We now wish to consider the situation in which a p-type and an n-type semiconductor are brought into contact
with one another to form a pn junction.
Most semiconductor devices contain at least one junction between p-type and n-type semiconductor regions.
Semiconductor device characteristics and operation are intimately connected to these pn junctions, so considerable attention is devoted initially to this basic device.
The pn junction diode itself provides characteristics that are used in rectifiers and switching circuits.
In addition, the analysis of the pn-junction device establishes some basic terminology and concepts that are
used in the discussion of other semiconductor devices.
The fundamental analysis techniques used for the pn junction will also be applied to other devices.
1
Basic Structure of the pn Junction
Basic Structure of the pn Junction
,
Figure 1a schematically shows the pn junction.
It is important to realize that the entire semiconductor is single-crystal material in which one region is doped
with acceptor impurity atoms to form the p region and the adjacent region is doped with donor atoms to form
the n region.
7 • 1 Basic Structure 0: the po
The interface separating the n and p regions is referred to as the metallurgical junction
N... negativ
charge
p
I
I
l
.'
" - Metallurgical
junction
P
I
I
.I I
I
(a)
: - Sp<lce
I
N,
Hole
diO'usion
- - -
I
:
f - - - - - - N.
- +-
EICCI"(lll
I .
"Diffusion
:
force" on ----.:
hole...
I
I
x =o
(b)
:
li,·ncl
i-4--
(ur.:e
I
hQles
I (.) Simplified
Figure 1: (a) Simplified geometry"Igure
of a pn7,1
junction.
(b) dopinggeometry
profile of of
an ideal uniformly doped pn junction.
a po junction: (b) doping profile of
ideal unifonnly doped pn junction.
Figure 7.2 111\e
charg
forces acting on the charg(.-d
The impurity doping concentrations in the p and n regions are shown in Figure 1b.
For simplicity, we will consider a step junction in which the doping concentration is uniform in each region and
there is an abrupt change in doping at the junction.
11le impurity doping concentrations in the p and n regions are shown
Initially, at the metallurgical junction, there is a very large density gradient in both the electron and hole
ure 7.1 b. For simplicity. we will consider a step jUlJction in which
concentrations.
uniform
ininto
each
there is carrier
an abrupt
change
Majority carrier electrons in the ncentration
region will is
begin
diffusing
theregion
p regionand
and majority
holes in
the
p region will begin diffusing into the
n region.
Initiall y, at the meUlllul'gical jUnction, there is a very large
junction.
the dop
in dopin
density
If we assume there are no externalin
connections
the semiconductor,
this diffusion process
cannot
continue
both the to
electron
and hole then
concentrations.
Majority
carrier
elect rons in
indefinitely.
gion will begin diffusing into the p region and majorily carrier holes in the
will begi n diffusing into the n region. If we ",sume there are no external con
to the semiconductOr,
then this diffusion process cannot continue indefin
1
electrons diffuse from the n region, positively charged donor atoms are lef
As electrons diffuse from the n region, positively charged donor atoms are left behind.
Similarly, as holes diffuse from the p region, they uncover negatively charged acceptor atoms.
The net positive and negative charges in the n and p regions induce an electric field in the region near the
metallurgical junction, in the direction from the positive to the negative charge, or from the n to the p region.
The net positively and negatively charged regions are shown in Figure 2.
These two regions are referred to as the space charge region.
Essentially all electrons and holes are swept out of the space charge region by the electric field.
Since the space charge region is depleted of any mobile charge, this region is also referred to as the depletion
239
7 • 1 Basic Structure 0: the po Junction
region.
Density gradients still exist in the majority carrier concentrations at each edge of the space charge
N... negative
Not pos.itj\'e
charge
thargc
I
" - Metallurgical
junction
P
I
I
.I I
I
- - - -
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
I
: - Sp<lce dl<lrgt' n:gioll - :
I
I
:
f - - - - - - N.
+-
EICCI"(lll
=o
(b)
E·fic ld
I .
:
I
"Diffusion
:
force" on ----.:
hole...
I
:
I
I
I
:
li,·ncld
E-licld
i-4--
(ur.:e Oil
force ,)1}
I
hQles
dcctrnns
"Oiffu-"ion
forcc" on
cleeu'Ons
:
l
I
I
mplified geometry of
Figure 7.2 111\e
charge region, the electric field, and the
oping profile of
Figure 2: The space charge region, the electric field, and the forces acting on the charged carriers.
forces
acting
on
the
charg(.-d
carriers.
ped pn junction.
region.
We can think of
density
gradient
as producing
a ”diffusion
y doping concentrations
in athe
p and
n regions
are shown
in Fig- force” that acts on the majority carriers.
Theseconsider
diffusionaforces,
acting on in
thewhich
electrons
holes at the edges of the space charge region, are shown in
mplicity. we will
step jUlJction
the and
doping
the
figure.
orm in each region and there is an abrupt change in doping at the
The electric
field in the
space
regiondensity
produces
another force on the electrons and holes which is in the
, at the meUlllul'gical
jUnction,
there
is a charge
very large
gradient
opposite direction to the diffusion force for each type of particle.
on and hole concentrations. Majority carrier elect rons in the n rethermal
equilibrium,
the diffusion
force exactly balance each other.
iffusing intoInthe
p region
and majorily
carrierforce
holesand
in the
the E-field
p region
ng into the n region. If we ",sume there are no external connections
uctOr, then this
process cannot
continue indefinitely. As
2 diffusion
Zero Applied
Bias
from the n region, positively charged donor atoms are left behind.
es diffuse from
p region,
Zerothe
Applied
Biasthey uncover negatively charged ace net pOSitiveWe
and
negative
charges
in the
p regions
an
have
considered
the basic
pn nand
junction
structureinduce
and discussed
briefly how the space charge region is formed.
he region near
the
metallurgical
junction,
in
the
d
ireClion
from
the
In this section we will examine the properties of the step junction in thermal equilibrium, where no current
gative charge,
or and
fromnothe
n to Ihe
p reg ion.
exist
external
excitation
is applied.
ilivcly and We
negatively
charged
regions
are
shownwidth,
in Figure
will determine the space charge region
electric7.2.
field, and potential through the depletion region.
s are referred to as the space clUlrge region, Essentially all electrons
ept out of the space charge region by the electric field. Since the
ion is depleted of any mob ile charge, this regio n is also referred to
egioll; these two tenos will be used interchangeably. Density gradithe majority carrier concentrations at each edge of the space charge
hink of a density gradienc as producing a "ditTusion force" that2 acts
arriers. These diffusion forces, acting On the electrons and holes at
lO mOve into the conduction band of the p region. This potential barrier is referred 10
as thebuill·ill potential barrier and is denoted by Vbj . The built-in potential barrier
maintCJins equilibrium between majorilY carrierelectJ'ons in the n region and minority
l
carrier electrons in the p region, and also between majority carrier holes in the
2.1 Built-in Potential Barrier
p region and minority carrier holes ill the n region. This potential difference across the
Built-in Potential
Barrier
junction
cannot be
whh <\ voltmeter bet:3use new potemial baff\el'S wiU be ,
If we assume that no voltage is applied across the pn junction, then the junction is in thermal equilibrium and
formed
between the probes and the semiconductor th at will cancel Vo,. The potential
the Fermi energy level is constant throughout the entire system.
Vb'
maintains
equilibrium,
so noforcurrent
is produced
byequilibrium.
this Voltage.
'
Figure
3 shows the
energy-band diagram
the pn junction
in thermal
intrinsic
Fermiband
level
is equidistant
conduction
band region,
edge since
through
The The
conduction
and valance
energies
must bend asfrom
we gothe
through
the space charge
the
relative
position thu
of the
and valence
bandsbarrier
with respect
thedetermined
Fermi energy changes
p and
rhe
junction,
s conduction
the built-in
potential
cantobe
as thebetween
difference
I
I
n regions.
p
n
E,
Ii,·,
Ef
"',
-
I
,
... - - - - - - ..
..
,
--t-..
__ m
__
Ef
:'!: '"
Figure
7.3
I Energy-band
diagram
of a pninjunction
in
Figure 3:
Energy-band
diagram
of a pn junction
thermal equilibrium.
thermal equilibrium.
Electrons in the conduction band of the n region see a potential barrier in trying to move into the conduction
band of the p region.
This potential barrier is referred to as the built-in potential barrier and is denoted by Vbi .
The built-in potential barrier maintains equilibrium between majority carrier electrons in the n region and
minority carrier electrons in the p region, and also between majority carrier holes in the p region and minority
carrier holes in the n region.
This potential difference across the junction cannot be measured by a voltmeter because new potential barriers
will be formed between the probes and the semiconductor that will cancel Vbi .
The potential Vbi maintains equilibrium, so no current is produced by this voltage.
The intrinsic Fermi level is equidistant from the conduction band edge through the junction, thus the built-in
potential barrier can be determined as the difference between the intrinsic Fermi level in the p and n regions.
We can define the potentials φF n and φF p as shown in Figure 3 so we have
Vbi = |φF n | + |φF p |
(1)
In the n region, the electron concentration in the conduction band is given by
Ec − EF
n0 = Nc exp −
kT
which can also be written in the form
n0 = ni exp
EF − EF i
kT
(2)
(3)
where ni and EF i are the intrinsic carrier concentration and the intrinsic Fermi energy, respectively. We may
define the potential φF n in the n region as
eφF n = EF i − EF
3
(4)
Equation (3) may then be written as
eφF n
n0 = ni exp −
kT
(5)
Taking the natural logarithm of both sides, setting n0 = Nd and solving for to the potential we obtain
Nd
kT
ln
φF n = −
e
ni
Similarly, in the p region, the hole concentration is given by
EF i − EF
p0 = Na = ni exp
kT
(6)
(7)
where Na is the acceptor concentration.
We can define the potential φF p in the p region as
eφF p = EF i − EF
(8)
Combining Equations (7) and (8), we find that
φF p
kT
=+
ln
e
Na
ni
(9)
Finally, the built-in potential barrier for the step junction is found by substituting Equations (6) and (8) into
Equation (1), which yields
Built-in potential
kT
ln
Vbi =
e
Na Nd
n2i
= Vt ln
Na Nd
n2i
(10)
where Vt = kT /e and is defined as the thermal voltage.
Previously in the discussion of a semiconductor material, Nd and Na denoted donor and acceptor impurity
concentrations in the same region, thereby forming a compensated semiconductor.
From this point on in the text, Nd and Na will denote the net donor and acceptor concentrations in the individual
n and p regions, respectively. If the p region, for example, is a compensated material, then Na will represent
the difference between the actual acceptor and donor impurity concentrations.
The parameter Nd is defined in a similar manner for the n region.
2.2
Electric Field
Electric Field
An electric field is created in the depletion region by the separation of positive and negative space charge
densities.
Figure 4 shows the volume charge density distribution in the pn junction assuming uniform doping and assuming
an abrupt junction approximation.
We will assume that the space charge region abruptly ends in the n region at x = +xn and abruptly ends in
the p region at x = −xp (xp is a positive quantity).
The electric field is determined from Poisson’s equation which, for a one-dimensional analysis, is
d2 φ(x)
−ρ(x)
dE(x)
=
=−
dx2
s
dx
(11)
where φ(x) is the electric potential, E(x) is the electric field, ρ(x) is the volume charge density, and s is the
permittivity of the semiconductor. From Figure 4 the charge densities are
ρ(x) = −eNa
− xp < x < 0
(12)
0 < x < xn
(13)
and
ρ(x) = eNd
4
7. 2
Zero
.ed Bi
p (Oem' )
p
"
+
7.41 The space charge density In
Figure 4: The space charge density in a uniformly doped pn junction assuming the abrupt junction approximation.
a unifonnly doped pn junction assuming
the abrupt junction approximation.
The electric field in the p region is found by integrating Equation (11)
Z
Z
ρ(x)
eNa
eNa
E(x) =
dx = −
dx = −
x + C1
s
s
s
(14)
The where
electric
field is detennined from Poisson's equation which. for a oo
C is a constant of integration.
1
dimensional
ana\ysis,
is to be zero in the neutral p region for x < −x
The electric
field is assumed
p
since the currents are zero in thermal
equilibrium.
As there are no surface charge 2
densities within the pn junction structure, the electric field is a continuous
function.
d tP(x)
-pCx)
dE(x)
----;Rl = -E-, - = - ---;t;("
(711
The constant of integration is determined by setting E = 0 at x = −xp .
The electric field in the p region is then given by
where ",(x) is the electric potential.
eNa E(x) is the e lectri c field. pIx) is the volum
E=−
(x + xp ) − xp ≤ x ≤ 0
(15)
s
density. and E, is the permittivity
of the semico nductor. From Figure 7 .4 . th
In the n region, the electric field is determined from
charge densities
are
Z
and
E(x) =
ρ(x)
dx =
s
Z
p(x) = - eN"
eNd
eNd
dx =
x + C2
s
s
- Xp
< x <
where C2 is again a constant of integration.
(16)
0
The constant C2 is determined by setting E = 0 at x = xn , since the E-field is assumed to be zero in the n
region and is a continuous function. Then
p(x) = eN,1
eNd
E=−
s
(xn − x)
0<x
< Xn
0 ≤ x ≤ xn
(7. [ 2
(7. 12
(17)
The The
electric
field
the pat region
is found
by
Equation (7.1 I). W
electric field
is alsoin
continuous
the metallurgical
junction, or
at xintegrating
= 0.
have thatSetting Equations (15) and (17) equal to each other at x = 0 gives
E=
f
p(x)
- - dx = -
f
Na xp = Nd xn
eN.
-
-eN.
d.t = -- x
(18)
+ C,
Equation (18) states that the number of negative charges per unit area in the p region is equal to the number
of positive charges per unit area
f .l in the n region. Figure 5fsis a plot of the electric
{fJ field in the depletion region.
(7. 1
where C, is a constant of integrati on. The5 elec tri c field is assumed to be zero in th
neUlral p region for x < - xp since the currents are zero in thermal equilibrium. A
C;
here
is again a constant of integration. The potential difference through th
n junction
thefield
important
pan,
rather
thexabsolute
poremial,
so we ma
The is
electric
direction is from
the nmeter,
to the p region,
or in than
the negative
direction for this
geometry.
the uniformly doped pn junction, the E-field is a linear function of distance through the junction, and the
bitrarilyFor
set
the potential equal to zero at x = -xp. The constant of integration
maximum (magnitude) electric field occurs at the metallurgical junction.
An electric field exists in the depletion region even when no voltage is applied between the p and n regions.
p
n
Figure 7.5 I Electric field in the ' pace
Figure 5: Electric field in the space charge region of a uniformly doped pn junction.
charge region of a unifonnly doped pn
.
In the p region then, wejUllcti()n
have
Z
Z
The potential in the junction is found by integrating the electric field.
eNa
(x + xp )dx
s
eNa x2
φ(x) =
+ xp x + C10
s
2
φ(x) = −
or
E(x)dx =
(19)
(20)
where C10 is again a constant of integration.
The potential difference through the pn junction is the important parameter, rather than the absolute potential,
so we may arbitrarily set the potential equal to zero at x = −xp .
The constant of integration is then found as
C10 =
eNa 2
x
2s p
(21)
so that the potential in the p region can now be written as
φ(x) =
eNa
(x + xp )2
2s
− xp ≤ x ≤ 0
(22)
In the n region then, we have
Z
or
Z
eNd
(xn − x)dx
s
eNd
x2
φ(x) =
xn x −
+ C20
s
2
φ(x) = −
E(x)dx =
(23)
(24)
where C20 is another constant of integration.
The potential is a continuous function, so setting Equation (22) equal to Equation (24) at the metallurgical
junction, or at x = 0, gives
eNa 2
C20 =
x
(25)
2s p
6
The potential in fhe n region Can thus be wrillen as
X')
eNd ( x"·x - - +-X
eN. 2
4>(x)=p
2
2€,
The potential in the n region canf.tthus be written as
2
eNdpotentialxthro
eNa 2 junction and
of=the
Figure 7.6 is a plot
φ(x)
xn x −
+ugh the
x
0 ≤ x ≤ xn
s
2
2s p
(7.25)
shows fhe quadratic (26)
dependence on di stance. The magnitude of the potential at x = x" is equal to the
Figure 6 is a plot of the potential trough the junction and shows the quadratic dependence on the distance.
p
n
, =0
" .1,.
F'iKUrl' 7.61 Electric potenlial through Ihe space charge
region of a uniformly doped pn junction.
Figure 6: Electric potential through the space charge region of a uniformly doped pn junction.
The magnitude of the potential at x = xn is equal to the built-in potential barrier.
Then from Equation (26), we have
e
Nd x2n + Na x2p
(27)
2s
The potential energy of an electron is given by E = −eφ, which means that the electron potential energy also
varies as a quadratic function of distance through the space charge region.
Vbi = |φ(x = xn )| =
2.3
Space Charge Width
Space Charge Width
We can determine the distance that the space charge region extends into the p and n regions from the metallurgical junction.
This distance is known as the space charge width.
From Equation (18), we may write, for example,
xp =
Nd x n
Na
Then, substituting Equation (18) into Equation (27) and solving for xn we obtain
1/2
1
2s Vbi Na
xn =
e
Nd
Na + Nd
(28)
(29)
Equation (29) gives the space charge width, or the width of the depletion region xn extending into the n-type
region for the case of zero applied voltage. Similarly, if we solve for xn from Equation (18) and substitute into
Equation (27), we find
1/2
1
2s Vbi Nd
xp =
(30)
e
Na
Na + Nd
where xp is the width of the depletion region extending into the p region for the case of zero applied voltage.
The total depletion or space charge width W is the sum of the two components, or
W = xn + xp
(31)
Using Equations (29) and (30), we obtain
W =
1/2
2s Vbi NA + Nd
e
Na Nd
7
(32)
3
Reverse Applied Bias
,
Reverse Applied Bias
If we apply a potential between the p and n regions, we will no longer be in an equilibrium condition: the Fermi
energy level will no longer be constant through the system.
CHAPTBR
7 The pn Junction
Figure 7 shows the energy-band diagram of the pn junction for the case when a positive voltage is applied to
the n region with respect to the p region.
p
£,.
"
--------------r-·
eV'IUotlII
------ £,.
Figure
Energy-band
djagrnm
a po under
junction
under
Figure7.71
7: Energy-band
diagram
of a pn of
junction
reverse
bias.
reverse bias.
As the positive potential is downward, the Fermi level on the n side is below the Fermi level on the p side.
The difference between the two is equal to the applied voltage in units of energy.
when
a potential
positivebarrier,
volt"ge
is 'lpplied
n region with respect to the p region. As the
The total
indicated
by Vtotal , to
hasthe
increased.
positive
the Fermi level on the n side is below the Fermi level
The appliedpotential
potential is is
thedownward)
reverse bias condition.
TheOle
totalppotential
barrier
is now given between
by
011
side. The
difference
Ihe twOis equal to Ihe applied ,'ollag< in units
ofencl'gy.
Vtotal = |φF n | + |φF p | + VR
(33)
The total potential batTier. indicated by V,,,,,). has increased. The applied poten·
where VR is the magnitude of the applied reverse-bias voltage.
tial
is the
The total potential barrier is now given by
Equation
(33) can be rewrittencondition.
as
Vtotal = Vbi + VR
where Vbi is the same built-in potential barrier we had defined in thermal equilibrium .
(34)
(7.32)
where
V8 isCharge
the magnitude
of Electric
the applied
reverse-bias voltage. Equation (7.32) can
3.1 Space
Width and
Field
be rewritten as
Space Charge Width and Electric Field
Figure 8 shows a pn junction with an applied reverse-bias voltage VR .
(7.33) I
Also indicated in the figure are the electric field in the space charge region and the electric field Eapp induced
by
the applied
wbore
V voltage.
is Ihe same built-in pOlenlial \)<lrrier we had defined in thermal
M
The electric fields in the neutral p and n regions are essentially zero, or at least very small, which means that
equilibrium.
the magnitude of the electric field in the space charge region must increase above the thermal-equilibrium value
due to the applied voltage. The electric field originates on positive charge and terminates on negative charge.
This means that the number of positive and negative charges must increase if the electric field increases.
7.3.1
Space Charge Width and Electric Field
8
Figure 7.8 shows a pnjunctioll with an applied
reverse-bias voltage IIR . Also indicated
in the figure arc the electric field in the space cMrge region and the electric field E. •
7 • 3 Reverse App!'e<l Bias
•
,, : E: i t ,,
, -- :;: ++ •,
-- + + ,
p
a
..
'.
I
I
1-11'-1
v.
-
Figure 7.8 I A pn junction. with
+
appUcd reverse-bias
Figure 8: A pn junction, with an applied reverse-bias voltage, showing the directions of the electric field induced
voltage,
showing
by VR and the space
charge electric
field.the diret:lions of the electric field induced
by VH and the space charge electri c field.
For given impurity doping concentrations, the number of positive and negative charges in the depletion region
can be increased only if the space charge width W increases.
therefore,
with an increasing reverse .. bias vollage VR _ \Ve are assuming that the elecThe space charge width W increases, therefore, with an increasing reverse-bias voltage VR .
tric field in the bulk nand p regions is zero. This assumption will become clearer in the
We are assuming that the electric field in the bulk n and p regions is zero.
next chapter when we discuss the current-voltage characteristics.
In all of the previous equations, the built·in potential barrier can be replaced by the total potential barrier.
In allspace
of the
previous
the
bui lt·in
barrier can be replaced by
The total
charge
width can equations,
be written from
Equation
(32)potential
as
the total potential barrier. The IOtal
space charge
width
1/2 Can be written from Equa·
2 (V + V ) N + N
tion (7.3 1)
W =
s
bi
R
e
a
d
(35)
Na Nd
showing that the total space charge widths increase as we apply a reverse bias voltage. By substituting the
total potential barrier Vtotal into Equations (29) and (30), the space charge widths in the n and p regions,
respectively, can be found as a function of applied reverse-bias voltage.
(7.34)
The magnitude of the electric field in the depletion region increases with an applied reverse-bias voltage.
The electric field is still given by Equations (15) and (17) and is still a linear function of distance through the
space charge region.
showillg
that
lora}
space
wjdrtheh magnitude
increase.')of as
apply
a increases.
Since xn and
xp lhe
increase
with
reversecharge
bias voltage,
the we
electric
field also
age.
By substituting
thestill
total
pote
barrier junction.
into Equations
(7.28)
The maximum
electric field
occurs
at ntial
the metallurgical
The maximum
electricand
field (7.29),
at the
metallurgical
junction,
from
Equations
(15)
and
(17)
is
the space charge width s in the n and p regions. respectively, can be found as H
−eNd xn
−eNa xp
lic)n of applied reverse· bim; voltage.
Emax =
=
(36)
\f,.",
s
s
If we use either Equation (29) or (30) in conjunction with the total potential barrier Vbi + VR then
1/2
2e(Vbi + VR )
Na Nd
Emax = −
s
+N
d
space charge region
in NaaI'll
juncli()n
Objective
To <:illculate
Iht
width of the
(37)
when a reverse· bias voltage
can show that the maximum electric field can also be written as
isWeapplied.
)
Ago'lin consider a silicon Pl1 jUI)cljon−2(V
III biT +==VR300
K with doping Ci)nCelllrDrions
of
(38)
Emax =
t
W II ,
Nfl =
crn-) and Nd
1015 cm- · •
th:tl
1,5 X 1011) cm-) and lei VR = 5 V.
=
=
where W is the total space charge width.
• Solution
The
potential
barrier was calculated in Example 7. I for this case and is v",,' = 0.635 V.
3.2 built-in
Junction
Capacitance
The
space charge width is determined from Equation (7.34) . We have
Junction Capacitance
12(J
Since we have a separation of positive and negative charges in the depletion region, a capacitance is associated
IV '"
1.7)(8.85 " 10- ")(0.635 5) ( 10" 10" ]
with the pn junction.
+
1.6 x 10-'·
(10'6)(10")
9
:-00
that
+
j'/2
CHAPT . A 7 ThBpnJunction
2
p
n
p
('
+eNd
dX
"-I_l
IQ'
+
+x
li,\'"
-
,,
eNo
- --
-
Withapplil.:d
,,
,,
,
,,,
,
,,,
,
V"'- -- _
t'igUrf: 7.9 I Differential change in the
charge width with
Figure 9: Differential change in the space charge width with a differential change in reverse-bias voltage for a
a differential change in rc"crse·bias voltage {or a uniformly
uniformly doped pn junction.
doped pn junction.
so that
Figure 9 shows the charge densities in the depIetion region for applied reverse-bias voltages of VR and VR + dVR .
An increase in the reverse-bias voltage dVR will uncover additional positive charges in the n region and additional
(7.4
negative charges in the p region.
The junction capacitance is defined as
dQ0
0
C
=
Exactly the same capacitance expression is obtained
by con.idering the
dVR
where
(39)
region extending into the p region xp. The junction capacitance is also referred to at
the depletioll layer capacitallce.dQ0 = eNd dxn = eNa dxp
t
0
(40)
0
2
The differential charge dQ is in units of C/cm so that the capacitance C is in units of farads per square
(F/cm2 ), or capacitance per unit area.
For the total potential barrier, Equation (29) may be
AMPLE 7.5 centimeter
Objective
written as
1/2
2s (Vof
VRjunction.
) Na
1
To calculate the junction capacitance
bi a+po
xn =
(41)
e in Example
Nd 1.3.
Na Again
+ Nd assume th at VR = 5 V.
Consider the same: pnjuDction as mat
The junction capacitance can be written as
• Solution
dQ0 (7.42)dx
n
The junction capacitance is found from
C 0 E<luation
=
= eNd as
So we obtain
•=
C
I
dVR
dVR
( 1.6 x 10- ")(1 1.7}(S.85 x 10-")(10'.)(10" )
2(0.635 + 5)(10'.
+ 10" )
(42)
1'/2
JunctionorCapacitance
Na NFlcrn'
d
C' = 3.66 e
x s10-'
C =
0
2(Vbi + VR )(Na + Nd )
1/2
(43)
Exactly the same capacitance expression is obtained by considering the space charge region extending into the
p region xp .
The junction capacitance is also referred to as the depletion layer capacitance. If we compare Equation (35)
for the total depletion width W of the space charge region under reverse bias and Equation (43) for the junction
capacitance C 0 , we find
s
C0 =
(44)
W
10
Equation (44) is the same as the capacitance per unit area of a parallel plate capacitor.
The space charge width is a function of the reverse bias voltage so that the junction capacitance is also a
function of the reverse bias voltage applied to the pn junction.
3.3
One-Sided Junctions
One-Sided Junctions
Consider a special pn junction called the one-sided junction.
If, for example, Na Nd , this junction is referred to as p+ n junction.
The total space charge width, from Equation (35), reduces to
W ≈
2s (Vbi − VR )
eNd
1/2
(45)
Considering the expressions for xn and xp we have for the p+ n junction
xp xn
(46)
W ≈ xn
(47)
and
Almost the entire space chargeCHAPTER
layer extends into
region of the junction.
264
7 the
Thelow-doped
pn Jur.ct:On
This effect can be seen in Figure 10
The junction capacitance of the p+ n junction reduces to
P.
II
+ eNd 1------,
+
I
,
I
I
I
- V"j 0
Figure
10: Space
charge
density
of one-sided
p+ nof.
junction.
Figure
7.10
I Space
charge
density
one-sided p+njullclion.
0
C ≈
es Nd
2(Vbi + VR )
Figure 7.111 (I/C) '
unifonnly doped pn ju
1/2
(48)
\Vhich shows that the inverse capacilance squared is a linear fun
The depletion layer capacitance of a one-sided junction is a function of the doping concentration in the low-doped
reverse-bias voltage.
region.
Figure 7.11 shows a pial of Equation (7.48). The built-in poten
tion can be determined by extrapolaling the Curve 10 the point whe
The slope of the 11
curve is inversely propOrtional to the doping con
low-doped region in (he junction; thu s. this dopi ng concen trati on ca
Equation (48) may be manipulated to give
1
C0
2
2(Vbi + VR )
es Nd
=
(49)
pn Jur.ct:Onwhich shows that the inverse capacitance squared is a linear function of applied reverse-bias voltage.
Figure 11
shows a plot of Equation (49).
The built-in potential of the junction can be determined by extrapolating the curve to the point where (1/C 0 )2 =
0.
II
The slope of the curve is inversely proportional to the doping concentration of the low-doped region in the
junction; thus, this doping concentration can be experimentally determined.
1------,
The assumptions used in the derivation of this capacitance include uniform doping in both semiconductor
regions, the abrupt junction approximation and a planar junction.
+
I
,
I
I
I
- V"j 0
charge density of.
Figure
' ve"usdoped
V.ofa
Figure 11:
(1/C 0 )27.111
versus (I/C)
VR of a uniformly
p+ n junction.
unifonnly doped pn junction.
on.
4
pn Junction Current
the inverse capacilance squared is a linear funclion of appli(-d
pn Junction Current
e.
When a forward-bias voltage is applied to a pn junction, a current will be induced in the device.
ows a pial
Equation
The
built-in
potential
junco
We of
initially
consider a(7.48).
qualitative
discussion
of how
charges flowof
in the
the pn
junction and then consider the
mathematical
derivation
of
the
current-voltage
relationship.
ined by extrapolaling the Curve 10 the point where (1/ C')2 ""
urve is inversely propOrtional to the doping concentration of the
4.1 Qualitative Description of Charge Flow in a pn Junction
n (he junction; thu s. this dopi ng concen trati on can be experimenQualitative Description of Charge Flow in a pn Junction
he assumptions
used in the derivation of this capacitance include
We can qualitatively understand the mechanism of the current in a pn junction by again considering the energy
bOlh semiconduclor
band diagrams. regions, the abrupi junction approximation.
Figure 12a shows the energy band diagram of a pn junction in thermal equilibrium.
on.
o.
We argued that the potential barrier seen by the electrons, for example, holds back the large concentration of
electrons in the n region and keeps them from flowing into the p region.
Similarly, the potential barrier seen by the holes holds back the large concentration of holes in the p region and
keeps them from flowing into the n region.
The potential barrier, then, maintains thermal equilibrium.
purity doping concentrations in a p+ n junction given [he parameters 110m
n p+ n junction at T
12
10
= 300 K with n, = 1.5 X 10
cm- J . Assume thaL
agram of a pn junclion in thermal equilibrium Ihat was developed in the last chapter.
We argued that the pOlential barrier seen by the electrons. for example. holds back
the large concentratioll of electrons in the n region and keeps them from flowing into
the p region. Similarly, the potential barrier seen by the holes holds back the large
- v.
,----I! 11 - - - ,
p
p
p
•
04-
E
rv
Electron flow
T
1
1
1
1
V )
1.; - "
__
['"'-.;:::,l...I-,-__
I
t
!
I
£1-1' _ •• ---:
£1"1· - - - -
-' 1
EI'N
I
1
1
r-- - 1
eF..
Hole now
1
I
1
(a)
(b)
(e)
Figure 8.1 IA pnjuncti on and its associated energy band diagram for (a) zero bias. (b) reverse bias. and (c) forward bias.
Figure 12: A pn junction and its associated energy band diagram for (a) zero bias, (b) reverse bias, and (c)
forward bias.
Figure 12b shows the energy band diagram of a reverse-biased pn junction.
The potential of the n region is positive with respect to the p region so the Fermi energy in the n region is lower
than that in the p region.
The total potential barrier is now larger than for the zero-bias case.
We argued in the previous sections that the increased potential barrier continues to hold back the electrons and
holes so that there is still essentially no charge flow and hence essentially no current. Figure 12c now shows
the energy band diagram for the case when a positive voltage is applied to the p region with respect to the n
region.
The Fermi level in the p region is now lower than that in the n region.
The total potential barrier is now reduced.
The smaller potential barrier means that the electric field in the depletion region is also reduced.
The smaller electric field means that the electrons and holes are no longer held back in the n and p regions,
respectively.
There will be a diffusion of holes from the p region across the space-charge region where they now will flow into
the n region.
Similarly, there will be a diffusion of electrons from the n region across the space-charge region where they will
flow into the p region.
The flow of charge generates a current through the pn junction. The injection of holes into the n region means
that these holes are minority carriers.
The injection of electrons into the p region means that these electrons are minority carriers.
The behavior of these minority carriers is described by the ambipolar transport equations.
There will be diffusion as well as recombination of excess carriers in these regions.
The diffusion of carriers implies that there will be diffusion currents.
The mathematical derivation or the pn junction current-voltage relationship is considered in the next section.
4.2
Ideal Current-Voltage Relationship
Ideal Current-Voltage Relationship
The current-voltage relationship is based on four assumptions:
13
=
n;/Nd
lip
1.
p,
op(- x,)
p region
Thcmlai equilibrium minority carrier hole concentration in the n region
Total minority camer electron concentration in the. p region
The abrupt Total
depletionminori
layer approximation
The space charge regions
abrupt boundaries and
ty carrier applies.
hole concentration
in thehave
n regioll
the semiconductor is neutral outside of the depletion region.
Minority carri er electron concentration in the p region al lhe spacc-
2. The Maxwell-Boltzmann approximation applies to carrier statistics.
charge edge
3. The concept of low injection applies.
Minority carrier hole concentration in the n region at the space charge
4. As concerning
the current:
edge
The total
current millOrity
is a constantcarrier
throughout
the entireconcentration
pn structure.
Snp ::::: np - (a)
Il pO
Excess
electron
in the p region
(b) The individual electron and hole currents are continuous functions through the pn structure.
Excess minority carrier hole concentmlion in (he n region
:'1/'11 = p" - p"(J
(c) The individual electron and hole currents are constant throughout the depletion region.
Table 1 lists some of the various electron and hole concentration that appear.
appear. Many lerms term
have already
been u<cd in prcviou< chaplers bUI are repealed
meaning
N
Acceptor concentration in the p region of the pn junction
here for convenience.
N
Donor concentration in the n region of the pn junction
a
d
8.1.3
nn0 = Nd
pp0 = Na
np0 = n2i /Na
pn0 = n2i /Nd
np
pn
np (−xp )
pn (xn )
δnp = np − np0
δpn = pn − pn0
Thermal equilibrium majority carrier electron concentration in the n region
Thermal equilibrium majority carrier hole concentration in the p region
Thermal equilibrium minority carrier electron concentration in the p region
Thermal equilibrium minority carrier hole concentration in the n region
Total minority carrier electron concentration in the p region
Total minority carrier hole concentration in the n region
Minority carrier electron concentration in the p region at the space-charge edge
Minority carrier hole concentration in the n region at the space-charge edge
Excess minority carrier electron concentration in the p region
Excess minority carrier hole concentration in the n region
Boundary Conditions
Figure 8.2 shows Ihe conduction· band energy through the pn junclion in thennal
equilibrium. The n region contains many more eleclrons in Ihe conduclion band Ihan
Ihe P region; Ihe buill-in
pOlenlial
barrier
prevenls
Ihisforlarge
densilY of eleclrons from
Table
1: Commonly
used terms
and notations
this section.
flowing inlo Ule p region. The built-in pOlential barrier maintai ns equilibrium beIween Ihe carrier distributions on either side of Ihe junclion.
Boundary Conditions
An4.3expression
for Ihe built-i n potential barrier was derived in Ihe lasl chapter
Boundary
and wa<
gi ven Conditions
by Equalion (7. 10) as
Figure 13 shows the conduction-band energy through the pn junction in thermal equilibrium.
The n region contains many more electrons in the conduction band than the p region.
The built-in potential barrier prevents this large density of electrons from flowing into the p region.
The built-in potential barrier maintains equilibrium between the carrier distributions on either side of the
junction. An expression for the built-in potential barrier is
Electron
J
p
ent:rgy
n
Figure M.2 I Conduction-band energy (hro\lgh a pll junction.
Figure 13: Conduction-band energy through a pn junction.
14
(f we divide the equation by V,
then take the reciproc"I, we obtain
=k T / e, take the exponential of both sides, J
-eVbi
Vbi = Vt ln
Na Nd'" exp ( - kT
n2i
)
(8
(50)
If we assume complete ionizalion, we can wri te
If we divide the equation by Vt = kT /e, take the exponential of both sides, and then take the reciprocaI, we
(8.
obtain
n2i
−eVbi
= exp
where IJnO is the thermal-equilibrium
concentration of majority carrier elcc((onS in (51)
Na Nd
kT
the n region. In the p region, we can write
If we assume complete ionization, we can write
nn0 ≈ Nd
(52)
where nn0 is the thermal-equilibrium
of majority carrier
electrons
the n region.
where n"oconcentration
is the thermal-equilibrium
concentration
of in
minority
carrier electrons.
(&.2)
and
(8.3)
into
Equation
(8.
J)
yields
Substituting
Equations
In the p region, we can write
n2
np0 ≈ i
(53)
(8.4)
Na
where np0 is the thermal-equilibrium concentration of minority carrier electrons.
This (53)
equation
the minority
carrier electron concenttation on the p side of the
Substituting Equations (52) and
into relates
Equation
(51) yields
junction to the majority carrier electron
concentration
on the n side of the junction in
−eVbi
thermal equilibrium.
np0 = nn0 exp
(54)
If a positive voltage is applied to
kTthe p region with respect [0 the n region, the porenti,tl barrier is reduced. Figure 8.3a shows ,t pn junction with an appHed voltage VoThis equation relates the minority
carrier
electron
concentration
on the
p side of
thesmall.
junction
to theallmajority
The electric
field
in the bulk
p and n regions
is nomlally
vcry
Essentially
of
carrier electron concentrationtheonapplied
the nvoltage
side ofisthe
junction
in
thermal
equilibrium.
If
a
positive
voltage is
across the juncrion region. The elecrric field E,pp induced by the
applied to the p region with respect to the n region, the potential barrier is reduced.
applied voltage is in the opposite direction to the thermal equilibrium space
Figure 14a shows a pn junction
withfield,
an applied
voltage
. in the
electric
so the net
electricVafield
charge region is reduced below tl.e
equili
brium
The
delicate
balance
hetween
diffusion and the E-field force
The electric field in the bulk p and n regions is normally very small.
Essentially all of the applied voltage is across the junction region.
I-IV,-\
p
E""
,,,
I
+
p
•
n
Ii:,
_____
E,."
- + + +
..
... - ......eVa
EFp -- - -
-- - - - -
to,.
v.
I
-
----T--------------
£F.
A
-..............
-----
(0)
FIgure 8.3 } (3) A pn jum:tlon with a.n applied
\foliage showing the: dirct::Iion." of the electric field induced
I
Figure by
14:Va (a)
junction
with an
applied
forward-bias
voltage
showing thepodirection
and A
thepn
space
<.: harge e1ectric
fi eld.
(b) Energy-band
diagram
of the forward-biased
junction. of the electric
1 field
induced by Va and the space charge electric field. (b) Energy-band diagram of the forward-biased pn junction.
The electric field Eapp induced by the applied voltage is in the opposite direction to the thermal equilibrium
space charge electric field, so the net electric field in the space charge region is reduced below the equilibrium
value.
The delicate balance between diffusion and the E-field force achieved at thermal equilibrium is upset.
electric field force that prevented majority carriers from crossing the space charge region is reduced.
The
Majority carrier electrons from the n side are now injected across the depletion region into the p neutral, and
majority carrier holes from the p side are injected across the depletion region into the n material.
As long as the bias Va is applied, the injection of carriers across the space charge region continues and a current
is created in the pn junction.
This bias condition is known as forward bias; the energy-band diagram of the forward-biased pn junction is
shown in Figure14b. The potential barrier Vbi in Equation (54) can be replaced by (Vbi − Va ) when the junction
is forward biased.
Equation (54) becomes
np = nn0 exp
−e(Vbi − Va )
kT
= nn0 exp
15
−eVbi
kT
exp
+eVa
kT
(55)
If we assume low injection, the majority carrier electron concentration nn0 for example, does not change significantly. However, the minority carrier concentration, np can deviate from its thermal-equilibrium value np0 by
orders of magnitude.
Using Equation (54), we can write Equation (55) as
np = np0 exp
eVa
kT
(56)
When a forward-bias voltage is applied to the pn junction, the junction is no longer in thermal equilibrium.
The left side of Equation (56) is the total minority carrier electron concentration in the p region, which is now
greater than the thermal equilibrium value.
The forward-bias voltage lowers the potential barrier so that majority carrier electrons from the n region are
injected across the junction into the p region, thereby increasing the minority carrier electron concentration.
We have produced excess, minority carrier electrons in the p region. When the electrons are injected into the
p region, these excess carriers are subject to the diffusion and recombination processes.
Equation (56) then, is the expression for the minority carrier electron concentration at the edge of the space
charge region in the p region. Exactly the same process occurs for majority carrier holes in the p region which
are injected across the space charge region into the n region under a forward-bias voltage:
eVa
(57)
pn = pn0 exp
kT
where pn is the concentration of minority carrier holes at the edge of the space charge region in the n region.
Figure 15 shows these results.
By applying a forward-bias voltage, we create excess minority carriers in each region of the pn junction.
The minority carrier concentrations at the space charge edges, given by Equations (56) and (57), were derived
CHAPTER
8 ThepnJunctionO:ode
assuming
a forward-bias voltage (Va > 0) was applied across the pn junction.
However, nothing in the derivation prevents Va from being negative (reverse bias).
r..,.
p
Hole injection
-+\---
Ei I!':lnln injtclic.n
1
- - - - - - - - - - - - P'IO
" pO - - - - - - - - - - - - -Xi'
.r - 0 .r"
Figure 8.4 I Ex<.:ess minority carrier
lhe
space charge edges generated by the focwurd·bias voltage.
Figure 15: Excess minority carrier concentration at the space charge
edges generatedSby31the forward-bias voltage.
COllccnlrali()I1
,
If a reverse-bias voltage greater than a few tenths of a volt is applied to the pn junction, then we see from
Equations (56) and (57) that the minority carrier concentrations at the space charge edge are essentially zero.
The minority carrier concentrations for the reverse-bias condition drop below the thermal-equilibrium values.
Objective
'1'0 calculate the minority carrier hole concentration at the edge of the space charge region of a
pn junction when a forward bias is "PI)lied.
16
1
4.4
Minority Carrier Distribution
Minority Carrier Distribution
We developed previously the ambipolar transport equation for excess minority carrier holes in an n region:
Dp
∂ 2 (δpn )
∂(δpn )
δpn
∂(δpn )
− µp E
+ g0 −
=
2
∂x
∂x
τp0
∂t
(58)
where δpn = pn − pn0 is the excess minority carrier hole concentration and is the difference between the total
and thermal equilibrium minority carrier concentrations.
The ambipolar transport equation describe the behavior of excess carriers as a function of time and spatial
coordinates.
We calculated previously the drift current densities in a semiconductor.
We determined that relatively large currents could be created with fairly small electric fields.
As a first approximation, we will assume that the electric field is zero in both the neutral p and n regions.
In the n region for x > xn , we have that E = 0 and g 0 = 0.
If we also assume steady state so ∂(δpn )/∂t = 0 then Equation (58) reduces to
d2 (δpn ) δpn
− 2 = 0 (x > xn )
dx2
Lp
(59)
where L2p = Dp τp0 .
For the same set of conditions, the excess minority carrier electron concentration in the p region is determined
from
d2 (δnp ) δnp
− 2 = 0 (x < xp )
(60)
dx2
Ln
where L2n = Dn τn0 .
The boundary conditions for the totaI minority carrier concentrations are
eVa
pn (xn ) = pn0 exp
kT
eVa
np (−xp ) = np0 exp
kT
(61)
(62)
pn (x → +∞) = pn0
(63)
np (x → −∞) = np0
(64)
As minority carriers diffuse from the space charge edge into the neutral semiconductor regions, they will
recombine with majority carriers.
We will assume that lengths Wn and Wp shown in Figure 14a are very long, meaning in particular that Wn Lp
and Wp Ln .
The excess minority carrier concentrations must approach zero at distances far from the space charge region.
The structure is referred to as long pn junction. The general solution to Equation (59) is
δpn (x) = pn (x) − pn0 = Aex/Lp + Be−x/Lp
(x ≥ xn )
(65)
(x ≤ −xp )
(66)
and the general solution to Equation (60) is
δnp (x) = np (x) − np0 = Cex/Ln + De−x/Ln
Applying the boundary conditions from Equations (63) and (64), the coefficients A and D must be zero.
The coefficients B and C may be determined from the boundary conditions given by Equations (61) and (62).
The excess carrier concentrations are then found to be, for (x ≥ xn ),
eVa
xn − x
δpn (x) = pn (x) − pn0 = pn0 exp
− 1 exp
(67)
kT
Lp
17
hrough Ihe depletion region. Since the electron and hole currents arc continuous fun
OIlS Ihrough Ihe pn j unclion. Ihe IOlal pn junction current will be the minority carr
hnle diffusion curren I at x = Xn plus the minorily carrier e1cctron diffusion current
and for (x ≤ −x )
= -x p • The gradients in
the
minority
carriereoncentrations,
as shown in
Figure 8
eV
x +x
δn (x) = n (x) − n = n
exp
− 1 exp
(68)
kT
L
produce diffusion currents, and since we are assuming tile eleclric field 10 be zero
p
a
p
p
p0
p
p0
n
minorityedges,
carrier concentrations
exponentially
away
from the junction
their thermalhe space The
Charge
we candecay
neglect
anywith
midistance
nority
carrier
drift tocurrent
compone
equilibrium values.
This approach
delermining
Ihe pn junction current is shown in Figure 8.6.
Figure 16in
shows
these results.
Again, we have assumed that both the n-region and the p-region lengths are long compared to the minority
carrier diffusion lengths.
p
n
- Pnil
ft(.IJ - - - -
Figure 16: 8.51
Steady-state
minority carrier
concentrations
in a pn cOllccntrations
junction under forward in
bias.B
Figure
Steady-state
minority
camer
pn j unction under forward bias.
To review, a forward-bias voltage lowers the built-in potential barrier of a pn junction so that electrons from
the n region are injected across the space charge region, creating excess minority carriers in the p region.
These excess electrons begin diffusing into the bulk p region where they can recombine with majority carrier
holes.
Current 1
The excess minority carrier electron concentration then decreases with distance from the junction.
density :
The same discussion applies to holes injected across the space charge region into the n region.
p
I
D
I
4.5 Ideal pn Junction Current
I
I
Ideal pn Junction Current
The approach we use to determine the current in a pn Ijunction is based on the three parts of the fourth
I
assumption stated earlier in this section.
I
The total current in
is the sum) of the individual
electron and hole currents which are constant
= the
J,I.junction
....,,) J,,(-x
I
p
...-- "p(x,, )
through the depletion region.
I
Since the electron and hole currents are continuous functions through the pn junction, the total pn junction
I x = xn plus the minority carrier electron diffusion
current will be the minority carrier hole diffusion current at
current at x = −xp
I
J.(carrier
- Xp )
/
The gradients in the minority
concentrations,
as shown
in Figure 16, produce diffusion currents, and
I
since we are assuming the electric field to be zero at the space
charge
edges, we can neglect any minority carrier
I
drift current component.
I
./+
I
This approach in determining the pn junction current is shown
in Figure 17.
Figure 8.6 , Electron and hole18 curreRI densities through the
spilce charge region of a pn jum;(ion.
pn j unction under forward bias.
Current 1
density :
p
=
I
I
I
I
./
J,I.....,,) + J,,(-x
D
I
I
I
I
I
p)
...-- "p(x,, )
I
I
I
I
J.( - Xp ) /
I
I
densities through the
Figure 8.6 , Electron and hole
Figure 17: Electron and hole current densities through
the space charge region of a pn junction.
curreRI
spilce charge region of a pn jum;(ion.
We can calculate the minority carrier hole diffusion current density at x = xn from the relation
dpn (x)
Jp (xn ) = −eDp
dx
x=xn
(69)
Since we are assuming uniformly doped regions, the thermal-equilibrium carrier concentration is constant, so
the hole diffusion current density may be written as
d(δpn (x))
Jp (xn ) = −eDp
(70)
dx
x=xn
Taking the derivative of Equation (67) and substituting into Equation (70), we obtain
eDp pn0
eVa
Jp (xn ) =
−1
exp
Lp
kT
(71)
The hole current density for this forward-bias condition is in the +x direction, which is, from the p to the n
region. Similarly, we may calculate the electron diffusion current density at x = −xp .
This may be written as
Jn (−xp ) = −eDn
d(δnp (x))
dx
(72)
x=−xp
Using Equation (68), we obtain
eDn np0
eVa
Jn (−xp ) =
exp
−1
Ln
kT
(73)
The electron current density is also in the +x direction. An assumption we made at the beginning was that the
individual electron and hole currents were continuous functions and constant through the space charge region.
The total current is the sum of the electron and hole currents and is constant through the entire junction.
Figure 17 again shows a plot of the magnitudes of these currents.
The total current density in the pn junction is then
eDp pn0
eDn np0
eVa
exp
−1
J = Jp (xn ) + Jn (−xp ) =
+
Lp
Ln
kT
19
(74)
Equation (74) is the ideal current-voltage relationship of a pn junction.
We may define a parameter Js as
eDp pn0
eDn np0
Js =
+
Lp
Ln
(75)
so that Equation (75) may be written as
Ideal characteristic of a pn junction diode
eVa
J = Js exp
−1
kT
(76)
Equation (76), known as the ideal-diode equation, gives a good description of the current-voltage characteristics
of the pn junction over a wide range of currents and voltages.
Although Equation (76) was derived assuming a forward-bias voltage (Va > 0), there is nothing to prevent Va
from being negative (reverse bias).
Equation (76) is plotted in Figure 18 as a function of forward-bias voltage Va .
If the voltage Va becomes negative (reverse bias) by a few kT /e V, then the reverse-bias current density becomes
independent of the reverse-bias voltage.
The parameter Js is then referred to as the reverse saturation current density.
.. t
8. 1
The current-voltage characteristics of the pn junction diode are obviously not bilateral.
J_
-1+
p
V.
J_
+
pn Junction Current
i
[;PI
v.
.
-- - ]
.'
v._
18: Ideal l-V
characteristic of a(If.(
pn junction
diode.
Figure 8.7Figure
Ildeall-V
dmracleristic
ptl junction
diode.
If the forward-bias voltage in Equation (76) is positive by more than a few kT /e volts, then the (−1) term in
Equation (76) becomes negligible.
so thaI Figure
Equation
) llIay current-voltage
be wrinencharacteristic
as
19 shows(8.21
the forward-bias
when the current is plotted on a log scale.
Ideally, this plot yields a straight line when Va is greater than are few kT /e volts.
The forward-bias current is an exponential function of the forward-bias voltage.
(8.23)
: Equalioll (8.23), known as the idcal-diode20 equation, gi ves a good description of the
current- voltage characteristics of the pn junction over a wide range of currents and
GN
MPLES.3
CHAPTER 8 The pnJl.lncticn DIode
which may be rt\,,,rittcn ,\S
Substituting the paramelers. we obtai!)
== 4. \S
X
lQ- IJ A/cml .
• Comment
"The idea) rcvfJ'se-hias salUr3tion curren! density is very small. ]f the pn junction
sectional area were A = 10- 4 cml, for example. then the ideal reverse-bias diode currer.l
would be I , = 4.15 X 10- 1•1 A.
tS
.s
,
,
,,
J '
T
Figure K.S 1Ideal 1-V
charJctcris(ic of a
pn juntliOfl diode with
the current plotlcd on
a iog sca1e.
Figure 19: Ideal I-V characteristic of a pn junction diode with the current plotted on a log scale.
If the forward-bias voltage in Equation (8.23) i. positive by more than a few
kTjevolts. then the (-I) term in Equation (8.23) becomes negligible. Figure 8.8
shows the forward -bias current-voltage characteristic when the Current is plOtted on
a log scale. Ideally. this plot yields a straight lillc when V" is greater than are'll
kTjevolts. The forward-bias current is an exponential function of the forward-bias
voltage.
Objective
To design a po junclion diode t'O produce particu);tr eleclron and hole <.:um!nt densities at a
given forward-bias VO ltage.
Consider a silicon pn junclion diode 3t T = 300 K. Design the diode such that J... =
20 AJcn,l and JI • = 5
at V II = 0.65 V. Assume the remaining semtconducto( parameter$
are as given in Example 8.2.
21