Chapter 6
Nonequilibrium excess carrier
in semiconductor
W.K. Chen
„
„
1
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 mechanisms
‰
‰
„
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Drift: 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
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2
Outline
„
„
„
„
„
„
„
Carrier generation and recombination
Characteristics of excess carrier
Ambipolar transport
Quasi-Fermi energy levels
Excess carrier lifetime
Surface effect
Summary
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3
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
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4
6.1.1 The semiconductor in equilibrium
„
„
Thermal-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
Gno = G po
Rno = R po
Gno = G po = Rno = R po
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6.1.2 Excess carrier generation and recombination
„
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
g 'n = g ' p (direct bandgap)
n = no + δn
p = po + δp
np ≠ no po = ni2
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W.K. Chen
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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
Rn' = R p' (direct bandgap)
Using the concept of collision model, we assume the rate of pair
recombination obeys
Recombination rate of electrons and holes under nonequilibrium
R = α r np
Recombination rate of electrons and holes at equilibrium
Ro = α r no po = α r ni2
Recombination rate of excess carriers
R ' = R − Ro = α r (np − ni2 )
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dn(t )
= α r [ni2 − n(t ) p (t )]
dt
n(t ) = no + δn(t )
p (t ) = po + δp (t )
dn(t )
= α r [ni2 − (no + δn(t ))( po + δp (t ))]
dt
= −α rδn(t )[(no + po ) + δn(t ))]
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Low-level injection
Low-level injection
assume p - type material ( p o >> no )
dn (t )
= α r [ ni2 − n (t ) p (t )] (Q δn (t ) << p o )
dt
dδn(t )
= −α r poδn(t ) (low - level injection)
dt
The solution to this equation is an exponential decay from initial excess
carrier concentration
δn(t ) = δn(0)e −α
r
po t
= δn(0)e − t /τ no
Excess minority lifetime τ =
no
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1
α r po
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The recombination rate of excess carriers
R' = −
dδn(t )
δn(t )
= +α r poδn(t ) =
(low - level injection)
dt
τ no
Rn' = R p' = R ' =
δn(t )
τ no
( p-type, low level injection)
n-type material, no>>po
Rn' = R p' = R ' =
W.K. Chen
δp (t )
τ po
( p-type, low level injection)
11
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6.2.1 Continuity equation
From the calculus, the Taylor expansion gives
Fpx+ ( x + dx) = Fpx+ ( x) +
+
px
∂F ( x)
∂x
g
R
⋅ dx
The net increase in the number of holes in the differential volume per unit time
Flux in
Flux out
dA
∂p
p
dxdydz = [ Fpx+ ( x) − Fpx+ ( x + dx)]dydz + g p dxdydz −
dxdydz
τ pn
∂t
Hole flux
generation recombination
∂Fpx+ ( x)
p
∂p
dxdydz + g p dxdydz −
dxdydz
dxdydz = −
τ pn
∂x
∂t
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p
: The recombination rate holes including
τpt thermal-equilibrium recombination and
g
excess recombination
R
τpt : The recombination lifetime which
includs thermal-equilibrium carrier
lifetime and excess carrier lifetime
∂Fpx+ ( x)
∂p
p
=−
+ gp −
(holes/cm2 - s)
Continuity equation for holes
∂t
∂x
τ pt
+
Continuity equation for electrons ∂n = − ∂Fn ( x) + g − p (electrons/cm 2 - s)
n
∂t
W.K. Chen
τ nt
∂x
13
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6.2.2 Time-dependent diffusion equation
∂Fpx+ ( x)
∂p
p
=−
+ gp −
∂t
∂x
τ pt
+
px
+
px
F ( x + dx) = F ( x) +
∂Fn+ ( x)
p
∂n
=−
+ gn −
∂t
∂x
τ nt
∂Fpx+ ( x)
∂x
⋅ dx
The current density in material is
J p = eμ p pE − eD p
∂p
∂n
, J n = eμ n nE + eDn
∂x
∂x
By dividing current density the charge of each individual particle, we obtain
particle flux
Jp
( + e)
W.K. Chen
= Fp+ = μ p pE − D p
∂p
,
∂x
Jn
∂n
= Fn− = − μ n nE − Dn
( − e)
∂x
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14
Thus the continuity equations can be rewritten as
∂p
∂ ( pE)
∂2 p
p
= −μ p
+ Dp 2 + g p −
∂t
∂x
∂x
τ pt
∂n
∂ (nE)
∂ 2n
n
= +μn
+ Dn 2 + g n −
∂t
∂x
∂x
τ nt
Q
∂ ( pE)
∂p
∂n
∂E
∂E
∂ (nE)
and
=E + p
=E +n
∂x
∂x
∂x
∂x
∂x
∂x
∂2 p
∂E ⎞
p ∂p
⎛ ∂p
=
Dp 2 − μ p ⎜ E + p
⎟+ gp −
∂x
∂x ⎠
τ pt ∂t
⎝ ∂x
∂ 2n
∂E ⎞
n ∂n
⎛ ∂n
Dn 2 + μ n ⎜ E + n ⎟ + g n −
=
∂x
∂
∂
x
x
τ
∂t
⎝
⎠
nt
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The thermal equilibrium concentrations, no and po, are not function of time.
For homogeneous semiconductor, no and po are also independent of space
coordinates
n = no + δn(t )
p = po + δp (t )
∂ 2 (δp )
∂E ⎞
p ∂ (δp )
⎛ ∂ (δp )
− μp⎜E
+p
=
Dp
⎟+ gp −
2
∂x
∂x
∂x ⎠
∂t
τ pt
⎝
∂ 2 (δn)
∂E ⎞
n ∂ (δn)
⎛ ∂ (δn)
E
+
+
Dn
n
+
g
−
=
μ
⎜
⎟
n
n
∂x 2
∂x
∂x ⎠
∂t
τ nt
⎝
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Homogeneous
semiconductor
16
6.3 Ambipolar transport
„
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.
E int << E app
E = E app − E int ≈ E app
∇ ⋅ E int =
e(δp − δn)
εs
=
∂E int
≈0
∂t
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For ambipolar transport, the excess electrons and excess holes generate
and recombine together
⎧gn = g p ≡ g
⎪
n
p
⎪
= Rp =
=R
⎨ Rn =
τ
τ
nt
pt
⎪
⎪δn = δp
⎩
n = no + δn
p = po + δp
τ nt ≠ τ pt
∂ 2 (δp )
∂E ⎞
p ∂ (δp )
⎛ ∂ (δp )
Dp
− μp⎜E
+p
=
⎟+ gp −
2
τ pt
∂x
∂x
∂x ⎠
∂t
⎝
∂ 2 (δn)
∂E ⎞
n ∂ (δn)
⎛ ∂ (δn)
μ
Dn
+
+
n
+
g
−
=
E
⎜
⎟
n
n
τ nt
∂x 2
∂x
∂x ⎠
∂t
⎝
W.K. Chen
∂ 2 (δn)
∂E ⎞
∂ (δn)
⎛ ∂ (δn)
−
E
+
+
−
=
μ
Dp
p
g
R
⎜
⎟
p
∂x ⎠
∂x 2
∂x
∂t
⎝
∂ 2 (δn)
∂E ⎞
∂ (δn)
⎛ ∂ (δn)
E
+
+
μ
Dn
n
+
g
−
R
=
⎜
⎟
n
∂x 2
∂x Electrophysics,
∂x ⎠NCTU
∂t
⎝
18
∂ 2 (δn)
⎛ ∂ (δn) ⎞
( μ n nD p + μ p pDn )
(
)(
)
μ
μ
−
p
−
n
⎜E
⎟ + ( μ n n + μ p p )( g − R)
p n
∂x 2
∂x ⎠
⎝
∂ (δn)
= ( μ n n + μ p p)
∂t
∂ (δn)
∂ 2 (δn)
⎛ ∂ (δn) ⎞
D'
− μ'⎜ E
⎟ + ( g − R) =
2
∂t
∂x ⎠
∂x
⎝
μ'=
D' =
Q
( μ p μ n )( p − n)
( μ n n + μ p p)
ambipolar transport
equation
(ambipolar mobilty)
μ n nD p + μ p pDn
( μ n n + μ p p)
μn
Dn
=
W.K. Chen
μp
Dp
=
e
⇒
kT
D' =
Dn D p (n + p)
Dn n + D p p
(ambipolar diffusion coefficient)
19
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∂ (δn)
∂ 2 (δn)
⎛ ∂ (δn) ⎞
D'
+
g
−
R
=
μ
−
'
E
(
)
⎜
⎟
∂t
∂x 2
∂x ⎠
⎝
ambipolar transport
equation
∂ 2 (δn)
∂E ⎞
n ∂ (δn)
⎛ ∂ (δn)
+ μn ⎜ E
Dn
+ n ⎟ + gn −
=
2
∂x
∂x
∂x ⎠
∂t
τ nt
⎝
continuity equation
for electrons
∂ 2 (δp )
∂E ⎞
p ∂ (δp ) continuity equation
⎛ ∂ (δp )
E
−
Dp
+
p
+
g
−
=
μ
⎜
⎟
p
p
∂x 2
∂x
∂x ⎠
∂t for holes
τ pt
⎝
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6.3.2 Ambipolar transport under low injection
∂ (δp ) Ambipolar transport
∂ 2 (δp)
⎛ ∂ (δp ) ⎞
'
E
(
)
D'
−
+
g
−
R
=
μ
⎜
⎟
∂x 2
∂x ⎠
∂t
⎝
Dn D p (n + p)
D' =
μ'=
Dn n + D p p
( μ p μ n )( p − n)
( μ n n + μ p p)
For n − type no >> po
δn = δp << no (low injection)
D' =
μ'=
Dn D p [(no + δn) + ( po + δp )]
Dn (no + δn) + D p ( po + δp )
( μ p μ n )[ p − n)
[ μ n n + μ p p]
≈
≈
( μ p μ n )(− n)
W.K. Chen
( μ n n)
Dn D p no
Dn no
= Dp
= −μ p
21
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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
Qτ pt = τ p
(minority carrier hole lifetime)
So the net generation rate,
Thermal-equilibrium
hole generation rate
excess hole
generation rate
g − R = g p − R p = (G po + g 'p ) − ( R po + R p' )
Thermal-equilibrium hole
recombination rate
Q G po = R po
W.K. Chen
g − R = g 'p − R p' = g 'p −
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excess hole
recombination rate
δp
τp
22
∂ 2 (δp)
∂ (δp )
⎛ ∂ (δp ) ⎞
−
+
−
=
D'
g
R
μ
'
E
(
)
⎜
⎟
∂x 2
∂x ⎠
∂t
⎝
D' = D p
g − R = g 'p − R p' = g 'p −
μ ' = −μ p
∂ 2 (δp )
∂ (δp )
δp ∂ (δp)
Dp
−
+
g
−
=
μ
E
'
p
∂x 2
∂x
τ po
∂t
„
„
δp
τp
n-type, homogeneous and lowinjection ambipolar transport
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
23
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For p-type semiconductor
∂ (δn) p-type, homogeneous
∂ 2 (δn)
⎛ ∂ (δn) ⎞
D'
−
'
E
+
(
g
−
R
)
=
μ
⎜
⎟
ambipolar transport
∂x ⎠
∂t
∂x 2
⎝
D' =
Dn D p (n + p)
Dn n + D p p
μ'=
( μ p μ n )( p − n)
( μ n n + μ p p)
For p − type po >> no
δp = δn << po (low injection)
D' =
μ'=
W.K. Chen
Dn D p [(no + δn) + ( po + δp)]
Dn (no + δn) + D p ( po + δp )
( μ p μ n )[ p − n)
[ μ n n + μ p p]
≈
( μ p μ n )( p)
( μ p p)
≈
Dn D p po
D p po
= Dn
= μn
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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
Qτ nt = τ n
(minority carrier electron lifetime)
So the net generation rate,
Thermal-equilibrium
excess electron
electron generation rate generation rate
g − R = g n − Rn = (Gno + g n' ) − ( Rno + Rn' )
Thermal-equilibrium hole
recombination rate
Q Gno = Rno
W.K. Chen
g − R = g n' − Rn' = g n' −
excess hole
recombination rate
δn
τn
25
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∂ (δn)
∂ 2 (δn)
⎛ ∂ (δn) ⎞
D'
g
R
−
'
E
+
(
−
)
=
μ
⎜
⎟
∂x 2
∂x ⎠
∂t
⎝
D' = Dn
μ ' = μn
g − R = g n' − Rn' = g n' −
∂ 2 (δn)
∂ (δn)
δn ∂ (δn)
Dn
+ μn E
+ g' −
=
2
∂x
∂x
∂t
τ no
„
„
δn
τn
p-type, homogeneous and lowinjection ambipolar transport
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
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6.3.3 Applications of the ambipolar transport
equation
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27
Example 6.1
Homogenous n - type semiconductor
zero applied electrical field
under low - injection condition
At t = 0, uniform concentration of excess carriers
g' = 0 for t > 0
⇒ Calculate the excess carrier concentration as a function of time
Solution:
E=0
g’=0
∂ 2 (δp )
∂ (δp )
δp ∂ (δp)
Dp
−
E
+
g
'
−
=
μ
p
∂x 2
∂x
∂t
τ po
∂ 2 (δp ) δp ∂ (δp )
Dp
−
=
∂x 2
∂t
τ po
Uniform excess carriers
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∂ (δp )
δp
=−
∂t
τ po
δp(t ) = δp(0)e
− t / τ po
From the charge neutrality condition, the excess majority electron
concentration is given by
δn(t ) = δn(0)e
W.K. Chen
− t / τ po
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Example 6.2
Homogenous n - type semiconductor
zero applied electrical field
under low - injection condition
At t > 0, uniform generation of excess carriers
⇒ Calculate the excess carrier concentration as a function of time
Solution:
Uniform generation
δp ∂ (δp)
∂ 2 (δp )
∂ (δp )
μ
Dp
−
E
+
g
'
−
=
p
∂x 2
∂x
∂t
τ po
g '−
∂ (δp ) δp
δp ∂ (δp)
=
⇒
+
− g'= 0
∂t
∂t
τ po
τ po
δp(t ) = τ po g ' (1 − e
W.K. Chen
− t / τ po
)
At steady state,
∂δp(t )
= 0, δp (t = ∞) = τ po g '
∂t
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Example 6.3
Homogenous p - type semiconductor
zero applied electrical field
the excess carriers are being generated at x = 0 only, under low - injection
condition
then diffuse in bith the + x and − x directions
⇒ Calculate the steady state excess carrier concentration as a function of x
Solution:
E=0
Steady state
∂ 2 (δn)
∂ (δn)
δn ∂ (δn)
'
μ
Dn
+
E
+
g
−
=
n
∂x 2
∂x
∂t
τ no
W.K. Chen
31
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δn
∂ 2 (δn)
'
At x = 0, Dn
+
−
=0
g
τ no
∂x 2
∂ 2 (δn) δn
At x ≠ 0 Dn
−
=0
∂x 2
τ no
∂ 2 (δn)
δn
−
=0
2
∂x
Dnτ no
Minority carrier diffusion length
p-type
log scale
L2n = Dnτ no
The general solution
δn( x) = Ae − x / Ln + Be − x / Ln
⎧⎪δn( x) = δn(0)e − x / Ln x ≥ 0
⇒ ⎨
⎪⎩δn( x) = δn(0)e + x / Ln x ≤ 0
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Example 6.4
Homogenous n - type semiconductor
constant applied electrical field E o in the x - direction
finite numbers of electron - hole pairs is generated instantaneously at t = 0
and x = 0
g' = 0 for t > 0
⇒ Calculate the excess carrier concentration as a function of x and t
Solution:
δp ∂ (δp)
∂ 2 (δp )
∂ (δp )
μ
Dp
−
E
+
g
'
−
=
p
∂x 2
∂x
∂t
τ po
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33
∂ 2 (δp )
∂ (δp) δp ∂ (δp )
Dp
− μ pEo
−
=
2
∂x
∂x
∂t
τ po
δp( x, t ) = p' ( x, t )e
− t / τ po
∂ 2 p ' ( x, t )
∂p ' ( x, t ) ∂p ' ( x, t )
μ
Dp
=
−
E
p
o
∂x
∂t
∂x 2
⎡ − ( x − μ p E ot ) 2 ⎤
1
p ' ( x, t ) =
exp ⎢
⎥
4D pt
4πD p t
⎢⎣
⎥⎦
δp( x, t ) = p' ( x, t )e
W.K. Chen
−t / τ po
− t /τ
⎡ − ( x − μ p E ot ) 2 ⎤
e po
=
exp ⎢
⎥
4D pt
4πD p t
⎢⎣
⎥⎦
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Zero applied field
W.K. Chen
constant applied field
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35
6.3.4 Dielectric relaxation time constant
„
How 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
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36
Poisson’s equation
∇⋅E =
ρ
ε
ρ = (+e)δp
J = σE
Continuity equation
∇⋅ J = −
∇⋅ J = −
∂ρ
∂t
(neglecting the effects of
generation and recombination)
∂ρ
ρ
∂ρ
⇒ σ (∇ ⋅ E) = σ = −
ε
∂t
∂t
∂ρ σ
+ ρ =0
∂t ε
ρ (t ) = ρ (0)e − t /τ
Dielectric relaxation time constant
W.K. Chen
τd =
ε
σ
d
( the time constant is related
to dielectric constant)
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37
Example 6.5 Dielectric relaxation time constant
Homogenous n - type semiconductor
N d = 1016 cm −3
⇒ Calculate the dielectric relaxation time constant
Solution:
ε rε o
ε
(11.7)(8.85 × 10 −14 )
τd = =
=
= 0.539 ps
σ eμ n N d (1.6 ×10 −19 )(1200)(1016 )
„
„
„
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
38
6.3.5 Haynes-Shockley experiment
„
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
δp( x, t ) = p' ( x, t )e
Zero applied field
W.K. Chen
−t / τ po
− t /τ
⎡ − ( x − μ p E ot ) 2 ⎤
e po
exp ⎢
=
⎥
4D pt
4πD p t
⎥⎦
⎢⎣
constant applied field
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39
Haynes-Shockley experiment
‰
‰
‰
‰
Excess-carrier pulse are
effectively injected at contact A
Contact B is rectifying contact
and 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
40
‰
‰
‰
‰
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
x − υ p t = 0 ⇒ x − μ p Eo t = 0
μp =
d
E oto
W.K. Chen
δp( x, t ) = p' ( x, t )e
‰
‰
41
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−t / τ po
− t /τ
⎡ − ( x − μ p E ot ) 2 ⎤
e po
exp ⎢
=
⎥
4D pt
4πD p t
⎥⎦
⎢⎣
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
( x − μ p E ot ) 2 = 4 D p t , t = t1 or t 2
From the broadened pulse width, we can obtain diffusion coefficient
diffusion coefficient D p =
W.K. Chen
( μ p E o ) 2 (Δt ) 2
16to
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, Δt = t 2 − t1
42
The area S undrer the curve is proportional to the number of excess holes
that have not recombined with majority carrier electrons
δp ( x, t ) = p ' ( x, t )e
S = K exp(
−t / τ po
− t /τ
⎡ − ( x − μ p E ot ) 2 ⎤
e po
exp ⎢
=
⎥
4
D
t
4πD p t
⎢⎣
p
⎦⎥
⎛ − d /( μ p E o ) ⎞
⎛
⎞
⎟ = K exp⎜ − d ⎟
) = K exp⎜
⎜
⎟
⎜μ E τ ⎟
τ po
τ po
⎝
⎠
⎝ p o po ⎠
− to
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
43
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We can determine the minority carrier lifetime
ln(S )
ln(S ) = ln( K ) +
W.K. Chen
1 ⎛⎜ − d ⎞⎟
⋅
τ po ⎜⎝ μ p E o ⎟⎠
1
τ po
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y=
d
μ p Eo
44
6.4 Quasi-Fermi energy levels
„
At 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
⎛ E − Ef
⎛ E f − E fi ⎞
⎟⎟ po = ni exp⎜⎜ f i
no = ni exp⎜⎜
⎝ kT ⎠
⎝ kT
no po = ni2
W.K. Chen
„
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⎞
⎟⎟
⎠
45
At non-thermal equilibrium
‰
‰
‰
If 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 nonthermal equilibrium conditions do not hold constants over the entire material
⎛ E fn − E fi ⎞
⎟⎟
n = no + δn = ni exp⎜⎜
kT
⎠
⎝
⎛ E f − E fp ⎞
⎟⎟
po = po + δp = ni exp⎜⎜ i
kT
⎝
⎠
np ≠ ni2
W.K. Chen
Electrophysics, NCTU
46
Example 6.6 Quasi-Fermi level
Homogenous n - type semiconductor at T = 300 K
no = 1015 cm −3 , ni = 1010 cm −3 and po = 105 cm −3
In nonequilibrium, δn = δp = 1013 cm −3
⇒ Calculate the quasi - Fermi energy
Solution:
⎛ E f − E fi ⎞
n
⎟⎟ ⇒ E f − E fi = kT ln( o ) = 0.2982 eV
no = ni exp⎜⎜
ni
⎝ kT ⎠
⎛ E fn − E fi ⎞
n + δn
⎟⎟ ⇒ E fn − E fi = kT ln( o
n = no + δn = ni exp⎜⎜
) = 0.2984 eV
ni
⎝ kT ⎠
⎛ E f − E fp ⎞
p + δp
⎟⎟ ⇒ E fi − E fp = kT ln( o
po = po + δp = ni exp⎜⎜ i
) = 0.179 eV
ni
⎝ kT ⎠
W.K. Chen
47
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n : 1.0 ×1015 cm −3 → 1.01×1015 cm −3
po = 105 cm −3 → 1013 cm −3
„
At non-thermal equilibrium
‰
‰
Since 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
W.K. Chen
Electrophysics, NCTU
48
6.5 Excess-carrier lifetime
„
In a perfect semiconductor
‰
„
The electronic energy states do not exist within the forbidden bandgap
In a real semiconductor
‰
‰
Defects (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
49
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6.5.1 Shockley-Read-Hall theory of recombination
„
Recombination 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 theory
‰
‰
Assume 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
electron
Neutral empty state
W.K. Chen
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50
„
Shockley-Read-Hall theory
There 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
51
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Process 1: electron capture
Electron capture rate
Rcn = Cn ⋅ [ N t ⋅ (1 − f F ( Et ))] ⋅ n
f F ( Et ) =
1
⎡E − Ef ⎤
1 + exp ⎢ t
⎥
⎣ kT ⎦
Process 2: electron emission
Ren = En ⋅ [ N t ⋅ f F ( Et )]
„
Is there relationship between capture constant and emission constant?
Rcn = Ren
(at thermal equilibrium)
Cn ⋅ [ N t ⋅ (1 − f Fo ( Et ))] ⋅ n = En ⋅ [ N t ⋅ f Fo ( Et )]
W.K. Chen
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52
Cn ⋅
(1 − f Fo ( Et )) ⋅ n
= En
f Fo ( Et )
⇒ Cn ⋅ (
1
− 1) ⋅ n = En
f Fo ( Et )
⎡ Et − E fo ⎤
⎡ Ec − E fo ⎤
1
)
exp
Cn ⋅ (1 + exp ⎢
−
⋅
N
c
⎥
⎢−
⎥ = En
kT
kT
⎣
⎦
⎣
⎦
⎡ E − Et ⎤
En = N c exp ⎢− c
Cn
kT ⎥⎦
⎣
En = n' C n
„
„
⎡ Et − Ei ⎤
⎡ E − Et ⎤
exp
=
n' = N c exp ⎢− c
n
i
⎢ kT ⎥
kT ⎥⎦
⎦
⎣
⎣
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
53
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Process 3: hole capture rate
Rcp = C p ⋅ ( N t ⋅ f F ( Et )) ⋅ p
Process 4: hole emission rate
Rep = E p ⋅ [ N t ⋅ (1 − f F ( Et ))]
(at thermal equilibrium)
Rcp = Rep
C p ⋅ ( N t ⋅ f F ( Et )) ⋅ p = E p ⋅ [ N t ⋅ (1 − f F ( Et ))]
Cp ⋅ p = Ep ⋅ (
f F ( Et ) =
1
⎡E − Ef ⎤
1 + exp ⎢ t
⎥
⎣ kT ⎦
1
− 1)
f F ( Et )
⎡ Eυ − E f ⎤
⎡ Et − E f ⎤
⎡ Et − E f ⎤
=
⋅
+
−
=
⋅
C p ⋅ Nυ exp ⎢
E
E
(
1
exp
1
)
exp
p
p
⎥
⎢
⎥
⎢
⎥
⎣ kT ⎦
⎣ kT ⎦
⎣ kT ⎦
⎡ E − Et ⎤
E p = C p Nυ exp ⎢ υ
⎥
⎣ kT ⎦
⎡ Eυ − Et ⎤
⎡ E − Et ⎤
= ni exp ⎢ i
E p = C p p ' p ' = Nυ exp ⎢
⎥
⎥
⎣ kT ⎦
⎣ kT ⎦
W.K. Chen
Electrophysics, NCTU
54
„
Capture constants
‰
The 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.
Cn = υthσ cn
‰
„
C p = υthσ cp
The capture constants for electrons and holes proportion to respective crosssectional area
Emission constants
‰
The electron emission rate from a trap is the product of electron capture rate and
free carrier concentration when Ef=Et.
En = n' C n
E p = C p p'
‰
‰
⎡ E − Et ⎤
n' = N c exp ⎢− c
kT ⎥⎦
⎣
⎡ E − Et ⎤
p ' = Nυ exp ⎢ υ
⎥
⎣ kT ⎦
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
W.K. Chen
55
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Rcn = Cn ⋅ [ N t ⋅ (1 − f F ( Et ))] ⋅ n
Rcp = C p ⋅ ( N t ⋅ f F ( Et )) ⋅ p
Ren = En ⋅ [ N t ⋅ f F ( Et )]
Rep = E p ⋅ [ N t ⋅ (1 − f F ( Et ))]
W.K. Chen
Electrophysics, NCTU
56
„
Under non-equilibrium condition
Net electron capture rate at acceptor trap
Rn = Rcn − Ren
Rn = Cn ⋅ [ N t ⋅ (1 − f F ( Et ))] ⋅ n − En ⋅ [ N t ⋅ f F ( Et )]
Rn = Cn N t ⋅ [n(1 − f F ( Et )) − n' f F ( Et )]
⎡ E − Et ⎤
n' = N c exp ⎢− c
kT ⎥⎦
⎣
Rn = Cn N t ⋅ [n(1 − f F ( Et )) − n' f F ( Et )]
Net hole capture rate at acceptor trap (Process 3 & 4)
R p = C p N t ⋅ [ pf F ( Et )) − p ' (1 − f F ( Et ))]
At steady state,
f F ( Et ) =
⎡ E − Eυ ⎤
p' = Nυ exp ⎢− t
kT ⎥⎦
⎣
Rn = R p
Cn n + C p p '
Cn ( n + n' ) + C p ( p + p ' )
W.K. Chen
57
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We may note that
⎡ E − Et ⎤
⎡ Et − Eυ ⎤
⎡ Ec − Eυ ⎤
exp
exp
n' p ' = N c exp ⎢− c
⋅
N
−
=
N
N
c
υ
υ
⎥
⎢
⎥
⎢− kT ⎥
kT
kT
⎣
⎦
⎣
⎦
⎣
⎦
n' p' = ni2
Rn = R p =
CnC p N t (np − ni2 )
C n ( n + n' ) + C p ( p + p ' )
≡R
Rn
Rp
For SRH-dominated recombination process, the recombination rate of
excess carriers is
CnC p N t (np − ni )
δn
=
R=
τ Cn ( n + n' ) + C p ( p + p ' )
2
W.K. Chen
Electrophysics, NCTU
58
SRH under low injection
Case 1: n-type semiconductor with deep trap energy at low injection
no >> po (n - type)
no >> δp (low injection)
no >> n' , no >> p' (deep trap) ( E f − Et > 3kT )
CnC p N t (np − ni )
CnC p N t nδp
δn
=
≈
= C p N tδp
R=
Cn n
τ Cn ( n + n' ) + C p ( p + p ' )
2
R=
„
δn
= C p N tδp
τ
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
W.K. Chen
„
„
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
R=
δn
δp
= C p N tδp =
τ po
τ
τ po =
„
59
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n-type
low injection
Rn
Rp
Ef
Et
1
C p Nt
If the trap concentration increases, the probability of excess carrier
recombination increases; thus the excess minority carrier lifetime decreases
W.K. Chen
Electrophysics, NCTU
60
Case 2: p-type semiconductor with deep trap energy at low injection
po >> no (p - type)
po >> δn (low injection)
Rn
Rp
po >> p' , po >> n' (deep trap)
R=
δp
δn
= Cn N tδn =
τ no
τ
τ no =
W.K. Chen
+
Et
Ef
p-type
low injection
1
Cn N t
61
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Case 3: n/p-type semiconductor with deep trap energy
CnC p N t (np − ni )
δn
R=
=
τ Cn ( n + n' ) + C p ( p + p ' )
2
(np − ni2 )
δn
R=
=
τ τ po (n + n' ) + τ no ( p + p' )
W.K. Chen
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SRH recombination process
General SRH
recombination rate
62
Example 6.7 Intrinsic semiconductor
no = po = ni , n = no + δn, p = po + δp
Assume deep trap is located at a energy equal to intrinsic energy level
n' = p ' = ni
2
(
np
−
n
)
δ
n
i
under low injection
R=
=
τ τ po (n + n' ) + τ no ( p + p' )
Solution:
(ni + δn) ⋅ (ni + δn) − ni2 )
2niδn + (δn) 2
R=
=
τ po (ni + δn + ni ) + τ no (ni + δn + ni ) (2ni + δn)(τ po + τ no )
Under low injection
δn << ni
R=
δn
δn
=
(τ po + τ no ) τ
τ = τ po + τ no
„
The excess-carrier lifetime increases as we change from an extrinsic to an
intrinsic semiconductor
W.K. Chen
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63
6.6 Surface effects
„
Surface 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
RB =
δp δpB
=
(in the bulk)
τ po τ po
Rs =
δps
(at the surface)
τ pos
τpos<<τpo
W.K. Chen
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64
For the case of uniform pumping
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
G = RB =
δpB
(in the bulk)
τ po
G = Rs =
δps
(at the surface)
τ pos
G = RB = Rs (uniform pumping)
τ pos < τ po ⇒ δps < δpB
W.K. Chen
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65
Example 6.8 Surface recombination
n - type semiconductor δ pB = 1014 cm -3 , τ po = 10 −6 s, τ p 0 s = 10 −7 s D p = 10 cm 2 /s
Assume E = 0 and uniform pumping
⇒ Steady state excess - carrier concentration distribution
Solution:
Excess - carrier concentration at surface ( x = 0)
−7
τ pos
δpB δps
14 (10 )
QG =
=
⇒ δps = δpB (
) = (10 ) −6 = 1013 cm -1
τ po τ pos
τ po
(10 )
Carrier transport equation (at E = 0)
d 2 (δp)
δp
Dp
g
+
'
−
= 0 (uniform pumping)
dt 2
τ po
δpB 1014
where g ' =
= −6 = 10 20 cm -3 - s -1
τ po 10
W.K. Chen
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66
δp( x) = g 'τ po + Ae
δp( x) = g 'τ po + Be
x / Lp
− x / Lp
+ Be
− x / Lp
(uniform pumping)
B.C. :
δp(+∞) = δpB = g 'τ po = 1014 cm −3
δp(0) = δps = 1014 cm −3 + B = 1014 cm −3 ⇒ B = −9 ×1013
δp( x) = δps = 1014 (1 - 0.9e
− x / Lp
)
where L p = D pτ po = (10)(10-6 ) = 0.00316 cm = 31.6μm
W.K. Chen
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67
6.6.2 Surface recombination velocity
A 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.
Surface recombination rate
Flux D p
W.K. Chen
d (δp)
= sδp (0) = s( p (0) − po )
dx surface
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68
For the case of uniform pumping
Carrier transport equation (at E = 0)
Uniform pumping
E=0
Steady state
δp
d (δp )
+
'
−
=0
g
τ po
dt 2
2
Dp
δp( x) = g 'τ po + Be
− x / Lp
Q δp(0) = g 'τ po + B
d (δp)
d (δp)
B
=
=−
dx surface
dx x =0
Lp
⇒B=
− sg 'τ po
(Dp / Lp ) + s
δp( x) = g 'τ po (1 −
W.K. Chen
s=
sg ' L pτ po
D p + sL p
e
− x / Lp
)
Uniform pumping
E=0
Steady state
69
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D p ⎛ g 'τ po ⎞
⎜
− 1⎟ cm/s (uniform pumping)
L p ⎜⎝ δp(0) ⎟⎠
δp(x)
δp( x) = g 'τ po if s = 0
s=0
x
„
Surface recombination velocity is sensitive to the surface conditions
‰
‰
For 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
W.K. Chen
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70
Example 6.10 Surface recombination velocity
For the case in Example 6.8 (uniform oumping)
n - type semiconductor δ pB = 1014 cm -3 , τ po = 10 −6 s, τ p 0 s = 10 −7 s
g'τ po = 1014 cm −3 , D p = 10 cm 2 /s, L p = 31.6 μm and δp (0) = 1013 cm -3
⇒ Determine the surface recombination velocity
Solution:
uniform pumping ⇒
s=
D p ⎛ g 'τ po ⎞
⎜
− 1⎟ cm/s
L p ⎜⎝ δp(0) ⎟⎠
⎛ 1014 ⎞
10
⎜ 13 − 1⎟⎟ = 2.85 ×10 4 cm/s
s=
−4 ⎜
31.6 ×10 ⎝ 10
⎠
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71
Short briefs for surface recombination
RB =
δp δpB
=
(in the bulk)
τ po τ po
Rs =
δps
(at the surface)
τ pos
Flux D p
W.K. Chen
d (δp)
= sδp (0) = s( p (0) − po )
dx surface
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72
W.K. Chen
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73
Figure 6.19 Figure for problems 6.18 and 6.20
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74
Figure 6.20 Figure for Problem 6.25
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75
Figure 6.21 Figure for Problem 6.38
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76
Figure 6.22 Figure for Problem 6.39
W.K. Chen
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77
Figure 6.23 Figure for Problem 6.40
W.K. Chen
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78
Figure 6.24 Figure for Problem 6.41
W.K. Chen
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79