Linearly Graded Junction
zThe Poisson equation for the case is
d 2Ψ − q
=
ax
2
εs
dx
−W
W
≤x≤
2
2
zWhere a is impurity gradient and W is
depletion width area.
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Figure 4-11.
Linearly graded junction in
thermal equilibrium.
(a) Impurity distribution.
(b) Electric-field distribution.
(c) Potential distribution with
distance.
(d) Energy band diagram.
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Linearly graded junction
zBuilt in Voltage for Linearly Graded
Junction
W ⎤
⎡ W
(
a
)(
a
)⎥
kT ⎢ 2
2kT ⎡ aW ⎤
2
=
Vbi =
ln ⎢
ln ⎢
⎥
⎥
2
q ⎢
q
2
n
ni
i
⎣
⎦
⎥
⎥⎦
⎢⎣
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Depletion Capacitance
The depletion layer capacitance
per unit area is defined
dQ
Cj =
dV
Where dQ is incremental charge
per unit area as for an incremental
charge per unit voltage.
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Figure 4.13.
(a) p-n junction with an
arbitrary impurity profile under
reverse bias. (b) Change in
space charge distribution due
to change in applied bias. (c)
Corresponding change in
electric-field distribution.
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Depletion Capacitance
zWhenever voltage dv is applied. The
charge filed distribution will expand. Figure
shows the region bounded by dashed line.
zThis increment bring increase in electric
field by an amount of dε = dQ
εs
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Capacitance-Voltage Characteristic
zSeparation between two plates represents
the depletion width. Refer to the figure 13.
zFor forward bias large current can flow
across the junction corresponding to large
number of mobile carrier.
zThe incremental change of these mobile
carriers with respect to the biasing voltage
contributes an additional term called
diffusion capacitance.
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Capacitance-Voltage Characteristic.
Abrupt Junction case
Cj =
εs
W
=
qε s N B
2(Vbi − V )
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Varactor
z Used in many circuit applications.
zCapacitance varies with frequency.
zConcept:
zReverse biased application depletion
capacitance:
Cj∞(Vbi + VR ) − n
or
Cj∞(VR ) − n
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Varactor
zVoltage sensitivity with respect to
capacitance is greater in abrupt junction
that linear grade.
zThis because of n factor which is higher in
abrupt as oppose to linear grade.
zFrom Poisson condition establish the
relationship between the depletion region
and VR.
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Varactor
zBy choosing different value of m we can
approximate the relationship between the
voltage and junction capacitance
W∞(V R )
1
m+2
C j = (VR )
−1
m+2
zIn RF circuit you can adjust the voltage to
control the capacitive value therefore you
can control resonant value.
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Current Voltage Characteristic
zA voltage applied to a p-n junction will
disturb the balance diffussion and drift
current.
zForward bias voltage will increase the
diffusion current and decrease the drift
current.
zReverse biased voltage will increase the
drift current and reduced the diffusion
current.
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Ideal characteristic.
zWe based on these assumptions:
zAbrupt boundaries ,outside the boundary
the semiconductor assume to be neutral.
zCarrier density related by potential
different.
zLow density carrier is smaller compare to
higher density.
zElectron and hole are constant throughout
the depletion region.
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Figure 4-16.
Depletion region, energy
band diagram and carrier
distribution.
(a) Forward bias.
(b) Reverse bias.
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Ideal characteristic.
zBuilt in Voltage
kT pponno
Vbi = ln 2
q
ni
zThe electron density and hole density are
related through the electrostatic potential
different.
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Ideal Characteristic
zForward bias cause the electrostatic
potential different reduced to Vbi-VF.
zReverse bias electrostatic potential
increase Vbi+VR
z
nn = n p e
q
(Vbi −V )
kT
zWhere nn and np are non-equilibrium
electron densities at the boundary.
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Ideal Characteristic
zJs saturation current Lp and Ln are
diffusion length.
Js ≡
qD p p no
Lp
+
qDn n po
Ln
zThe above equation is the ideal diode
equation.
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Figure 4-18.
Ideal current-voltage
characteristics.
(a) Cartesian plot.
(b) Semilog plot.
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Temperature Effect
zTemperature effect has an effect on the
device performance.
zIn reverse bias as well as forward bias the
magnitude of diffusion and generation
recombination depend strong on the
temperature.
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Charge storage and transient behavior
zUnder forward bias the electrons are
injected into p-type and holes are injected
into the n-type.
zOnce injected across junction the minority
carriers recombine with majority carrier
and decay exponentially with distance.
zThese lead to current flows and charge
storage in the pn junction.
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Minority Carrier Storage
zThe charge minority carrier can be found
by integrating excess hole in neutral
region.
∞
Qp = q ∫ ( p n − p no )dx
xn
= qL p p no (e
qV
−1
kT
)
Lp 2
Qp =
Jp( x n ) = τpJp( x n )
Dp
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Figure 4-17.
Injected minority carrier
distribution and electron and
hole currents. (a) Forward
bias. (b) Reverse bias. The
figure illustrates idealized
currents. For practical
devices, the currents are not
constant across the space
charge layer.
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Diffusion Capacitance
zWhen the junction is forward biased the is
additional significant contribution to the
junction capacitance.
zThis is called the diffusion capacitance
denoted by Cd. A is area of a device.
qV
Aq 2 Lpp no kT
Cd =
e
kT
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Transient Behavior
zFor switching behavior the forward to
reverse biased transition must be transient
time short.
zUnder the forward biased condition the
stored minority carrier
IF
A
⎛ I
QpA
toff ≅
= τp⎜⎜ F
I Rave
⎝ I Rave
Qp = τpJp = τp
⎞
⎟⎟
⎠
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Junction Breakdown
zWhen large reverse voltage is applied to
p-n junction, the junction breaks down and
conducts a very large current.
zThe important breakdown mechanism:
zTunneling process and avalance
mechanism.
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Tunneling Effect
zHigh electric field is applied in reverse
direction valence electron make transition
from valence band to conduction band.
zOnly occurs if the electric field is high
about 106V/cm
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Avalance multiplication
z This occurs under the reverse bias condition.
z The thermally generated electron gain kinetic
energy from the electric field.
z If the field is sufficiently high it can gain enough
kinetic energy to break the bond creating
electron hole pair.
z The newly created electron hole pair requires
energy thus create more electron hole pair and
this is called the avalance multiplication.
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