Physical Electronics Slides of Chapter 6 (All...
Transcript of Physical Electronics Slides of Chapter 6 (All...
PHYSICAL ELECTRONICS(ECE3540)
Wednesday, October 09, 2013Tennessee Technological University 1
CHAPTER 6 – CARRIER GENERATION AND RECOMBINATION
Chapter 6 – Carrier Generation and Recombination
Chapter 4: we considered the semiconductorin equilibrium and determined electron and holeconcentrations in the conduction and valencebands, respectively.
The net flow of the electrons and holes in asemiconductor will generate currents. The processby which these charged particles move is calledtransport.
Chapter 5: we considered the two basic transportmechanisms in a semiconductor crystal: drift themovement of charge due to electric fields, anddiffusion the flow of charge due to densitygradients.
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Chapter 6 – Carrier Generation and Recombination
Chapter 6: we will discuss the behavior ofnon-equilibrium electron and holeconcentrations as functions of time and space.
We will develop the ambi-polar transportequation which describes the behavior of theexcess electrons and holes.
We can define two new parameters that apply tothe non-equilibrium semiconductor: the quasi-Fermi energy for electrons and the quasi-Fermienergy for holes.
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Carrier Generation and Recombination Generation: is the process whereby electrons and holes
are created. Recombination: is the process whereby electrons and
holes are annihilated. Any deviation from thermal equilibrium will tend to
change the electron and hole concentrations in asemiconductor.
Examples: A sudden increase in temperature will increase the rate at which
electrons and holes are thermally generated so that theirconcentrations will change with time until new equilibrium valuesare reached.
An external excitation, such as light (a flux of photons), can alsogenerate electrons and holes, creating a non-equilibriumcondition.
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The Semiconductor in Equilibrium Since the net carrier concentrations are
independent of time in thermal equilibrium, therate at which electrons and holes are generatedand the rate at which they recombine must beequal.
Let Gn0 and Gp0 be the thermal-generation rateof electrons and holes in #/cm3-s. For a directband-to-band generation, the electrons andholes are created in pairs, therefore: Gn0 = Gp0.
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The Semiconductor in Equilibrium
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Let Rn0 and Rp0 be the recombination rates of electronsand holes, respectively, for a semiconductor in thermalequilibrium, again given in units of #/cm3-s.
In direct band-to-band recombination, electrons andholes recombine in pairs, so that Rn0 = Rp0.
In thermal equilibrium, the concentrations of electronsand holes are independent of time; therefore, thegeneration and recombination rates are equal, so we haveGn0 = Gp0 = Rn0 = Rp0
Fig. 6.1: Electron-hole generation and recombination
-
+ +
-
Electron-Hole Generation
Electron-Hole Recombination
X
Excess Carrier Generation and Recombination Electrons in the valence band may be excited into the
conduction band when, for example, high-energyphotons are incident on a semiconductor.
When this happens, not only is an electron created inthe conduction band, but a hole is created in thevalence band; thus an electron-hole pair is generated.
The additional electrons and holes created are calledexcess electrons and excess holes.
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Excess Carrier Generation and Recombination The excess electrons and holes are generated by an
external force at a particular rate. Let gn’ be the generation rate of excess electrons and gp’
be the generation rate of excess holes in units of #/cm3s For the direct band-to-band generation, the excess
electrons and holes are also created in pairs, so we musthave gn’ = gp’.
When excess electrons and holes are created, theconcentration of electrons in the conduction band andof holes in the valence band increase above their thermalequilibrium value.
We may write n = n0 + n and p = p0 + p. n0 and p0 arethe thermal-equilibrium concentrations, and n and pare the excess electron and hole concentrations.
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Excess Carrier Generation and Recombination
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Symbol Definition
no, po Thermal equilibrium electron and hole concentration (independent of time and position)
n, p Total electron and hole concentrations(may be functions of time and/or position)
n = n – n0 Excess electron concentration(may be function of time and/or position)
p = p – p0 Excess hole concentration(may be function of time and/or position)
gn’ , gp’ Excess electron and hole generation rates.
Rn’ , Rp’ Excess electron and hole recombination rates.
n0, p0 Excess minority carrier electron and hole lifetimes.Table. 6.1: Notations used in Carrier Generation and Recombination
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 ofthermal equilibrium, an electron in the conduction band may "falldown" into the valence band, leading to the process of excesselectron-hole recombination.
The recombination rate for excess electrons is denoted by Rn’, and forexcess holes by Rp’.
Both parameters have units of #/cm3-s. The excess electrons andholes recombine in pairs, so the recombination rates must be equal.We can then write Rn’ = Rp’.
In the direct band-to-band recombination that we are considering, therecombination occurs spontaneously: thus, the probability of anelectron and hole recombining is constant with time.
The rate at which electrons recombine must be proportional to theelectron concentration and must also be proportional to the holeconcentration.
If there are no electrons or holes, there can be no recombination.
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Excess Carrier Generation and Recombination The net rate of change in the electron concentration can
be written as:(eq. 6.1)
where
rni2 (eq. 6.1) is thermal-equilibrium generation rate.
Since excess electrons and holes are created andrecombine in pairs, n(t) = p(t). (Excess electron andhole concentrations are equal and termed excess carriers.)
The thermal-equilibrium parameters, n0 and p0,independent of time, become: (eq. 6.2)
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)]()([)( 2 tptnndt
tdnir
)()( 0 tnntn )()( 0 tpptp
)]())[(())]())((([))((0000
2 tnpntntpptnnndt
tndrir
Excess Carrier Generation and Recombination (Eq. 6.2) can easily be solved if we impose the condition of
Low Level Injection. In an extrinsic n-type material, we generally have no >> po
and, in an extrinsic p-type material, we generally have po>>no.
Low-level injection means that the excess carrierconcentration is much less than the thermal equilibriummajority carrier concentration.
Conversely, high-level injection occurs when the excesscarrier concentration becomes comparable to or greater thanthe thermal equilibrium majority carrier concentrations.
If we consider a p-type material (po >> no) under low-levelinjection (n(t) <<po), then (eq. (6.2)) is:
(eq. (6.2)Wednesday, October 09, 2013Tennessee Technological University 12
)())((0 tnp
dttnd
r
Excess Carrier Generation and Recombination The solution to the equation is an exponential decay
from the initial excess concentration, or(eq. 6.3)
where n0 = (rp0)-1 and is a constant for the low-levelinjection.
The above equation describes the decay of excessminority carrier electrons so that n0 is often referredto as the excess minority carrier lifetime.
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00 /)0()0()( nr ttp enentn
Excess Carrier Generation and Recombination The recombination rate defined as a positive quantity--of excess minority carrier
electrons can be written as:
For the direct band-to-band recombination, the excess majority carrier holesrecombine at the same rate, so that for the p-type material
For an n-type material (n0 >> p0) under low-level injection (n(t) <<no), the decayof minority carrier holes occurs with a time constant p0= (rno)-1 where p0 is alsoreferred to as the excess minority carrier lifetime. The recombination rate of themajority carrier electrons is the same as that of the minority carrier holes, so we have
The generation rates of excess carriers are not functions of electron or holeconcentrations. In general, the generation and recombination rates may befunctions of the space coordinates and time.
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00
' ))(()())((
nrn
tntnpdt
tndR
0
'' )(
npn
tnRR
0
'' )(
ppn
tpRR
Exercise1. Consider a semiconductor in which n0 = 1013
cm-3 and ni = 1010 cm-3. Assume that theexcess-carrier lifetime is 10-6 s. Determine theelectron-hole recombination rate if the excess-hole concentration is p = 5 x 1013 cm-3.
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0
'' )(
ppn
tpRR
Solution1. n-type semiconductor, low-injection so that
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13196
13
0
10510105' scmxxpR
p
Exercise2. A semiconductor in thermal equilibrium, has a
hole concentration of p0= 1016 cm-3 and anintrinsic concentration of ni = 1010 cm-3. Theminority carrier lifetime is 2 x 107 s.
(a) Determine the thermal-equilibrium recombinationrate of electrons.
(b) Determine the change in the recombination rate ofelectrons if an excess electron concentration of n =1012 cm-3 exists.
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0
'' )(
npn
tnRR
Solutiona) thermal-equilibrium recombination rate of
electrons
and
Then
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0
00
nn
nR
3416
210
0
2
0 1010
)10( cmpnn i
13107
4
0 105)102(
)10( scmx
xRn
Solutionb) the change in the recombination rate of
electrons if an excess electron concentration ofn = 1012 cm-3 exists
Therefore,
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13187
12
00 105
10210
scmxx
nRn
n
10180 105105 xxRRR nnn
1318105 scmxRn
Continuity Equations The continuity equations for electrons and holes are
shown as a differential volume element in which aone-dimensional hole particle flux is entering thedifferential element at r and is leaving 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:
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dxx
FxFdxxF px
pxpx .)()(
Fig. 6.2: Differential volume showing x component of the hole-particle flux.
Fpx+(x) Fpx
+(x+dx)
x x+dxdy
dz
Continuity Equations This equation is a Taylor expansion of Fpx
+(x + dx).where the differential length dx is small, so that onlythe first two terms in the expansion are significant.The net increase in the number of holes per unit timewithin the differential volume element due to the x-component of hole flux is given by:
If Fpx+(x) > Fpx
+ (x + dx), for example, there will bea net increase in the number of holes in thedifferential volume element with time.
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dxdydzx
FdydzdxxFxFdxdydz
tp px
pxpx
)]()([
Continuity Equations The generation rate and recombination rate of holes will also affect
the hole concentration in the differential volume. The net increase inthe number of holes per unit time in the differential volume elementis then given by
where p is the density of holes. The first term on the right side of the equation is the increase in the
number of holes per unit time due to the hole flux, the second termis the increase in the number of holes per unit time due to thegeneration of holes, and the last term is the decrease in the numberof holes per unit time due to the recombination of holes.
The recombination rate for holes is given by p/pt where ptincludes the thermal equilibrium carrier lifetime and the excess carrierlifetime.
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dxdydzPdxdydzgdxdydzx
Fdxdydz
tp
ptp
px
Continuity Equations If we divide both sides of the equation by the differential
volume dx.dy.dz, the net increase in the holeconcentration per unit time is :
The above equation is known as the continuity equationfor holes.
Similarly, the one-dimensional continuity equation forelectrons is given by:
where Fn- is the electron-particle flow, or flux, also given in units
of number of electrons/cm2-s.
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plp
p pgx
Ftp
nln
n ngx
Ftn
Time-Dependent Diffusion Equations The hole and electron current densities are given as:
Dividing the hole current density by (+e) and the electron current density by (-e) to find the flux of each particle:
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xpeDpEeJ ppp
xneDnEeJ nnn
xpDpEF
eJ
pppp
)(
xnDnEF
eJ
nnnn
)(
Time-Dependent Diffusion Equations Taking the divergence of the equations, and
substituting back into the continuity equations:
Keeping in mind that we are limiting ourselves to aone-dimensional analysis, we can expand thederivative of the product as
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ptppp
pgxpD
xpE
tp
2
2)(
ntnnn
ngxnD
xnE
tn
2
2)(
xEp
xpE
xpE
)(
Time-Dependent Diffusion Equations In a more generalized three-dimensional analysis, the
above equation would have to be replaced by a vector identity.
The above equations 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, the equations describe the space and time behavior of the excess carriers.
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tppg
xEp
xpE
xpD
ptppp
))()((2
2
tnng
xEn
xnE
xnD
ptnnn
))()((2
2
Time-Dependent Diffusion Equations The hole and electron concentrations are functions of
both the thermal equilibrium. The thermal equilibriumconcentrations, no and po. are not functions of time.
For the special case of a homogeneous semiconductor,no and po are also independent of the space coordinates,therefore:
Note that the Equations (6.29) and (6.30) contain termsinvolving the total concentrations, p and n, and termsinvolving only the excess concentrations, p and n.
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tppg
xEp
xpE
xpD
ptppp
)())()(()(
2
2
tnng
xEn
xnE
xnD
ptnnn
)())()(()(
2
2
Ambipolar Transport If a pulse of excess electrons and a pulse of excess holes are
created at a particular point in a semiconductor with anapplied electric field, the excess holes and electrons will tendto drift in opposite directions.
Electrons and holes are charged particles, any separation willinduce an internal electric field between the two sets ofparticles. This internal electric field will create a forceattracting the electrons and holes back toward each other.
The electric field is then composed of the externally appliedfield plus the induced internal field.
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intEEE app
Eapp
appn
x
---
+ + +
Fig. 6.3: The creation of an internal electric fieldas excess electrons and holes tend to separate.
Ambipolar Transport Since the internal E-field creates a force attracting the
electrons and holes, this E-field will hold the pulsesof excess electrons and excess holes together.
The negatively charged electrons and positivelycharged holes then will drift or diffuse together witha single effective mobility or diffusion coefficient.
This phenomenon is called ambipolar diffusion orambipolar transport:
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tnRg
xnE
xnD
)()()( '
2
2'
Ambipolar Transport
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tnRg
xnE
xnD
)()()( '
2
2'
pnnp
pn
pn
)('
pDnDpnDD
Dpn
pn
)('
The ambipolar diffusion coefficient D’ and the ambipolar mobilitycoefficient μ’ are defined from Einstein’s relation as follows:
Applications of Ambipolar Transport: Illustration of the behaviorof excess carriers in semiconductors, in PN junction diodes and otherdevices.
Exercise3. Consider an infinitely large, homogeneous n-
type semiconductor with a zero applied electricfield. Assume that, for t < 0, thesemiconductor is in thermal equilibrium andthat, for t≥0, a uniform generation rate existsin the crystal. Calculate the excess carrier concentration as a
function of time assuming the condition of lowinjection (where excess carrier concentration is muchlower than thermal equilibrium majority carrierconcentration n<<p0 (p type) or p<<n0 (n type) )
Wednesday, October 09, 2013Tennessee Technological University 31tnRg
xnE
xnD
)()()( '
2
2'
Solution The condition of a uniform generation rate and a
homogeneous semiconductor implies that:
Therefore, the ambipolar transport equationreduces to:
The solution of the differential is:
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0)()(2
2
xp
xp
tppg
p
)(
0
'
)1()( 00
' p
t
p egtp
tpRg
xpE
xpD
)()()( '
2
2'
Exercise4. Consider an n-type Silicon at T = 300K doped
at Nd = 2 x 1016cm-3. Assume that p0 = 10-7sand g' = 5 x 1021 cm-3s-l. Determine the timedependence of excess carriers in reaching asteady-state condition. Does low injectionapply?
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)1()( 00
' p
t
p egtp
Solution4. Using the equation:
Therefore,
Note 1: for t ∞, the steady state excess hole andelectron concentration of 5x1014cm-3 exists.
Note 2: p<<n0 therefore low injection is valid.
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)1()( 00
' p
t
p egtp
3101410721 ]1[105)1)(10()105()( 77
cmexextp
tt
Dielectric Relaxation Time Constant
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• Quasi-neutrality condition: the concentration of excess holes is balanced by an equal concentration of excess electrons.
• If a uniform concentration of holes p is suddenly injected into aportion of the surface of a semiconductor, we will instantly havea concentration of excess holes and a net positive charge densitythat is not balanced by a concentration of excess electrons.
How is charge neutrality achieved and how fast?• Dielectric Relaxation Time Constant
N typep holes
Fig. 6.4: The injection of a concentration of holes into a small region at the surface of an n-type semiconductor
Dielectric Relaxation Time Constant There are three defining equations to be considered. Poisson's equation is: (gradient of electric field equals net
charge density divided by permittivity)
From Ohm’s Law : (current density equals conductivitytimes electric field)
And from continuity equation, neglecting the effects ofgeneration and recombination:
The parameter is the net charge density and the initialvalue is given by e(p). We will assume that p isuniform over a short distance at the surface. Theparameter is the permittivity of the semiconductor.
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E.
EJ
tJ
.
Dielectric Relaxation Time Constant Taking the divergence of Ohm's law and using
Poisson's equation: Substituting the above equation into the continuity
equation,which can be rearranged into the first order equation:
Whose solution is:
Dielectric Relaxation Time Constant is:
Wednesday, October 09, 2013Tennessee Technological University 37
EJ ..
dtd
t
0
dtd
d
t
et
)0()(
d
Exercise5. Assume an n-type Silicon semiconductor with a
donor impurity concentration of Nd =1016cm-3
and mobility n = 1200cm2/V.sec . Calculate the dielectric relaxation time constant.
Wednesday, October 09, 2013Tennessee Technological University 38
Solution5. The conductivity is found as
The permittivity of Silicon is calculated as:
The dielectric relaxation time constant is then:
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11619 )(92.1)10)(1200)(10*6.1( cmNe dn
cmFr /)10*85.8)(7.11( 140
scm
cmFd
131
14
10*39.5)92.1(/)10*85.8)(7.11(
Picture Credits Semiconductor Physics and Devices, Donald Neaman, 4th
Edition, McGraw Hill Publications. B. Van Zeghbroeck, Principles of Semiconductor Devices,
Dept. of ECE, University of Colorado, Boulder, 2011http://ecee.colorado.edu/~bart/book/book/toc2.htm
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