not majority carriers
(as is the case in forward bias).
The energy band diagrams correspondingly change as illustrated
below:
42
44

Electron Energy
E C
Ei EF EV VD 0
p type n type
qV D
– Lp
0
potential barrier for electrons
EC EF Ei potential barrier for holes
EV x
Note: the applied potential of VD 0 volts results in the Fermi
level being qVD volts higher in the p region than the n region.
L n
energy barrier = qj– VD The first assumption is consistent with moderate doping levels in
the n and p type regions such these regions act as regions of relatively
low resistance. The further approximation is then made:
The magnitude of the electric field in the regions outside of the
depletion region is negligible when compared with the magnitude
of the electric field in the depletion region:
anode
depletion region
p type
- - - +++
n type
- - - +++
- - - +++
EXx E ˜
E
˜
x
cathode
0
zero potential drop
45
47
4.0 The Diode Equation
(Nuedeck, ch. 3; P. E. Gray, ch. 2)
To determine the relationship between the applied pn junction
voltage, and the resulting current flow, the following assumptions
are made:
1. Low level injection: The majority carrier concentrations are not
significantly affected by the bias voltage applied to the junction.
2. The depletion approximation is valid.
3. In the depletion region there is no generation or recombination
of carriers. This assumption implies that carriers entering the
depletion region, either due to drift or diffusion mechanisms, transit
across the depletion region either due to the strong electric field
or the strong ‘diffusion pressure’.
4. Full ionization: Away from the junction p NAin the p type
semiconductor and nNDin the n type semiconductor.
A consequence of this assumption is that the applied potential VD appears across the depletion region and not in the regions outside
x
the depletion region. This follows as x = – Exd. –
Preliminary Considerations
For both the forward and reverse bias cases the total electron and
hole current densities are given by
Je = Jedrift + Jediffusion Jh = Jhdrift + Jhdiffusion For the zero bias case the electron and hole current densities are
zero: the drift current of minority carriers is matched by the diffusion
current of majority carriers.
With the depletion approximation, and with zero bias, the carrier
concentrations at the edge of the depletion junction are the equilibrium
concentrations that exist in separated p and n
regions:
46
48

npo =
Carrier Concentrations for Zero Bias Case - Depletion Approx.
N A
N D
2
ni ------
NA np pp = NA np pn p type n type
– Lp
0
L n
nn = ND linear-log graph
pno =
2
ni -------
ND x
In this, and subsequent diagrams the carrier concentrations in the
depletion region are not show. Consistent with the third approximation
noted above, carriers that enter the depletion region are
assumed to transit across the depletion region.
d
Jedrift = qnnx EXx Jediffusion = qDn nx
dx
the assumption of Je 0 implies
EXx – Dn
---------------d
= ( nx )
nnx dx
Using the Einstein relationship Dn n = kT q it follows that
EXx kT 1
= – ------ ----------
d
nx
q nx dx
Utilizing this result, the electrostatic potential across the depletion
region can be determined according to:
Ln = – EXxdx =
– Lp Ln kT 1
------ --------d
q nx dx
=
nx dx
– Lp kT
Ln ------ lnnx
q
– Lp =
kT
-----ln q
nL n
---------------n–
Lp
= VT ln
nL n
---------------n–
Lp
VT =
kT
-----q
49
51
4.1 Modelling Carrier Concentration Close to Depletion Region
The diode current-voltage relationship is determined from the carrier
concentrations at the edge of the depletion region. To determine
the concentrations at the edge of the depletion region for the
forward, and reverse, bias cases the following is assumed:
a) The high electric field, and the large change in carrier densities,
in the depletion region results in large drift and diffusion current
flows thought the depletion region.
b) The low level injection assumption is consistent with the net current
flow through the pn junction being at a much lower level than
the drift and diffusion currents in the depletion region.
Hence, to a first order approximation:
Je = Jedrift + Jediffusion 0
Jh = Jhdrift + Jhdiffusion 0
Utilizing the definitions for the drift and diffusion currents:
assuming x = 0 at the pn junction.
With low level injection all the applied potential appears across the
junction. Hence = j– VD . Further, for the case of low level
injection, and full ionization, it is the case that nL n
then follows that
= ND . It
n– Lp
nL ne
– V T
= =
NDe j V – – D
VT A similar argument can be used to ascertain the relationship
between the hole concentrations at the depletion region edges:
pL n
NAe j V – – D
VT =
As j that
NAND = VT ln---------------
,
which implies e
2
n
i
, it follows
– j VT =
2
ni ---------------
NAND 50
52

n– Lp
2
ni ------ e
NA VD V T
= pL n
=
2
ni ------- e
ND VD V T
Further, as the minority carrier densities in equilibrium are given
by npo =
2
ni NA and pno =
2
ni ND it follows that
n– Lp
npoe VD V T
= pL n
pnoe VD V T
=
Hence, the following carrier concentrations hold for the forward
and reverse bias cases:
npo =
N A
N D
2
ni ------
NA Carrier Concentrations for Reverse Bias Case VD0 np linear-log graph
pp = NA nn = ND n p
diffusion of electrons
npoe VD V T
p n
pno =
pnoe VD V T
p type
– Lp
0
n type
Ln force on electrons
x
E
EXx ˜
x
drift of holes
net current flow
drift of electrons
diffusion of holes
2
ni ND
53
55
npo =
Carrier Concentrations for Forward Bias Case VD0 N A
N D
2
ni ------
NA np pp = NA linear-log graph
nn = ND npoe VD V T
pnoe VD V T
np pn pno 2
= ni ND
p type
– Lp
0 Ln n type
x
force on electrons
E
EXx ˜
x
diffusion of holes
diffusion of electrons
drift of holes
drift of electrons
net current flow
Summary:
For a forward biased pn junction the net effect is diffusion of
majority carriers (holes from the p region and electrons from the
n region) across the pn junction.
For a reverse biased pn junction the net effect is drift of minority
carriers (electrons in p region and holes from the n
region) across
the pn junction. Minority carriers that reach the edge of the depletion
region are swept, by the depletion region electric field, across
the junction.
Summary of Dominant Mechanisms:
Forward Bias - Diffusion of majority carriers across junction
Reverse Bias - Drift of minority carriers across junction
54
56

4.2 Modelling Carrier Concentrations away from Junction
Consistent with the above diagrams, and assuming exponential
decay with distance of the carrier concentrations, the minority carrier
concentrations can be written as:
pnz pno pno e VD V T
– 1e
z l – h
= +
n type
npz npo npo e VD V T
– 1e
z l – n
= +
p type
where z is zero at the edge of the depletion region and is positive
away from the depletion region edge in the specified semiconductor
type. Here lh is the diffusion length of holes in the n region and
ln is the diffusion length of electrons in the p region. By definition
the diffusion length is the distance where the concentration
decreases by 1 e.
Clearly, these currents vary appreciably with distance away from
the depletion region edge as illustrated below for the forward bias
case:
diffusion current
p type
– Je
– Lp
0
L n
hole movement
electron movement
J = – Je+
Jh n type
x
linear-log graph
To determine the total current flow it is easiest to consider the sum
of the hole and electron diffusion current densities at the edge of
the depletion regions (where, with the depletion approximation,
the electric field, and hence the drift current, is negligible). At the
depletion region edge:
J h
diffusion current
57
59
4.3 Determining the Diode Equation
Away from the depletion region the drift current is negligible
E0 and the current flow is that due to diffusion. In the p
region:
˜
Jediffusion qD
d – qDnn
po
n
npz --------------------- e
dz
ln VD V T
– 1e
z l – n
= =
As z is in the opposite direction to x it is the case that
– Jediffusion has to be considered.
Similarly, in the n region:
Jhdiffusion = – qD
d
h
pnz =
dz
qDhpno ------------------ e
lh VD V T
– 1e
z l – h
As z is in the direction of x it is the case that Jhdiffusion has to
be considered.
– Jediffusion
Jhdiffusion The total current in the x direction ( ID being positive with current
flow from the p to the n region) then is
With the definition
it follows that
qDnnpo ------------------ e
ln VD V T
= – 1
p type
qDhpno ------------------ e
lh VD V T
= – 1
n type
ID AJ A qDnnpo qDhpno ------------------ + ------------------ e
ln lh VD V T
= =
– 1
IS =
A qDnnpo ----------------ln
+
qDhpno -----------------lh
58
60

ID IS e VD V T
= – 1
This equation is called the diode equation. IS is called the saturation
current.
The graph of the diode current versus diode voltage has the form:
I S
I D
V D
The following approximations are valid:
+
V D
ID ISe VD V T
= VD » VT VT -
=
I D
cathode
kT
-----q
ID = – IS
VD « – VT
VT = 0.0259 at 300K
61
vt
V B
+
V D
-
First, consider the case where VB 0 and vt = 0.
The diode is
operating on the ID vs VD curve as illustrated below. The diode
voltage equals the bias voltage VB and the resulting current flow,
IB , is determined by the diode characteristic curve.
I D
V B for Bias Voltage
63
5.0 Modelling Diode - Part 1: Small Signal Operation
It is the case that many electronic devices are operated in a manner
such that two distinct current (or voltage) components can be
defined:
a) The first component is a DC component and this component
determines the region of operation of the component. This component
is usually the dominant component.
b) The second component, usually a fraction of the level of the first
component, is usually a time varying signal that contains information
to be modified (usually amplified) by the device. This component
results in small variations in the properties of the device; the
properties essentially being determined by the first DC component.
To illustrate the two components consider a diode which is driven
by a DC voltage source VB and a AC signal source vt as
illustrated
below:
I B
I D
Definition: Operating Point, Bias Point
The operating point, or bias point, of a device is the current-voltage
pair - IBVB for the diode circuit illustrated above - that is
determined by the DC conditions applied to the device.
V B
Operating Point
For most electronic devices correct operation is dependent on a
correct, or appropriate, bias point being established. The major
exception are devices that are operated digitally (these devices can
be considered to be operating at either of one of two possible operating
points).
V D
62
64

Second, consider the case of VB 0 and vt « VB . For this case
the voltage vt results in small changes, as illustrated below,
around the operating point. In this diagram it = ID t – IB .
I B
I D
V B
t
V D
vt
it
linear approx.
5.1 Small Signal Equivalent Model: Part 1 - Equivalent Resistance
The diode equation is ID IS e . For the case where
it follows that
VD V T
= – 1
VD t = VB + vt
ID t IS e VB vt + V T
= – 1
The bias current corresponding to VB is
IB IS e VB V T
= – 1
To establish a small signal model, which is valid around the operating
point VBIB, consider a first order Taylor series expansion
(i.e. a linear curve) around the operating point:
d
IDVD = IDVB + VD– VB IDV dVD
D
VD = VB t
65
67
Note: for the case where the maximum magnitude of vt is
small,
relative to the bias voltage VB , the diode characteristic curve, as
given by the ID – VD relationship, is close to being affine (linear)
around the bias point defined by VBIB. Definition: Small Signal Operation (Linear Operation)
A device with a set bias point is said to be operating in a ‘small signal
manner’, i.e. small signal operation, if the signal variation
around the operating point is small enough such that linear operation
is valid.
Small signal operation is consistent with linear operation and as
far as the small input signal is concerned the device can be
replaced by an equivalent ‘small signal model’.
In this equation the time dependence of the parameters has been
suppressed. It then follows that
IS IDVD IDVB VD– VB ------e
VT VB V T
= +
IB + IS = IDVB + VD– VB ----------------
V
T
Incorporating the time variables, and with IB = IDVB, it then
follows that
IDVD t – IB vt IB + IS = ----------------
V
T
Defining the difference between the diode bias current and the
total current as it , i.e. it = IDVD t – IB , ( it is
the small
signal diode current) it then follows that
it vt IB + IS =
----------------
V
T
66
68

This relationship between the small signal driving voltage and the
resultant small signal current is a linear relationship consistent
with that of a resistance
VT rD = ----------------
IB + IS and the following model is valid:
+ it
VT vt rD = ----------------
IB + IS -
Notes: For forward bias where IB » IS it is the case that
VT rD ------
IB For the case of reverse bias, where IB – IS , the resistance
approaches infinity and the diode appears, in a small signal sense,
as an open circuit.
5.2 Part 2: Depletion Capacitance of a PN Junction
Consider the charge in the depletion region in a PN junction:
– Lp
qN D
x – qNA Such charge separation is consistent with a capacitor.
CDVD CD0 ,
1 V D
– ------
j
m
j = ------------------------- VD --- C
2 D0 A 2qNAND 1
= ------------------------ --------
NA + ND j where m is the grading coefficient. Usually m 0.33 0.5
and
m = 0.5 for an abrupt junction and modelling consistent with the
depletion approximation. See Appendix 1 for details.
L n
x
69
71
Note: Most devices are non-linear. The above approach can be
used, and is widely used, to determine the small signal linear performance
of a non-linear device around a set operating point.
5.3 Part 3: Diffusion Capacitance of a Forward Biased PN Junction
Consider the carrier concentrations for the forward bias case:
npo =
N A
N D
2
ni ------
NA Carrier Concentrations for Forward Bias Case
np pp nn linear-log graph
np pn pnoe pno 2
= ni ND
p type
– Lp
Ln n type
x
VD V npoe T
VD V T
0
Additional charge
The additional charge leads to an additional capacitance - the diffusion
capacitance. The diffusion capacitance can be written in the
form
70
72

CdifVD kdID VD qA
= ----------------------- kd = ------ n I V poln + pnolp S
T
Clearly, the diffusion capacitance is proportional to the diode current.
See Appendix 2 for details.
6.1 Large Signal Model for Diode
The following is a large signal model for a diode that is valid at low
frequencies. More complicated models that incorporate non-linear
capacitances are required to predict high frequency performance.
+
V D
-
anode
ID cathode
circuit symbol
+
V D
-
model
ID ID IS e VD V T
= – 1
73
75
6.0 Diode Models
For most electronic devices the following two types of models are
defined and used. First, a large signal model. Second a small signal
model.
Definition: Large Signal Model
A large signal model is a model that accurately accounts for the
large signal behaviour of the device. As most devices exhibit nonlinear
characteristics the model usually contains elements with
non-linear characteristics.
Definition: Small Signal Model
A small signal model is a model that accurately accounts for small
signal variations around the bias, or operating point, of the device.
First order low frequency models for restricted regions of operation
are:
+
V D
ID ID =
0
-
reverse bias
+
V D
-
forward bias
See Exercise 11 for a simplified model that is useful for the forward
bias case.
I D
ID ISe VD V T
=
74
76

6.2 Small Signal Model for Diode
Assuming the diode is biased with an operating point defined by
VBIB, the following small signal models for a diode are valid:
In these models
rD CD + CDif forward bias
C D
reverse bias
VT CD0 rD = ------ CDVD IB 1 V D
– ------
j
m
= ------------------------- VD j 2
CdifVD kdID VD = -----------------------
IS 6.3 Enhanced Small Signal Model
A more accurate small signal model for a diode, as shown below,
can be proposed:
CC r S
CJ rJ Here rJ is the resistance associated with the junction, CJ is the
junction capacitance and the other parameters are defined based
on the following observations:
1) The finite conductivity in the p and n regions lead to a small
resistance rS in series with the junction resistance and junction
capacitance.
2) The two metal contacts result in a small capacitance CC in parallel
with the components modelling the interior of the diode.
L L
77
79
Note: the parameters in the small signal model depend on the bias
or operating point, of the diode.
In general, parameters in a small signal model of a device depend
on the bias or operating point of the device.
3) The leads of the diode result in lead inductance and this is
accounted for by the inductor LL .
For example, the model for a Schottky diode manufactured by
Hewlett Packard, and suitable for GHz operation, has the following
parameters with a 50 A bias current.
r J
C C
= 618 CJ = 0.12pF rS = 4.7
= 0.02pF LL =
0.1nH
78
80

7.0 Other Issues
7.1 Switching Time
Consider the case where a diode is to be switched from operating
with forward bias to operating with reverse bias. This switching,
ideally, is instantaneous and high speed switching of devices, in
general, is desirable.
Q: What limits the switching speed of a diode?
A: The charge stored in the junction.
Implication: devices that are suitable for high speed operation
need to be designed with low levels of capacitance. Over many decades
significant research, and technological advances, have underpinned
the development of high speed devices. Such devices
underpin the modern computing-communications revolution.
One problem with such devices is that they exhibit a high level of
noise.
7.3 Special Types of Diodes
See, for example, section 3.8 of Sedra, 2004, or section 3.7 of Sedra
2011.
81
83
7.2 Breakdown
Breakdown is due to one of two phenomena: avalanching or tunnelling
(called the Zener process for the case of diodes) See Neudeck
p. 75 f. for more details.
The result of either of these phenomena is a a diode - voltage characteristic
shown below:
ID V BR
When operating in breakdown the diode acts like an ‘ideal’ voltage
source. Special diodes - Zener diodes - are manufactured to utilize
the breakdown phenomena. These diodes have many applications.
Appendix 1: Modelling Diode - Part 2: Depletion Capacitance
Consider a reverse biased diode:
anode
+
p type n type
where VD 0.
Consistent with the depletion approximation, and
low level injection (all applied potential appears across the depletion
region), the depletion regions contain the charges as illustrated
below where the depletion region widths Lp and Ln vary
with the reverse bias.
V D
VD -
+
cathode
V D
-
anode
ID cathode
82
84

– Lp
qN D
x – qNA Such charge separation is consistent with a capacitor. To establish
the capacitance of this charge separation consider the relationships:
EXx where EXx is the component of the electric field in the x direction.
The change in electric field and electrostatic potential is illustrated
below:
L n
x
x
= ----------- d
x = – E
Xd –
–
Charge neutrality requires that the charge in the depletion region
to the left of the junction equals the charge in the depletion region
to the right of the junction, i.e. NALp = NDLn . It follows that the
electrostatic potential across the junction can be written as
j – VD 2 2
qNALp qNDLn q 2 2
= ---------------- + ----------------- = ----- N 2 2 2 ALp + NDLn
By definition, the capacitance of a structure is the change in charge
required per unit of applied potential, i.e.
C
dQ
=
dV
Consistent with the definition of capacitance, to establish the
depletion capacitance of the pn junction, a requirement is to relate
the potential across the depletion region to the charge in the depletion
region. To this end note that the magnitude of the charge
stored in the p side of the junction and the n side of the junction is
Q = qANALp Q = qANDLn x
85
87
– Lp
– Lp
– Lp
qN D
j – VD x – qNA EXx x L n
L n
x
– qNAL
p
--------------------
L n
x
x
2
qNALp ----------------
2
+
qNALpL n
-----------------------
2
where A is the cross sectional area of the junction. It then follows
that the potential across the depletion region can be written as
Hence
j – VD 1 2 2
= ----- qN 2 ALp + qNDLn =
Q 2
2qA 2
--------------- 1
= ------ +
1
-------
NA ND Q A 2qNAND = ------------------------ j – VD NA + ND 1
-----
2
Q 2
qA 2 Q
-----------------
NA 2
qA 2 + -----------------
ND It then follows that the depletion capacitance is given by
CDVD dQ
A
dj–
VD 2qNAND 1
= = ------------------------ --------------------- VD
0
NA + ND j – VD Noting that the depletion capacitance for zero bias is
86
88

CD0 A 2qNAND 1
= ------------------------ --------
NA + ND j the depletion region capacitance can be written as
CDVD CD0 1 V = -------------------- VD 0
D
– ------
j The following figure graphs the relationship between the applied
diode voltage VD and the depletion region capacitance. Also in this
Figure is the actual depletion capacitance using more sophisticated
analysis for the forward bias case.
has been derived assuming an abrupt change in the doping levels
in the p and n regions as well as the depletion approximation.
When these assumptions are not valid a more general model that
closely approximates the change in the depletion capacitance with
diode voltage is:
CDVD CD0 1 V D
– ------
j
m
j = ------------------------- VD ---
2
where m is the grading coefficient. Usually m 0.33 0.5.
Application: A useful method of implementing a variable capacitor
is to vary the bias on a reverse biased PN junction.
89
91
CD0 The equation given above for the depletion capacitance is reasonably
accurate for diode voltages up to about half the built in junction
potential.
Note: The formula
CDVD CDVD j
CD0 1 V j = -------------------- VD ---
2
D
– ------
j V D
actual
Appendix 2: Modelling Diode - Part 3: Diffusion Capacitance
Consider a forward biased diode:
anode
+
VD -
+
p type n type
cathode
anode
ID cathode
where VD
0.
Under forward bias the carrier concentrations can
be approximated as shown below:
V D
-
90
92

npo =
N A
N D
2
ni ------
NA Carrier Concentrations for Forward Bias Case
np pp nn linear-log graph
np pn pnoe pno 2
= ni ND
p type
– Lp
Ln n type
x
VD V npoe T
VD V T
0
Additional charge
The additional charge leads to an additional capacitance - the diffusion
capacitance.
To determine the diffusion capacitance consider the expression for
the minority carrier charge density in the n region and outside the
depletion region:
Utilizing the definition of capacitance
dQVD C = ------------------dVD
the capacitance associated with the diffusion charge in the
region is
dQnVD Cn = ---------------------- =
dVD qApnolh --------------------e
VT VD V T
Similarly, the capacitance associated with the diffusion charge in
the p region is
dQpVD Cp = ---------------------- =
dVD qAnpoln --------------------e
VT VD V T
The total diffusion capacitance is associated with the sum of the
capacitances in the n and the p regions (this arises as the origin of
the charges leading to the two diffusion capacitances are different
and independent of one another), i.e.
n
93
95
pnz pno pno e VD V T
– 1e
z l – h
= +
n type
As before the origin of z is the edge of the depletion region. It then
follows that the total minority charge in the n region is
QnVD qA pno pno e VD V T
– 1e
z l =
– h
+
dz 0
qApnoLnn qApnolh e VD V T
=
+ – 1
where Lnn is the length of the n type region excluding the depletion
width. Similarly, the minority charge in the p type region outside
the depletion region is
QpVD – qA
npo npo e VD V T
– 1e
z l
– n
= +
dz 0
– qAnpoLpp
qAnpoln e VD V T
=
– – 1
qA
CdifVD Cn + C ------e p VT VD V T
= = npoln + pnolp With the definition of kd =
qA
------ n, and by noting, for
V poln + pnolp T
the forward bias case, that the diode current can be approximated
according to IDVD ISe , it follows that the diffusion
capacitance can be written in the form
VD V T
=
CdifVD kdID VD =
-----------------------
IS Clearly, the diffusion capacitance is proportional to the diode current.
94
96

8.0 Exercises
The following exercises are provided to assist your education. It is
expected that you are proactive with respect to your education and
are progressing towards the standard where you learn independently,
attempt problems prior to a tutorial, and know why your
answer to a set problem is correct.
Unless specified assume, for the following exercises, a temperature
of 300 Kelvin. Where appropriate utilize the constant values tabulated
in the previous set of notes.
Exercise 2
Consider a PN junction without external bias and in equilibrium.
For such a PN junction the following relationships apply:
NALp = NDLn q 2 2
----- j = N 2 ALp + NDLn
Use these relationships to determine an expression for Lp , Ln and
the depletion region width Wd = Lp + Ln . Write your expression
for the depletion region width in the form
Wd =
2NA+ NDj ------------------------------------qNAND
1 1 + x
The following relationship x + ------ = ----------- is useful.
x x
97
99
Exercise 1
Consider a Silicon pn junction which has been constructed with
the following parameters:
NA 10 17 cm 3 –
= ND 10 16 cm 3 –
=
ni 1.5x10 10 cm 3 –
=
a) Determine the built in junction potential.
b) If NA is changed to 10 then determine the built in potential.
15 cm 3 –
Note the logarithmic (i.e. slow) change in the built in junction
potential with doping density. This is the reason that the built in
junction potential can be approximated by a set value. Typical values
that are used are 0.65 V or 0.7V.
Exercise 3
Consider a Silicon PN junction with abrupt doping, with no external
bias, and with the following parameters:
ni 10 10 cm 3 –
= NA 10 16 cm 3 –
= ND 10 15 cm 3 –
= = 11.8o a) Determine the built in junction potential, the depletion region
widths in the n and the p regions and the total depletion region
width.
b) Note the large variation in the depletion region widths. What is
the source of the large variation?
c) Determine the charge density levels, the maximum Electric field
strength in the depletion region and the potential change across the
p type depletion region. Does most of the potential change occur in
the p region or the n region?
Note: an electric field of around 3x10 causes breakdown in
air.
6 Vm 98
100

Exercise 4
A linearly graded PN junction is a junction with the following
charge density profile:
– Lp
qN D
x – qNA charge density
x
Ln negative charge
positive charge
a) For such a junction sketch the Electric Field variation, and the
potential variation, across the depletion region.
b) Establish an expression for the maximum electric field
c) Determine an expression for the electric field that is valid for all
values of x.
Exercise 6
Consider the reverse bias case and the following energy band diagram
for the conduction band:
Energy
E
drift
˜
--
diffusion
E -
C
qj– VD zero bias case
– Lp
p type n type
L n
-----
EC x
Draw a corresponding diagram for the valence band and hole
movement.
---
q j
– qVD
101
103
d) Use your expression for the electric field to establish the potential
that exists across the junction.
Exercise 5
Consider the forward bias case and the following energy band diagram
for the conduction band:
Energy
E C
zero bias level
drift
-
p type n type
– Lp
diffusion---
L n
-----
qj– VD Draw a corresponding diagram for the valence band and hole
movement.
Exercise 7
E C
qV D
Consider a pn junction with a bias voltage of VD volts. Utilize the
approximation:
Jh Jh ˜ ˜
and the assumptions of low level injection and full ionization to
show that
drift J + hdiffusion 0
˜
=
pL n
NAe j V – – D
VT =
Exercise 8
Consider the relations
j – VD 1 2 2
= ----- qN 2 ALp + qNDLn Q A 2qNAND =
------------------------ j – VD NA + ND x
q j
102
104

Here Q = qANALp= qANDLn . Use these relations to determine
an expression for the depletion junction width Wd terms of the potential j – VD .
= Ln + Lp in
Exercise 9
For the case of reverse bias, the depletion width is given by
If
Wd =
2NA+ ND -------------------------------- j – VD qN AND ni 10 10 cm 3 –
= NA 10 16 cm 3 –
= ND 10 15 cm 3 –
= = 11.8o then determine the deletion region width for the case of reverse
bias voltages of 0125V .
Graph the depletion region width for reverse bias voltages in the
range of -5 V to 0 V.
Exercise 11
Consider the case of an ideal diode, which is forward biased with a
voltage VD , and with a current flow of ID . What increase in the
voltage is required for the diode current to double at T = 300K ?
The small variation of the diode voltage for large increases in the
diode current leads to the following simplified model for a diode
when operating with forward bias and for the case where the diode
current can be reasonably approximated in advance:
+
V D
-
anode
ID cathode
circuit symbol
Typical values for Von are 0.65 V or 0.7 V.
+
V D
ID Von -
model
105
107
Exercise 10
On a linear-log graph plot the current through a forward biased
ideal diode for the case of IS 10 , and for forward
bias voltages in the range of to .
14 –
= T = 300K
0.1V 0.7V
Exercise 12
A practical diode is modelled according to
for the case of forward bias. Here, n is a number in the range of 1
to 2.
Assuming this model, determine an expression for the equivalent
small signal resistance around a bias point VBIB. Exercise 13
ID ISe nVD V T
=
A diode with the model ID ISe , for the forward bias case,
has the following parameters: , , and .
a) Calculate the small signal diode resistance for diode currents of
, , , and .
nVD V =
T
IS 10 12 –
= n = 1.5 T = 300K
1A 10A 100A 1mA 10mA
b) Graph, on a log-log graph, the small signal diode resistance for
diode bias currents in the range of 1A to 10mA.
106
108

Exercise 14
Given
CDVD A 2qNAND 1
= ------------------------ --------------------- VD j 2
NA + ND j – VD show that CDVD can be written in the form
CDVD Define CD0. CD0 1 V = --------------------
D
– ------
j Exercise 17
In electronic circuits a current mirror is widely used (see standard
electronic textbooks for the structure). An equivalent model for the
current mirror is shown below:
ID V B
R B
+
VD -
kI D
The goal of the circuit is to produce a current source, modelled on
the right of the circuit, with a current kID where k 1.
a) Using the model ID ISe establish a non-linear equation
that the diode voltage must satisfy. This equation should be in
terms of and . There is no analytical solution to this
equation.
VD V T
=
VBRB IS VT 109
111
Exercise 15
Consider a Silicon pn junction with the following parameters:
ni 10 , , ,
10 cm 3 –
= NA 10 16 cm 3 –
= ND 2x10 15 cm 3 –
= = 11.8o T = 300K , cross-sectional dimensions of 10m 100m
a) Determine the built in junction potential.
b) Evaluate the depletion capacitance for reverse bias voltages of
-5, -4, -3, -2 -1 and 0 Volts.
c) Graph the diode depletion capacitance for reverse bias voltages
in the range of – 5V to zero volts.
Exercise 16
What is the easiest parameter to vary to reduce the capacitance of
a reverse biased PN junction?
b) To find an approximation for the current ID , and the diode voltage
VD , the following iterative procedure can be used:
(i) Use VD = 0.6V as a first estimate for VD .
(ii) Establish a first estimate for ID by using Ohm’s law and the
first estimate for VD .
(iii) Establish a second estimate for VD by using the first estimate
for ID and the diode equation.
(iv) Establish a second estimate for ID by using Ohm’s law and the
second estimate for VD .
(v) Iterate until reasonable convergence of values is achieved.
Follow this procedure to determine reasonable estimates for ID and VD for the case of VB .
= 12V , RB = 12k , IS 10 and
14 –
T =
300K
=
110
112

Exercise 18
Modern communication technology is underpinned by high speed
optical communication technology where communication of information
is via light. In an optical communication system a photodetector
is used to detect, at the receiver, the transmitted optical
signal. A photodetector is essentially a reverse biased PN junction
and has the small signal equivalent model as shown below:
I S
C D
In this model CD is the depletion capacitance of the PN junction
and IS is the current generated in the PN junction due to incident
light ( IS is not to be confused, here, with the diode saturation current).
113
A simple optical receiver can be constructed by placing the photodetector
in parallel with a resistor R resulting in the following
small signal equivalent circuit:
I S
C D
R
+
V
-
a) Establish a first order differential equation for the output voltage
V.
b) Solve this equation for the time interval 0 T
when IS has the
form shown below and V0 = 0.
Sketch your solution.
Specify where is the time constant associated with the rising
output voltage.
IS t
IS T
t
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