ELL211: Physical Electronics
Madhusudan Singh
IIT Delhi
Winter 2022
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Lecture 12: Time dependent response of pn junctions.
Metal semiconductor junctions.
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Recap
Materials used here are bound by Fair Use for educational purpose.
Class problem with solved example.
Capacitance of pn junctions: junction/depletion capacitance, and storage/diffusion
capacitance. Comments on measuring capacitance in forward bias. Storage in injected
carriers: differing behavior of short and long diodes.
Varactor diodes: tuning of LC circuits.
Graded junctions: linear approximation in the region of the junction for compensation
doping. Higher breakdown voltages with graded junctions.
One-sided junctions. Simplification of expressions for reverse saturation current and
depletion width.
Thermally generated carriers in depletion region. Production of additional carriers due to
ionization under high reverse biases. Chynoweth’s law. Multiplication factor. Avalanche
breakdown. Breakdown voltage. Effective ionization rates. Shields-Fulop power law.
Temperature dependence of ionization rates.
Blocking/reverse currents. Modification of saturation current. Requirements for low
blocking currents. Competing requirements for base doping. Trade-offs. Methods of
achieving lower carrier lifetimes with RCs. Effects of heat on aggressively scaled MOS
devices and power devices.
Currents during breakdown. Empirical dependence of multiplication factors on breakdown
voltage.
Zener diodes: direct tunneling.
Designing around a breakdown. Critical electric field.
Diode behavior for different semiconductors. Threshold / turn-on voltage specifications.
Trade-off between low reverse saturation current, and low turn-on voltage. High injection:
recombination in the depletion region. Ideality
factor.
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Class problem
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Question: If the effective ionization rate prefactor (aeff ) is 1 × 106 cm−1 , and the effective field
coefficient is 1.5 MV cm−1 , estimate the field at which the effective ionization rate is 100 cm−1 .
Assume n = 7.
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Class problem
Materials used here are bound by Fair Use for educational purpose.
Question: If the effective ionization rate prefactor (aeff ) is 1 × 106 cm−1 , and the effective field
coefficient is 1.5 MV cm−1 , estimate the field at which the effective ionization rate is 100 cm−1 .
Assume n = 7.
The expression for the effective ionization rate is given by the Shields-Fulop power law:
beff /E0
E
αeff (E ) ≈ aeff e −beff /E0
E0
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Class problem
Materials used here are bound by Fair Use for educational purpose.
Question: If the effective ionization rate prefactor (aeff ) is 1 × 106 cm−1 , and the effective field
coefficient is 1.5 MV cm−1 , estimate the field at which the effective ionization rate is 100 cm−1 .
Assume n = 7.
The expression for the effective ionization rate is given by the Shields-Fulop power law:
beff /E0
E
αeff (E ) ≈ aeff e −beff /E0
E0
Since n ≡ beff /E0 = 7, we can set up the following equation to estimate E :
7
E
100 cm−1 = 1 × 106 cm−1 e −7
1.5 MV cm−1
e
⇒ E = 1.5 MV cm−1 × √
≈ 0.4862 MV cm−1
7
1 × 104
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Time-dependent response of pn junctions
Materials used here are bound by Fair Use for educational purpose.
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∂Jp (z, t)
∆p(z, t)
∂p(z, t)
=q
+q
∂z
τp
∂t
Continuity
equation.
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Time-dependent response of pn junctions
Materials used here are bound by Fair Use for educational purpose.
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∂Jp (z, t)
∆p(z, t)
∂p(z, t)
=q
+q
∂z
τp
∂t
Zz Continuity
equation.
Jp (z, t) − Jp (0, t) = −q
∆p(z ′ , t) ∂p(z ′ , t)
+
dz ′
τp
∂t
0
In a one-sided
junction, we can
integrate to
“infinity”.
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Time-dependent response of pn junctions
Materials used here are bound by Fair Use for educational purpose.
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∂Jp (z, t)
∆p(z, t)
∂p(z, t)
=q
+q
∂z
τp
∂t
Zz Continuity
equation.
In a one-sided
junction, we can
integrate to
“infinity”.
Two sources of
time-dependent
response.
ELL211 Instructors (IIT Delhi)
Jp (z, t) − Jp (0, t) = −q
∆p(z ′ , t) ∂p(z ′ , t)
+
dz ′
τp
∂t
0
i(t) =
Qp (t)
τp
| {z }
Recombination
ELL211
+
dQp
dt
|{z}
Response to time-dependent potentials
Winter 2022
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Step response
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Initial steady state
current. Switched off at
t = 0− .
I
t
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Step response
Materials used here are bound by Fair Use for educational purpose.
Initial steady state
current. Switched off at
t = 0− .
Qp (t) = I τp e −t/τp
I
Exponential solution
(Qp (t) = I τp e −t/τp ).
I = 0 but V ̸= 0. V =?
t
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Step response
Materials used here are bound by Fair Use for educational purpose.
Initial steady state
current. Switched off at
t = 0− .
Qp (t) = I τp e −t/τp
I
Exponential solution
(Qp (t) = I τp e −t/τp ).
I = 0 but V ̸= 0. V =?
Method: find excess hole
charge (we expect a delay
due to the capacitance).
t
qv (t)
kB T
∆pn (0, t) = pn e
−1
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Step response
Materials used here are bound by Fair Use for educational purpose.
Initial steady state
current. Switched off at
t = 0− .
Qp (t) = I τp e −t/τp
I
Exponential solution
(Qp (t) = I τp e −t/τp ).
I = 0 but V ̸= 0. V =?
Method: find excess hole
charge (we expect a delay
due to the capacitance).
Problem: I = 0 ⇒ slope
of Qp (0, t = 0+ ) should
be zero. It isn’t.
t
qv (t)
kB T
∆pn (0, t) = pn e
−1
I (0, 0) ∼
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∂p(z, t)
∂t
≡0
z=zn ,t=0
Winter 2022
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Step response
Materials used here are bound by Fair Use for educational purpose.
Initial steady state
current. Switched off at
t = 0− .
Qp (t) = I τp e −t/τp
I
Exponential solution
(Qp (t) = I τp e −t/τp ).
I = 0 but V ̸= 0. V =?
Method: find excess hole
charge (we expect a delay
due to the capacitance).
Problem: I = 0 ⇒ slope
of Qp (0, t = 0+ ) should
be zero. It isn’t.
Exact solution is
complicated. Assume: an
exponential at every t.
Quasi-steady state.
t
qv (t)
kB T
∆pn (0, t) = pn e
−1
I (0, 0) ∼
∂p(z, t)
∂t
∆pn (zn , t) = ∆pn (t)e
kB T
log
v (t) =
q
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≡0
z=zn ,t=0
−zn /Lp
I τp
e −t/τp + 1
qALp pn
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Step response
Materials used here are bound by Fair Use for educational purpose.
Initial steady state
current. Switched off at
t = 0− .
Qp (t) = I τp e −t/τp
I
Exponential solution
(Qp (t) = I τp e −t/τp ).
I = 0 but V ̸= 0. V =?
Method: find excess hole
charge (we expect a delay
due to the capacitance).
Problem: I = 0 ⇒ slope
of Qp (0, t = 0+ ) should
be zero. It isn’t.
Exact solution is
complicated. Assume: an
exponential at every t.
Quasi-steady state.
A p-n diode cannot be
switched off
instantaneously.
One-sided diodes with
short bases are used to
reduce
the delay.
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Instructors
(IIT Delhi)
t
qv (t)
kB T
∆pn (0, t) = pn e
−1
I (0, 0) ∼
∂p(z, t)
∂t
∆pn (zn , t) = ∆pn (t)e
kB T
log
v (t) =
q
ELL211
≡0
z=zn ,t=0
−zn /Lp
I τp
e −t/τp + 1
qALp pn
Winter 2022
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Square wave response
Materials used here are bound by Fair Use for educational purpose.
Vf
+
−
Vf
Initial steady forward bias (stored
minority carrier charge). i = If ≈
Vf
R
t
.
R
−Vf
If
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T
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Square wave response
Materials used here are bound by Fair Use for educational purpose.
Vf
+
−
Vf
Initial steady forward bias (stored
minority carrier charge). i = If ≈
Vf
.
R
−
0 .
t
R
Switched to negative bias at t =
Sudden flow of large anomalous reverse
current. i = Ir ≈ − VRf ≫ Is .
−Vf
If
T
Source: Streetman
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Square wave response
Materials used here are bound by Fair Use for educational purpose.
Vf
+
−
Vf
Initial steady forward bias (stored
minority carrier charge). i = If ≈
Vf
.
R
−
0 .
t
R
Switched to negative bias at t =
Sudden flow of large anomalous reverse
current. i = Ir ≈ − VRf ≫ Is .
−Vf
If
T
Stored charge cannot dissipate
instantaneously. Large current flows until
the injected minority carriers are swept
away by drift.
Source: Streetman
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Winter 2022
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Square wave response
Materials used here are bound by Fair Use for educational purpose.
Vf
+
−
Vf
Initial steady forward bias (stored
minority carrier charge). i = If ≈
Vf
.
R
−
0 .
t
R
Switched to negative bias at t =
Sudden flow of large anomalous reverse
current. i = Ir ≈ − VRf ≫ Is .
−Vf
If
T
Stored charge cannot dissipate
instantaneously. Large current flows until
the injected minority carriers are swept
away by drift.
Deep level transient spectroscopy
(DLTS): storage delay time ⇒ τp .
Source: Streetman
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Winter 2022
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Square wave response
Materials used here are bound by Fair Use for educational purpose.
Vf
+
−
Vf
Initial steady forward bias (stored
minority carrier charge). i = If ≈
Vf
.
R
−
0 .
t
R
Switched to negative bias at t =
Sudden flow of large anomalous reverse
current. i = Ir ≈ − VRf ≫ Is .
−Vf
If
T
Stored charge cannot dissipate
instantaneously. Large current flows until
the injected minority carriers are swept
away by drift.
Deep level transient spectroscopy
(DLTS): storage delay time ⇒ τp .
Reduction of storage time:
Source: Streetman
ELL211 Instructors (IIT Delhi)
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Winter 2022
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Square wave response
Materials used here are bound by Fair Use for educational purpose.
Vf
+
−
Vf
Initial steady forward bias (stored
minority carrier charge). i = If ≈
Vf
.
R
−
0 .
t
R
Switched to negative bias at t =
Sudden flow of large anomalous reverse
current. i = Ir ≈ − VRf ≫ Is .
−Vf
If
T
Stored charge cannot dissipate
instantaneously. Large current flows until
the injected minority carriers are swept
away by drift.
Deep level transient spectroscopy
(DLTS): storage delay time ⇒ τp .
Reduction of storage time:
RCs (Au/Pt).
Source: Streetman
ELL211 Instructors (IIT Delhi)
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Winter 2022
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Square wave response
Materials used here are bound by Fair Use for educational purpose.
Vf
+
−
Vf
Initial steady forward bias (stored
minority carrier charge). i = If ≈
Vf
.
R
−
0 .
t
R
Switched to negative bias at t =
Sudden flow of large anomalous reverse
current. i = Ir ≈ − VRf ≫ Is .
−Vf
If
T
Stored charge cannot dissipate
instantaneously. Large current flows until
the injected minority carriers are swept
away by drift.
Deep level transient spectroscopy
(DLTS): storage delay time ⇒ τp .
Reduction of storage time:
RCs (Au/Pt).
Narrow base diode, WB < Lp .
Source: Streetman
ELL211 Instructors (IIT Delhi)
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Winter 2022
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Square wave response
Materials used here are bound by Fair Use for educational purpose.
Vf
+
−
Vf
Initial steady forward bias (stored
minority carrier charge). i = If ≈
Vf
.
R
−
0 .
t
R
Switched to negative bias at t =
Sudden flow of large anomalous reverse
current. i = Ir ≈ − VRf ≫ Is .
−Vf
If
T
Stored charge cannot dissipate
instantaneously. Large current flows until
the injected minority carriers are swept
away by drift.
Deep level transient spectroscopy
(DLTS): storage delay time ⇒ τp .
Reduction of storage time:
RCs (Au/Pt).
Narrow base diode, WB < Lp .
Schottky diodes. Metal-semiconductor
junctions. No storage one side of the junction.
Source: Streetman
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Junctions
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Semiconductor insulator: MOS
devices.
Semiconductor metal: MS
junctions: Schottky and Ohmic
contacts.
G
B
S
p +-Si
n+-Si
Semiconductor-semiconductor:
pn junctions, heterojunctions,
p-i-n diodes, power pin diodes,
etc.
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Dielectric
D
n+-Si
p-Si
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Energetics of metals
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Photoelectric effect: below a certain
photon energy, no electrons are dislodged
from metal.
e−
Em = hν − qΦm
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Energetics of metals
Materials used here are bound by Fair Use for educational purpose.
Photoelectric effect: below a certain
photon energy, no electrons are dislodged
from metal.
e−
Em = hν − qΦm
The energy threshold: work function.
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Energetics of metals
Materials used here are bound by Fair Use for educational purpose.
Photoelectric effect: below a certain
photon energy, no electrons are dislodged
from metal.
e−
Em = hν − qΦm
The energy threshold: work function.
The energy needed to remove an electron
from the metal to a point in vacuum
immediately outside the metal at rest.
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Energetics of metals
Materials used here are bound by Fair Use for educational purpose.
Photoelectric effect: below a certain
photon energy, no electrons are dislodged
from metal.
e−
Em = hν − qΦm
The energy threshold: work function.
The energy needed to remove an electron
from the metal to a point in vacuum
immediately outside the metal at rest.
This is a property of the surface, not the
metal as such. Effect of image charge
dipole.
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Vacuum level
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Vvac : the potential in vacuum close to
the interface.
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Vacuum level
Materials used here are bound by Fair Use for educational purpose.
Vvac : the potential in vacuum close to
the interface.
−eVvac : potential energy of the electron
at rest near the metal surface.
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Vacuum level
Materials used here are bound by Fair Use for educational purpose.
Vvac : the potential in vacuum close to
the interface.
−eVvac : potential energy of the electron
at rest near the metal surface.
EF : work needed to remove an electron
from its average energy state to a state of
zero total energy.
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Vacuum level
Materials used here are bound by Fair Use for educational purpose.
Vvac : the potential in vacuum close to
the interface.
−eVvac : potential energy of the electron
at rest near the metal surface.
EF : work needed to remove an electron
from its average energy state to a state of
zero total energy.
Cannot be calculated purely from kF .
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Vacuum level
Materials used here are bound by Fair Use for educational purpose.
Vvac : the potential in vacuum close to
the interface.
−eVvac : potential energy of the electron
at rest near the metal surface.
EF : work needed to remove an electron
from its average energy state to a state of
zero total energy.
Vi =
q X Zj r0
4πϵ0 r0
rij
j,j̸=i
| {z }
Mi
Cannot be calculated purely from kF .
Account for electrostatic energy of ions.
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Vacuum level
Materials used here are bound by Fair Use for educational purpose.
Vvac : the potential in vacuum close to
the interface.
−eVvac : potential energy of the electron
at rest near the metal surface.
EF : work needed to remove an electron
from its average energy state to a state of
zero total energy.
Vi =
q X Zj r0
4πϵ0 r0
rij
j,j̸=i
| {z }
Mi
Cannot be calculated purely from kF .
Account for electrostatic energy of ions.
Similar to calculation of Madelung constants:
calculation of net electric potential of all ions
with charge state Zj in a lattice experienced by
an ion at a given position.
ELL211 Instructors (IIT Delhi)
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Winter 2022
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Vacuum level
Materials used here are bound by Fair Use for educational purpose.
Vvac : the potential in vacuum close to
the interface.
−eVvac : potential energy of the electron
at rest near the metal surface.
EF : work needed to remove an electron
from its average energy state to a state of
zero total energy.
Cannot be calculated purely from kF .
Account for electrostatic energy of ions.
Similar to calculation of Madelung constants:
calculation of net electric potential of all ions
with charge state Zj in a lattice experienced by
an ion at a given position.
Work function is not the same as EF .
Reference level shift.
ELL211 Instructors (IIT Delhi)
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Vi =
q X Zj r0
4πϵ0 r0
rij
j,j̸=i
| {z }
Mi
qΦm = −eVvac − EF
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