chapter 6 nonequilibrium excess carrier in...
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W.K. Chen Electrophysics, NCTU 1
Chapter 6Nonequilibrium excess carrier in semiconductor
W.K. Chen Electrophysics, NCTU 2
Ambipolar transport
Excess electrons and excess holes do not move independently of each other. They diffuse, drift, and recombine with same effective diffusion coefficient, drift mobility and lifetime. This phenomenon is called ambipolar transport.
Two basic transport mechanismsDrift: movement of charged due to electric fields
Diffusion: the flow of charges due to density gradient
We implicitly assume the thermal equilibrium during the carrier transport is not substantially disturbed
W.K. Chen Electrophysics, NCTU 3
Outline
Carrier generation and recombination
Characteristics of excess carrier
Ambipolar transport
Quasi-Fermi energy levels
Excess carrier lifetime
Surface effect
Summary
W.K. Chen Electrophysics, NCTU 4
6.1 Carrier generation and recombination
Generation
Generation is the process whereby electrons and holes are created
Recombination
Recombination is the process whereby electrons and holes are annihilated
Any deviation from thermal equilibrium will tend to change the electron and hole concentration in a semiconductor.
(thermal exitation, photon pumping, carrier injection)
When the external excitation is removed, the concentrations of electron and hole in semiconductor will return eventually to their thermal-equilibrium values
W.K. Chen Electrophysics, NCTU 5
pono GG =
pono RR =
ponopono RRGG ===
6.1.1 The semiconductor in equilibriumThermal-equilibrium concentrations of electron and hole in conduction and valence bands are independent of time.
Since the net carrier concentrations are independent of time, the rate at which the electrons and holes are generated and the rate at which they recombine must be equal.
For direction band-to-band transition
Direct bandgap semiconductor
W.K. Chen Electrophysics, NCTU 6
6.1.2 Excess carrier generation and recombination
bandgap)(direct '' pn gg =
ppp
nnn
o
o
δ
δ
+=
+=
Excess electrons and excess holes
When external excitation is applied, an electron-hole pair is generated. The additional electrons and holes are called excess electrons and excess holes.
Generation rate of excess carriers
For direct band-to-band generation, the excess electrons and holes are created in pairs
2ioo npnnp =≠
W.K. Chen Electrophysics, NCTU 7
W.K. Chen Electrophysics, NCTU 8
Excess carriers recombination rate
In the direct band-to-band recombination, the excess electrons and holes recombine in pairs, so the recombination rate must be equal
bandgap)(direct ''pn RR =
Using the concept of collision model, we assume the rate of pair recombination obeys
)('
carriers excess of rateion Recombinat
mequilibriuat holes and electrons of rateion Recombinat
riumnonequilibunder holes and electrons of rateion Recombinat
2
2
iro
irooro
r
nnpRRR
npnR
npR
−=−=
==
=
α
αα
α
W.K. Chen Electrophysics, NCTU 9
)]()([)( 2 tptnn
dt
tdnir −=α
)()( )()( tpptptnntn oo δδ +=+=
))]())[((
))]())((([)( 2
tnpntn
tpptnnndt
tdn
oor
ooir
δδα
δδα
++−=
++−=
W.K. Chen Electrophysics, NCTU 10
noor ttp enentn τα δδδ /)0()0()( −− ==
orno pα
τ 1=
Low-level injectionLow-level injection
))(( )]()([)(
)( material type-p assume
2oir
oo
ptδntptnndt
tdn
np
<<−=
>>
Qα
injection) level-(low)()(
tnpdt
tndor δαδ
−=
The solution to this equation is an exponential decay from initial excess carrier concentration
Excess minority lifetime
W.K. Chen Electrophysics, NCTU 11
The recombination rate of excess carriers
injection) level-(low )(
)()(
'no
or
tntnp
dt
tndR
τδδαδ
=+=−=
nopn
tnRRR
τδ )(
''' ===
popn
tpRRR
τδ )(
''' ===
( p-type, low level injection)
( p-type, low level injection)
n-type material, no>>po
W.K. Chen Electrophysics, NCTU 12
6.2.1 Continuity equation
dxx
xFxFdxxF px
pxpx ⋅∂
∂+=+
+++ )(
)()(
dxdydzp
dxdydzgdydzdxxFxFdxdydzt
p
pnppxpx τ
−++−=∂∂ ++ )]()([
The net increase in the number of holes in the differential volume per unit time
From the calculus, the Taylor expansion gives
g R
Hole flux generation recombination
Flux in Flux out
dxdydzp
dxdydzgdxdydzx
xFdxdydz
t
p
pnp
px
τ−+
∂
∂−=
∂∂ + )(
dA
W.K. Chen Electrophysics, NCTU 13
s)-(holes/cm )( 2
ptp
px pg
x
xF
t
p
τ−+
∂
∂−=
∂∂ +
s)-/cm(electrons )( 2
ntn
n pg
x
xF
t
n
τ−+
∂∂
−=∂∂ +
Continuity equation for holes
Continuity equation for electrons
:pt
p
τThe recombination rate holes including thermal-equilibrium recombination and excess recombination
:ptτ The recombination lifetime which includs thermal-equilibrium carrier lifetime and excess carrier lifetime
g R
W.K. Chen Electrophysics, NCTU 14
6.2.2 Time-dependent diffusion equation
x
neDneJ
x
peDpeJ nnnppp ∂
∂+=
∂∂
−= E , E μμ
x
nDnF
e
J
x
pDpF
e
Jnnn
nppp
p
∂∂
−−==−∂
∂−==
+−+ E
)( , E
)(μμ
ptp
px pg
x
xF
t
p
τ−+
∂
∂−=
∂∂ + )(
ntn
n pg
x
xF
t
n
τ−+
∂∂
−=∂∂ + )(
dxx
xFxFdxxF px
pxpx ⋅∂
∂+=+
+++ )(
)()(
The current density in material is
By dividing current density the charge of each individual particle, we obtain particle flux
W.K. Chen Electrophysics, NCTU 15
ntnnn
ptppp
ng
x
nD
x
n
t
n
pg
x
pD
x
p
t
p
τμ
τμ
−+∂∂
+∂
∂+=
∂∂
−+∂∂
+∂
∂−=
∂∂
2
2
2
2
E)(
E)(
Thus the continuity equations can be rewritten as
xn
x
n
x
n
xp
x
p
x
p
∂∂
+∂∂
=∂
∂∂∂
+∂∂
=∂
∂ EE
E)( and
EE
E)(Q
t
nng
xn
x
n
x
nD
t
ppg
xp
x
p
x
pD
ntnnn
ptppp
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
+∂∂
+∂∂
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
+∂∂
−∂∂
τμ
τμ
EE
EE
2
2
2
2
W.K. Chen Electrophysics, NCTU 16
t
nng
xn
x
n
x
nD
t
ppg
xp
x
p
x
pD
ntnnn
ptppp
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
+∂
∂+
∂∂
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
+∂
∂−
∂∂
)(E)(E
)(
)(E)(E
)(
2
2
2
2
δτ
δμδ
δτ
δμδ
The thermal equilibrium concentrations, no and po, are not function of time. For homogeneous semiconductor, no and po are also independent of space coordinates
)(
)(
tppp
tnnn
o
o
δδ+=+=
Homogeneous semiconductor
W.K. Chen Electrophysics, NCTU 17
6.3 Ambipolar transport
appint EE <<
Ambipolar transport When excess carriers are generated, under external applied field the excess holes and electrons will tend to drift in opposite directions
However, because the electrons and holes are charged particles, any separation will induce an internal field between two sets of particles, creating a force attracting the electrons and holes back toward each other
Only a relatively small internal electric field is sufficient to keep the excess electrons and holes drifting and diffusing together
The excess electrons and holes do not move independently of each other, but they diffuse and drift together, with the same effective diffusion coefficient and with the same effective mobility. This phenomenon is called ambipolar transport.
appintapp EEEE ≈−=
0E)(
E intint ≈
∂∂
=−
=⋅∇t
npe
sεδδ
W.K. Chen Electrophysics, NCTU 18t
nRg
xn
x
n
x
nD
t
nRg
xp
x
n
x
nD
nn
pp
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
+∂
∂+
∂∂
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
+∂
∂−
∂∂
)(E)(E
)(
)(E)(E
)(
2
2
2
2
δδμδ
δδμδ
⎪⎪
⎩
⎪⎪
⎨
⎧
=
====
≡=
pn
Rp
Rn
R
ggg
ptp
ntn
pn
δδ
ττ
For ambipolar transport, the excess electrons and excess holes generate and recombine together
ptnt ττ ≠
t
nng
xn
x
n
x
nD
t
ppg
xp
x
p
x
pD
ntnnn
ptppp
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
+∂
∂+
∂∂
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
+∂
∂−
∂∂
)(E)(E
)(
)(E)(E
)(
2
2
2
2
δτ
δμδ
δτ
δμδ
ppp
nnn
o
o
δδ+=+=
W.K. Chen Electrophysics, NCTU 19
t
npn
Rgpnx
nnp
x
npDnD
pn
pnnpnppn
∂∂
+=
−++⎟⎠⎞
⎜⎝⎛
∂∂
−−∂
∂+
)()(
))(()(
E))(()(
)(2
2
δμμ
μμδμμδμμ
t
nRg
x
n
x
nD
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
−∂
∂ )()(
)(E'
)('
2
2 δδμδ
)('
pn
pDnDD
pn
nppn
μμμμ
+
+=
)(
))(('
pn
np
pn
np
μμμμ
μ+
−=
⇒== kT
e
DD p
p
n
nμμ
QpDnD
pnDDD
pn
pn
+
+=
)(' (ambipolar diffusion coefficient)
(ambipolar mobilty)
ambipolar transport equation
W.K. Chen Electrophysics, NCTU 20
t
nng
xn
x
n
x
nD
ntnnn ∂
∂=−+⎟
⎠⎞
⎜⎝⎛
∂∂
+∂
∂+
∂∂ )(E)(
E)(
2
2 δτ
δμδ
t
nRg
x
n
x
nD
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
−∂
∂ )()(
)(E'
)('
2
2 δδμδ ambipolar transport equation
continuity equation for electrons
t
ppg
xp
x
p
x
pD
ptppp ∂
∂=−+⎟
⎠⎞
⎜⎝⎛
∂∂
+∂
∂−
∂∂ )(E)(
E)(
2
2 δτ
δμδ continuity equation for holes
W.K. Chen Electrophysics, NCTU 21
6.3.2 Ambipolar transport under low injection
pon
opn
opon
oopn DnD
nDD
ppDnnD
ppnnDDD =≈
+++
+++=
)()(
)]()[('
δδδδ
injection) (low
typenFor
o
o
npn
pn o
<<=>>−δδ
t
pRg
x
p
x
pD
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
−∂
∂ )()(
)(E'
)('
2
2 δδμδ
)(
))(('
pn
np
pn
np
μμμμ
μ+
−=
pDnD
pnDDD
pn
pn
+
+=
)('
pn
np
pn
np
n
n
pn
npμ
μμμ
μμμμ
μ −=−
≈+
−=
)(
))((
][
))[('
Ambipolar transport
W.K. Chen Electrophysics, NCTU 22
)()( ''ppoppopp RRgGRgRg +−+=−=−
popo RG =Qp
ppp
pgRgRg
τδ
−=−=− '''
excess hole generation rate
Thermal-equilibrium hole generation rate
Thermal-equilibrium hole recombination rate
excess hole recombination rate
For n-type,Under low injection, the concentration of majority carriers electrons will be essentially constant. Then the probability per unit time of a minority carrier hole encountering a majority carrier electron will remain almost a constant
ppt ττ =Q (minority carrier hole lifetime)
So the net generation rate,
W.K. Chen Electrophysics, NCTU 23
t
ppg
x
p
x
pD
popp ∂
∂=−+
∂∂
−∂
∂ )('
)(E
)(2
2 δτδδμδ n-type, homogeneous and low-
injection ambipolar transport
t
pRg
x
p
x
pD
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
−∂
∂ )()(
)(E'
)('
2
2 δδμδ
pDD ='
pμμ −=' pppp
pgRgRg
τδ
−=−=− '''
For ambipolar transport , the transport and recombination parameters are governed by minority carriers
The excess majority carriers diffuse and drift with the excess minority carriers; thus the behavior of the excess majority carriers is determined by the minority carrier parameters
W.K. Chen Electrophysics, NCTU 24
injection) (low
typepFor
o
o
pnp
np o
<<=>>−δδ
For p-type semiconductor
t
nRg
x
n
x
nD
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
−∂
∂ )()(
)(E'
)('
2
2 δδμδ
)(
))(('
pn
np
pn
np
μμμμ
μ+
−=
pDnD
pnDDD
pn
pn
+
+=
)('
p-type, homogeneous ambipolar transport
nop
opn
opon
oopn DpD
pDD
ppDnnD
ppnnDDD =≈
+++
+++=
)()(
)]()[('
δδδδ
np
np
pn
np
p
p
pn
npμ
μμμ
μμμμ
μ =≈+
−=
)(
))((
][
))[('
W.K. Chen Electrophysics, NCTU 25
)()( ''nnonnonn RRgGRgRg +−+=−=−
nono RG =Qn
nnn
ngRgRg
τδ
−=−=− '''
excess electron generation rate
Thermal-equilibrium electron generation rate
Thermal-equilibrium hole recombination rate
excess hole recombination rate
For p-type,Under low injection, the concentration of majority carriers holes will be essentially constant. Then the probability per unit time of a minority carrier electron encountering a majority carrier hole will remain almost a constant
nnt ττ =Q (minority carrier electron lifetime)
So the net generation rate,
W.K. Chen Electrophysics, NCTU 26
n
nDD
μμ ==
'
'
t
nRg
x
n
x
nD
∂∂
=−+⎟⎠⎞
⎜⎝⎛
∂∂
−∂
∂ )()(
)(E'
)('
2
2 δδμδ
t
nng
x
n
x
nD
nonn ∂
∂=−+
∂∂
+∂
∂ )()(E
)( '2
2 δτδδμδ p-type, homogeneous and low-
injection ambipolar transport
nnnn
ngRgRg
τδ
−=−=− '''
For ambipolar transport , the transport and recombination parameters are governed by minority carriers
The excess majority carriers diffuse and drift with the excess minority carriers; thus the behavior of the excess majority carriers is determined by the minority carrier parameters
W.K. Chen Electrophysics, NCTU 27
6.3.3 Applications of the ambipolar transport equation
W.K. Chen Electrophysics, NCTU 28
Example 6.1
timeoffunction a asion concentratcarrier excess theCalculate
0for 0g'
carriers excess ofion concentrat uniform 0,At
conditioninjection -lowunder
field electrical applied zero
tor semiconduc type-n Homogenous
⇒>=
=t
t
Solution:
t
pp
x
pD
pop ∂
∂=−
∂∂ )()(
2
2 δτδδ
Uniform excess carriers
t
ppg
x
p
x
pD
popp ∂
∂=−+
∂∂
−∂
∂ )('
)(E
)(2
2 δτδδμδ
E=0 g’=0
W.K. Chen Electrophysics, NCTU 29
po
p
t
p
τδδ
−=∂
∂ )(
poteptp τδδ /)0()( −=
potentn τδδ /)0()( −=
From the charge neutrality condition, the excess majority electron concentration is given by
W.K. Chen Electrophysics, NCTU 30
Example 6.2
timeoffunction a asion concentratcarrier excess theCalculate
carriers excess of generation uniform 0,At
conditioninjection -lowunder
field electrical applied zero
tor semiconduc type-n Homogenous
⇒>t
t
ppg
x
p
x
pD
popp ∂
∂=−+
∂∂
−∂
∂ )('
)(E
)(2
2 δτδδμδ
Solution:
0')(
)(
' =−+∂
∂⇒
∂∂
=− gp
t
p
t
ppg
popo τδδδ
τδ
Uniform generation
)1(')( / potpo egtp ττδ −−=
' )( ,0)(
state,steady At
gtpt
tppoτδδ
=∞==∂
∂
W.K. Chen Electrophysics, NCTU 31
Example 6.3
offunction a asion concentratcarrier excess statesteady theCalculate
directions and bith thein diffuse then
condition
injection -lowunder only, 0at generated being are carriers excess the
field electrical applied zero
tor semiconduc type-p Homogenous
x
xx
x
⇒−+
=
Solution:
t
nng
x
n
x
nD
nonn ∂
∂=−+
∂∂
+∂
∂ )()(E
)( '2
2 δτδδμδ
E=0 Steady state
W.K. Chen Electrophysics, NCTU 32
⎪⎩
⎪⎨⎧
≤=
≥=⇒
+=
+
−
−−
0 )0()(
0 )0()(
)(
/
/
//
xenxn
xenxn
BeAexn
Lnx
Lnx
LnxLnx
δδ
δδ
δ
0)(
0At
0)(
0,At
2
2
'2
2
=−∂
∂≠
=−+∂
∂=
non
non
n
x
nDx
ng
x
nDx
τδδ
τδδ
0)(
2
2
=−∂
∂
nonD
n
x
n
τδδ
nonn DL τ=2
Minority carrier diffusion length
The general solution
p-typelog scale
W.K. Chen Electrophysics, NCTU 33
Example 6.4
tx
tg'
x
t
xi
and offunction a asion concentratcarrier excess theCalculate
0for 0
0 and
0at ously instantane generated is pairs hole-electron of numbers finite
direction- n the E field electrical appliedconstant
tor semiconduc type-n Homogenous
o
⇒>=
==
Solution:
t
ppg
x
p
x
pD
popp ∂
∂=−+
∂∂
−∂
∂ )('
)(E
)(2
2 δτδδμδ
W.K. Chen Electrophysics, NCTU 34
t
pp
x
p
x
pD
popp ∂
∂=−
∂∂
−∂
∂ )()(E
)(o2
2 δτδδμδ
potetxptxp τδ /),('),( −=
t
txp
x
txp
x
txpD pp ∂
∂=
∂∂
−∂
∂ ),('),('E
),('o2
2
μ
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−=
tD
tx
tDtxp
p
p
p 4
)E(exp
4
1),('
2oμ
π
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−==
−−
tD
tx
tD
eetxptxp
p
p
p
tt
po
po
4
)E(exp
4),('),(
2o
// μ
πδ
ττ
W.K. Chen Electrophysics, NCTU 35
Zero applied field constant applied field
W.K. Chen Electrophysics, NCTU 36
6.3.4 Dielectric relaxation time constantHow is the charge neutrality achieved and how fast ?
In previous analysis, we have assume a quasi-neutrality condition exists-that is, the concentration of excess holes is balanced by an equal concentration of excess electrons
Suppose that we have a situation in which a uniform concentration of δp is suddenly injected into a portion of the surface of a semiconductor. We will have instantly have a concentration of excess holes and a net positive charge density δp that is not balance by a concentration of excess electrons. How is the charge neutrality achieved and how fast ?
W.K. Chen Electrophysics, NCTU 37
ερ
=⋅∇ E
tJ
∂∂
−=⋅∇ρ
Eσ=J
Poisson’s equation
Continuity equation(neglecting the effects of generation and recombination)
ttJ
∂∂
−==⋅∇⇒∂∂
−=⋅∇ρ
ερσσρ
E)(
0=+∂∂ ρ
εσρ
tdtet τρρ /)0()( −=
σετ =d
Dielectric relaxation time constant
pe δρ )(+=
( the time constant is related to dielectric constant)
W.K. Chen Electrophysics, NCTU 38
Example 6.5 Dielectric relaxation time constant
constant timerelaxation dielectric theCalculate
cm 10
tor semiconduc type-n Homogenous316
⇒= −
dN
Solution:
ps 0.539 )10)(1200)(106.1(
)1085.8)(7.11(1619
14
=×
×=== −
−
dn
ord Neμ
εεσετ
In approximately four time constants (2 ps), the net charge density is essentially zero
The relaxation process occur very quickly (τd ≈0.5 ps) compared to the normal excess carrier lifetime (τ =0.1 μs).
That is the reason why the continuity equation in calculating relaxation time does not contain any generation or recombination terms.
W.K. Chen Electrophysics, NCTU 39
6.3.5 Haynes-Shockley experiment
Zero applied field constant applied field
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−==
−−
tD
tx
tD
eetxptxp
p
p
p
tt
po
po
4
)E(exp
4),('),(
2o
// μ
πδ
ττ
The Haynes-Shockley experiment was one of the first experiment to actually measure excess-carrier behavior, which can determine
The minority carrier lifetime
The minority carrier diffusion coefficient
The minority carrier lifetime
W.K. Chen Electrophysics, NCTU 40
Haynes-Shockley experiment
Excess-carrier pulse are effectively injected at contact A
Contact B is rectifying contactand is under reverse bias (do not perturb the electric field)
A fraction of excess carriers will be collected by contact B
The collect carriers will generate an output voltage Vo when flow through output resistance R2
W.K. Chen Electrophysics, NCTU 41
The idealized excess minority carrier (hole) pulse is injected at contact A at time t=0
The excess carriers (holes) will drift along the semiconductor producing an output voltage as a function of time
The peak of pulse will arrive at contact B at time to
During the time period, the occurs diffusion and recombination
0 0 =−⇒=− tExtx opp μυ
oop t
d
E=μ
W.K. Chen Electrophysics, NCTU 42
At t=to, the peak of pulse reaches contact B. where times t1 and t2, the magnitude of the excess concentration is e-1
If the time difference between t1
and t2, is not too large, the prefactor do not change appreciably during this time
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−==
−−
tD
tx
tD
eetxptxp
p
p
p
tt
po
po
4
)E(exp
4),('),(
2o
// μ
πδ
ττ
212
o or ,4)E( ttttDtx pp ==− μ
12
22o ,
16
)()E(ttt
t
tD
o
pp −=Δ
Δ=
μ
From the broadened pulse width, we can obtain diffusion coefficient
diffusion coefficient
W.K. Chen Electrophysics, NCTU 43
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟
⎟⎠
⎞⎜⎜⎝
⎛ −=
−=
pooppo
op
po
o dK
dK
tKS
τμτμ
τ Eexp
)E/(exp)exp(
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−==
−−
tD
tx
tD
eetxptxp
p
p
p
tt
po
po
4
)E(exp
4),('),(
2o
// μ
πδ
ττ
The area S undrer the curve is proportional to the number of excess holes that have not recombined with majority carrier electrons
By varying the electric field, the area under the curve will change
A plot of ln(S) as a function of (d/μpEo) will yield a straight line whose slope is (1/τpo)
W.K. Chen Electrophysics, NCTU 44
⎟⎟⎠
⎞⎜⎜⎝
⎛ −⋅+=
oppo
dKS
E
1)ln()ln(
μτ
)ln(S
opEμ
dy =
poτ1
We can determine the minority carrier lifetime
W.K. Chen Electrophysics, NCTU 45
6.4 Quasi-Fermi energy levelsAt thermal equilibrium,
the electron and hole concentrations are functions of the Fermi-level.
The Fermi level remains constant throughout the entire material
The carrier concentrations is exponentially determined by the Fermi-level
2
exp exp
ioo
ffio
fifio
npn
kT
EEnp
kT
EEnn i
=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=
W.K. Chen Electrophysics, NCTU 46
At non-thermal equilibriumIf excess carriers are created, thermal equilibrium no longer exists and Fermi energy is strictly no longer defined
We may define quasi-Fermi levels for electrons and holes to relate the concentrations for non-equilibrium semiconductor in the same form of equation as that in thermal equilibrium
In such a way, the quasi-Fermi levels for electrons and holes specified for non-thermal equilibrium conditions do not hold constants over the entire material
2
exp
exp
i
fpfioo
fifnio
nnp
kT
EEnppp
kT
EEnnnn
i
≠
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=+=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=+=
δ
δ
W.K. Chen Electrophysics, NCTU 47
Example 6.6 Quasi-Fermi level
energy Fermi-quasi theCalculate
cm 10pn rium,nonequilibIn
cm 10 and cm 10 ,cm 10
K 300Tat tor semiconduc type-n Homogenous
313
35310315
⇒==
===
=
−
−−−
δδoio pnn
Solution:
eV 2982.0)ln( exp ==−⇒⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
i
ofif
fifio n
nkTEE
kT
EEnn
eV 179.0)ln( exp
eV 2984.0)ln( exp
=+
=−⇒⎟⎟⎠
⎞⎜⎜⎝
⎛ −=+=
=+
=−⇒⎟⎟⎠
⎞⎜⎜⎝
⎛ −=+=
i
ofpfi
fpfioo
i
ofifn
fifnio
n
ppkTEE
kT
EEnppp
n
nnkTEE
kT
EEnnnn
iδδ
δδ
W.K. Chen Electrophysics, NCTU 48
At non-thermal equilibriumSince the majority carrier electron concentration does not change significantly for low-injection condition, the quasi-Fermi level for electrons here is not much different from the thermal-equilibrium Fermi level.
The quasi-Fermi level for minority carrier holes is significantly the Fermi level and illustrate the fact that we have deviate from the thermal equilibrium significantly
31335
315315
cm 10cm 10
cm 1001.1cm 100.1 :−−
−−
→=
×→×
op
n
W.K. Chen Electrophysics, NCTU 49
6.5 Excess-carrier lifetimeIn a perfect semiconductor
The electronic energy states do not exist within the forbidden bandgap
In a real semiconductorDefects (traps) occur within the crystal, creating discrete electronic energy states or impurity energy bands within the forbidden-energy band
These defect energy states may be the dominant effect in determining the mean carrier lifetime in the real semiconductor
W.K. Chen Electrophysics, NCTU 50
6.5.1 Shockley-Read-Hall theory of recombinationRecombination center
A trap within the forbidden bandgap may act as a recombination center, capturing both electrons and holes with almost equal probability.
Shockley-Read-Hall theoryAssume a single recombination center exists at an energy Et within the bandgap.
If the trap is an acceptor-like trap; that is , it is negatively charged when it contains an electron and is neutral when it does not contain an electron.
negatively charged electronNeutral empty state
W.K. Chen Electrophysics, NCTU 51
Shockley-Read-Hall theoryThere are four basic processes in SRH theory
Process 1: The capture of an electron from the CB by a neutral empty trap
Process 2 The emission of an electron back into the CB
Process 3 The capture of a hole from the VB by a trap containing an electron
Process 4 The emission of a hole from a neutral trap into the VB
W.K. Chen Electrophysics, NCTU 52
Process 1: electron capture
nEfNCR tFtncn ⋅−⋅⋅= ))](1([Electron capture rate
⎥⎦
⎤⎢⎣
⎡ −+
=
kT
EEEf
fttF
exp1
1)(
Process 2: electron emission
)]([ tFtnen EfNER ⋅⋅=
(at thermal equilibrium)encn RR =
)]([))](1([ tFotntFotn EfNEnEfNC ⋅⋅=⋅−⋅⋅
Is there relationship between capture constant and emission constant?
W.K. Chen Electrophysics, NCTU 53
ntc
cn
nfoc
cfot
n
ntFo
nntFo
tFon
CkT
EENE
EkT
EEN
kT
EEC
EnEf
CEEf
nEfC
⎥⎦⎤
⎢⎣⎡ −−=
=⎥⎦
⎤⎢⎣
⎡ −−⋅−⎥
⎦
⎤⎢⎣
⎡ −+⋅
=⋅−⋅⇒=⋅−
⋅
exp
exp)1exp1(
)1)(
1(
)(
))(1(
nn CnE '= ⎥⎦⎤
⎢⎣⎡ −
=⎥⎦⎤
⎢⎣⎡ −−=
kT
EEn
kT
EENn it
itc
c expexp'
The relation between emission constant and capture constant is valid at all conditions including when Fermi level is right located at trap energy, in which the trap is the dominant process to provide the free electrons in the conduction band
The electron emission rate is increased exponentially as the trap energy level closes to the conduction band
W.K. Chen Electrophysics, NCTU 54
pEfNCR tFtpcp ⋅⋅⋅= ))((
⎥⎦
⎤⎢⎣
⎡ −+
=
kT
EEEf
fttF
exp1
1)(
))](1([ tFtpep EfNER −⋅⋅=
Process 3: hole capture rate
Process 4: hole emission rate
epcp RR =
⎥⎦⎤
⎢⎣⎡ −
=
⎥⎦
⎤⎢⎣
⎡ −⋅=−⎥
⎦
⎤⎢⎣
⎡ −+⋅=⎥
⎦
⎤⎢⎣
⎡ −⋅
−⋅=⋅
−⋅⋅=⋅⋅⋅
kT
EENCE
kT
EEE
kT
EEE
kT
EENC
EfEpC
EfNEpEfNC
tpp
ftp
ftp
fp
tFpp
tFtptFtp
υυ
υυ
exp
exp)1exp1(exp
)1)(
1(
))](1([))((
(at thermal equilibrium)
'pCE pp = ⎥⎦⎤
⎢⎣⎡ −
=⎥⎦⎤
⎢⎣⎡ −
=kT
EEn
kT
EENp ti
it expexp' υ
υ
W.K. Chen Electrophysics, NCTU 55
Capture constantsThe electron capture constant comes from a electron with velocity υth must come within a cross-sectional area σcn of a trap to be captured and thus sweep out an effective trap volume per second. The same is for hole capture constant.
The capture constants for electrons and holes proportion to respective cross-sectional area
Emission constantsThe electron emission rate from a trap is the product of electron capture rate and free carrier concentration when Ef=Et.
The electron emission rate is increased exponentially as the trap energy level closes to the conduction band
The hole emission rate is increased exponentially as the trap energy level closes to the valence band
'pCE pp =
nn CnE '=
cnthnC συ= cpthpC συ=
⎥⎦⎤
⎢⎣⎡ −−=
kT
EENn tc
c exp' ⎥⎦⎤
⎢⎣⎡ −
=kT
EENp tυ
υ exp'
W.K. Chen Electrophysics, NCTU 56
nEfNCR tFtncn ⋅−⋅⋅= ))](1([
)]([ tFtnen EfNER ⋅⋅=
pEfNCR tFtpcp ⋅⋅⋅= ))((
))](1([ tFtpep EfNER −⋅⋅=
W.K. Chen Electrophysics, NCTU 57
Under non-equilibrium condition
)]('))(1([
)]([))](1([
tFtFtnn
tFtntFtnn
encnn
EfnEfnNCR
EfNEnEfNCR
RRR
−−⋅=⋅⋅−⋅−⋅⋅=
−=
Net electron capture rate at acceptor trap
)]('))(1([ tFtFtnn EfnEfnNCR −−⋅=
))](1('))([ tFtFtpp EfpEpfNCR −−⋅=
⎥⎦⎤
⎢⎣⎡ −−=
kT
EENn tc
c exp'
⎥⎦⎤
⎢⎣⎡ −−=
kT
EENp t υ
υ exp'
)'()'(
')(
ppCnnC
pCnCEf
pn
pntF +++
+=
Net hole capture rate at acceptor trap (Process 3 & 4)
At steady state, pn RR =
W.K. Chen Electrophysics, NCTU 58
We may note that
⎥⎦⎤
⎢⎣⎡ −−=⎥⎦
⎤⎢⎣⎡ −−⋅⎥⎦
⎤⎢⎣⎡ −−=
kT
EENN
kT
EEN
kT
EENpn c
cttc
cυ
υυ
υ expexpexp''
2'' inpn =
RppCnnC
nnpNCCRR
pn
itpnpn ≡
+++
−==
)'()'(
)( 2
For SRH-dominated recombination process, the recombination rate of excess carriers is
)'()'(
)( 2
ppCnnC
nnpNCCnR
pn
itpn
+++
−==
τδ
nRpR
W.K. Chen Electrophysics, NCTU 59
SRH under low injection
trap)(deep ' ,'
injection) (low
type)-(n
pnnn
pn
pn
oo
o
oo
>>>>>>>>
δ
Case 1: n-type semiconductor with deep trap energy at low injection
pNCnC
pnNCC
ppCnnC
nnpNCCnR tp
n
tpn
pn
itpn δδ
τδ
=≈+++
−==
)'()'(
)( 2
pNCn
R tp δτδ
==
For n-type semiconductor with deep trap energy at low injection, the net recombination rate is limited by the hole capture process during SRH recombination
)3( kTEE tf >−
W.K. Chen Electrophysics, NCTU 60
For n-type semiconductor with deep trap energy at low injection, the net recombination rate is limited by the hole capture process during SRH recombination
The recombination is related to the mean minority carrier lifetime
nRpR
fEtEpo
tp
ppNC
nR
τδδ
τδ
===
tppo NC
1=τ
If the trap concentration increases, the probability of excess carrier recombination increases; thus the excess minority carrier lifetime decreases
n-typelow injection
W.K. Chen Electrophysics, NCTU 61
Case 2: p-type semiconductor with deep trap energy at low injection
trap)(deep ' ,'
injection) (low
type)-(p
nppp
np
np
oo
o
oo
>>>>>>>>
δ
tnno NC
1=τ
nRpR
fEtE
+
notn
nnNC
pR
τδδ
τδ
=== p-typelow injection
W.K. Chen Electrophysics, NCTU 62
)'()'(
)( 2
ppCnnC
nnpNCCnR
pn
itpn
+++
−==
τδ
)'()'(
)( 2
ppnn
nnpnR
nopo
i
+++−
==τττ
δ
SRH recombination process
Case 3: n/p-type semiconductor with deep trap energy
General SRH recombination rate
W.K. Chen Electrophysics, NCTU 63
Example 6.7 Intrinsic semiconductor
Solution: )'()'(
)( 2
ppnn
nnpnR
nopo
i
+++−
==τττ
δ
))(2(
)(2
)()(
))()( 22
nopoi
i
iinoiipo
iii
nn
nnn
nnnnnn
nnnnnR
ττδδδ
δτδτδδ
+++
=+++++
−+⋅+=
injection lowunder
''
levelenergy intrinsic toequalenergy aat located is trapdeep Assume
, ,
i
ooioo
npn
δpppδnnnnpn
==
+=+===
τδ
ττδ nn
Rnopo
=+
=)(
Under low injection inn <<δ
The excess-carrier lifetime increases as we change from an extrinsic to an intrinsic semiconductor
nopo τττ +=
W.K. Chen Electrophysics, NCTU 64
6.6 Surface effectsSurface states
At the surface, the semiconductor is abruptly terminated. This disruption of periodic potential function results in allowed electronic energy states within the energy bandgap, which is called surface states
Since the density of traps at the surface is larger than the bulk, the excess minority carrier lifetime at the surface will be smaller than the corresponding lifetime in the bulk material
For n-type material
bulk) (in the po
B
poB
ppR
τδ
τδ
==
surface) (at the pos
ss
pR
τδ
=
popos ττ <<
W.K. Chen Electrophysics, NCTU 65
Assume excess carriers are uniformly generated throughout the entire semiconductor material
At steady state, the generation rate is equal to recombination rate at a given position either in the bulk or at the surface
surface) (at the
bulk) (in the
pos
ss
po
BB
pRG
pRG
τδ
τδ
==
==
Bspopos pp δδττ <⇒<
pumping) (uniform sB RRG ==
For the case of uniform pumping
W.K. Chen Electrophysics, NCTU 66
Example 6.8 Surface recombination
Solution:
ondistributiion concentratcarrier -excess stateSteady
pumping uniform and 0E Assume
/scm 10 s10 ,s10 ,cm10tor semiconduc type-n 270
6-314
⇒=
==== −−psppopB Dττδ
1-136
714 cm 10
)10(
)10()10()(
)0( surfaceat ion concentratcarrier -Excess
===⇒==
=
−
−
po
posBs
pos
s
po
B pppp
G
x
ττ
δδτδ
τδ
Q
pumping) (uniform 0')(
0)E(at equation ansport Carrier tr
2
2
=−+
=
pop
pg
dt
pdD
τδδ
1-3-206
14
s-cm 1010
10' where === −
po
Bpg
τδ
W.K. Chen Electrophysics, NCTU 67
pp LxLxpo BeAegxp //')( −++= τδ
pumping) (uniform ')( / pLxpo Begxp −+= τδ
13314314
314
109 cm 10cm 10)0(
cm 10')(
:B.C.
×−=⇒=+==
===+∞−−
−
BBpp
gpp
s
poB
δδ
τδδ
)0.9-(110)( /14 pLxs epxp −== δδ
m31.6cm 31600.0)(10)(10 where -6 μτ ==== popp DL
W.K. Chen Electrophysics, NCTU 68
6.6.2 Surface recombination velocityA gradient in the excess-carrier concentration existing near the surface leads to a diffusion of excess carriers from the bulk region toward the surface where they recombine.
))0(()0()(
Flux osurface
p ppspsdx
pdD −== δδ
Surface recombination rate
W.K. Chen Electrophysics, NCTU 69
For the case of uniform pumping
pLxpo Begxp /')( −+= τδ
0')(
0)E(at equation ansport Carrier tr
2
2
=−+
=
pop
pg
dt
pdD
τδδ
)'
1(')( / pLx
pp
poppo e
sLD
Lsggxp −
+−=
ττδ
sLD
sgB
L
B
dx
pd
dx
pd
Bgp
pp
po
pxsurface
po
+
−=⇒
−==
+=
=
)/(
'
)()(
')0(
0
τ
δδ
τδQ
Uniform pumpingE=0Steady state
Uniform pumpingE=0Steady state
W.K. Chen Electrophysics, NCTU 70
x
)(xpδ
0=s
Surface recombination velocity is sensitive to the surface conditionsFor sand-blasted surfaces, the typical values of s may be as high as 105 cm/s
For clean etched surfaces, this value may be as low as 10 to 100 cm/s states
pumping) (uniform cm/s 1)0(
'⎟⎟⎠
⎞⎜⎜⎝
⎛−=
p
g
L
Ds po
p
p
δτ
0if ')( == sgxp poτδ
W.K. Chen Electrophysics, NCTU 71
Example 6.10 Surface recombination velocity
Solution:
tyion velocirecombinat surface theDetermine
m 10(0) and m 6.13 /s,cm 10,cm 10
s10 ,s10 ,cm10tor semiconduc type-n
oumping)(uniform 6.8 Examplein case For the
3-132314
70
63-14
⇒
====
===−
−−
cδp L Dg'τ pppo
sppopB
μ
ττδ
cm/s 1)0(
'⎟⎟⎠
⎞⎜⎜⎝
⎛−=
p
g
L
Ds po
p
p
δτ
cm/s 1085.2110
10
106.31
10 413
14
4×=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
×= −s
⇒ pumping uniform
W.K. Chen Electrophysics, NCTU 72
Short briefs for surface recombination
bulk) (in the po
B
poB
ppR
τδ
τδ
==
surface) (at the pos
ss
pR
τδ
=
))0(()0()(
Flux osurface
p ppspsdx
pdD −== δδ
W.K. Chen Electrophysics, NCTU 73
W.K. Chen Electrophysics, NCTU 74
Figure 6.19 Figure for problems 6.18 and 6.20
W.K. Chen Electrophysics, NCTU 75
Figure 6.20 Figure for Problem 6.25
W.K. Chen Electrophysics, NCTU 76
Figure 6.21 Figure for Problem 6.38
W.K. Chen Electrophysics, NCTU 77
Figure 6.22 Figure for Problem 6.39
W.K. Chen Electrophysics, NCTU 78
Figure 6.23 Figure for Problem 6.40
W.K. Chen Electrophysics, NCTU 79
Figure 6.24 Figure for Problem 6.41