Download 1 Carrier Drift

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.
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.
1
Carrier Drift
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.
1.1
Drift Current Density
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
where Jp|drf is the drift current density due to holes and vdp is the average drift velocity of the holes.
equation of motion of a positively charged hole in the presence of an electric field is
(2)
The
F = m∗p a = eE
(3)
where e is the magnitude of the electronic charge, a is the acceleration, E is the electric field, and m∗p is the
effective mass of the hole.
If the electric field is constant, then we expect the velocity to increase linearly with time. 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. 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. 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.
1
The unit of mobility is usually expressed in terms of cm2 /(Vs). 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.
to the electric field for small fields.
The average drift velocity of an electron is also proportional
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. 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.
Silicon
Gallium arsenide
Germanium
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.
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
1.2
(9)
Mobility Effects
Mobility Effects
Equation 3 related the acceleration of a hole to a force such as an electric field.
We may write this equation as
dv
= eE
(10)
dt
where v is the velocity of the particle due to the electric field and does not include the random thermal velocity.
If we assume that the effective mass and electric field are constants, then we may integrate Equation (10) and
obtain
eEt
v= ∗
(11)
mp
F = m∗p
where we have assumed the initial drift velocity to be zero. 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. If we use the mean time between collisions τcp
2
C HAP T E R
5 Cruner Transport po l€nomena
.' . . .
. ... . ;, .., ,..,
3
-
E fleld
(h}
(a }
Figure 1: SOl
Typical
randomr.mdon\
behavior h<!hav\or
of a hole in of
a semiconductor
(a) without an electric
field and (b)
Figure
ITypical
a hole in a sem,condUClOf
(a) without
an with an ,
electric field.
clcotric field .nd (b) with an electric field.
.J,
in place of the time t in Equation (11) then the mean peak velocity just prior to a collision or scattering event
applied
as indicated in Figure 5.1b. there will
is:
be
a net drift of the hole in the directioo
eτcp
vd|peak =
(12)
of the E··field. and the net drift velocity
willm∗be Ea small pel1urbation on the random
p
Ihermal "elocily, so the time between collisions will not be altered appreciably. If'>t
The average drift velocity is one half the peak value so that we can write
use the mean time between collisions tt'l ' in place
of the time I in Equacion (5.111
1 eτcp
hvd ito
= ,tcollision
E
(13)
{hen [he mean peak velocity just prior
or scattering eVent is
2 m∗
p
(-"".Ill;-, )
However, the collision process is not as simple as this model, but is statistical in nature.
(5.123)
udlpcak
E
In a more accurate model including the effect
of a statistical distribution,
the factor 1/2 in Equation (13)
does
not appear. The hole mobility is then given by
=
mobility
TheHole
average
drift velocity is one half Ihevpeakeτvalue so that we Can write
µp =
The same analysis applies to electrons:
dp
E
=
cp
m∗p
(} =,.I(er"")E
-.Vd
...
(14)
(5.12b)
Ill, )
Electron mobility
eτcn
However. the collision process is
notvdn =simple
a1i (his mode1, but is statistical
ill
(15)
µn =
E
m∗n
nature. In a more accurate model including the effect of a statistical
factor
inisEquation
(5.12b)
doe, not
appear.
hole
mobility
isscattering
Ihen given
by
where τcn
the mean time
between collisions
for an
electron.The
There
are two
collision or
mechanisms
that dominate in a semiconductor and affect the carrier mobility: phonon or lattice scattering, and ionized
impurity scattering.
UII"
(5.1l)
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
E their lattice position within the crystal.
The lattice vibrations cause a disruption in the perfect periodic potential function. A perfect periodic potential
Thein same
analysis
applies
e)ec,rons:
Ihus
we can through
write the
electron mobility as
a solid allows
electrons
to movetounimpeded
or with
no scattering
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.
(5.14)
This lattice scattering is also referred to as phonon scattering. 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
where
r is
thestates
mean
between
collisions for an electron.
scattering
theory
thattime
to first
order
−3/2
µL ∝ Tmechanism!'.
(16)
There :ire two
or scattering
that dominate in a semicool ·"
duclor and affect the t arrier mobility: phonon
or lauice
3
purity scattering.
and
in>
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. 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.
3 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.
5000
1000
4000
•
N"
Nf)
=
•
100
I
"'" "'"
I
I
I
I
I
I
N• • I Off'!
Nt>"' 10" ,' A: '
= 10\(1
I
JO I4
T
IIXlIl
2000
I
N... - 10\1
111M
?
:>
'"..,
. , tV" -
1\
2000
••
1000
,
II
NO .
:>
""§
10 11
500
!VA
100
,t
N" ". U) 19
I
UI::<I
Nt) == lOll!
IIIIII I
11111111
No . 10 19
JW L
UK)
r
•
2<10
N ., 10u(u\
' I
1m
Ill()
50
- 50
J='iguno
S. 2 I ( ;, 1 I '"
.. lIl ,-,rc .1.
u
J:
\
" I'
10
50
100
)50
- 50
200
o
50
100
7 ("0
T( OC}
(a)
( b}
;11111 ( h i h<\k II I1.hi l il i\·.. i ll ..
.' , ... ";'''11, ... , ''; . • .. j, ... , . " .
_-'
_
200
J
I I
SIIQ
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 dependence for almost intrinsic silicon.
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. If we denote µI as the mobility that would be observed if only ionized impurity scattering
existed, then to first-order we have
T +3/2
µI ∝
(17)
NI
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. 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 .
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. If τL is the mean time between collisions due to lattice scattering, then dt/τL is the probability of
4
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
Figure 3: Electron and hole mobilities
versus
impurity
concentrations
for germanium, silicon, and gallium
arsenide 31 T = 300 K .
arsenide at T = 300 K.
fFromSu (121.)
TEST YOUR UNDERSTANDING
a lattice scattering event occurring in a differential time dt.
ES.J (a} Using FigufC 5.2. find the cl<."Clron Jtlobilil)' for (i) N,t
= J().1 cm- J . l ' = }5(fC
Likewise, if τI is the mean time
collisions
toFind
ionized
scattering,
then dt/τI is the probability
and between
(;i) N" = 10"
em-·' . T = due
O' C. (u)
Ihe holeimpurity
mobililies for
(i) N" =
lO"c,.-'.
T = SO'
C; and (ii) N"in=the
10" em-'
. T = ISO"C.
of an ionized impurity scattering
event
occurring
differential
time dt. If these two scattering processes
(s·"" probability
UJ'OOo- (!I ) 's'N,W'
(il (q) :S'N" UJ OOS 1- (!II 'S'N,W' 00, <!J (/J) '<UV)
are independent, then the total
of aOS\:scattering
event occurring in the differential time dt is the sum
ESA Using F'iRurc 5.3. determine the eleclroll and holt mobilities in (0) silicon fOf
of the individual events or ,vol = JOI5
N" =: 0: (b) siti.:-on
lOii cm- ). Nfl == 5 X 10 1t! cm- 3 ;
dt for
dtNil = dt
(18)
(e) silicon for Nd = tO I6 COl-;', N<,4=
= 10" +
ern ); and (d)
fur
τ
τL '" ',I (p)
IV_ = N" = 10" cm- '. ["N, 'UJOLZ
"τI"'I'OOSt
:01£ '" d" any
'OOR '"scattering
NTI (,) '00£ '"event.
- n 'OOL '" " ,I (q) :OR1> = ,In 'OS£I '" "71 (u) ·suv]
where τ is the mean time between
Comparing Equation (18) with the definitions of mobility given by Equations (14) or (15), we can write
If r" is the mean ti me between collisions Jue to lattice scattering. then ,II Ir, is
Matthiessen rule
the probability of a lattice scattering event occurring in a differentia l time dt.
Likewise. if rl is the mean time between
due to ionized impurity scattering,
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.
1.3
Conductivity
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. 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
5
Resistivity
ρ=
1
1
=
σ
e(µn n + µp p)
(21)
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.
If we have a bar of
•
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
Figure 4: Resistivity versus impurity concentration at T = 300 K in (a) silicon and (b) germanium, gallium
j
163
arsenide, and gallium phosphide.
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
We can now rewrite Equation (20) as
or
V
I
=σ
A
L
L
ρL
V =
I=
I = RI
σA
A
(24)
(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.
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
(26)
If we also assume complete ionization, then Equation (26) becomes
σ ≈ eµp Na ≈
6
1
ρ
(27)
CHAPT.R 5
Carcier TransportPheoomel'..a
Figure
5.515:Bar
Figure
Barofof semiconductor matedal
material as aa resistor.
resistor.
We can now rewrite Equation (5.19) as
The conductivity and resistivity of an extrinsic semiconductor are a function primarily of the majority carrier
parameters.
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 of6inverse
for
. 1 Catemperature
ier Orift
the case when Nd = 1015 cm−1 . In the midtemperature range, or extrinsic range, as shown, we have complete
or
500
300
J(
10 11 r-"1:'
,
T(K)
200
100
75
Equation (S.22b) is OhmI,'s law for a semiconductor. The resistance is a function 01
I
I'esisti vity, or conductivity.
as well as the geometry of the semiconductor.
I
16
If we
a p-Iype semiconductor with10an, acceptor doping
10 fOr example.
:
Nn(Nd = 0) in which Nil » Ilj, and if we assume thut the electron and hole mobili·
.§ orderII of magnitude. then the conductivity becomes
ties are of the same
I
......
10'"
I
............
1.0
c:
I
(5.23)
"
' I ionization.
'
If we also <lssume complete
thcn Equation (5.23) becomes
'\ I, ' "
'.'
,
I
0.1
I
,
The conduc{\vity and resistivity of an cx,trlnsic semiconductor are a functiOI\ priI Ilj
I
marily of (he majority carrier parameters.
I
We may plot the carrier concentration and conductivity of a emiconductor as I
{uncti on of temperature for a paJ'licular doping concentration. Figure 5.6 shows the
electron concentration and conductivity of si licon as a funclion of inverse temperature
Figureconcentration
5.6 1Eleclron
concenlration
and conductivity versus
l conductivity versus inverse temperature for silicon.
6: Electron
and
forFigure
the case
when
Nd remperalure
= 10 15 cm-·
In the midtemperature range. or extri nsic range,
io\'crse
for. .o;ilicon.
as shown, we fAfter
have Sr.
cl)rllplete
' " 12}. ) ionization-lhe elec[n)1l l:oncentradon remains essenionization
the constant.
electron concentration
remains
essentially
constant.of tempcC'dture so the conductil'ity
tially
However, the
mObility
is a function
However, the mobility is a function of temperature so the conductivity varies with temperature in this range.
varies
with temperature
in this
range.
At higherincreases
temperatures.
the to
intrinsic
carrier
con- conAt higher
temperatures,
the intrinsic
carrier
concentration
and begins
dominate
the electron
centration
as well as
the conductivity.
centration
increases
and begins to dominate the electron concentration as well as the
conductivity. In the lower temperature range, freeze-oUi begins to occur; the electron
7 decreasing temperarure.
concentration and conductivity decrease with
In the lower temperature range, freeze-out begins to occur; the electron concentration and conductivity decrease
with decreasing temperature. 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,
1.4
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
1
3
3
2
mvth
= kT = (0.0259) = 0.03885 eV
2
2
2
(29)
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. 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.
there
is a linear
variation of velocity with electric field, the slope of the drift velocity
C HA P T
. R 15where
Ca!rier
Transport
Pheoorna'1a
versus electric field curve is the mobility. The behavior of the drift velocity of carriers at high electric fields
e
i!'2
H)T
<:
:§
" lU'
"
'E
0
ElcctriC field (V/cm)
Carrierdrifl
velo<:ity
versus eleclric
field
for
Figure 7: Carrier driftFigure
velocity5.71
versus
electric field
for high-purity
silicon,
germanium,
and gallium arsenide.
high-purilY silicon.
and gallium
( F",,,, 05" (/1/.)
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.
This energy
translates into a mean thermal velocity of approximately 10' cmls foran
2/V-s in low·
If the drift
velocity
of a charge
carrier
saturates,
then the
drift current
also
and becomes
electron
in silicon.
Jf we
asSume
an electron
mobility
of ttl' density
J350
cmsaturates
independent
of
the
applied
electric
field.
The
drift
velocity
versus
electric
field
characteristic
doped silicon. a drift velocity of lOS cmh, or I percent of the thermal velocity, i,of gallium
arsenide is more complicated than for silicon or germanium.
=
nchievcd if the applied electric field is approximately 75 V/cm . This applied elecuk
field doe< not appreciably alter the energy of Ihe electron.
8
Figure 5.7 is a plot of average drift velocity as a function of appl ied eiectric field
for electrons and holes in silicon. gallium arsenide. and germanium. At low eJectO:
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.
electron drift velocity in gallium arsenide reaches a peak and then decreases.
As the field increases, the
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.
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 m∗n = 0.067m0 .
The small effective mass leads to a large mobility.
5 . 2 Csnie.- Oitfusitln
As the E-field increases, the energy of the electron increases
O:IA:;
Conduction
5.8 field
I Encrgy-bnnd
Figure 8: Carrier drift velocity versus electric
for high-purity silicon, germanium, and gallium arsenide.
for gall ium arsenide showing the
upper \'ulley and lower valley in
and the electron can be scattered into the upper valley, where the density of states effective mass is 0.55m0 .
the conducliull hallu.
The larger effective mass in the upper valley yields a smaller mobility.
(From S!e / 13/.)
This intervalley transfer mechanism results in a decreasing average drift velocity of electrons with electric field,
or the negative differential mobility characteristic.
The negati ve diffcrcntialillobility can be understood by considering the £ versus
2 kCarrier
Diffusion
diagram for
galliull1 arsenide, which is shown agai n in I'igure 5.8. The density of
states effective mass of the electron in the lower valley is III:' = 0.0671110. The small
Carrier Diffusion
effective mass leads to a large mobility. As the E-ficld increases, the energy of the
There is a second mechanism that can induce a current in a semiconductor.
electron increases and the electron Can be scauered into the upper valley. where the
We may consider a classic physics example in which a container, as shown in Figure 9, is divided into two parts
density of statcs effecti ve mass is 0.55/110' The larger effecti ve mass in Ihe upper
by a membrane.
a .gas
"maller
mobility.
mechanism
resuhs
in n devalley
The left
side yields
contains
molecules
at a particularintervalley
temperature rransfer
and the right
side is initially
empty.
creasing
average
velocity
of electrons
withsoelectric
lield,
the negati
ve differThe gas
molecules
are indrift
continual
random
thermal motion
that, when
the or
membrane
is broken,
the gas
enlialflow
mobility
molecules
into the right side of the. container.
•
5.2 ICARRIER DIFFUSION9
TaR!5 Carrier Transpon Ph&rtOma......a
170
• •• •.,
• •• ••
•
1
1
I
• 1
. . -0
Figure 9: Container divided by a membrane with gas molecules on one side.
Figure 5.91 Container
divided by a Olen'lbraJle with
Diffusion is the process whereby particles flow from a region of high concentration toward a
region of low concentration.
gas molecules on one side.
If the gas molecules were electrically charged, the net flow of carriers would result in a diffusion current.
CHAPTaR!5 Carrier Transpon Ph&rtOma......a
2.1
Diffusion Current Density
• •• •.,
• • ••
To begin to understand the diffusion process•in •a semiconductor,
we will consider a simplified analysis.
Diffusion Current Density
1
1
I
• 1
. . -0
Assume that an electron concentration varies in one dimension as shown in Figure 10.
The temperature is assumed to be uniform
so that
average thermal velocity of electrons is independent of
Figure
5.91the
Container
x.
divided by a Olen'lbraJle with
gas the
molecules
on of
oneelectrons
side.
To calculate the current, we will determine
net flow
per unit time per unit area crossing the
plane x = 0.
If the distance l shown in Figure 10 is the mean-free path of an electron, that is, the average
n(+ I)
- - - - - - - - - - - - - - - - - - -- - - - n(+ I)
11(0)
- - - - - - - - - - - - - - - - - - -- - - - -
- - - - - - - - - - --- -- - - 11(0)
- - - - - - - - - - --- -- - - -
II( - I)
II( - I)
.( = -I
;t
=0
.(
= +1
.,--
Figure S.10 l Electron concentration versus distance.
Figure 10: Electron concentration versus distance.
.,--
the gascollisions
molecules
were
the distance
nel ftow an
of celectron moving to
distance an concentrdtion.
electron travels If
between
(l =
vth τelectrically
the average,
cn ), then on charged,
.(
=
;t
in a diffusion
currenl.
the right at would
x = −lresult
and electrons
moving
to the left at x = +l will cross the x = 0 plane.
-I
=0
.(
= +1
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 5.2.1
x = +l will
be traveling
to the Density
left at any given time. The net rate of electron flow, Fn , in the +x
Diffusion
Current
direction at x = 0 is given by
To begi n to understand Ihe diffusion process in " semiconductor, we will consider
1
1
1
Fn Assume
= n(−l)v
− electron
n(+l)vthconcentration
= vth [n(−l)varies
− n(+l)]
(30)
th an
simplified analysis.
that
in one dimensi
2
2
2
Figure S.10 l Electron concentration versus distance.
shown in Figure S.l O. The temperature is assumed to be uniform so that the al'e
thennal velocity of electrons is independen
10t of x. To calculate the current, we will
tennine the net flow of electrons per unit time per unit atea crossing the plane.
x = 0.1f the distance / shown in Figure S.l is the mean-free path of an
trdtion. If the gas molecules were electrically
charged, the nel ftow of
a
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
1
dn
dn
Fn = vth n(0) − l
− n(0) + l
(31)
2
dx
dx
which becomes
Fn = −vth l
dn
dx
(32)
Each electron has a charge (−e), so the current is
J = −eFn = +evth l
dn
dx
(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. 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.
172
C HAP
T E R one-dimensional
5 Carriet Transportflux
Phenomel"la
Figure 11a shows
these
and current directions.
We may write the electron diffusion
,
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
Figure 11: (a) Diffusion of electrons
to a density
gradient. (b) Diffusion of holes due to a density gradient.
gradienl. due
(b) Diffusion
of holes due lO a densi ty gradient
current densityfofor
this
one-dimensionalcase,
caseThe
in the
form Dp is cal.led the hole diffllsim. <'01
r the
one-dimensional
parameter
'
"ielll, has units of cm2 /s, and is a positive quantity, If the hole density gradient
Electron diffusion current density
comes negati ve, the hole diffusion cutTCnt density will be in the positive x direcri ,
dn
Jnx|dif = eDn
dx
EXAMPLE 5,4
1
Objective
(34)
where Dn is called the electron diffusion coefficient, has units of cm2 /s, and is a positive quantity.
To calculate the diffusion current density given'l dcnsicy gradien!.
If the electron density gradient becomes negative, the electron diffusion current density will be in the negative
thai. an
in an
n-Iype gaHium arsenide semiconductor 3t T = 300 K. the eJec
x direction. Figure 11b shows
example
of a hole concentration as a function of distance in a semiconductor.
concentration
linearly from I x 10"1 to 7 X 10 11 em -l over a distance orO.10 em.e
The diffusion of holes, from a region of high concentration to a region of low concentration produces a flux of
cuialc the diffusion current density if the electron diffusion coerficient is 0,. = 225 em!/..,
holes in the negative x direction.
Solution charged particles, the conventional diffusion current density is also in the negative x
Since holes are• positively
The
diffusion
current
density
is givenis by
direction. The hole
diffusion
current
density
proportional to the hole density gradient and to the electronic
charge, so we may write
dn
D l!.n
J,, 'di/ = eD" e /1 dx
= (1.6 x
Ax
10- ")(225) (
11
_ 7XIOI7)
-I X-JOI8
-=-=--='
0,10
,
Alem-
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.
2.2
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,
dn
dp
− eDp
dx
dx
(36)
J = enµn E + epµp E + eDn ∇n − eDp ∇p
(37)
J = enµn E + epµp E + eDn
This equation may he generalized to three dimensions as
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. 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.
3
Graded Impurity Distribution
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.
3.1
Induced Electric Field
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
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. 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. In most cases of interest, the
12
------------------- E,.
-- ---
_------- E"
----- --E,.
Figure
S.12 1Energy-band
diagram with
ror a nonuniform donor impurity
Figure 12: Energy-band diagram
for a semiconductor
in thermal equilibrium
concentration.
a scmiconlim:to r in thermal equilibrium
wilh a nonuniform donnr impurity
concentration.
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. The electric potential φ is related
to electron potential energy by the charge −e:
1
φ = + (EF − EF i )
e
(38)
The electric field for the one-dimensional situation is defined as
Ex = −
1 dEF i
dφ
=
dx
e dx
(39)
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 − EF i
n0 = ni exp
≈ Nd (x)
(40)
kT
Solving for EF − EF i we obtain
EF − EF i = 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
−
kT dNd (x)
dEF i
=
dx
Nd (x) dx
The electric field can then be written, combining Equations (42) and (39), as
kT
1 dNd (x)
Ex = −
e
Nd (x) dx
(42)
(43)
Since we have an electric field, there will be a potential difference through the semiconductor due to the
nonuniform doping.
3.2
The Einstein Relation
The Einstein Relation
13
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
dn
dx
(44)
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)
0 = −eµn Nd (x)
+ eDn
e
Nd (x) dx
dx
(46)
Equation (46) is valid for the condition
Dn
kT
=
µn
e
(47)
The hole current must also be zero in the semiconductor. From this condition we can show that
Dp
kT
=
µp
e
(48)
Dp
kT
Dn
=
=
µn
µp
e
(49)
Combining Equations (47) and (48) gives
Einstein Relations
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. 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).
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.
4
The Hall Effect
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.
14
The Hall effect device is used to experimentally measure semiconductor parameters.
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. 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.
178
C HAP T E R
5 Carri9l' Transport Phenomena
Electrons and holes flowing in the semiconductor will experience a force as indicated in the figure.
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
The force on bothgiven
electrons
by and holes is in the (−y) direction.
F=qv
x !J of positive charge on the y(5.46)
In a p-type semiconductor (p0 > n0 ), there will be
a buildup
= 0 surface of the
semiconductor and, in an n-type semiconductor (n0 > p0 ), there will be a buildup of negative charge on the
where the cross product is la.ken between velocity and magnetic field so that rhe force
y = 0 surface. This net charge induces an electric field in the y-direction as shown in the figure.
vector is perpendicular to both the velocity and magnetic field.
In steady state, the magnetic
force willthe
be Hall
exactly
balanced
by the induced
field
Figure 5.field
13 illustrates
effect.
A semiconductor
withelectric
a current
I, force.
in a magnetic
field perpendicular to the current. In this case. the magnetic fiehl
This balance mayplaced
be written
as
is in the z direction. Electrons
andq(E + vflowing
F=
× B) =ill0 the semiconductor will experi·
(51)
ence a force as indicated in the figure. The force on both eleClrOIlS and hules is in the
(- y) direction. In a p·typ. semiconductor (Po > "0). there will be a buildup of po;.
qEy = qvx Bz
itive charge on the y = 0 surface of the semiconductor and, in an n-type semiconducior
(110in>the
Po).
there will isbecalled
a buildup
of negati
The induced electric
field
y-direction
the Hall
field.ve charge on rhe ." = 0 surf"".
This nCt charge induces an electric field in the y-dinection as shown in the figure. In
The Hall field produces a voltage across the semiconductor which is called the Hall voltage.
steady state. the magnetic field force will be exactl y balanced by the indnced electric
We can write
field force. This balallce may be writlen as
which becomes
(52)
VH = +EH W
(53)
(s,47a)
F = q(E + v x 81 = 0
where EH is assumed
whichpositive
becomesin the +y-direction and VH is positive with the polarity shown. In a p-type
semiconductor in which holes are the majority carrier, the Hall voltage will be positive as defined in Figure 13.
(5.41b)
In an n-type semiconductor it will be negative.
The induced electric field in the y-direc!;on is called !he Hall lield. The Hall fiehl
The polarity of the Hall voltage is used to determine wether an extrinsic semiconductor is n-type or p-type.
produces a voltage ,lcross the semiconduClor which is ("lied the H(lfl voltage. Wec311.
Substituting Equation (53) into Equation (52) gives
write
VH = vx W Bz
15
(5.48)
(54)
For a p-type semiconductor we can write
vdx =
Jx
Ix
=
ep
(ep)(W d)
(55)
where e is the magnitude of the electronic charge. Combining Equations (54) and (55) we have
VH =
Ix Bz
epd
(56)
or, solving for the hole concentration
p=
Ix Bz
edVH
(57)
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
VH = −
so that electron concentration is
n=−
Ix Bz
end
Ix Bz
edVH
(58)
(59)
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 à p-type semiconductor we can write
Jx = epµp Ex
(60)
The current density and electric field can be converted to current and voltage so that Equation (60) becomes
epµp Vx
Ix
=
Wd
L
(61)
Ix L
epVx W d
(62)
The hole mobility is then given by
µp =
Similarly for an n-type semiconductor, the low-field electron mobility is determined from
µn =
5
Ix L
enVx W d
(63)
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.
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.
16
electron and hole. Since the net can·jer concentrations are independent of time in
thennal equili brium. the race at which electrons and holes a re gencwted and the rate
at which they recombine must be equal. The generatiun and recomb ination
5.1 The Semiconductor in Equilibrium
are schemati cally , hoWII in Figure 6. 1.
The Semiconductor in Equilibrium
Let
GIIO are
and
Gpo be the thermaJ ..gcllcrat ioJ1 r:'ltes of I! lec.:tnms and
Electrons
continually being thermally excited from the valence band into the conduction band by the random
of the
thermal of
process.
tive ly,nature
given
in un)\:-'
#/cm·1 -s. For the dire ct ba nd-tO·band gene rati (m. the electrons
At the same time, electrons moving randomly through the crystal in the conduction band may come in close
and holes
are created in pairs, so we
have that
proximity to holes and ”fall” into the empty states in the valence band.
This recombination process annihilates both the electron and hole. Since the net carrier concentrations are
independent of time in thermal equilibrium, the rate at which electrons and holes are generated and the rate(6.1)
at which they recombine must be equal.
The generation and recombination processes are schematically shown in Figure 14.
'.
e
e-
-rr---------:: . -Ir---
Electron-h(llc
f:,.
EleClrl)n-hnlc
,<comb;,,''';ol>
E:,
-,
Figure 14: Electron-hole generation and recombination.
Figure fl. I I Eleclron-hole ge llt!ratioll and rccu lilhimu ion,
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
(64)
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
(65)
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
5.2
(66)
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.
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.
17
Symbol
n0 , p 0
CHAPTER 8
Definition
Thermal equilibrium electron and hole concentrations
(independent of time and also usually position)
n, p
Total electron
and
Excess
C3frle;S
in hole concentrations
(may be functions of time and/or position)
δn = n − n0 Excess electron and hole concentrations
δp = p − p0
(may be functions of time and/or position)
gn0 , gp0
Excess electron and hole generation rates
Rn0 , Rp0
Excess electron and hole recombination rates
τn0 , τp0
Excess minority carrier electron and hole lifetimes
where /1<) and Po are the thernlal-equilibrium
and b/l and J" are the
excess electron and hole concentrations. Figure 6.2 .,hows the excess electron-hole
generation process and the resulting carrier concentrations. The external force has
Table 3: Relevant notation used in this chapter.
pcnurbed the equilibrium condition sO that the semiconductor is 11 0 longer in thermal
equilibrium. We may note from Equations (6.5a) and (6.5b) that. in a nooequilibrium
·The
. additional
)
holes
con dIlIon.
np .t.
-relectrons
noP{) =andni
. created are called excess electrons and excess holes.
The excess electrons and holes are generated by an external force at a particular rate.
A 0
generation of exc.ess dectrons and ho1es win n()t cause a cont\llual
Let gn be the generation rate of excess electrons and gp0 be that of excess holes.
buildup
ofgeneration
the carrier
concentrations.
As
ca'eband-to-band
of thermal
equilibrium.
an ele<:- I
These
rates also
have units of cm−3 s−1
, soin
for the
the direct
generation,
the excess electrons
andthe
holesconduction
are also created band
in pairs may "fall down" into the valence band. leadillg to tile '
tron in
gn0 = gp0
(67)
process of "M;ess electron-hole recombination.
Figure 6.3 show' this pr<)cess. Tile !
When excess electrons and holes are created, the concentration of electrons in the conduction band and of holes
recombination
for exC
esS their
clecuons
is denoted
by R;, and for excess holes by
in the valence rate
band increase
above
thermal-equilibrium
values:
Both parameters hnve units of #/cm' -s. nThe
excess electrons and holes recombine
in
= n0 + δn
(68)
pairs. so the recombinarion rares must be equal, \Ve can then write
and
p = p0 + δp
(69)
(6.6)
where n0 and p0 are the thermal-equilibrium concentrations, and δn and δp are the excess electron and hole
concentrations. Figure 15 shows the excess electron-hole generation process and the resulting carrier concenIn
the:: direct
recombination that we are considering. the re<.·ombj·
trations.
nationTheoccurs
Ihus.
Ihe prObability
ofthe
ansemiconductor
eleclron and
externalspontaneously;
force has perturbed the
equilibrium
condition so that
is nohule
longer recombinin thermal
ing isequilibrium.
constant with time. The rate at which electrons recombine must be proportional
We may note from Equations (68) and (69) that, in a nonequilibrium condition np 6= n0 p0 = n2i
+
E,.
+ ... +
+ +,
o/>
Creationof
of excess
electron
and hole
by photons.by
FigureFigure
6.21 15:
Creation
CXl'CSS
electron
anddensities
hole densitie.s
photons.
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.
18
-
-
E._
photons.
-
-
+ + +,
E._
E•.
Figure 16:
Recombination
of excess carriers
reestablishing
thermal equilibrium.
Figure
6.31
Recombination
of excess
carriers
reestablishing thermal equilibrium.
Figure 16 shows this process.
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
(70)
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.
concentration can be written as
dn(t)
= αr n2i − n(t)p(t)
dt
where n(t) = n0 + δn(t) et p(t) = p0 + δp(t).
The net rate of change in the electron
(71)
The first term, αr n2i , 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.
The thermal-equilibrium parameters, n0 and p0 , being independent of time, Equation (71) becomes
d(δn(t))
= αr n2i − (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.
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.
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.
19
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
(74)
where τn0 = (αr p0 )−1 and is constant for low-level injection.
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
(75)
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.
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 =
δn(t)
τp0
(77)
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.
6
Characteristics of Excess Carriers
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. 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.
20
of holes in the differentiaJ yolume! elemcm with (jme. If we generJlJize to a
men<ional hole flux, then the right side of Equation (6.16) I\lay be wrill
: 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.
6.1
Continuity Equation
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.
+
The parameter Fpx
is the hole-particle flux, or flow, and has units of number of holes/cm2 s.
For the x component of the particle current density shown, we may write
+
+
Fpx
(x + dx) = Fpx
(x) +
+
∂Fpx
dx
∂x
(78)
+
(x) where the differential length dx is small so that only the first two
This equation is a Taylor expansion of Fpx
terms in the expansion are significant.
The net increase in the number of holes per unit time within the differential volume element due to the xcomponent of hole flux is given by
+
+
∂Fpx
∂p
+
dxdydz = Fpx
(x) − Fpx
(x + dx) dydz = −
dxdydz
∂t
∂x
(79)
+
+
If Fpx
(x) > Fpx
(x + dx), for example, there will be a net increase in the number of holes in the differential
yolume element with time. If we generalize to a three-dimensional hole flux, then the right side of Equation (79)
+
+
may be written as −∇Fpx
dxdydz where ∇Fpx
is the divergence of the flux vector.
We will limit ourselves to a one-dimensional analysis.
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
(80)
where p is the density of holes. 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.
21
The recombination rate for holes is given by p/τpt , where τpt includes the thermal equilibrium carrier lifetime
and the excess carrier lifetime. 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 one·dimensional continuity equation for electrons is given by
∂n
∂F −
n
= − n + gn −
∂t
∂x
τnt
(82)
where Fn− is the electron-particle flow, or flux, also given in units of number of electrons/cm2 s.
6.2
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
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)
Taking the divergence of Equations (85) and (86), and substituting back into the continuity Equations (81) and
(82) we obtain
∂p
∂(pE)
∂2p
p
= −µp
+ Dp 2 + gp −
(87)
∂t
∂x
∂x
τpt
and
∂n
∂(nE)
∂2n
n
= +µn
+ Dn 2 + gn −
∂t
∂x
∂x
τnt
(88)
Keeping in mind that we are limited to a onedimensional analysis we can expand the derivatives as
∂p
∂E
∂(pE)
=E
+p
∂x
∂x
∂x
Equations (87) and (88) can be written in the form
∂2p
∂E
p
∂p
∂p
Dp 2 − µp E
+p
+ gp −
=
∂x
∂x
∂x
τpt
∂t
and
∂2n
∂E
n
∂n
∂n
Dn 2 + µn E
+n
+ gn −
=
∂x
∂x
∂x
τnt
∂t
(89)
(90)
(91)
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.
The hole and electron
concentrations are functions of both the thermal equilibrium and the excess values are given in Equations (69)
and (68).
22
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.
Equations (90) and (91) may be written in the form:
∂ 2 (δp)
∂(δp)
∂E
p
∂(δp)
Dp
−
µ
E
+
p
+ gp −
=
p
∂x2
∂x
∂x
τpt
∂t
and
Dn
∂(δn)
∂E
n
∂(δn)
∂ 2 (δn)
+
µ
E
+
n
+ gn −
=
n
∂x2
∂x
∂x
τnt
∂t
(92)
(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.
7
Ambipolar Transport
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. However,
because the electrons and holes are charged particles, any separation will induce an internal electric field between
the two sets of particles.
. PT. R 6
NoneQuilibrium
Csrtiers
inthe
Semcondu{.
1ors
This internal
electric field willExcess
create a force
attracting
electrons and holes
back toward each other.
This effect is shown in Figure 18.
The electric field term in Equations (92) and (93) is then composed of the
ElIl'P
,
,+
-
:
I
+
Elm
I
: -
: +
Figure (•.51 The cre(uion of an internal cicCIric f1ckl
,ISexcess electrons and holes lend In separate.
Figure 18: The creation of an internal electric fields as excess electrons and holes tend to separate.
externally applied field plus the induced internal field.
This E-field may be written as
E = Eapp + Eint
(94)
where E and
is theaapplied
electric
and E holes
is the induced
internal electric
ess electrons
pulse
of field
excess
nrc created
atfield.
a particular point in a sem
Since the internal E-field creates a force attracting the electrons and hole, this E-field will hold the pulses of
ductor with
an applied
electric
excess electrons
and excess holes
together. field, the excess holes and electron, will telld
The negatively charged electrons and positively charged holes then will drift or diffuse together with a single
t in opposite
directions. However. because the electrons and holes are charge
effective mobility or diffusion coefficient.
This separation
phenomenon is called
diffusion
ambipolar transport.
icles, any
wiambipolar
ll ind uce
anor internal
electric field between the two scts
icles. This illlernal electric field will create a force attmcting the electrons a
es back toward each o ther. This effecI 23is shown in Figure 6 .5. The electric fie
m in Equati ons (6.29) and (6.30) is the n composed of the externally applied fie
app
int
It is possible to show that, in the presence of ambipolar transport the time-dependent diffusion equation becomes
D0
∂ 2 (δn)
∂(δn)
∂(δn)
+ µ0 E
+g−R=
2
∂x
∂x
∂t
where
D0 =
(95)
µn nDp + µp pDn
µn n + µp p
(96)
µn µp (p − n)
µn n + µp p
(97)
and
µ0 =
and
R = Rn =
p
n
= Rp =
.
τnt
τpt
(98)
Equation (95) is called ambipolar transport equation and describes the behavior of the excess electrons and
holes in time and space.
The parameter D0 is called the ambipolar diffusion coefficient and µ0 is called the ambipolar mobility.
Einstein relation relates the mobility and diffusion coefficient by
µp
e
µn
=
=
Dn
Dp
kT
The
(99)
Using these relations, the ambipolar diffusion coefficient may be written in the form
D0 =
Dn Dp (n + p)
Dn n + Dp p
(100)
The ambipolar diffusion coefficient D0 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.
7.1
Limits of Extrinsic Doping and Low Injection
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
D0 =
Dn Dp [(n0 + δn) + (p0 + δn)]
Dn (n0 + δn) + Dp (p0 + δn)
(101)
where n0 and p0 are the thermal-equilibrium electron and hole concentrations, respectively, and δn is the excess
carrier concentration. 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 . 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
D0 = Dn
(102)
If we apply the conditions of an extrinsic p-type semiconductor and low injection to the ambipolar mobility,
Equation (97) reduces to
µ0 = µn
(103)
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.
24
The ambipolar transport equation reduces to a linear differential equation with constant coefficients. 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
D0 = Dp
(104)
µ0 = −µp
(105)
and the ambipolar mobility reduces to
The ambipolar parameters again reduce to the minority-carrier values, which are constants.
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). 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
R = Rn =
n
p
= Rp =
τnt
τpt
(106)
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.
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.
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.
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.
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 )
where Gn0 and
terms Rn0 and
(107)
gn0 are the thermal-equilibrium electron and excess electron generation rates, respectively. The
Rn0 are the thermal-equilibrium electron and excess electron recombination rates, respectively.
For thermal equilibrium, we have that
Gn0 = Rn0
(108)
So Equation (107) reduces to
g − R = gn0 − Rn0 = gn0 −
δn
τn
(109)
δp
τp
(110)
where τn is the excess minority carrier electron lifetime.
For the case of holes we obtain
g − R = gp0 − Rp0 = gp0 −
where τp is the excess minority carrier hole lifetime.
generation rate for excess holes.
The generation rate for excess electrons must equal the
We may then define a generation rate for excess carriers as g 0 , so that gn0 = gp0 = g 0 .
25
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. The ambipolar transport equation, given by Equation (95), for a p-type semiconductor under low
injection then becomes
∂(δn)
∂ 2 (δn)
δn
∂(δn)
+ µn E
Dn
+ g0 −
=
(111)
∂x2
∂x
τn0
∂t
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.
Similarly for the case of an extrinsic n-type semiconductor under low injection we have
Dp
∂ 2 (δp)
∂(δp)
δp
∂(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.
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. 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
Specification
Effect
Steady state
∂(δn)
∂(δp)
∂t = 0, ∂t = 0
2
∂ 2 (δn)
(δp)
Dn ∂x2 = 0, Dp ∂ ∂x
2
Uniform distribution of excess carriers
(uniform generation rate)
Zero electric field
No excess carrier generation
No excess carrier recombination
(infinite lifetime)
=0
∂(δp)
E ∂(δn)
∂x = 0, E ∂x = 0
0
g =0
δp
δn
τn0 = 0, τp0 = 0
Table 4: Common ambipolar transport equation simplifications.
7.2
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.
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? 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
∇·J=−
26
∂ρ
∂t
(115)
'\I - J= - 8t
Figure 6.10 \ The injectlon of a concentration of
hole... into a Slllall region O1lthe
of an n-type
Figure 19: The injection of a concentration of holes into a small region at the surface of an n-type semiconductor.
The parameter ρ is the net charge density and the initial value is given by e(δp).
semiconductor.
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
∇ · J = σ∇ · E =
σρ
(116)
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.
Equation (117) can be rearranged as
dρ σ =
ρ=0
dt
Equation (118) is a first-order differential equation whose solution is
ρ(t) = ρ(0)e−t/τd
(118)
(119)
where
σ
and is called the dielectric relaxation time constant.
τd =
8
(120)
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 − EF i
kT
EF i − EF
kT
(121)
(122)
where EF and EF i are the Fermi energy and intrinsic Fermi energy, respectively, and ni is the intrinsic carrier
concentration.
Figure 20(a) shows the energy-band diagram for an n-type semiconductor in which EF > EF i .
For this case, we may note from Equations (121) and (122) that n0 > ni and p0 < ni as we would expect.
Similarly, Figure 20(b) shows the energy-band diagram for a p-type semiconductor in which EF < EF i .
27
(6.853)
t----F.r
>.
- - - - - - - - - - - - - £F
gu ----- --------------- £FI
-gt' --------------------
-
?
e
.. -,.
.•:--•.-: ,,;
,
..
•.
E·
iLi
.." . -
'"
'. - "'
...•- .-. £.
.
EFt
- - - - - - - - - - - - - E,.
:.: r
f l'
(a)
Figure 6.1 41 Thcnnal-cqu ilibriultl energY-band diagr.t.mfl f()f (a) n-type
Figure 20: Thermal-equilibrium energy-band diagram for (a) n-type semiconductor and (b) p-type semiconducscmicunduc\ur
ant! (lJ) P-\)'pe
tor.
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. 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.
If δn and δp are the excess electron and hole concentrations, respectively, we may write:
EF n − EF i
n0 + δn = ni exp
kT
and
p0 + δp = ni exp
EF i − EF p
kT
(123)
(124)
where EF n and EF p 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.
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.
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.
28
CHAPT.R 8
N'onequilibfium Excess Carners in SemiConductors
O.2984cV
tI--------T--------.------'------'"
tl
t ,,,,-:1,.,,,.,.. , .
;; ----l---------------___ -___ t _________
===",..!'f!
En
b
•g
tE;1
.
1
B
.... ... . .
w
±
0.29&2 cV
0.2982 eV
10.171) eV .".
t.
til
(a)
(b)
fIgure 6.15 I (a) 'n1ermal-equilibtium energy-band diagram for N. = 10" cm- 1 and
Figure 21: (a) Thermal-equilibrium
energy-band diagram for Nd = 1015 cm−3 and n0 = 1010 cm−3 . (b) Quasi10 cm- .1. (b) Quasi-Fermi
no·
10
levels
forcarriers
electrons
and holes jf lOll cm -:'I excess
Fermi levels for electrons and holes if 1013 cm−3
excess
are present.
=
carriers arc 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,
TEST YOUR UNDERSTANDING
£ 6. 10
Silicon.u T ::: 300 K is doped at impurity t:oncentrations of Nil
N<,l
= 101(,
and
=- O. Exces.s carriers are generated such th2a\ the steady-stale va\ues are
=5 X
cm- J . (a) Calculate the thermal equilibrium Fenni level with
respect to En. (b) Detennine £1'" and E,.·p with respect 10 En .
[AO L69Z'O d" 3 - 1$.'1 A' 98P£'O 1i3 - "'3 (q)
6n = 8p
',\ '
£6.11
=
[L.[ ·O = "'3 -
J :] (v)
" "VI
ImpuritycQncenlfat;ons of Nd::: lUI S cm) "md NIJ -::::6 x
C\l\- ·l are added to
silicon at T 300 K. Excess carriers are generated in the material such that the
steady-stale COnCCOlf'dlions are 8n ;: lJp 2 X 10 14 c m - l . (a) Find the ,henna)
29
equilibrium Ftrmi leve l with respect 10 En · (b) Calculate £"11 and F. rp with respect
10 EN· LA' W[£'O
JJ3 - " 3 'A. O'>.Z·O= "3 - "<llq)
=
=
=