Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Physics of Semiconductor Devices
Part IV - Carrier Transport Phenomena
Luca Varani
University Montpellier 2
November 8, 2011
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The net flow of electrons and holes in a semiconductor will
generate currents.
The process by which these charged particles move is called
transport.
In this chapter we will consider the two basic transport
mechanisms in a semiconductor crystal: drift the movement of
charge due to electric fields, and diffusion the flow of charge due
to density gradients.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The carrier transport phenomena are the foundation for finally
determining the current-voltage characteristics of semiconductor
devices.
We will implicitly assume that, though there will be a net flow of
electrons and holes due to the transport processes, thermal
equilibrium will not be substantially disturbed.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Outline
1
Carrier Drift
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
2
Carrier Diffusion
3
Graded Impurity Distribution
4
The Hall Effect
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Carrier Drift
An electric field applied to a semiconductor will produce a force on
electrons and holes so that they will experience a net acceleration
and net movement, provided there are available energy states in
the conduction and valence bands.
This net movement of charge due to an electric field is called drift.
The net drift of charge give, rise to a drift current.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Drift Current Density
If we have a positive volume charge density ρ moving at an
average drift velocity vd , the drift current density is given by
Jdrf = ρvd
(1)
where J is in units of A/cm2 .
If the volume charge density is due to positively charged holes, then
Jp|drf = epvdp
(2)
where Jp|drf is the drift current density due to holes and vdp is the
average drift velocity of the holes.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
The equation of motion of a positively charged hole in the presence
of an electric field is
F = mp∗ a = eE
(3)
where e is the magnitude of the electronic charge, a is the
acceleration, E is the electric field, and mp∗ is the effective mass of
the hole.
If the electric field is constant, then we expect the velocity
to increase linearly with time.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
However, charged particles in a semiconductor are involved in
collisions with ionized impurity atoms and with thermally vibrating
lattice atoms.
These collisions, or scattering events, alter the velocity
characteristics of the particle.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
As the hole accelerates in a crystal due to the electric field, the
velocity increases.
When the charged particle collides with an atom in the crystal, for
example, the particle loses most or all of its energy.
The particle will again begin to accelerate and gain energy until it
is again involved in a scattering process.
This continues over and over again.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Throughout this process the particle will gain an average drift
velocity which, for low electric fields, is directly proportional to the
electric field.
We may then write
vdp = µp E
(4)
where µp is the proportionality factor and is called the hole
(ohmic) mobility.
The mobility is an important parameter of the semiconductor since
it describes how well a particle will move due to an electric field.
The unit of mobility is usually expressed in terms of cm2 /(Vs).
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
By combining Equations (2) and (4), we may write the drift
current density due to holes as
Jp|drf = (ep)vdp = eµp pE
(5)
The drift current due to hole, is in the same direction as the
applied electric field.
The same discussion of drift applies to electrons:
Jn|drf = ρvdn = (−en)vdn
(6)
where Jn|drf is the drift current density due to electrons and vdn is
the average drift velocily of electrons.
The net charge density of electrons is negative.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
The average drift velocity of an electron is also proportional to the
electric field for small fields.
However, since the electron is negatively charged, the net motion
of the electron is opposite to the electric field direction. We can
then write
vdn = −µn E
(7)
where µn is the electron mobility and is a positive quantity.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Equation (6) may be written as
Jn|drf = (−en)(−µn E ) = enµn E
(8)
The conventional drift current due to electrons is also in the same
direction as the applied electric field even though the electrons
movement is in the opposite direction.
Electron and hole mobilities are functions of temperature and
doping concentration.
Table 1 shows some typical mobility values at T = 300 K for low
doping concentrations.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Silicon
Gallium arsenide
Germanium
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
µn (cm2 /Vs)
1350
8500
3900
µp (cm2 /Vs)
480
400
1900
Table 1: Typical mobility values at T = 300 K and low doping
concentrations.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Since both electrons and holes contribute to the drift current, the
total drift current density is the sum at the individual electron and
hole drift current densities:
Total Drift Current Density
Jdrf = e(nµn + pµp )E
L. Varani
Physics of Semiconductors Devices
(9)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Mobility Effects
Equation 3 related the acceleration of a hole to a force such as an
electric field.
We may write this equation as
F = mp∗
dv
= eE
dt
(10)
where v is the velocity of the particle due to the electric field and
does not include the random thermal velocity.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
If we assume that the effective mass and electric field are
constants, then we may integrate Equation (10) and obtain
v=
eEt
mp∗
where we have assumed the initial drift velocity to be zero.
L. Varani
Physics of Semiconductors Devices
(11)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Figure 1 shows a schematic model of the random thermal velocity
and motion of a hole in a semiconductor with zero electric field.
There is a mean time between collisions which may be denoted by
τcp .
If a small electric field (E-field) is applied as indicated in Figure 1b,
there will be a net drift of the hole in the direction of the E-field,
and the net drift velocity will be a small perturbation on the
random thermal velocity, so the time between collisions will not be
altered appreciably.
L. Varani
Physics of Semiconductors Devices
C HAP T E R
5
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi
Levels
Cruner Transport
po Energy
l€nomena
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
.' . . .
. ... . ;, .., ,..,
3
-
E fleld
(h}
(a }
Figure
Typical r.mdon\
h<!hav\or
of aof
hole
a sem,condUClOf
(a) without
an
FigureSOl
1: ITypical
random
behavior
a in
hole
in a semiconductor
(a)
clcotric field .nd (b) with an electric field.
without an electric field and (b) with an electric field.
,
.J,
applied as indicated in Figure 5.1b. there will be a net drift of the hole in the directioo
of the E··field. and the net drift velocity will be a small pel1urbation on the random
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
If we use the mean time between collisions τcp in place of the time
t in Equation (11) then the mean peak velocity just prior to a
collision or scattering event is:
eτcp
E
(12)
vd|peak =
mp∗
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
The average drift velocity is one half the peak value so that we can
write
1 eτcp
hvd i =
E
(13)
2 mp∗
However, the collision process is not as simple as this model, but is
statistical in nature.
In a more accurate model including the effect of a statistical
distribution, the factor 1/2 in Equation (13) does not appear.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
The hole mobility is then given by
Hole mobility
µp =
vdp
eτcp
= ∗
E
mp
(14)
The same analysis applies to electrons:
Electron mobility
µn =
vdn
eτcn
= ∗
E
mn
where τcn is the mean time between collisions for an electron.
L. Varani
Physics of Semiconductors Devices
(15)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
There are two collision or scattering mechanisms that dominate in
a semiconductor and affect the carrier mobility: phonon or lattice
scattering, and ionized impurity scattering.
The atoms in a semiconductor crystal have a certain amount of
thermal energy at temperatures above absolute zero that causes
the atoms to randomly vibrate about their lattice position within
the crystal.
The lattice vibrations cause a disruption in the perfect periodic
potential function.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
A perfect periodic potential in a solid allows electrons to move
unimpeded or with no scattering through the crystal.
But the thermal vibrations cause a disruption of the potential
function, resulting in an interaction between the electrons or holes
and the vibrating lattice atoms.
This lattice scattering is also referred to as phonon scattering.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Since lattice scattering is related to the thermal motion of atoms,
the rate at which the scattering occurs is a function of
temperature.
If we denote µL as the thermal mobility that would be observed if
only lattice scattering existed, then the scattering theory states
that to first order
µL ∝ T −3/2
(16)
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Mobility that is due to lattice scattering increases as the
temperature decreases.
Intuitively we expect the lattice vibrations to decrease as the
temperature decreases, which implies that the probability of a
scattering event also decreases, thus increasing mobility.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Figure 2 shows the temperature dependence of electron and hole
mobilities in silicon.
In lightly doped semiconductors, lattice scattering dominates and
the carrier mobility decreases with temperature as we have
discussed.
The temperature dependence of mobility is proportional to T −n .
The inserts in the figure show that the parameter n is not equal to
3/2 as the first-order scattering theory predicted.
However, mobility does increase as the temperature decreases.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
-
3
5000
1000
4000
••
•
N"
'"..
Nf)
=
100
"'" "'"
I
N• •
•
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I
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I
I
JO I4
T
Nt>"' 10" ,' A: '
= 10\(1
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I
N... - 10\1
111M
?
:>
,
. , tV" -
IIXlIl
2000
,
II
1\
2000
1000
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
NO .
:>
10 11
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500
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100
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I
UI::<I
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\
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- 50
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100
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(a)
( b}
;11111 ( h i h<\k II I1.hi l il i\·.. i ll ..
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_-'
200
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lell ll i
1'I K}
150
200
••••••
Figure 2: (a) Electron and (b) hole mobilities in silicon versus
temperature for various doping concentrations. Insert show temperature
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
The second interaction mechanism affecting carrier mobility is
called ionized impurity scattering.
We have seen that impurity atoms are added to the semiconductor
to control or alter its characteristics.
These impurities are ionized at room temperature so that a
coulomb interaction exists between the electrons or holes and the
ionized impurities.
This coulomb interaction produces scattering or collisions and also
alters the velocity characteristics of the charge carrier.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
If we denote µI as the mobility that would be observed if only
ionized impurity scattering existed, then to first-order we have
µI ∝
T +3/2
NI
(17)
where NI = Nd+ + Na− is the total ionized impurity concentration.
If temperature increases, the random thermal velocity of carriers
increases thus reducing the time the carrier spends in the vicinity
of the ionized impurity center.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
The less time spent in the vicinity of a coulomb force, the smaller
the scattering effect and the larger the expected value of µI .
If the number of ionized impurity centers increases, then the
probability of a carrier carrier encountering an ionized impurity
center increases, implying a smaller value of µI .
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Figure 3 is a plot of electron and hole mobilities in germanium,
silicon, and gallium arsenide at T = 300 K as a function of
impurity concentration.
These curves are of mobility versus ionized impurity concentration
NI .
As the impurity concentration increases, the number of impurity
scattering centers increases, thus reducing mobility.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
15 • 1 Carrier Drift
181
104
I {}'
10::
t
.!,
10'
".
:>
;;-
E I(}'
:E
:>:=
10'
10'
Impuril)' concemration (em-.l )
Figure 5.3 1Electron amI hule mobiJi(ics versus im»urity
concentr.ui(ms for germanium. silicon. and gallium
arsenide 31 T = 300 K .
Figure 3: Electron and hole mobilities versus impurity concentrations for
germanium, silicon, and gallium arsenide at T = 300 K.
fFromSu (121.)
TEST YOUR UNDERSTANDING
L. the
Varani
Physics
ES.J (a} Using FigufC 5.2. find
cl<."Clron Jtlobilil)'
for (i) N,tof=Semiconductors
J().1 cm- J . l ' = }5(fCDevices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
If τL is the mean time between collisions due to lattice scattering,
then dt/τL is the probability of a lattice scattering event occurring
in a differential time dt.
Likewise, if τI is the mean time between collisions due to ionized
impurity scattering, then dt/τI is the probability of an ionized
impurity scattering event occurring in the differential time dt.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
If these two scattering processes are independent, then the total
probability of a scattering event occurring in the differential time
dt is the sum of the individual events or
dt
dt
dt
=
+
τ
τI
τL
where τ is the mean time between any scattering event.
L. Varani
Physics of Semiconductors Devices
(18)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Comparing Equation (18) with the definitions of mobility given by
Equations (14) or (15), we can write
Matthiessen rule
1
1
1
=
+
µ
µI
µL
(19)
where µI is the mobility due to the ionized impurity scattering
process and µL is the mobility due to the lattice scattering process.
The parameter µ is the net mobility with two or more independent
scattering mechanisms, the inverse mobilities add which means
that the net mobility decreases.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Conductivity
The drift current density, given by Equation (9), may be written as
Jdrf = e(nµn + pµp )E = σE
(20)
where σ is the conductivity of the semiconductor material.
The conductivity is given in units of (Ω cm)−1 and is a function of
the electron and hole concentrations and mobilities.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
We have just seen that the mobilities are functions of impurity
concentration: conductivity, then is a somewhat complicated
function of impurity concentration.
The reciprocal of conductivity is resistivity, which is denoted by
ρ and is given in units of (Ω cm).
We can write the formula for resistivity as
Resistivity
ρ=
1
1
=
σ
e(µn n + µp p)
L. Varani
Physics of Semiconductors Devices
(21)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Figure 4 is a plot of resistivity as a function of impurity
concentration in silicon, germanium, gallium arsenide, and gallium
phosphide at T = 300 K.
Obviously, the curves are not linear functions of Nd or Na because
of mobility effects.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
•
Impurity conctnlration (cm-')
10"
f"igure 5.41
versus impurity concentration at T
=::
300 K in (a) ilicon
and {b) gennaJ)i urn. gallium ;)rscllide. and gaJHum phosphide.
rF"""S", {12/.J
j impurity
Figure 4: Resistivity versus
concentration
at T = 300 K in (a)
L. Varani
Physics of Semiconductors Devices
163
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
If we have a bar of semiconductor material as shown in Figure 5
with a voltage applied that produces a current then we can write
J=
I
A
(22)
E=
V
L
(23)
and
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
CHAPT.R 5
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Carcier TransportPheoomel'..a
Figure 5.51 Bar of
matedal
a resistor.
Figure 5: Bar of semiconductor material as a resistor.
We can now rewrite Equation (5.19) as
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
We can now rewrite Equation (20) as
I
V
=σ
A
L
or
V =
L
σA
I =
ρL
A
(24)
I = RI
(25)
Equation (25) is Ohm’s law for a semiconductor.
The resistance is a function of resistivity, or conductivity, as well as
the geometry of the semiconductor.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
If we consjder, for example, a p-type semiconductor with an
acceptor doping Na (Nd = 0) in which Na ni , and if we assume
that the electron and hole mobilities are of the same order of
magnitude, then the conductivity becomes
σ = e(µn n + µp p) ≈ eµp p
L. Varani
Physics of Semiconductors Devices
(26)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
If we also assume complete ionization, then Equation (26) becomes
σ ≈ eµp Na ≈
1
ρ
(27)
The conductivity and resistivity of an extrinsic semiconductor are a
function primarily of the majority carrier parameters.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
We may plot the carrier concentration and conductivity of a
semiconductor as a function of temperature for a particular doping
concentration.
Figure 6 shows the electron concentration and conductivity of
silicon as a function of inverse temperature for the case when
Nd = 1015 cm−1 .
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
500
6 . 1 Ca ier Orift
T(K)
300
J(
10 11 r-"1:'
,
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
200
100
75
,
I
I
10
I
16
.§
I
10
:
I
I
......
10'"
............
I
'
1.0
,
c:
"
I
'\ I, ' "
'.'
,
I
'
0.1
I
,
I Ilj
I
Figure 5.6 1Eleclron concenlration and conductivity versus
Figure 6: Electron
concentration
and conductivity versus inverse
io\'crse
remperalure for .o;ilicon.
Sr.' " 12}. )
L. Varani
Physics of Semiconductors Devices
temperature for fAfter
silicon.
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
In the midtemperature range, or extrinsic range, as shown, we have
complete ionization the electron concentration remains essentially
constant.
However, the mobility is a function of temperature so the
conductivity varies with temperature in this range.
At higher temperatures, the intrinsic carrier concentration increases
and begins to dominate the electron concentration as well as the
conductivity.
In the lower temperature range, freeze-out begins to occur; the
electron concentration and conductivity decrease with decreasing
temperature.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
For an intrinsic material, the conductivity can be written as
σi = e(µn + µp )ni
(28)
The concentrations of electrons and holes are equal in an intrinsic
semiconductor, so the intrinsic conductivity includes both the
electron and hole mobility.
Since, in general. the electron and hole mobilities are not equal,
the intrinsic conductivity is not the minimum value possible at a
given temperature,
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Velocity Saturation
So far in our discussion of drift velocity, we have assumed that
mobility is not a function of electric field, meaning that the drift
velocity will increase linearly with applied electric field.
The total velocity of a particle is the sum of the random thermal
velocity and drift velocity.
At T = 300 K, the average random thermal energy is given by
3
3
1 2
mv = kT = (0.0259) = 0.03885 eV
2 th
2
2
L. Varani
Physics of Semiconductors Devices
(29)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
This energy translates into a mean thermal velocity of
approximately 107 cm/s for an electron in silicon.
If we assume an electron mobility of µn = 1350cm2 /(Vs) in low
doped silicon, a drift velocity of 105 cm/s, or 1 percent of the
thermal velocity, is achieved if the applied electric field is
approximately 75 V/cm.
This applied electric field does not appreciably alter the energy of
the electron.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
Figure 7 is a plot of average drift velocity as a function of applied
electric field for electrons and holes in silicon, gallium arsenide, and
germanium.
At low electric fields. where there is a linear variation of velocity
with electric field, the slope of the drift velocity versus electric field
curve is the mobility.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi
Levels
C H A P T . R 15 Ca!rier
TransportEnergy
Pheoorna'1a
e
i!'2
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
H)T
<:
:§
" lU'
"
'E
0
ElcctriC field (V/cm)
Figure 5.71 Carrierdrifl velo<:ity versus eleclric field for
Figure 7: Carrierhigh-purilY
drift velocity
electric
silicon. versus and
galliumfield for high-purity silicon,
05" (/1/.)arsenide.
germanium, and( F",,,,
gallium
This energy translates into a mean thermal velocity of approximately 10' cmls foran
L. Varani
Physics of Semiconductors Devices
2
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
The behavior of the drift velocity of carriers at high electric fields
deviates substantially from the linear relationship observed at low
fields.
The drift velocity of electrons in silicon, for example, saturates at
approximately 107 cm/s at an electric field of approximately
30 kV/cm.
If the drift velocity of a charge carrier saturates, then the drift
current density also saturates and becomes independent of the
applied electric field.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
The drift velocity versus electric field characteristic of gallium
arsenide is more complicated than for silicon or germanium.
At low fields, the slope of the drift velocity versus E-field is
constant and is the low-field electron mobility, which is
approximately 8500 cm2 /(Vs) for gallium arsenide.
The low-field electron mobility in gallium arsenide is much larger
than in silicon.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
As the field increases, the electron drift velocity in gallium arsenide
reaches a peak and then decreases.
A differential mobility is the slope of the vd versus E curve at a
particular point on the curve and the negative slope of the drift
velocity versus electric field represents a negative differential
mobility.
The negative differential mobility produces a negative differential
resistance; this characteristic is used in the design of oscillators.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
The negative differential mobility can be understood by considering
the E versus k diagram for gallium arsenide, which is shown again
in Figure 8.
The density of states effective mass of the electron in the lower
valley is mn∗ = 0.067m0 .
The small effective mass leads to a large mobility.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
O:IA:;
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
5 . 2 Csnie.- Oitfusitln
Conduction
5.8Varani
I Encrgy-bnnd
L.
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Drift Current Density
Mobility Effects
Conductivity
Velocity Saturation
As the E-field increases, the energy of the electron increases and
the electron can be scattered into the upper valley, where the
density of states effective mass is 0.55m0 .
The larger effective mass in the upper valley yields a smaller
mobility.
This intervalley transfer mechanism results in a decreasing average
drift velocity of electrons with electric field, or the negative
differential mobility characteristic.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
Outline
1
Carrier Drift
2
Carrier Diffusion
Diffusion Current Density
Total Current Density
3
Graded Impurity Distribution
4
The Hall Effect
5
Carrier Generation and Recombination
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
Carrier Diffusion
There is a second mechanism that can induce a current in a
semiconductor.
We may consider a classic physics example in which a container, as
shown in Figure 9, is divided into two parts by a membrane.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
aR!5 Carrier Transpon Ph&rtOma......a
• •• •.,
• •• ••
•
1
1
I
• 1
. . -0
Figure 9: ContainerFigure
divided 5.91
by a membrane
Containerwith gas molecules on one
side.
divided by a Olen'lbraJle with
gas molecules on one side.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
The left side contains gas molecules at a particular temperature
and the right side is initially empty.
The gas molecules are in continual random thermal motion so
that, when the membrane is broken, the gas molecules flow into
the right side of the container.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
Diffusion is the process whereby particles flow from a region
of high concentration toward a region of low concentration.
If the gas molecules were electrically charged, the net flow of
carriers would result in a diffusion current.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
Diffusion Current Density
To begin to understand the diffusion process in a semiconductor,
we will consider a simplified analysis.
Assume that an electron concentration varies in one dimension as
shown in Figure 10.
The temperature is assumed to be uniform so that the average
thermal velocity of electrons is independent of x.
To calculate the current, we will determine the net flow of
electrons per unit time per unit area crossing the plane x = 0.
L. Varani
Physics of Semiconductors Devices
• •••
• •• ••
•
Carrier Drift 11
Carrier Diffusion I
Graded Impurity Distribution • 1
The Hall Effect
Diffusion Current Density
Carrier Generation and Recombination
Total Current Density
Figure
5.91 Container
Characteristics of Excess
Carriers
Ambipolar
Transport
divided
by a Olen'lbraJle with
Quasi-Fermi Energy
Levels on one side.
gas molecules
. . -0
n(+ I)
11(0)
- - - - - - - - - - - - - - - - - - -- - - - -
- - - - - - - - - - --- -- - - -
II( - I)
.( = -I
;t
=0
.(
= +1
.,--
Figure S.10 l Electron concentration versus distance.
Figure 10: Electron concentration versus distance.
concentrdtion. If the gas molecules were electrically charged, the nel ftow of c
would result in a diffusion currenl.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
If the distance l shown in Figure 10 is the mean-free path of an
electron, that is, the average distance an electron travels between
collisions (l = vth τcn ), then on the average, distance an electron
moving to the right at x = −l and electrons moving to the left at
x = +l will cross the x = 0 plane.
One half of the electrons at x = −l will be traveling to the right or
at any instant of time and one half of the electrons at x = +l will
be traveling to the left at any given time.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
The net rate of electron flow, Fn , in the +x direction at x = 0 is
given by
1
1
1
Fn = n(−l)vth − n(+l)vth = vth [n(−l) − n(+l)]
2
2
2
(30)
If we expand the electron concentration in a Taylor series about
x = 0 keeping only the first two terms, then we can write
Equation (30) as
dn
dn
1
− n(0) + l
(31)
Fn = vth n(0) − l
2
dx
dx
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
which becomes
dn
dx
Each electron has a charge (−e), so the current is
Fn = −vth l
J = −eFn = +evth l
dn
dx
(32)
(33)
The current described by Equation (33) is the electron diffusion
current and is proportional to the spatial derivative, or density
gradient, of the electron concentration.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
The diffusion of electrons from a region of high concentration to a
region of low concentration produces a flux of electrons flowing in
the negative x direction for this example.
Since electrons have a negative charge, the conventional current
direction is in the positive x direction.
Figure 11a shows these one-dimensional flux and current directions.
L. Varani
Physics of Semiconductors Devices
172
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
C HAP T E R 5 Carriet Transport Phenomel"la
Diffusion Current Density
Total Current Density
,
c,
/ -
E'
5:
S:i I
c. I
Ele<tron nil'
Elet.'lfOJl
",
c,
g,
z_
Gj:
CUfTCnt
ta)
,
.."..
0.'
I
e,
c
/-
..
>tole n",
Hole di(tusion
t
BI
current dc::maty
5:
c'
:t',
"( b)
Figure S.l1l (a) Diffusion of electrons due 10 a density
gradienl. (b)
of holesdue
due lOto
a densi
gradient
Diffusion
ofDiffusion
electrons
a tydensity
Figure 11: (a)
Diffusion of holes due to a density gradient.
gradient. (b)
fo r the one-dimensional case,L.
The
parameterPhysics
Dp is cal.led
the hole diffllsim.
<'01
'
Varani
of Semiconductors
Devices
2
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
We may write the electron diffusion current density for this
one-dimensional case in the form
Electron diffusion current density
Jnx|dif = eDn
dn
dx
(34)
where Dn is called the electron diffusion coefficient, has units of
cm2 /s, and is a positive quantity.
If the electron density gradient becomes negative, the electron
diffusion current density will be in the negative x direction.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
Figure 11b shows an example of a hole concentration as a function
of distance in a semiconductor.
The diffusion of holes, from a region of high concentration to a
region of low concentration produces a flux of holes in the negative
x direction.
Since holes are positively charged particles, the conventional
diffusion current density is also in the negative x direction.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
The hole diffusion current density is proportional to the hole
density gradient and to the electronic charge, so we may write
Hole diffusion current density
Jpx|dif = −eDp
dp
dx
(35)
for the one-dimensional case. The parameter Dp is called the hole
diffusion coefficient and has units of cm2 /s, and is a positive
quantity.
If the hole density gradient becomes negative, the hole diffusion
current density will be in the positive x direction.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
Total Current Density
We now have four possible independent current mechanisms in a
semiconductor.
These components are electron drift and diffusion currents and
hole drift and diffusion currents.
The total current density is the sum of these four components, or,
for the one-dimensional case,
J = enµn E + epµp E + eDn
L. Varani
dp
dn
− eDp
dx
dx
Physics of Semiconductors Devices
(36)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
This equation may he generalized to three dimensions as
J = enµn E + epµp E + eDn ∇n − eDp ∇p
L. Varani
Physics of Semiconductors Devices
(37)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
The electron mobility gives an indication of how well an electron
moves in a semiconductor as a result of the force of an electric
field.
The electron diffusion coefficient gives an indication of how well an
electron moves in a semiconductor as a result of a density gradient.
The electron mobility and diffusion coefficient are not independent
parameters.
Similarly, the hole mobility and diffusion coefficient are not
independent parameters.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Diffusion Current Density
Total Current Density
The relationship between mobility and the diffusion coefficient will
be developed in the next section.
The expression for the total current in a semiconductor contains
four terms.
Fortunately in most situations, we will only need to consider one
term at anyone time at a particular point in a semiconductor.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
Outline
1
Carrier Drift
2
Carrier Diffusion
3
Graded Impurity Distribution
Induced Electric Field
The Einstein Relation
4
The Hall Effect
5
Carrier Generation and Recombination
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
Graded Impurity Distribution
In many semiconductor devices there may be regions that are
nonuniformly doped.
We will investigate how a nonunifomly doped semiconductor
reaches thermal equilibrium and, from this analysis, we will derive
the Einstein relation, which relates mobility and the diffusion
coefficient.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
Induced Electric Field
Consider a semiconductor that is nonuniformly doped with donor
impurity atoms. If the semiconductor is in thermal equilibrium, the
Fermi energy level is constant through the crystal so the
energy-band diagram may qualitatively look like that shown in
Figure 12
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
The electric field for the one-dimensional
situation is defi ned as
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
------------------- E,.
----------
_------- E"
E,.
Figure S.12 1Energy-band diagram ror
Figure 12: Energy-band
diagram for a semiconductor in thermal
a scmiconlim:to r in thermal equilibrium
equilibrium with a nonuniform
donordonnr
impurity
concentration.
wilh a nonuniform
impurity
concentration.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
The doping concentration decreases as x increases in this case.
There will be a diffusion of majority carrier electrons from the
region of high concentration to the region of low concentration,
which is in the +x direction.
The flow of negative electrons leaves behind positively charged
donor ions.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
The separation of positive and negative charge induces an electric
field that is in a direction to oppose the diffusion process.
When equilibrium is reached, the mobile carrier concentration is
not exactly equal to the fixed impurity concentration and the
induced electric field prevents any further separation of charge.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
In most cases of interest, the space charge induced by this diffusion
process is a small fraction of the impurity concentration thus the
mobile carrier concentration is not too different from the impurity
dopant density.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
The electric potential φ is related to electron potential energy by
the charge −e:
1
(38)
φ = + (EF − EFi )
e
The electric field for the one-dimensional situation is defined as
Ex = −
dφ
1 dEFi
=
dx
e dx
L. Varani
Physics of Semiconductors Devices
(39)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
If the intrinsic Fermi level changes as a function of distance
through a semiconductor in thermal equilibrium, an electric field
exists in the semiconductor.
If we assume a quasi-neutrality condition in which the electron
concentration is almost equal to the donor impurity concentration,
then we can still write
EF − EFi
n0 = ni exp
≈ Nd (x)
(40)
kT
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
Solving for EF − EFi we obtain
EF − EFi = kT ln
Nd (x)
ni
(41)
The Fermi level is constant for thermal equilibrium so when we
take the derivative with respect to x we obtain
−
dEFi
kT dNd (x)
=
dx
Nd (x) dx
L. Varani
Physics of Semiconductors Devices
(42)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
The electric field can then be written, combining Equations (42)
and (39), as
kT
1 dNd (x)
(43)
Ex = −
e
Nd (x) dx
Since we have an electric field, there will be a potential difference
through the semiconductor due to the nonuniform doping.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
The Einstein Relation
If we consider the nonunifornly doped semiconductor represented
by the energy band diagram shown in Figure 12 and assume there
are no electrical connections so that the semiconductor is in
thermal equilibrium, then the individual electron hole currents
must be zero.
We can write
Jn = 0 = enµn Ex + eDn
L. Varani
dn
dx
Physics of Semiconductors Devices
(44)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
If we assume quasi-neutrality so that n ≈ Nd (x), then we can
rewrite Eqution (44) as
Jn = 0 = eµn Nd (x)Ex + eDn
dNd (x)
dx
(45)
Substituting the expression for the electric field from Equation (43)
into Equation (45), we obtain
kT
1 dNd (x)
dNd (x)
+ eDn
(46)
0 = −eµn Nd (x)
e
Nd (x) dx
dx
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
Equation (46) is valid for the condition
kT
Dn
=
µn
e
(47)
The hole current must also be zero in the semiconductor. From
this condition we can show that
Dp
kT
=
µp
e
L. Varani
Physics of Semiconductors Devices
(48)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
Combining Equations (47) and (48) gives
Einstein Relations
Dp
kT
Dn
=
=
µn
µp
e
(49)
The diffusion coefficient and mobility are not independent
parameters.
This relation between the mobility and diffusion coefficient,
given by Equation (49), is known as the Einstein relation.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
Table 2 shows the diffusion coefficient values at T = 300 K
corresponding to the mobilities listed in Table 1 for silicon, gallium
arsenide, and germanium.
Silicon
Gallium arsenide
Germanium
µn
1350
8500
3900
Dn
35
220
101
µp
480
400
1900
Dp
12.4
10.4
49.2
Table 2: Typical mobility and diffusion coefficient values at T = 300 K
and low doping concentrations. Mobility is given in units of (cm2 /Vs)
and diffusion coefficient in units of (cm2 /s).
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Induced Electric Field
The Einstein Relation
The relation between the mobility and diffusion coefficient givenn
by Equation (49) contains temperature.
It is important to keep in mind that the major temperature effects
are a result of lattice scattering and ionized impurity scattering
processes.
As the mobilities are strong functions of temperature because of
the scattering processes, the diffusion coefficients are also strong
functions of temperature.
The specific temperature dependence given in Equation (49) is a
small fraction of the real temperature characteristic.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Outline
1
Carrier Drift
2
Carrier Diffusion
3
Graded Impurity Distribution
4
The Hall Effect
5
Carrier Generation and Recombination
6
Characteristics of Excess Carriers
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Hall Effect
The Hall effect is a consequence of the forces that are exerted on
moving charges by electric and magnetic fields.
The Hall effect is used to distinguish whether a semiconductor is
n-type or p-type and to measure the majority carrier concentration
and majority carrier mobility.
The Hall effect device is used to experimentally measure
semiconductor parameters.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The force on a particle having a charge q and moving in a
magnetic field is given by
F = qv × B
(50)
where the cross product is taken between velocity and magnetic
field so that the force vector is perpendicular to both the velocity
and magnetic field.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Figure 13 illustrates the Hall effect.
A semiconductor with a current I , placed in a magnetic field
perpendicular to the current.
In this case, the magnetic field is in the z direction.
Electrons and holes flowing in the semiconductor will experience a
force as indicated in the figure.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
178
C HAP T E R
5 Carri9l' Transport Phenomena
Figure 5.131 Geol1letry for measuring the HaJI effect.
Figure 13: Geometry for measuring the Hall effect.
The force on a partic1e having a charge q and moving in a magnetic field is
given by
F=qv x !J
(5.46)
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The force on both electrons and holes is in the (−y ) direction.
In a p-type semiconductor (p0 > n0 ), there will be a buildup of
positive charge on the y = 0 surface of the semiconductor and, in
an n-type semiconductor (n0 > p0 ), there will be a buildup of
negative charge on the y = 0 surface.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
This net charge induces an electric field in the y -direction as
shown in the figure.
In steady state, the magnetic field force will be exactly balanced by
the induced electric field force.
This balance may be written as
F = q(E + v × B) = 0
(51)
qEy = qvx Bz
(52)
which becomes
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The induced electric field in the y -direction is called the Hall field.
The Hall field produces a voltage across the semiconductor which
is called the Hall voltage.
We can write
VH = +EH W
where EH is assumed positive in the +y -direction and VH is
positive with the polarity shown.
L. Varani
Physics of Semiconductors Devices
(53)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
In a p-type semiconductor in which holes are the majority carrier,
the Hall voltage will be positive as defined in Figure 13.
In an n-type semiconductor it will be negative.
The polarity of the Hall voltage is used to determine wether an
extrinsic semiconductor is n-type or p-type.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Substituting Equation (53) into Equation (52) gives
VH = vx WBz
(54)
For a p-type semiconductor we can write
vdx =
Jx
Ix
=
ep
(ep)(Wd)
where e is the magnitude of the electronic charge.
L. Varani
Physics of Semiconductors Devices
(55)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Combining Equations (54) and (55) we have
VH =
Ix Bz
epd
(56)
or, solving for the hole concentration
p=
L. Varani
Ix Bz
edVH
Physics of Semiconductors Devices
(57)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The majority carrier hole concentration is determined from the
current, magnetic field and Hall voltage.
For an n-type semiconductor the Hall voltage is given by
Ix Bz
end
(58)
Ix Bz
edVH
(59)
VH = −
so that electron concentration is
n=−
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Note that the Hall voltage is negative for the n-type
semiconductor, therefore the electron concentration determined
from Equation (59) is actually a positive quantity.
Once the majority carrier concentration has been determined, we
can calculate the low-field majority carrier mobility.
For p-type semiconductor we can write
Jx = epµp Ex
L. Varani
Physics of Semiconductors Devices
(60)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The current density and electric field can be converted to current
and voltage so that Equation (60) becomes
epµp Vx
Ix
=
Wd
L
(61)
The hole mobility is then given by
µp =
L. Varani
Ix L
epVx Wd
Physics of Semiconductors Devices
(62)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Similarly for an n-type semiconductor, the low-field electron
mobility is determined from
µn =
L. Varani
Ix L
enVx Wd
Physics of Semiconductors Devices
(63)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Outline
1
Carrier Drift
2
Carrier Diffusion
3
Graded Impurity Distribution
4
The Hall Effect
5
Carrier Generation and Recombination
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Carrier Generation and Recombination
Generation is the process whereby electrons and holes are
created, and recombination is the process whereby electrons
and holes are annihilated.
Any deviation from thermal equilibrium will lend to change the
electron and hole concentrations in a semiconductor.
A sudden increase in temperature, for example, will increase the
rate at which electrons and holes are thermally generated so that
their concentrations will change with time until new equilibrium
values are reached.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
An external excitation, such as light (a flux of photons), can also
generate electrons and holes, creating a nonequilibrium condition.
To understand the generation and recombination processes, we will
first consider direct band-to-band generation and recombination
and then, later, the effect of allowed electronic energy states within
the bandgap, referred to as impurity or recombination centers.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
The Semiconductor in Equilibrium
Electrons are continually being thermally excited from the valence
band into the conduction band by the random nature of the
thermal process.
At the same time, electrons moving randomly through the crystal
in the conduction band may come in close proximity to holes and
”fall” into the empty states in the valence band.
This recombination process annihilates both the electron and hole.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Since the net carrier concentrations are independent of time in
thermal equilibrium, the rate at which electrons and holes are
generated and the rate at which they recombine must be equal.
The generation and recombination processes are schematically
shown in Figure 14.
L. Varani
Physics of Semiconductors Devices
are schemati cally , hoWII in Figure
6. 1.Drift
Carrier
Carrier Diffusion
Let GIIO and Gpo be
the
thermaJ
..gcllcrat ioJ1 r:'ltes of I! lec.:tnms and
Graded Impurity Distribution
1 The Hall Effect
in Equilibrium
tive ly, given in un)\:-'
of #/cm· -s. For the dire ctThe
baSemiconductor
nd-tO·band
gene rati (m. the electrons
Carrier Generation and Recombination
Excess Carrier Generation and Recombination
Characteristics
of
Excess
Carriers
and holes are created in pairs,
so
we
have
that
Ambipolar Transport
Quasi-Fermi Energy Levels
(6.1)
'.
e
e-
-rr---------:: . -Ir---
Electron-h(llc
f:,.
EleClrl)n-hnlc
,<comb;,,''';ol>
E:,
-,
Figure
I I Eleclron-hole
llt!ratioll andand
rccurecombination.
lilhimu ion,
Figurefl.14:
Electron-holegegeneration
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Let Gn0 and Gp0 be the thermal-generation rates of electrons and
holes, respectively, given in units of cm−3 s−1 .
For the direct band-to-band generation, the electrons and holes are
created in pairs, so we must have that
Gn0 = Gp0
L. Varani
Physics of Semiconductors Devices
(64)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Let Rn0 and Rp0 be the recombination rates of electrons and holes,
respectively, for a semiconductor in thermal equilibrium, again
given in units of cm−3 s−1 .
In direct band-to-band recombination, electrons and holes
recombine in pairs, so that
Rn0 = Rp0
L. Varani
Physics of Semiconductors Devices
(65)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
In thermal equilibrium. the concentrations of electrons and holes
are independent of time: therefore, the generation and
recombination rates are equal:
Gn0 = Gp0 = Rn0 = Rp0
L. Varani
Physics of Semiconductors Devices
(66)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Excess Carrier Generation and Recombination
Additional notation is introduced in this chapter.
Table 3 lists some of the more pertinent symbols used throughout
the chapter.
Other symbols will be defined as we advance through the chapter.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Symbol
n0 , p0
n, p
δn = n − n0
δp = p − p0
gn0 , gp0
Rn0 , Rp0
τn0 , τp0
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Definition
Thermal equilibrium electron and hole concentrations
(independent of time and also usually position)
Total electron and hole concentrations
(may be functions of time and/or position)
Excess electron and hole concentrations
(may be functions of time and/or position)
Excess electron and hole generation rates
Excess electron and hole recombination rates
Excess minority carrier electron and hole lifetimes
Table 3: Relevant notation used in this chapter.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Electrons in the valence band may be excited into the conduction
band when, for example, high-energy photons are incident on a
semiconductor.
When this happens, not only is an electron created in the
conduction band, but a hole is created in the valence band; thus
an electron-hole pair is generated.
The additional electrons and holes created are called excess
electrons and excess holes.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
The excess electrons and holes are generated by an external force
at a particular rate.
Let gn0 be the generation rate of excess electrons and gp0 be that of
excess holes.
These generation rates also have units of cm−3 s−1 , so for the
direct band-to-band generation, the excess electrons and holes are
also created in pairs
gn0 = gp0
(67)
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
When excess electrons and holes are created, the concentration of
electrons in the conduction band and of holes in the valence band
increase above their thermal-equilibrium values:
n = n0 + δn
(68)
p = p0 + δp
(69)
and
where n0 and p0 are the thermal-equilibrium concentrations, and
δn and δp are the excess electron and hole concentrations.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Figure 15 shows the excess electron-hole generation process and
the resulting carrier concentrations.
The external force has perturbed the equilibrium condition so that
the semiconductor is no longer in thermal equilibrium.
We may note from Equations (68) and (69) that, in a
nonequilibrium condition np 6= n0 p0 = ni2
L. Varani
Physics of Semiconductors Devices
pairs. so the recombinarion rares must
be equal, \Ve can then write
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
The Semiconductor in Equilibrium
Carrier Generation and Recombination
Excess Carrier Generation and Recombination
Carriers
direct Characteristics of Excess
recombination
that we are considering. the re<.·ombj·
Ambipolar Transport
Quasi-Fermi Energy Levels
(6.6)
In the::
nation occurs spontaneously; Ihus. Ihe prObability of an eleclron and hule recombining is constant with time. The rate at which electrons recombine must be proportional
+
+ +,
+ ... +
E,.
o/>
6.21 Creation
of CXl'CSS
electron
andhole
holedensities
densitie.s by
FigureFigure
15: Creation
of excess
electron
and
by photons.
photons.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
A steady-state generation of excess electrons and holes will not
cause a continual buildup of the carrier concentrations.
As in the case of thermal equilibrium, an electron in the
conduction band may ”fall down” into the valence band, leading to
the process of excess electron-hole recombination.
Figure 16 shows this process.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
CXl'CSS
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
o/>
Figure 6.21 Creation of
photons.
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
electron
and hole densitie.s by
-
-
+ + +,
E._
E•.
Figure
6.31 Recombination
excessreestablishing
carriers
Figure 16:
Recombination
of excess of
carriers
thermal
reestablishing
thermal
equilibrium.
equilibrium.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
The recombination rate for excess electrons is denoted by Rn0 and
for excess holes by Rp0 .
Both parameters have units of cm−3 s−1 .
The excess electrons and holes recombine in pairs, so the
recombination rates must be equal:
Rn0 = Rp0
L. Varani
Physics of Semiconductors Devices
(70)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
In the direct band-to-band recombination that we are considering,
the recombination occurs spontaneously: thus the probability of an
electron and hole recombinating is constant with time.
The rate at which electrons recombine must be proportional to the
electron concentration and must also be proportional to the hole
concentration.
If there are no electrons or holes, there can be no recombination.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
The net rate of change in the electron concentration can be
written as
dn(t)
= αr ni2 − n(t)p(t)
dt
where n(t) = n0 + δn(t) et p(t) = p0 + δp(t).
(71)
The first term, αr ni2 , in Equation (71) is the thermal-equilibrium
generation rate.
Since excess electrons and holes are created and recombine in
pairs, we have that δn(t) = δp(t).
Excess electron and hole concentrations are equal so we can simply
use the phrase excess carriers to mean either.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
The thermal-equilibrium parameters, n0 and p0 , being independent
of time, Equation (71) becomes
d(δn(t))
= αr ni2 − (n0 + δn(t))(p0 + δp(t))
dt
= −αr δn(t) [(n0 + p0 ) + δn(t)]
(72)
Equation (72) can easily be solved if we impose the condition of
low-level injection.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Low-level injection puts limits on the magnitude of the excess
carrier concentration compared with the thermal equilibrium carrier
concentrations.
In an extrinsic n-type material, we generally have n0 p0 and in
an extrinsic p-type material, we generally have p0 n0 material.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Low-level injection means that the excess carrier concentration is
much less than the thermal equilibrium majority carrier
concentration.
Conversely, high-level injection occurs when the excess carrier
concentration becomes comparable to or greater than the thermal
equilibrium majority carrier concentrations.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
If we consider a p-type material (p0 n0 ) under low-Ievel
injection (δn(t) p0 ), then Equation (72) becomes
d(δn(t))
= −αr p0 δn(t)
dt
(73)
The solution is an exponential decay from the initial excess
concentration
δn(t) = δn(0)e −αr p0 t = δn(0)e −t/τn0
where τn0 = (αr p0 )−1 and is constant for low-level injection.
L. Varani
Physics of Semiconductors Devices
(74)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
Equation (74) describes the decay of excess minority carrier
electrons so that τn0 is often referred to as the excess minority
carrier Iifetime.
The recombination rate, which is defined as a positive quantity, of
excess minority carrier electrons can be written, using
Equation (72), as
Rn0 =
−d(δn(t))
δn(t)
= +αr p0 δn(t) =
dt
τn0
L. Varani
Physics of Semiconductors Devices
(75)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
For the direct band-to-band recombination, the excess majority
carrier holes recombine at the same rate, so that for the p-type
material
Excess recombination rates in p-type material
Rn0 = Rp0 =
δn(t)
τn0
(76)
In the case of an n-type material (n0 p0 ) under low-level
injection (δn(t) n0 ), the decay of minority carrier holes occurs
with a time constant τp0 = (αr n0 )−1 , where τp0 is referred to as
the excess minority carrier lifetime.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
The recombination rate of the majority carrier electrons will be the
same as that of the minority carrier holes, so we have
Excess recombination rates in n-type material
Rn0 = Rp0 =
L. Varani
δn(t)
τp0
Physics of Semiconductors Devices
(77)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The Semiconductor in Equilibrium
Excess Carrier Generation and Recombination
The generation rates of excess carriers are not functions of electron
or hole concentrations.
In general, the generation and recombination rates may be
functions of the space coordinates and time.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
Outline
1
Carrier Drift
2
Carrier Diffusion
3
Graded Impurity Distribution
4
The Hall Effect
5
Carrier Generation and Recombination
6
Characteristics of Excess Carriers
Continuity Equation
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
Characteristics of Excess Carriers
The generation and recombination rates of excess carriers are
important parameters, but how the excess carriers behave with
time and in space in the presence of electric fields and density
gradients is of equal importance.
The excess electrons and holes do not move independently of each
other, but they diffuse and drift with the same effective diffusion
coefficient and with the same effective mobility.
This phenomenon is called ambipolar transport.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
The question is what is the effective diffusion and what is the
effective mobility that characterizes the behavior of these excess
carriers.
The final results show that, for an extrinsic semiconductor under
low injection the effective diffusion coefficient and mobility
parameters are those of the minority carrier.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
Continuity Equation
Figure 17 shows a differential volume element in which a
one-dimensional hole-particle flux is entering the differential
element at x and is leaving the element at x + dx.
+ is the hole-particle flux, or flow, and has units
The parameter Fpx
of number of holes/cm2 s.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
If F;-' (x) > F,':,(x
+ tlx),
Continuity
Equation
for example.
there
will be a net increase in th
Time-Dependent Diffusion Equations
of holes in the differentiaJ yolume! elemcm with (jme. If we generJlJize to
n<ional hole flux, then the right side of Equation (6.16) I\lay be wr
: FirV:}
•
I
.,
I
I
,.. ... L _ ____
...
",-,- '-
-
---""- /
.t -f
7 -
- --
.11·
.
dr
F'lgurt: 6.4
1Differential
volume showing
Figure 17: Differential
volume
showing x component
of the hole-particle
flux.
x component of the hole-pJlrticie flux.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
For the x component of the particle current density shown, we may
write
+
∂Fpx
+
+
Fpx
(x + dx) = Fpx
(x) +
dx
(78)
∂x
+ (x) where the differential
This equation is a Taylor expansion of Fpx
length dx is small so that only the first two terms in the expansion
are significant.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
The net increase in the number of holes per unit time within the
differential volume element due to the x-component of hole flux is
given by
+
+
∂Fpx
∂p
+
(x) − Fpx
(x + dx) dydz = −
dxdydz = Fpx
dxdydz (79)
∂t
∂x
+ (x) > F + (x + dx), for example, there will be a net increase
If Fpx
px
in the number of holes in the differential yolume element with time.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
If we generalize to a three-dimensional hole flux, then the right
+ dxdydz where
side of Equation (79) may be written as −∇Fpx
+
∇Fpx is the divergence of the flux vector.
We will limit ourselves to a one-dimensional analysis.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
The generation rate and recombination rate of holes will also affect
the hole concentration in the differential volume.
The net increase in the number of holes per unit time in the
differential volume element is then given by
∂Fp+
∂p
p
dxdydz = −
dxdydz + gp dxdydz −
dxdydz
∂t
∂x
τpt
where p is the density of holes.
L. Varani
Physics of Semiconductors Devices
(80)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
The first term on the right side of Equation (80) is the increase in
the number of holes per unit time due to the hole flux, the second
term is the increase in the number of holes per unit time due to
the generation of holes, and the last term is the decrease in the
number of holes per unit time due to the recombination of holes.
The recombination rate for holes is given by p/τpt , where τpt
includes the thermal equilibrium carrier lifetime and the excess
carrier lifetime.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
If we divide both sides of Equation (80) by the differential volume
dxdydz, the net increase in the hole concentration per unit time is
∂Fp+
∂p
p
=−
+ gp −
∂t
∂x
τpt
(81)
Equation (81) is known as the continuity equation for holes.
Similarly, the onedimensional continuity equation for electrons is
given by
∂n
∂F −
n
= − n + gn −
(82)
∂t
∂x
τnt
where Fn− is the electron-particle flow, or flux, also given in units
of number of electrons/cm2 s.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
Time-Dependent Diffusion Equations
We derived previously the hole and electron current densities,
which are given, in one dimension, by
Jp = eµp pE − eDp
∂p
∂x
(83)
Jn = eµn nE + eDn
∂n
∂x
(84)
and
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
If we divide the hole current density by +e and the electron
current density by −e, we obtain each particle flux:
and
Jp
∂p
= Fp+ = µp pE − Dp
+e
∂x
(85)
Jn
∂n
= Fn− = −µn nE − Dn
−e
∂x
(86)
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
Taking the divergence of Equations (85) and (86), and substituting
back into the continuity Equations (81) and (82) we obtain
and
∂(pE )
∂2p
∂p
p
= −µp
+ Dp 2 + gp −
∂t
∂x
∂x
τpt
(87)
∂(nE )
∂2n
∂n
n
= +µn
+ Dn 2 + gn −
∂t
∂x
∂x
τnt
(88)
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
Keeping in mind that we are limited to a onedimensional analysis
we can expand the derivatives as
∂(pE )
∂p
∂E
=E
+p
∂x
∂x
∂x
Equations (87) and (88) can be written in the form
∂2p
∂p
∂E
∂p
p
+p
=
Dp 2 − µ p E
+ gp −
∂x
∂x
∂x
τpt
∂t
and
∂E
∂2n
∂n
n
∂n
Dn 2 + µn E
+n
+ gn −
=
∂x
∂x
∂x
τnt
∂t
L. Varani
Physics of Semiconductors Devices
(89)
(90)
(91)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
Equations (90) and (91) are the time-dependent diffusion
equations for holes and electrons, respectively.
Since both the hole concentration p and the electron concentration
n contain the excess concentrations, Equations (90) and (91)
describe the space and time behavior of the excess carriers.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
The hole and electron concentrations are functions of both the
thermal equilibrium and the excess values are given in
Equations (69) and (68).
The thermal-equilibrium concentrations, n0 and p0 , are not
functions of time.
For the special case of a homogeneous semiconductor, n0 and p0
are also independent of the space coordinates.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Continuity Equation
Time-Dependent Diffusion Equations
Equations (90) and (91) may be written in the form:
∂(δp)
∂E
p
∂(δp)
∂ 2 (δp)
− µp E
+p
+ gp −
=
Dp
2
∂x
∂x
∂x
τpt
∂t
(92)
and
Dn
∂ 2 (δn)
∂(δn)
∂E
∂(δn)
n
+
µ
E
+
n
=
+ gn −
n
∂x 2
∂x
∂x
τnt
∂t
(93)
Note that the Equations (92) and (93) contain terms involving the
total concentrations, p and n, and terms involving only the excess
concentrations δp and δn.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Outline
1
Carrier Drift
2
Carrier Diffusion
3
Graded Impurity Distribution
4
The Hall Effect
5
Carrier Generation and Recombination
6
Characteristics of Excess Carriers
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Ambipolar Transport
Originally, we assumed that the electric field in the current
Equations (83) and (84) was an applied electric field.
This electric field term appears in the time-dependent diffusion
equations given by Equations (92) and (93).
If a pulse of excess electrons and a pulse of excess holes are
created at a particular point in a semiconductor with an applied
electric field, the excess holes and electron, will tend to drift in
opposite directions.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
However, because the electrons and holes are charged particles,
any separation will induce an internal electric field between the two
sets of particles.
This internal electric field will create a force attracting the
electrons and holes back toward each other.
This effect is shown in Figure 18.
L. Varani
Physics of Semiconductors Devices
PT. R 6
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
NoneQuilibrium Excess Csrtiers in Semcondu{.1ors
ElIl'P
,
,+
-
:
I
+
Elm
I
: -
: +
Figure (•.51 The cre(uion of an internal cicCIric f1ckl
Figure 18: The creation of an internal electric fields as excess electrons
excess
electrons and holes lend In separate.
and holes,IStend
to separate.
ss electrons and a pulse of excess holes nrc created at a particular point in a se
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
The electric field term in Equations (92) and (93) is then composed
of the externally applied field plus the induced internal field.
This E-field may be written as
E = Eapp + Eint
where Eapp is the applied electric field and Eint is the induced
internal electric field.
L. Varani
Physics of Semiconductors Devices
(94)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Since the internal E-field creates a force attracting the electrons
and hole, this E-field will hold the pulses of excess electrons and
excess holes together.
The negatively charged electrons and positively charged holes then
will drift or diffuse together with a single effective mobility or
diffusion coefficient.
This phenomenon is called ambipolar diffusion or ambipolar
transport.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
It is possible to show that, in the presence of ambipolar transport
the time-dependent diffusion equation becomes
D0
∂ 2 (δn)
∂(δn)
∂(δn)
+g −R =
+ µ0 E
2
∂x
∂x
∂t
where
D0 =
(95)
µn nDp + µp pDn
µ n n + µp p
(96)
µn µp (p − n)
µ n n + µp p
(97)
and
µ0 =
and
R = Rn =
n
p
= Rp =
.
τnt
τpt
L. Varani
Physics of Semiconductors Devices
(98)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Equation (95) is called ambipolar transport equation and describes
the behavior of the excess electrons and holes in time and space.
The parameter D 0 is called the ambipolar diffusion coefficient and
µ0 is called the ambipolar mobility.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
The Einstein relation relates the mobility and diffusion coefficient
by
µp
e
µn
=
=
(99)
Dn
Dp
kT
Using these relations, the ambipolar diffusion coefficient may be
written in the form
Dn Dp (n + p)
D0 =
(100)
Dn n + Dp p
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
The ambipolar diffusion coefficient D 0 and the ambipolar mobility
µ0 are functions of the electron and hole concentrations, n and p,
respectively.
Since both n and p contain the excess-carrier concentration δn, the
coefficient in the ambipolar transport equation are not constants.
The ambipolar transport equation, given by Equation (95), then, is
a nonlinear differential equation.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Limits of Extrinsic Doping and Low Injection
The ambipolar transport equation may be simplified and linearized
by considering an extrinsic semiconductor and by considering
low-level injection.
The ambipolar diffusion coefficient, from Equation (100), may be
written as
Dn Dp [(n0 + δn) + (p0 + δn)]
(101)
D0 =
Dn (n0 + δn) + Dp (p0 + δn)
where n0 and p0 are the thermal-equilibrium electron and hole
concentrations, respectively, and δn is the excess carrier
concentration.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
If we consider a p-type semiconductor, we can assume that
p0 n0 .
The condition of low-level injection, or just low injection, means
that the excess carrier concentration is much smaller than the
thermal-equilibrium majority carrier concentration.
For the p-type semiconductor, then, low injection implies that
δn p0 .
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Assuming that n0 p0 and δn p0 , and assuming that Dn and
Dp are on the same order of magnitude, the ambipolar diffusion
coefficient from Equation (101) reduces to
D 0 = Dn
L. Varani
Physics of Semiconductors Devices
(102)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
If we apply the conditions of an extrinsic p-type semiconductor and
low injection to the ambipolar mobility, Equation (97) reduces to
µ0 = µn
L. Varani
Physics of Semiconductors Devices
(103)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
It is important to note that for an extrinsic p-type semiconductor
under low injection, the ambipolar diffusion coefficient and the
ambipolar mobility coefficient reduce to the minority-carrier
electron parameter values, which are constant.
The ambipolar transport equation reduces to a linear differential
equation with constant coefficients.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
If we now consider an extrinsic n-type semiconductor under low
injection, we may assume that p0 n0 and δn n0 .
The ambipolar diffusion coefficient reduces to
D 0 = Dp
(104)
and the ambipolar mobility reduces to
µ0 = −µp
(105)
The ambipolar parameters again reduce to the minority-carrier
values, which are constants.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Note that, for the n-type semiconductor, the ambipolar mobility is
a negative value.
The ambipolar mobility term is associated with carrier drift,
therefore, the sign of the drift term depends on the charge of the
particle.
The equivalent ambipolar particle is negatively charged, as one can
see by comparing Equations (93) and (95).
If the ambipolar mobility reduces to that of a positively charged
hole, a negative sign is introduced as shown in Equation (105).
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
The remaining terms we need to consider in the ambipolar
transport equation are the generation rate and the recombination
rate.
Recall that the electron and hole recombination rates are equal and
were given by
n
p
R = Rn =
= Rp =
(106)
τnt
τpt
where τnt and τpt are the mean electron and hole lifetimes.
If we consider the inverse lifetime functions, then 1/τnt is the
probability per unit time that an electron will encounter a hole and
recombine.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Likewise, 1/τpt is the probability per unit time that a hole will
encounter an electron and recombine.
If we again consider an extrinsic p-type semiconductor under low
injection the concentration of majority carrier holes will be
essentially constant, even when excess carriers are present.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Then, the probability per unit time of a minority carrier electron
encountering a majority carrier hole will be essentially constant.
Hence τnt = τn , minority carrier electron lifetime, will remain a
constant for the extrinsic p-type semiconductor under low injection.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Similarly, if we consider an extrinsic n-type semiconductor under
low injection, the minority carrier hole lifetime, τpt = τp , wilI
remain constant.
Even under the condition of low injection, the minority carrier hole
concentration may increase by several orders of magnitude.
The probability per unit time of a majority carrier electron
encountering a hole may change drastically.
The majority carrier lifetime, then may change substantially when
excess carriers are present.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Consider, again, the generation and recombination terms in the
ambipolar transport equation.
For electrons we may write
g − R = gn − Rn = (Gn0 + gn0 ) − (Rn0 + Rn0 )
(107)
where Gn0 and gn0 are the thermal-equilibrium electron and excess
electron generation rates, respectively.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
The terms Rn0 and Rn0 are the thermal-equilibrium electron and
excess electron recombination rates, respectively.
For thermal equilibrium, we have that
Gn0 = Rn0
L. Varani
Physics of Semiconductors Devices
(108)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
So Equation (107) reduces to
g − R = gn0 − Rn0 = gn0 −
δn
τn
where τn is the excess minority carrier electron lifetime.
L. Varani
Physics of Semiconductors Devices
(109)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
For the case of holes we obtain
g − R = gp0 − Rp0 = gp0 −
δp
τp
where τp is the excess minority carrier hole lifetime.
L. Varani
Physics of Semiconductors Devices
(110)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
The generation rate for excess electrons must equal the generation
rate for excess holes.
We may then define a generation rate for excess carriers as g 0 , so
that gn0 = gp0 = g 0 .
We also determined that the minority carrier lifetime is essentially
a constant for low injection.
Then the term g − R in the ambipolar transport equation may be
written in terms of the minority-carrier parameters.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
The ambipolar transport equation, given by Equation (95), for a
p-type semiconductor under low injection then becomes
Dn
∂ 2 (δn)
∂(δn)
δn
∂(δn)
+ µn E
+ g0 −
=
∂x 2
∂x
τn0
∂t
(111)
The parameter δn is the excess minority carrier electron
concentration, the parameter τn0 is the minority carrier lifetime
under low injection, and the other parameters are the usual
minority carrier electron parameters.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Similarly for the case of an extrinsic n-type semiconductor under
low injection we have
Dp
∂(δp)
δp
∂(δp)
∂ 2 (δp)
+ µp E
+ g0 −
=
2
∂x
∂x
τp0
∂t
(112)
The parameter δp is the excess minority carrier hole concentration,
the parameter τp0 is the minority carrier lifetime under low
injection, and the other parameters are the usual minority carrier
hole parameters.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
It is important to note that the transport and recombination
parameters in Equations (111) and (112) are those of the minority
carrier.
Equations (111) and (112) describe the drift, diffusion, and
recombination of excess minority carriers as a function of spatial
coordinates and time.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Recall that we had imposed the condition of charge neutrality; the
excess minority carrier concentration is equal to the excess majority
carrier concentration.
The excess majority carriers, then, diffuse and drift with the excess
minority carriers: thus the behavior of the excess majority carrier is
determined by the minority carrier parameters.
Table 4 summarizes the possible simplifications of the ambipolar
transport equation
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Specification
Effect
Steady state
∂(δn)
∂(δp)
∂t = 0, ∂t = 0
2
2 (δp)
(δn)
Dn ∂ ∂x
= 0, Dp ∂ ∂x
2
2
Uniform distribution of excess carriers
(uniform generation rate)
Zero electric field
No excess carrier generation
No excess carrier recombination
(infinite lifetime)
∂(δp)
E ∂(δn)
∂x = 0, E ∂x = 0
g0 = 0
δp
δn
τn0 = 0, τp0 = 0
Table 4: Common ambipolar transport equation simplifications.
L. Varani
Physics of Semiconductors Devices
=0
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Dielectric Relaxation Time Constant
We have assumed in the previous analysis that a quasi-neutrality
conditions exists, that is, the concentration of excess holes is
balanced by an equal concentration of excess electrons.
Suppose that we have a situation as shown in Figure 19, in which
a uniform concentration of holes δp is suddenly injected into a
portion of the surface of a semiconductor.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
nuity equation. neglecting the effects
generation
and
recombim.t
Limitsof
of Extrinsic
Doping and Low
Injection
Dielectric Relaxation Time Constant
Dp
'\I - J= - 8t
\ Theofinjectlon
of a concentration
of region at
FigureFigure
19: The6.10
injection
a concentration
of holes into a small
the surface
an a
n-type
hole... of
into
Slllallsemiconductor.
region O1lthe
of an n-type
semiconductor.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
We will instantly have a concentration of excess holes and a net
positive charge density that is not balanced by a concentration of
excess electrons.
How is charge neutrality achieved and how fast?
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
There are three defining equations to be considered.
Poisson’s equation
∇·E=
ρ
(113)
The current equation, Ohm’s law, is
J = σE
(114)
The continuity equation neglecting the effect of generation and
recombination is
∂ρ
∇·J=−
(115)
∂t
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
The parameter ρ is the net charge density and the initial value is
given by e(δp).
We will assume that (δp) is uniform over a short distance at the
surface.
The parameter is the permittivity of the semiconductor.
Taking the divergence of Ohm’s law and using Poisson’s equation,
we find
σρ
(116)
∇ · J = σ∇ · E =
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Substituting Equation (116) into the continuity equation we have
σρ
∂ρ
dρ
=−
=−
∂t
dt
(117)
Since Equation (117) is a function of time only, we can write the
equation as a total derivative.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Limits of Extrinsic Doping and Low Injection
Dielectric Relaxation Time Constant
Equation (117) can be rearranged as
dρ σ =
ρ=0
dt
(118)
Equation (118) is a first-order differential equation whose solution
is
ρ(t) = ρ(0)e −t/τd
(119)
where
σ
and is called the dielectric relaxation time constant.
τd =
L. Varani
Physics of Semiconductors Devices
(120)
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Outline
1
Carrier Drift
2
Carrier Diffusion
3
Graded Impurity Distribution
4
The Hall Effect
5
Carrier Generation and Recombination
6
Characteristics of Excess Carriers
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Quasi-Fermi Energy Levels
The thermal-equilibrium electron and hole concentrations are
functions of the Fermi energy level.
We can write
n0 = ni exp
and
p0 = ni exp
EF − EFi
kT
EFi − EF
kT
(121)
(122)
where EF and EFi are the Fermi energy and intrinsic Fermi energy,
respectively, and ni is the intrinsic carrier concentration.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Figure 20(a) shows the energy-band diagram for an n-type
semiconductor in which EF > EFi .
For this case, we may note from Equations (121) and (122) that
n0 > ni and p0 < ni as we would expect.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
ftlle" quasi-Fermi le ve l for
electrons
and a quasi-Fermi level for holes Ihat apply for
Carrier Diffusion
noncquilibrium.Graded
If 611Impurity
and 01'Distribution
are the excess electron and hole concentrJtions. re-
The Hall Effect
spectively,
we Generation
may write:
Carrier
and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
(6.853)
- - - - - - - - - - - - - £F
t----F.r
>.
gu ----- --------------- £FI
-gt' --------------------
-
E·
iLi
e
?
.. -,.
.•:--•.-: ,,;
,
..
•.
.." . -
'"
'. - "'
...•- .-. £.
.
EFt
- - - - - - - - - - - - - E,.
f l'
:.: r
(a)
Figure 6.1 41 Thcnnal-cqu ilibriultl energY-band diagr.t.mfl
f()f (a)
n-type
ant! (lJ) P-\)'pe
Figurescmicunduc\ur
20: Thermal-equilibrium
energy-band diagram for (a) n-type
semiconductor and (b) p-type semiconductor.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Similarly, Figure 20(b) shows the energy-band diagram for a p-type
semiconductor in which EF < EFi .
Again we may note from Equations (121) and (122) that n0 < ni
and p0 > ni , as we would expect for the p-type material.
These results are for thermal equilibrium.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
If excess carriers are created in a semiconductor, we are no longer
in thermal equilibrium and the Fermi energy is strictly no longer
defined.
However, we may define a quasi-Fermi level for electrons and a
quasi-Fermi level for holes that apply for nonequilibrium.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
If δn and δp are the excess electron and hole concentrations,
respectively, we may write:
EFn − EFi
(123)
n0 + δn = ni exp
kT
and
p0 + δp = ni exp
EFi − EFp
kT
(124)
where EFn and EFp are the quasi-Fermi energy levels for electrons
and holes, respectively.
The total electron concentration and the total hole concentration
are functions of the quasi-Fermi levels.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
Figure 21a shows the energy-band diagram with the Fermi energy
level corresponding to thermal equilibrium.
Figure 21b shows the energy-band diagram under the
nonequilibrium condition.
Since the majority carrier electron concentration does not change
significantly for this low-injection condition, the quasi-Fermi level
for electrons is not much different from the thermal-equilibrium
Fermi level.
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
CHAPT.R 8
N'onequilibfium Excess Carners in SemiConductors
O.2984cV
tI--------T--------.------'------'"
tl
w
.... ... . .
(a)
±
0.29&2 cV
0.2982 eV
t ,,,,-:1,.,,,.,.. , .
;; ----l---------------___ -___ t _________
===",..!'f!
En
b
•g
tE;1
.
1
B
10.171) eV .".
t.
til
(b)
fIgure 6.15 I (a) 'n1ermal-equilibtium energy-band diagram for N. = 10" cm- 1 and
no· =
cm-Thermal-equilibrium
.1. (b) Quasi-Fermi levels for
electrons and holes
jf lOllfor
cm -:'I excess
Figure
21:10 10(a)
energy-band
diagram
−3
carriers
Nd = 1015 arc
cmpresent.
and n0 = 1010 cm−3 . (b) Quasi-Fermi levels for
electrons and holes if 1013 cm−3 excess carriers are present.
increased significantly so that the quasi-Fermi level for hole.' has
concenlration
moved much closer to Ihe valence band. We will consider the quasi-Fenni energy
le\'els agalo when we discuss forv.;ard-blased
pn junctions,
L. Varani
Physics of Semiconductors Devices
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
The Hall Effect
Carrier Generation and Recombination
Characteristics of Excess Carriers
Ambipolar Transport
Quasi-Fermi Energy Levels
The quasi-Fermi energy level for the minority carrier holes is
significantly different from the Fermi level and illustrates the fact
that we have deviated from thermal equilibrium significantly.
Since the electron concentration has increased, the quasi-Fermi
level for electrons has moved slightly closer to the conduction band.
The hole concentration has increased significantly so that the
quasi-Fermi level for holes has moved much closer to the valence
band.
L. Varani
Physics of Semiconductors Devices