Download Professor`s notes on pn junctions

PROFESSORS’s NOTES
8.1 SEMICONDUCTOR JUNCTIONS
Electronics is a philosophy and technology that is defined in terms of ’active devices’. The terminology ”active” device usually means that it is of a construction that controls the flow of currents by means of special layers, patterns,
grids, and terminals. For modern electronic circuits, the majority of active devices are semiconductor devices, and
almost all are constructed in terms of layers or layer patterns. These layers and patterns invariably include semiconductor junctions, most of which are intentional, some of which are not. In order to assess the characteristics and performance of an active device we need to understand the electrical characteristics of the semiconductor junctions embedded therein.
Although there are many interesting types of semiconductors and therefore many nifty and fascinating types of junctions that we can create, it is best to confine our attention to a single semiconductor material and let the others be an
extension of concepts. Our best semiconductor choice is silicon, since it is used for the majority of the present generation of circuits. Silicon is favored as a base material since its fabrication processes are relatively straightforward, it
does not require special handling, is non–toxic, and the raw material (SiO2) is readily available.
Semiconductor junctions are formed when two layers of different doping concentrations are metallurgically joined.
If we dope one layer as p–type and fuse it to one which is doped n–type, this junction is called a ”pn” junction. If
the two layers are of like impurities, we call ”isotype” junctions, but its electrical properties are not as pronounced
as they are for junctions of opposite gender.
We will confine our perspective to the simple pn junction and two representative profiles:
(a) abrupt junction
(b) linearly–graded junction.
There are probably as many different junction doping profiles as there are electrical engineers. For the sake of simplicity and an assessment of the basic electrical characteristics of the junction, we will focus only on the simpler of the
junction types and identify all other junctions as being either ’approximately abrupt’ or ’approximately linear’, or
some combination thereto.
The semiconductor junction is usually a result of either deposition of one layer on another, or, more likely, implantation
of a concentration of one type impurity into a substrate of the opposite gender, as indicated by figure 8.1–1, to form
a ’diffused’ junction.
(a)
(c)
p
(n–type) substrate
photoresist
(n–type) substrate
after thermal annealing
and stripping off photoresist layer
(b)
Boron implant
(d) completed pn junction
p
n (= substrate)
p
ion implant of acceptor impurities
through photoresist pattern
Figure 8.1–1: Planar pn junction construction
80
n
Naturally, variations in the implant and annealing process can make some very interesting junctions. But for the sake
of simplicity we will focus on the junction as being of the simple cross–sectional form as indicated by figure 8.1–1(d).
In terms of the electrical properties we tend to think of pn junctions as being of characteristic form and electrical properties as identified by figure 8.1–2
p
(a) As represented by
encyclopaedia
n
(b) As represented by
circuits textbook
p
n
(c) As fabricated by planar
implant process, ( figure 8.1–1)
Figure 8.1–2 pn junction representations.
Our mission, should we choose to undertake it, is to make a good, complete physical identification of the characteristics of this basic pn interface, with the greater view of it serving as a electrical component that exists within a number
of other important semiconductor devices, as well being as an important active component in its own right.
8.2 EQUILIBRIUM POTENTIALS
One of the things that we know about semiconductor materials with dopings of opposite gender is that the equilibrium
index for average electron electron energy, the Fermi–level, is placed very differently for the two types, as represented
by figure 8.2–1(a). For the p–type material, the Fermi level is close to the valence band edge, EV, whereas for the
n–type material it is close to the conduction band edge, EC . The difference between the Fermi levels represents a difference in the average electron energy. So when the materials are metallurgically joined, thermodynamics forces the
materials to come to equilibrium, as represented by figure 8.2–1(b).
In this case, the transfer of energy is manifested by a migration of electrons in the vicinity of the metallurgical junction,
migrating from the n–type material to the p–type material and creating a difference of potential across the junction.
The transfer of energy across the junction also may be represented in terms of a ’band–bending’ of the energies EC
and EV associated with the crystalline lattice. This difference in the lattice energy also is a means of identifying the
electrical potential that is developed across the junction.
81
EC
EC
(Separated)
EF
EV
EF
EV
p–type
n–type
(a) separated p– and n–type materials. EF represents the average electron energy
transition
EC
p–type
EF
q
0
EC
EV
n–type
EV
(b) Joined p– and n–type materials. Energy (electrons) must migrate from n– to p–
on order for thermal equilibrium to occur. Note that this creates a ‘band–bending’
energy change from p– to n– materials. When the materials are at thermal equilibrium then EF = same throughout.
Figure 8.2–1: Equilibrium processes in the fusion of n– and p–type materials.
We can identify this difference of potential by use of the relationships in which the charge–carrier densities are related
to energy by means of thermal statistics:
n
n i exp (E F E in) kT
(8.2–1a)
p n exp (E i
E ) kT ip
(8.2–1b)
F
Recognizing that the energies are electron energy levels, then every one can be replaced by an equivalent voltage potential, which relates to the electronic charge, –q, by:
E ( q)
or
E
q
(8.2–2)
As indicated by figure 8.2–2. In the figure, the drop in energy of crystalline lattice across the junction from left–to–
right indicates a positive increase in electrical potential, since the electrons are of negative polarity (!).
Using the association identified by equation (8.2–2), the free electron densities on each side of the junction can be
expressed as:
( ) V n exp ( ) V np
nn
n i exp
i
F
pi
T
(8.2–3a)
F
ni
T
(8.2–3b)
Take note of the syntax that we use in equation 8.2–3. Since we have two sides we must identify a notation for the
electron densities on each side. Density nn represents the electron density on n–side of the junction and np represents
the (minority–level) electron density on the p–side of the junction. We can similarly identify hole densities pp and
pn .
If we take the ratio of 8.2–3a to 8.2–3b
np
nn
) V exp (
pi
82
ni
T
(8.2–4)
where we have, for convenience defined the thermal potential as being associated with the thermal fugacity kT, by
kT
q
VT
thermal potential
(8.2–5)
Now, since the pn junction is in an equilibrium state, then
n 2i N A
np
and
nn
ND
assuming the we are operating at moderate temperatures for which ni << ND , NA . Then equation 8.2–4 will give
n 2i
N AN D
or a difference of potential across the junction,
0,
ni
pi
0
exp (
pi
ni
) VT
of
V T ln
N AN D
n 2i
(8.2–6)
Take note of the polarity of this potential. It is an equilibrium” potential, nominally on the order of 0.5 to 1.0 V. It
is due to a built–in E–field. It is a ’reverse’ built–in potential relative to the forward conductive direction of the pn
junction diode (figure 8.1–2b). We call it the built–in potential, and sometimes label it as BI . Only electrons have
the capability to ”see” this potential, but it is they that define the behavior of the pn junction anyway, not the human
observer, or other non–electron species. The built–in potential BI is not a visible circuit potential, it is an equilibrium
condition that is real to electrons and to holes and inhibits their ability to migrate across the junction.
Don’t expect to measure 0 ( or BI ) with a voltmeter. It is an electron potential that forms a natural potential barrier
to the continued diffusion of charge–carriers across the junction. It only manifests itself as a strong localized E–field,
as we will assess in the next few sections.
8.3 THE PN JUNCTION IN REVERSE BIAS
When we examine the natural potential barrier that exists as result of joining the two materials of different gender,
we recognize that the transition from p to n is dependent upon the process in which the junction is formed. The transition may be fairly abrupt, or it may be fairly gradual, usually depending upon the temperature processing of the junction. Junction profiles can and do change during the lifetime of the junction as it is subject to thermal stresses
associated with it appointed task.
CASE I: The Abrupt Junction
Assumption that the junction makes an abrupt transition from p to n is the simplest interpretation of junction profile
and is often sufficient to identify basic properties of existing pn junctions. As implied, the junction makes an ”abrupt”
change from (p–type) doping NA to (n–type) doping ND . Realistically, the pn–junction, abrupt or otherwise, is created
on a planer substrate, as shown by figure 8.2–1. The ’slice’ across the pn junction is just like that of figure 8.1–1(a),
except that dimensions, including layer thicknesses, are at the micron level.
p
n (= substrate)
Figure 8.3–1: Planar pn junction construction
83
NA
ND
p
n
A reasonably true ’abrupt’ junction is realized when the doping material is ’implanted’ by means of a high–energy
ion source and then annealed at high temperatures over a relatively short period. Long heating processes tend to drive
the doping material into the material and diffuse the junction boundary over a ’gradual’ transition profile.
metallurgical boundary
NA
ND
EC
p–type
EF
q
0
EC
EV
n–type
EV
Figure 8.3–2. Built–in potential due to thermal equilibrium.
As indicated by section 8.2 the electrical properties are a consequence of the thermal equilibrium process in which
a localized migration of electrons, from the n–side to the p–side, takes place in order to reach energy balance.
Naturally, such a migration of electrons also represents a transfer of charge. This transfer is only manifested in the
immediate vicinity of the junction since it is a localized diffusion across the junction.
Electrons are like restless teenagers. Influenced by the opposite gender on the other side of the junction, the electrons
will take leave of their home sites and cross over (diffuse across) the junction. Likewise the holes will leave their little
homes (acceptor sites) behind and run off with one of the interesting electrons that have come across the junction.
Doping sites in the vicinity of the junction are therefore left empty on both sides. As represented by figure 8.3–3, the
empty (or depleted) donor sites reflect the absence of an electron, with net (+) charge at the site, whereas the acceptor
sites on the p–side reflect the absence of a hole, or net (–) charge, at each acceptor site. The density of uncovered
sites in each case is the same as the respective doping densities.
We usually call this region of uncovered sites the ”depletion region” since the mobile charge carriers are all absent,
and therefore the region is ”depleted” of charge carriers. It is somewhat more descriptive to call it the ”space–charge
region” since the uncovered sites represent a separation of charge, and induce a powerful electric field that opposes
the intrusion of any free charge carriers and forms a barrier to any casual transport of charge across the junction.
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Figure 8.3–3: Diffusion of charge carriers creates a localized charge distribution. In this example,
the acceptor doping NA > donor doping ND . Uniform doping densities NA and ND are assumed.
84
The migration of electrons and holes across the junction also can be identified in terms of a ”diffusion pressure” that
encourages the mobile charges from the home region in which there is a greater density to emigrate to the opposite
region in which there is a lesser density. But when these mobile charges emigrate across the junction, they create a
localized imbalance of charge. This imbalance of charge then defines a strong localized electric field that inhibits
further emigration. The field correlates to the natural difference potential due to the difference of the work–functions
across the junction. This energy difference is on the order of 0.5 to 1.12 eV for the silicon pn junction.
The details of this behavior is represented by figure 8.3–4, which shows an assessment of the electrostatic characteristics of the (abrupt) pn junction. For convenience, we will call this region the space–charge region (SCR) since it is
a region of uncovered, ionized, doping sites. Thickness of the SCR is typically on the order of about 1 W m.
Since this E–field is a real electric field, it will push mobile charge carriers outside the space–charge region (SCR)
away from the transition region, and thereby considerably inhibit conduction. If we increase this E–field by an increase of the ”reverse–biased” potential X J = X J + VR , VR being the externally–applied ’reverse–bias’, then conduction
becomes even more inhibited, although there will always be a small thermally–generated conduction current.
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Figure 8.3–4: Analysis of the abrupt, uniformly–doped pn junction in reverse bias.
85
As indicated by Figure 8.3–4, the separation of charge in the junction region forms an electric field and potential thereto. The characteristics of this ’space–charge region’ can be analyzed straightforwardly by the application of Gauss’
law, which, in one–dimensional form, as is the case represented by figure 8.3–4, is:
dE
dx
(8.3–1)
s
where es is the permittivity of the semiconductor material. For silicon, the permittivity
density of the uncovered doping sites is given by
qN A
for
qN D
for
xp
0
x
x
s
1.045 pF/cm[1]. The
0
xn
where the dimensions of the space–charge region (which we will call the ’SCR’), are indicated by figure 8.3–4. The
SCR does not terminate at –xp and xn as abruptly as indicated by the figure. The transition from SCR to the neutral
regions are more on the order of a Fermi–Dirac distribution, but the distinction is not sufficiently different from the
uniform depletion approximation to merit the extra mathematical complexity.
Integrating equation (8.3–1) from the left,
E
dE
E(x)
qN A
s
(x
dx
s
0
for which we get
x
qN A
x p)
xp
for
xp
x
o
(8.3–2a)
If we integrate equation (8.3–1) from the right, using x’ = –x, which admittedly is backward from our usual way of
thinking, but is perfectly OK for the mathematics, the result is
E(x)
qN D
s
(x
x n)
for
xn
x
of course, at x = 0 and x’ = 0, the electric field is a maximum, and is
qN Ax p
qN Dx n
E(0)
s
Charge balance offers us some simplification since
qN Ax p
qN Dx n
s
0
(8.3–2b)
E MAX
(8.3–3)
(8.3–4)
QS
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
[1] The relative permittivity of silicon is
10–12 F/m =
r = 11.8. Since the vacuum permittivity 0 = 8.85
8.85 pF/cm, then
s
11.8
(8.85 pF m)
104.5 pF m
1.045 pF cm.
We might as well use this value, rather than retracing our computational process every time.
86
We have identified QS as charge/area that is uncovered on each side of the SCR. This equality also gives us a way to
identify the full thickness, W, of the space–charge region (SCR),
W
xp
(8.3–5)
xn
since equation (8.3–4) allows us to relate the boundaries xp and xn to each other, for which
NA
nD
xn
xp
so that
xn
xp
W
xp 1
xp
W
xn
W
for which
and similarly,
NA
ND
xp 1
NA
ND
1
(8.3–6a)
ND
NA
1
(8.3–6b)
We can apply (8.3–6) to equation (8.3–4) to put the charge/area, QS , in terms of depletion layer thickness W,
QS
qN Ax p
QS
qW
qN AW
1
NA
ND
which we can express as,
N AN D
(N D N A)
where
1
NB
1
ND
qWN B
(8.3–7)
1
NA
(8.3–8)
We have taken some extra pains to explain the development of equations (8.3–7) and (8.3–8) since they provide some
nice simplification options.
When we continue the analysis, to obtain the relationship between the potential across the junction and the extent of the
SCR, we use the definition of electric field as a gradient of the potential, which in one dimension, is:
E
dV
dx
Applying this definition to equation (8.3–2) and integrating, we get potential drop across the p–side of:
0
Vp
qN A
(x p
s
x)dx
xp
87
qN Ax 2p
2 s
(8.3–9a)
and if we likewise integrate from the right,
qN D
Vn
s
x )dx 0
(x n
qN Dx 2n
2 s
(8.3–9b)
xn
for which the total junction potential will be
J
Vp
qN2 x x qN2 x Q
(x x )
2 A p
Vn
A p
p
s
xn
s
S
p
n
s
since qNA xp = Qs = qND xn , as given by equation (8.3–4). Now, since W = xn + xp we can rewrite this equation very
simply as
J
qW 2N B
2 s
QS
W
2 s
(8.3–10)
We are usually interested on just how the thickness of the SCR layer relates to the junction potential, which, from
(8.3–10) will be given by
W
2
s
J
(8.3–11)
qN B
This equation is complete, but has a much more convenient form. Any time we associate electrostatic fields with an
extended space–charge layer within a material, a characteristic length[2], called the Debye length,
LB
V
s
T
(8.3–12)
qN B
can be defined. Using this characteristic length in the definition of depletion width, equation (8.3–11) can be written
as
W
LB 2
J
V
T
(8.3–13)
We sometimes call the ratio J /VT the normalized junction potential. Equation (8.3–13) is a form that will be useful in
the description of many devices for which the junctions are approximately abrupt and doping densities are approximately uniform.
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
[2] The Debye length is actually defined in terms of the total charge density level, and emerges any time we
make a Gauss–law analysis of a distributed charge density, whether at the molecular level, as indicated by this
treatment, or at the atomic level. The Debye length is
LB
V
q(n p)
s
T
but since, for isolated dopings within a semiconductor, (n + p) = either NA or ND . We have taken the liberty
of applying it to the junction with NB as defined by equation (8.3–13).
88
It is also convenient to take note that equation (8.3–3) will give
QS
E MAX
(8.3–14)
s
If we combine this equation with (8.3–10), we get the nice simplification
2 J
W
E MAX
(8.3–15)
To get a feeling for typical layer thicknesses and E–fields within the semiconductor junction, consider the following
example:
*************************************************************************************
EXAMPLE E8.3–1: An abrupt silicon pn junction is formed by creating an implant of NA = 5 10 16 #/cm3
into an n–type substrate of doping ND = 10 15 #/cm3. Determine: (a) Built–in potential 0, and (b) W and
(c) EMAX at this (zero bias) condition. Assume default temperature 300K.
SOLUTION:
(a) According to equation (8.2–3), the built–in junction potential is
N AN D
n 2i
V T ln
0
31
ln 5 10 20
2.25 10
.02585
0.675V
(b) As a matter of convenience, we will find the depletion layer thickness W by first applying equation
(8.3–12) to find Debye length. For the doping levels given, we get
L 2B
sV T
qN B
1.045pF cm .02585V
1.6 10 7pC
1.725
10
10
1
10 15# cm 3
5
1
10 16# cm 3
cm 2
for which we get Debye length:
LB
1.31
10
5
cm
This can also be expressed as:
LB
131 nm
0.131 m
Note that we need to pay careful attention to units of measure.
You might take note of the choices of the units of measure used in this calculation, for example, s = 1.045
pF/cm and q = 1.6 10–7 pC. This scheme may make the process of keeping track of unit magnitudes a little
simpler, and it makes sense to designate these quantities in terms of picosize magnitudes.
89
From this measure of Debye length we get SCR layer thickness of
W
LB 2
0
VT
0.131 m
1.35
.02585
0.9495 m
(c) Equation (8.3–15) gives us a quick convenient means of determining the electric field at the metallurgical
boundary once we have identified the layer thickness W. Application of this equation gives
2 0.675 V
0.9495 m
E MAX
1.42 V
m
1.42
10 4 V cm
Take note that the breakdown voltage of air, for which we are always able to see fairly spectacular effects, is E
= 10 4 V/cm. The E–fields in the pn junction are formidable fields indeed! In this case, the equilibrium E–
field, for which NO external potential has been applied, is greater than the E–fields that exist within natural
lightning storms!
*************************************************************************************
CASE II. The Gradual (Linearly–graded) Junction
For processes in which the junction is annealed over a long period of time, impurities will migrate and diffuse further
across the transition region, which tends to make the junction transition more gradual. To first–order, this ”diffused”
junction may be assumed to be approximately linear, for which we may identify charge regions and electric fields behavior much like that represented by figure 8.3–6
Often the pn junction may be assessed as a linear profile in the vicinity of the transition from p to n, with the doping
levels beyond the junction region being relatively uniform, as represented by figure 8.3–5. Therefore it would be appropriate to combine a linearly–graded analysis with an abrupt analysis.
ND
a = (ND – NA)/d
linearly graded
NA
Figure 8.3–4 Representation of gradual junction profile, assuming that the transition is approximately linear.
In the vicinity of the metallurgical junction, the transition is approximately linear, and therefore the junction may be
analyzed as a linear distribution of uncovered charge sites, with the junction boundary being defined either by the
change of polarity of uncovered sites or when ND = NA . For convenience of analysis, this point we should let this
point be the center of coordinates, as represented by the charge analysis represented by figure 8.3–6.
90
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J
F
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W
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Figure 8.3–5: Analysis of the linearly–graded pn junction in reverse–bias.
According to figure 8.3–4, we expect to have a grading coefficient in the vicinity of the junction that relates to the
doping levels NA and ND far from the junction, as follows
a
V
ND
X
NA
d
const
Y
(8.3–16)
So when we analyze the linearly–graded pn junction we need to acknowledge that there is a gradual transition from
p–type to n–type doping sites across the boundary of the junction of the form
X
N D(x)
N A(x)
V
ax
(8.3–16)
for which the junction boundary falls at x = 0.
When the junction is in equilibrium or in reverse bias, doping sites in the vicinity of the junction will be uncovered
with distribution as defined by equation (8.3–16), so that use of (the linear form) of Gauss’ law will be of the form:
dE
dx
V
UZ
V
s
Z
qax
s
with solution
91
qa
E(x)
s
12 x W4 x
qa
xdx
2
2
(8.3–17)
s
W 2
When x = 0, the electric field is at its maximum,
qa 2
W
8 s
E MAX
(8.3–18)
The relationship between the thickness of the SCR (= W) and the potential across the junction is readily obtained by
integration of equation (8.3–17) according to the definition of the electric field as a gradient in the electrostatic potential, for which
J
2qa x3 W 2
3
E(x)dx
s
W 2
xW 2
4
W 2
W 2
qaW 3
12 s
(8.3–19)
12
W
qa The width of the depletion (SCR) layer is then
1 3
s
The junction potential itself is still of the form
J
BI
J
(8.3–20)
VR
where BI is given by equation (8.2–6) and VR is the applied reverse bias. The use of equation equation (8.3–20) is
constrained by the fact that linear profile is not infinite in extent. Doping levels are expected to reach approximately
uniform limits ND and NA far from the junction, for which equation (8.2–6) is applicable. Equation (8.3–20) is a representation of electrical characteristics for the charge distribution that is uncovered by the built–in plus applied fields.
The relationship between EMAX and
J
is defined by taking the ratio of equations (8.3–18) and (8.3–19)
3 J
2W
E MAX
(8.3–21)
This result might be compared to the analogous result, for the abrupt junction, as given by equation (8.3.15).
*************************************************************************************
EXAMPLE E8.3–2 A diffused junction has a profile, as shown, that makes an approximately linear transition over a distance d = 5 m from NA = 4 10 16#/cm3 to ND = 1 10 16 #/cm3. (a) Determine the location of
the junction boundary by finding XP and XN . (b) Determine the equilibrium value of W. (c) Determine the
upper limit of voltage such that the extent of the SCR remains within the linear profile.
ND = 1016
–XP
XN
NA = 4 1016
92
SOLUTION:
XP
XN
(a) Using similar triangles,
Since
then
d
5 m
XP XP
d 1 N D N A
similarly,
d 1 XN
and grading coefficient a
XP 1 10 16 (4 5 m 1 (4 10 32) (2.25 10 16)|
1.0 4
(12 1.045 pF cm 0.729V )
(1.6 10 7 pC) (1 10 20 cm 3) J
1 m
0.729 V
10 20# cm 3
W 4 m
1 3
W
W
min(X N, X P)
2
then from equation (8.3–19)
10 16) 10 16) 10 16 10 20) 12 s J
qa (c)
ND
NA V T ln N AN D n 2i 10 16 ( 5 10 |4 Then, using equation (8.3–20)
5 m 1 BI
.02585 ln (5 BI
XN
XP XP 1 XN
NA ND
(b) Using equation (8.2–6)
NA
ND
1 3
0.8295 m
1 m
(1.6 qaW 3
12 s
10 7) (1.0 10 20) 12 1.045
(2 10 4) 3
10.22V
then
VR
10.22V 0.7292
9.492 V
This result tells us that if we apply a VR = 9.492 V to the junction then W/2 = 1 m = XN , which is the limit of
bias for which the bilateral linear junction characteristics remains valid. If we exceed this potential, then we
must evaluate the junction as if it were a linear profile on the NA side and a uniform profile on the ND side.
*************************************************************************************
Example E8.3–2 Also tells us that, for the one–sided junction, most of the linear gradient will lie on the heavily–doped
side of the junction boundary. In many cases it is best to assess the junction as if it is half linear and half uniform.
Then we must apply the results of both of these type profiles to analyze the junction behavior. For example, if we had
chosen NA = 1.73 1017 #/cm3, then XN would have been 0.273 m and W at J = BI would have been 0.546 m.
A number of interesting problems that combine linear and uniformly–graded junction profiles are available at the end
of the chapter.
93
8.4 JUNCTION CAPACITANCE
As noted by the previous sections, the pn junction in reverse bias is characterized by a space–charge region (SCR) in
which doping sites of opposite polarity are uncovered on both sides of the junction. As long as the junction is kept in
reverse bias, no current will flow. This aspect is exactly the same as that for a capacitance. The pn junction happens to
have considerable capacitance, since the thickness of the SCR is small, usually on the order of microns, as was represented by example E8.3–1.
p
n
_
+
Figure 8.4–1: Space–charge region and separation of charges = pn junction storage capacitance.
As might be expected, however, the pn junction capacitance is voltage dependent, which may make it unsatisfactory
for some applications but makes it invaluable for others. As a voltage variable capacitance it is usually referred to as a
varactor, although it is in fact just another durn pn junction.
To get a handle on the capacitance behavior of the junction in reverse bias, consider the junction slice represented by
figure 8.4–2. We should acknowledge that the effective capacitance to which time–varying signals will respond is
defined as
C
dQ
dV
(8.4–1)
where, as represented by the figure, dQ represents the increment of charge with dV, which we might note takes place at
the outside boundaries of the SCR as illustrated by figure 8.4–2.
dQ
p
n
W
dQ
Figure 8.4–2: pn junction incremental capacitance.
Since this incremental charge is added at the boundaries, then the separation between +dQ and –dQ is the SCR layer
thickness W, so that the junction capacitance/area is
94
CJ
s
(8.4–2)
W
If you don’t like this lazy (but accurate) argument, then recognize that junction capacitance can also be defined by
examining the E–field. Since an increment in charge also represents an increment in the E–field, then
dQ
dE
s
The corresponding change in the applied voltage is
dV
WdE
W
dQ
(8.4–3)
s
from which we get the same result for junction capacitance CJ as given by equation (8.4–2).
But since the increment of charge also represents more of the doping sites being uncovered, we might also recognize
that equations (8.4–2) and (8.4–3) represent a means for examining this profile, particularly if one side of the junction
is very heavily doped, so that there is relatively little effect on W due to the doping sites uncovered on its side. In this
respect we may approximate. If, for example, we consider a p +n junction (for which NA >> ND ) then
dQ
qN BdW
qN DdW
then
s
dQ
W
dV
qN D
s
(qN DdW )
W
s
qN D
d(W 2)
2 s
WdW
qN(W) d
2 s
2
s
C 2J so that the doping profile can be examined by means of the slope of 1/CJ 2 with respect to V.
1
2
d 1
q s dV C 2
J N(W)
(8.4–4)
For the abrupt junction the depletion capacitance will be
CJ
s
2(
LB 0
V R) V T
(8.4–5)
For the linearly–graded junction the depletion layer capacitance is
CJ
12(
qa
0
2
s
1 3
V R) (8.4–6)
We might take note that both of these equations may be written as
CJ
C J0 1
VR
0
95
1 (m 2)
(8.4–7)
for which CJ0 is a constant, corresponding to the zero–bias (VR = 0) capacitance. This is the form for junction capacitance that is used by the SPICE software. It default to m = 0, corresponding to the abrupt, uniform junction profile,
for which
V
s
C J0
2
LB
0
(8.4–8)
T
*************************************************************************************
EXAMPLE 8.4–1: Consider the circuit for which the capacitances are replaced by reverse–biased diodes,
as shown by figure E8.4–1. This circuit is a bandpass circuit and the peak frequencies are defined by
1
C R 1R 2
(E8.4–1)
When the diodes are reverse–biased they will behave as if they
are voltage–controlled capacitances of the form:
CJ
C J0
1 V vi
1 (m 2)
R2 = 1 M
+
+
vo
V
–
0
R
2R1 = 20 k
2R1 = 20 k
where CJ0 is the SPICE diode model parameter CJO.
Figure E8.4–1: Dellyannis–Friend Biquad
We execute the DF biquad in SPICE, choosing CJ0 to be 400 pF. Using either the .STEP command or some
other SPICE input option, we may apply bias sequence V = {0, 2, 6, 14}V. Since a voltage divider exists at the
input we will will get a diode reverse bias of VR = {0, 1, 3, 7}V applied to the two diodes. Using PROBE to
see all traces concurrently, then we will see a SPICE output something like that shown by figure E8.4–2.
|vO|
f1
1k
f2 f3
3k
f4
10k
Measured values, using PROBE cursor
f1 = 3.95 kHz (for VR = 0 )
f2 = 5.53 kHz (for VR = 1 )
f3 = 7.76 kHz (for VR = 3 )
f4 = 10.78 kHz (for VR = 7 )
30k
f(Hz)
Figure E8.4–2: Approximate transfer response for DF biquad with biased varicaps.
Using these measurements we can find the values of CJ from equation (E8.4–1), for which
CJ1
CJ2
CJ3
CJ4
1/C2
= 403 pF
= 288 pF
= 205 pF
= 148 pF
From the plot we get VBI = 1.2 V
VBI
VR
*************************************************************************************
96
The grading constant, m, defines the profile, which may be assumed to be of the form:
N 0 xx
0
N
m
(8.4–9)
When m = 0, the doping profile on each side of the junction is constant. When m = 1, the doping profile is linear.
For the special case in which m = –3/2, which we call the hyperabrupt doping profile, we then have a junction capacitance for which
CJ
C J0
1 VR
ND
2
(8.4–10)
0
N 0 xx
0
3 2
NA
Figure 8.4–3 Typical hyperabrupt junction profile
So that, should this junction be used in a resonant LC circuit,
f
2
1
LC
and if we use a hyperabrupt junction, the frequency will be directly proportional to the applied bias:
f
1 VR
0
giving us a frequency that is linearly–controlled by applied voltage. Pretty neat, huh?
8.5 PN JUNCTION IN FORWARD BIAS: LOW–LEVEL INJECTION
When the pn junction is forward biased, the inhibiting force of the electric field is reduced in magnitude. Charge carriers are more free to emigrate across the junction, and current flow will take place. The emigration of charge carriers
is primarily a diffusion process, which is a thermal process and is driven by thermal statistics.
Therefore we turn to our thermal statistics to identify the effects that are taking place and define the basic junction
electrical characteristics. If we look at the density of electrons on both sides of the junction, as defined by the thermal
statistics we see that
nn
N C exp (E Fn
np
N C exp (E Fp
) kT E Cn) kT
(8.5–1a)
E Cp
(8.5–1b)
where EFn, ECn are the energy levels on the n–side of the junction and EFp , ECp are the energy levels on the p–side
of the junction, respectively. If the junction is at equilibrium, then EFn = EFp , and we are back to equation (8.2–4),
with 0 defined by the difference in the lattice energy levels across the junction or the equivalent electron potentials,
for which
97
nn
np
exp (E Cp
E Cn) kT exp(
0
(8.5–2)
V T)
where
0
E Cn) q
(E Cp
E in) q
(E ip
in
(8.5–3)
ip
as defined by equation (8.2–6).
Solving equation (8.5–2) for np , for which the equilibrium value can be designated as np0 , we get
0
n n exp( n p0
(8.5–4)
VT )
V>0
V=0
NA
ND
NA
EC
p–type
ND
ECp
q
q
qV
0
ECn
EFn
EC
EF
EV
EFp
EVp
n–type
EVn
EV
Figure 8.5–1. Junction potentials (a) at equilibrium and (b) at forward bias
When the junction is at forward bias, for which EFn > EFp , then the junction potential is reduced by an amount,
V
E Fp) q
(E Fn
(8.5–5)
as represented by figure 8.5–1. We also see that
E Cp
E Cn
q
q(
0
V)
(8.5–6)
for which equation (8.5–2) becomes
nn
np
exp V T exp (
0
V ) VT
(8.5–7)
and equation (8.5–4) becomes
np
n n exp (
0
V) V T (8.5–8)
If we take the ratio of equations (8.5–7) and (8.5–4) we get
np
n p0 exp V V T 98
(8.5–9)
This result is the level of electrons that are ’injected’ across the junction by the thermal processes. The voltage V is
the applied bias that imbalances the thermal equilibrium and allows carriers to migrate across. Since the relationship
is exponential, the forward conduction current is very strong.
Similarly, holes will also emigrate from the p–side into the n–region and we will see an injected hole density of
pn
p n0 exp V V T (8.5–10)
Naturally, these injected carrier densities will upset thermal equilibrium on each side, and the recombination processes
will then begin to act to restore equilibrium.
If we make the approximation that the minority–carrier populations will be the primary densities that are affected, then
we can assess the junction current by following the action of the minority–carrier levels. This assumption is called
low–level injection, corresponding roughly to the condition
np < 0.1 pp
and pn < 0.1 nn
low–level injection
(8.5–11)
If we have a forward bias V such that the majority–carrier densities are also affected, then we are in a situation which
we identify as high–level injection, and the analysis becomes more complicated. However, in most cases, the normal
junction operation corresponds to low–level injection, so we will postpone high level injection to later entertainment.
The zones beyond the space charge region (SCR) include many charge carriers. As the excess enemy carriers of the
opposite gender invade these regions, war is declared, and recombination processes take place. A combination of diffusion processes and recombination processes make up the factors that drive the carrier levels toward equilibrium.
There will be relatively little free charge within these regions, so they would aptly be described as quasi–neutral regions (QNR), and are so represented by figure 8.5–2.
p
pn
np
n
QNR
SCR
QNR
x’
x
Figure 8.5–2: The quasi–neutral regions.
Since a non–equilibrium situation exists, recombination and diffusion processes will govern the fate of the injected
charge carriers. In the case of excess electrons injected into the p–side, the density will have a survival profile given by
np
n p(0) exp( x L n)
(8.5–12)
where np (0) is the level of excess carriers above equilibrium at the boundary between SCR and the QNR for the p–
type side. This excess level of carriers = np (0) – np0 . The parameter Ln is the recombination length, also called the
diffusion length, and is given by
Ln
D n
(8.5–13)
n
99
We need to note that Dn = n VT is the diffusion constant for n–type carriers in the p–type environment and recombination decay time constant for these carriers.
n
is the
The invading n–type carriers recombine as result of an encounter with the deadly holes, which results in an annihilation
of both, and emission of a photon or phonon as a marker of the encounter. The level np (0) is the level of carriers entering
the p–side (at x = 0) after having migrated across the somewhat diminished SCR barrier.
Similarly, the density of minority type carriers injected into the n–side will be
pn
p n(0) exp( (8.5–14)
x L p)
where pn (0) represents the excess injected carrier density at x = 0. These injected carriers will suffer the same recombination fate as their cousins, according to a diffusion and recombination process for holes, as defined by the recombination (diffusion) length
Lp
D p
(8.5–15)
p
We note that Dp = p VT is the diffusion constant for p–type carriers in the n–type environment and p is the recombination time constant for these carriers. Whether we consider the injected holes or electrons it is essential that we identify
the action charge carriers are the minority carriers, and therefore we must identify the diffusion constants or mobilities
for these minority carriers NOT the majority–carrier diffusion constants.
The minority–carrier levels for the junction in forward–bias are shown by figure (8.5–3). Typical recombination
lengths are on the order of 10 m.
n
p
pn(x)
np(x’)
np0
pn0
QNR
QNR
SCR
Figure 8.5–3: Injected carrier profiles.
Since equations (8.5–12) and (8.5–14), and figure 8.5–3 identify that the injected carriers will have a gradient in density, due to recombination, then also they also imply that we will have two components of diffusion current that occur:
qD p d [ p(x )]
x dx
Jn
qD n d [ n(x)]
x
dx
0
n p(0)
Ln qD n
Jp
0
qD p
p n(0)
Lp (8.5–16a)
(8.5–16b)
Since
n p(0)
n p(0)
n p0
n p0 exp(V V T)
n p0
(8.5–17a)
p n(0)
p n(0)
p n0
p n0 exp(V V T)
p n0
(8.5–17b)
100
then
J
Jn
qD n
J
J
Jp
n p0 Ln
qD p
J S exp(V V T) p n0
exp(V V T) L p 1
1
(8.5–18)
This equation is called the Shockley equation, or also the ideal diode equation, and is the effective description of current in the forward direction.
Or it is ALMOST the description of current in the forward direction. There is more, as we will see in coverage given in
section 8.7.
In equation (8.5–18), JS is called the (reverse) saturation current density, given by
JS
qD n
JS
or, simplifying, we get
qn
2
i
n p0 Ln
Dn L nN A
qD p
p n0
Lp Dp
L pN D (8.5–19)
since np0 = ni 2/NA and pn0 = ni 2/ND . As we see from the Shockley equation, this is the level of current that results when
V < 0, corresponding to reverse bias. It is small, on the order of fA/cm2. When biased in the forward direction, typical
junction current density levels are on the order of A/cm2.
8.6 QUASI–EQUILIBRIUM STATISTICS AND QUASI–FERMI LEVELS
Since, at forward bias, we clearly are in a non–equilibrium situation, and the equilibrium statistics that we used so
cheerfully with equation (8.2–1) are ruined. Equation (8.2–1) assumed that the index, EF, for equilibrium, had to be a
constant, from one type semiconductor, all the way across the junction, to the other type semiconductor, for which we
could readily identify a built–in voltage 0 as a consequence.
But when the semiconductor is in forward bias, the Fermi energies on the opposite sides of the junction are no longer
the same, so we might as well identify them as EFp and EFn . As indicated by figure (8.5–2b) the difference between
EFn and EFp is just
qV
E Fn E Fp
(8.6–1)
When we are well away from the junction, ’deep’ within the quasi–neutral region, we expect that thermal statistics,
such as is represented by equation (8.2–1), is fine. Equations (8.6–2) should be fine, and our use of the mass–action
law,
p n0
n 2i n n0
as assumed so cheerfully by equations (8.5–17), should also be fine and reasonable. Far from the junction, conduction
is entirely identifiable in terms of majority carrier flow, for which equilibrium thermal statistics is fine.
It is only in the vicinity of the junction that the thermal statistics may be compromised, because it is only in the vicinity
of the junction for which pn > pn0 and np > np0 .
We may retain all of the simplicity of the thermal statistics by assuming a state of quasi–equilibrium for which equations (8.2–1) are valid, provided that we define an EFn and EFp everywhere. We therefore define EFn and EFp as quasi–
101
Fermi energy levels since they represent a statement of quasi–equilibrium. For minority–carrier levels as well as majority carrier levels, equations (8.2–1) will apply, for which:
n exp (E np
E ip) kT
(8.6–2a)
E Fp(x)) kT
(8.6–2b)
n i exp (E Fn(x)
pn
i
in
where an EFn is now assumed to also be defined on the p–side of the junction concurrently with Eip , and is distinctly
different from EFp . Similarly, we expect an EFp (x), distinctly different from EFn which may be defined on the n–side
of the junction concurrently with Enp ). The quasi–Fermi levels
Note that, for low–level injection conditions, equations (8.2–1) are still just fine, and define the majority–carrier levels
n exp (E nn
n i exp (E Fn
pp
i
ip
) kT E in) kT
(8.6–3a)
E Fp
(8.6–3b)
If we take the product of equation (8.6–2a) and equation (8.6–3b), and assume that we are at the injection boundary
of the QNR, for which x = 0, we get
n np n
Using equation (8.6–1) and nn
ND, then
pn
n 2i exp (E Fn
E Fp) kT
Nn exp qV kT (8.6–4)
2
i
(8.6–5)
p n0 exp V V T
D
which is exactly the same as equation (8.5–9). Similarly, we could take the product of equations (8.6–2b) and (8.6–3a),
for which, in like manner, we would find
n 2i
(8.6–6)
np
exp qV kT
n p0 exp V V T
NA
which is the same as equation (8.5–10).
We expect that the behavior of the quasi–equilibrium levels will be something like that represented by figure 8.6–1.
qV
EFn
p
EFn
EFp
EFp
QNR
n
SCR
QNR
Figure 8.6–1: Non–equilibrium conditions, for which it is convenient to define quasi–equilibrium
Fermi levels. In the SCR and in the QNR region near to the junction, EF splits into EFn and EFp . Far from the
junction where there is little excess minority carrier levels, the quasi–equilibrium Fermi levels coincide.
The figure shows the approximate behavior of the quasi–Fermi levels, EFn and EFp across the junction, as continuous
levels extending from one side of the semiconductor to the other. Since we are in a non–equilibrium state of forward
bias, EF on the n–side is higher than EF on the p–side by bias energy qV, as is given by equation (8.6–1). Far from
the junction, the quasi–equilibrium levels are coincident, EFn = EFp = EF. We might take note of the fact that, in order
to satisfy equations (8.6–5) and (8.6–6), it is necessary that EFn and EFp extend all of the way across the SCR with
little or no change.
102
In the vicinity of the junction EFn and EFp will split into two levels according to equation (8.6–2). The split between
the levels is an indication of the levels of injected minority–carriers, np = np (x) and pn = pn (x’), as represented by equations (8.5–12) and (8.5–14). Since pn (x) asymptotically approaches pn0 , we expect that EFp (x) will asymptotically
approach EFn , as represented by the figure. Similarly, on the p–side, we expect that EFn (x) will asymptotically approach EFp , as governed by equation (8.5–12).
8.7 RECOMBINATION OF CHARGE–CARRIERS IN THE SPACE–CHARGE REGION.
Yes, brothers and sisters, recombination also take place in the SCR. After all, carriers that dare to attempt a transit of
this region are in a no–man’s land, in which charge–carriers of both types are present. The usual warfare takes place, in
which carriers annihilate each other and constitute a recombination current of the form
Qn
Jn
(8.7–1a)
n
for n–type carriers, where n represents the recombination time constant for electrons within the SCR. We should have
a similar form for the p–type carriers, given by
Qp
Jp
for which
p
(8.7–1b)
p
represents the recombination time constant for holes within the SCR.
Since the densities, n and p, of both type charge–carriers, vary across the space–charge region by several orders of
magnitude, the process for determining an appropriate Qn and Qp can be very mathematical if we so choose. In
order to gain insight without the mathematical encumbrance, it is reasonable to make a number of analytical approximations that will make the process a little more tame.
The behavior of energy levels within the SCR is reflected by figure 8.7–1. Since a built–in E–field exists, the lattice
energies undergo a ’band–bending’ as represented by the figure, while the quasi–equilibrium Fermi levels remain
approximately constant across the region, as discussed by section 8.6.
ECp
qV
ECn
Ei
EFn
EFp
EVp
EVn
QNR
SCR
QNR
Figure 8.6–2: Energy level behavior within the SCR
We can analyze the carrier densities by means of equation (8.6.2) with knowledge of Ei (x) as function of position within
the SCR. But we would find that the mathematics would be a task of unwelcome proportions, since we would be
entertaining ourselves with integrals of exponential functions. We can accomplish just as much by use of a few approximations gained from applying equations (8.6–2) to the SCR, for which
103
n
n i exp (E Fn
p
n i exp (E i(x)
E i(x)) kT
E Fp) kT
(8.7–3a)
(8.7–3b)
If we take the product of these two equations, we get:
np
n 2i exp(E Fp
E Fn) kT
n 2i exp(V V T)
(8.7–4)
As one of the expectations of low–level injection, we expect that somewhere within the space–charge region the carrier
densities will be n
p. Using equation (8.6–4), this level corresponds to
n2
np
p2
n 2i exp(V V T)
and therefore, taking the square root, the crossover carrier level will be
n
p
n i exp(V 2V T)
104
(8.6–5)
This concept is reinforced by the fact that the differences (EFn – Ei ) and (Ei – EFp ) change from one side of the SCR
to the other as Ei = Ei (x) changes, as represented by figure 8.6–2. The point within the SCR at which equation (8.6–5)
is met will not necessarily be at the metallurgical junction.
The approximate behavior of carrier densities from one side of the SCR to the other, with the junction in forward bias,
is represented by figure 8.6–3. If we also make the approximation that the regions within the SCR for which recombination takes place are approximately triangular, as also represented by figure 8.6–3, then
Ei
EFn
EFp
Ei
SCR
Figure 8.6–3: Region of recombination in the SCR for excess n–type carriers.
q W n(SCR)
J n(SCR)
(8.6–6)
n
2
If we assume that the capture cross–sections for electrons and holes are about the same, then cn = cp , and equation
(7.8–8) will be somewhat simplified,
c nN t(pn n 2i )
n p 2n i cosh
R
105
(7.8–10)
for which we make the definition of short–circuit resistance
V
R SCP
K (V R
t dt f
V TP
I DSAT
K P(V
1
V V V
y(2bdy y) C R dI(x, t) C V dx
t
dV(x, t) L I dx
t
I
V T1)
O
L
0.9
106
(15.3–5)
T2
0.1
C LR SC1
(15.5–16)
V TP)
SC1
y
1 ln
2b y
2b
0.1
(15.3–13)
0.9
(15.6.3a)
(15.6.3b)