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Chap 6. Nonequilibrium Excess Carriers in
Semiconductor
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Carrier Generation and Recombination
Continuity Equation
Ambipolar Transport
Quasi-Fermi Energy Levels
Excess-Carrier Lifertime
Surface Effects
Solid-State Electronics
Chap. 6
1
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Nonequilibrium
 When a voltage is applied or a current exists in a semiconductor device,
the semiconductor is operating under nonequilibrium conditions.
 Excess electrons/holes in the conduction/valence bands may be
generated and recombined in addition to the thermal equilibrium
concentrations if an external excitation is applied to the semiconductor.
 Examples:
1. A sudden increase in temperature will increase the thermal
generation rate of electrons and holes so that their concentration will
change with time until new equilibrium reaches.
2. A light illumination on the semiconductor (a flux of photons) can
also generate electron-hole pairs, creating a nonequilibrium condition.
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Generation and Recombination
 In thermal equilibrium, the electrons are continually being thermal
generated from the valence band (hereby holes are generated) to
conduction band by the random thermal process.
 At the same time, electrons moving randomly through the crystal may
come in close proximity to holes and recombine. The rate of
generation and recombination of electrons/holes are equal so the net
electron and hole concentrations are constant (independent of time).
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Excess Carrier Generation and Recombination
 When high-energy photons are incident on a semiconductor, electronhole pairs are generated (excess electrons/holes)  the concentration
of electrons in the conduction band and of holes in the valence band
increase above their thermal-equilibrium value. n = no +n, p = po+ p
where no/po are thermal–equilibrium concentrations, and n/p are the
excess electron/hole concentrations. np  nopo = ni2 ( nonequilibrium)
 For the direct band-to-band generation, the generation rates (in the unit
of #/cm3-sec) of electrons and holes are equal; gn’ = gp’ (may be
functions of the space coordinates and time)
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Excess Carrier Generation and Recombination
 An electron in conduction band may “fall down” into the valence band
and leads to the excess electron-hole recombination process.
 Since the excess electrons and holes recombine in pairs so the
recombination rates for excess electrons and holes are equal, Rn’ = Rp’.
(in the unit of #/cm3-sec).  n(t) = p(t)
 The direct band-to-band recombination is spontaneous, thus the
probability of an electron and hole recombination is constant with time.
 Rn’ = Rp’  the electron and hole concentration.
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Recombination Process
 Band-to-Band: direct thermal recombination.
This process is typically radiative,
with the excess energy released
during the process going into the
production of a photon (light)
 R-G Center: Induced by certain impurity atoms or crystal defects.
Electron and hole are attracted to the
R-G center and lead to the annihilation
of the electron-hole pair.
Or a carrier is first captured at the R-G
site and then makes an annihilating
transition to the opposite carrier band.
This process is indirect thermal recombination (nonradiative).
Thermal energy (heat) is released during the process (lattice
vibrations, phonons are produced)
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Recombination Process
 Recombination via Shallow Levels:
—induced by donor or acceptor sites.
At RT, if an electron is captured at a donor site,
however, it has a high probability of being re-emitted into
the conduction band before completing the recombination
process. Therefore, the probability of recombination via
shallow levels is quite low at RT.
It should be noted that the probability of observing shallowlevel processes increases with decreasing system temperature.
 Recombination involving Excitons:
It is possible for an electron and a hole to become bound
together into a hydrogen-atom-like arrangement which
moves as a unit in response to applied forces. This coupled
e-h pair is called an “exciton”. The formation of an exciton
can be viewed as introducing a temporary level into the bandgap slightly
above or below the band edge.
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Recombination Process
 Recombination involving Excitons: Recombination involving excitons is a
very important mechanism at low temperatures and is the major lightproducing mechanism in LED’s.
 Auger Recombinations:
In a Auger process, band-to-band recombination
at a bandgap center occurs simultaneously with
the collision between two like carriers. The
energy released by the recombination or trapping
subprocess is transferred during the collision to
the surviving carrier. Subsequently, this high
energetic carrier “thermalizes”-loses energy
through collisions with the semiconductor lattice.
Auger recombination increases with carrier concentration, becoming very
important at high carrier concentration. Therefore, Auger recombination
mmust be considered in treating degenerately doped regions (like solar cell,
junction lasers, and LED’s)
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Generation Process
 Band-to-Band generation:
 R-G center generation:
 Photoemission from band gap centers:
Solid-State Electronics
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Dept. of E. E. NCU
Generation Process
 Impact-Ionization:
An e-h pair is produced as a result of the
energy released when a highly energetic
carrier collides with the crystal lattice. The
generation of carriers through impact ionization
routinely occurs in the high e-filed regions of
devices and is responsible for the avalanche
breakdown in pn junctions.
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Momentum Consideration
 In a direct semiconductor where the kvalues of electrons and holes are all
bunched near k = 0, little change is
required for the recombination process
to proceed. The conservation of both
energy and crystal momentum is readily
met by the emission of a photon.
 In a indirect semiconductor, there is
a large change in crystal momentum
associated with the recombination
process. The emission of a photon
will conserve energy but cannot
simultaneously conserve momentum.
Thus for band-to-band recombination
to proceed in an indirect semiconductor a phonon must be emitted coincident
with the emission of a photon.
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Excess Carrier Generation and Recombination
 Low-level injection: the excess carrier concentration is much less than
the thermal equilibrium majority carrier concentration, e.g., for a ntype semiconductor, n = p << no.
 High-level injection: n  no or n >> no
 For a p-type material (po >> no) under low-level injection, the excess
carrier will decay from the initial excess concentration with time;
n(t )  n(t  0)et /
n0
where n0 is referred to as the excess minority carrier lifetime (n0 1/p0)
 n (t )
and the recombination rate of excess carriers Rn’ = Rp’=
 n0
 For a n-type material (no >> po) under low-level injection,
t /  pn 0
Rn’ = Rp’= p(t )
p(t )  p(t  0)e
 p0
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Continuity Equations
 Consider a differential volume element in which a 1-D hole flux, Fp+ (#
of holes/cm2-sec), is entering this element at x and is leaving at x+dx.
 So the net change in hole concentration per unit time is

Fp
p
p

 gp 
t
x
 pt
----continuity equation for holes
 Similarly, the continuity equation for electron flux is
Fn
n
n

 gn 
t
x
 nt
Solid-State Electronics
Chap. 6
13
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Ambipolar Transport
 If a pulse of excess electrons and holes are created at a particular point
due to an applied E-field, the excess e-s and h+s will tend to drift in
opposite directions. However, any separation of e-s and h+s will induce
an internal E-field and create a force attracting the e-s and h+s back.
 The internal E-field will hold the pulses of excess e -s and h+s together,
then the electrons and holes will drift or diffuse together with a single
effective mobility or diffusion coefficient. This is so called “ambipolar
diffusion” or “ambipolar transport”.
 Fig. Show the above situation
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Ambipolar Transport
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Ambipolar Transport
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Quasi-Fermi Levels
 At thermal-equilibrium, the electron and hole concentrations are
functions of the Fermi level by
 E  EFi 
 EFi  EF 
no  ni exp F
and
p

n
exp
o
i

 kT 
 kT 


 Under nonequilibrium conditions, excess carriers are created in a
semiconductor, the Fermi energy is strictly no longer defined. We may
define a quasi-Fermi level, EFn, for electrons and a quasi-Fermi level,
EFp, for holes that apply for nonequilibrium. So that the total electron
and hole concentrations are functions of the quasi-Fermi levels.
 E  EFi 
 EFi  EF 
no  n  ni exp Fn
and
p

p

n
exp
o
i

 kT 
 kT 


Solid-State Electronics
Chap. 6
17
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Quasi-Fermi Levels
 For a n-type semiconductor under thermal equilibrium, the band
diagram is
 Under low-level injection, excess carriers are created and the quasiFermi level for holes (minority), EFp, is significantly different from EF.
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Excess-Carrier Lifetime
 An allowed energy state, also called a trap, within the forbidden
bandgap may act as a recombination center, capturing both electrons
and holes with almost equal probability. (it means that the capture
cross sections for electrons and holes are approximately equal)
 Acceptor-type trap:
– it is negatively charged when it contains an electron and it is neutrall when it does
not contain an electron.
 Donor-type trap:
– it is positively charged when empty and neutral when filled with an electron
Solid-State Electronics
Chap. 6
19
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Shockley-Read-Hall Theory of Recombination
 Assume that a single recombination center exists at an energy Et within
the bandgap. And there are four basic processes that may occur at this
single trap.
 Process 1: electron from the
conduction band captured by an
initially neutral empty trap.
 Process 2: electron emission from a
trap into the conduction band.
 Process 3: capture of a hole from the
valence band by a trap containing an
electron.
 Process 4: emission of a hole from a
neutral trap into the valence band.
Solid-State Electronics
Chap. 6
20
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Shockley-Read-Hall Theory of Recombination
 In Process 1: the electron capture rate (#/cm3-sec):
Rcn = CnNt(1-fF(Et))n
Cn=constant proportional to electron-capture cross section
Nt = total concentration in the conduction band
n = electron concentration in the conduction band
fF(Et)= Fermi function at the trap energy
 For Process 2: the electron emission rate (#/cm3-sec):
Ren = EnNtfF(Et)
En=constant proportional to electron-capture cross section Cn
 In thermal equilibrium, Rcn = Ren, using the Boltzmann approximation
for the Fermi function,
  Ec  Et 
'
En  n Cn  N c exp

 In nonequilibrium, excess electrons exist,
kT
Cn


Rn  Rcn  Ren  Cn Nt n1  f F ( Et )  n' f F ( Et )
Solid-State Electronics
Chap. 6
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Shockley-Read-Hall Theory of Recombination
 In Process 3 and 4, the net rate at which holes are captured from the
valence band is given by Rp  Cp Nt pfF ( Et )  p' (1  f F ( Et ))
  Et  Ev 
p '  N v exp

kT

 In semiconductor, if the trap density is not too large, the excess
electron and hole concentrations are equal and the recombination rates
of electrons and holes are equal.
 f F ( Et ) 
Cn n  C p p'
Cn (n  n' )  C p ( p  p' )
and Rn  R p 
CnC p N t (np  ni2 )
Cn (n  n' )  C p ( p  p' )
R
 In thermal equilibrium, np = ni2  Rn = Rp = 0
Solid-State Electronics
Chap. 6
22
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Surface Effects
 Surface states are functionally equivalent to R-G centers localized at
the surface of a material. However, the surface states (or interfacial
traps) are typically found to be continuously distributed in energy
throughout the semiconductor bandgap.
Solid-State Electronics
Chap. 6
23
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Surface Recombination Velocity
 As the excess concentration at the surface becomes smaller than that in
the bulk, excess carriers from the bulk region diffuse toward the
surface where they recombine, and the surface recombination velocity
increases.
 An infinite surface recombination velocity implies that the excess
minority carrier concentration and lifetime are zero.
Solid-State Electronics
Chap. 6
24
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Homework
 6.14
 6.17
 6.19
Solid-State Electronics
Chap. 6
25
Instructor: Pei-Wen Li
Dept. of E. E. NCU