Chapter 4 Excess Carriers in semiconductor
4.1 Optical Absorption
4.2 Luminescence
4.3 Photoconductivity and carrier lifetime
4.4 Concentration gradient and the diffusion of carrier
ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)
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Preview •
Induce carriers in CB and VB such that carrier concentrations (n and p) > equilibrium concentrations (n0 and p0). •
“Excess” carriers can be created by processes:
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Optical excitation.
Energetic electron bombardment.
Carrier injection.
n  n  n0
p  p  p0
excess carriers δn and δp, :
excitation is removed, electron‐hole pair recombination acts to bring the carrier concentrations back to their equilibrium values.
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4.1 Optical Absorption
•
Photon with energy hν enters a semiconductor:
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transmitted if hν <Eg.
absorbed if hν >Eg.
Critical photon absorption energy can be used to determine Eg of semiconductors.
If a photon is absorbed by the semiconductor, an EHP is generated.
h ν >Eg, then electrons can be promoted to energy levels above the bottom of the CB. Such electrons lose energy to the lattice until their average velocity approaches the equilibrium velocity of the electrons in the CB.
When the supply of photon is removed, excess carriers will be removed from the system by EHP recombination.
Fig. 4.1
Optical absorption of a photon with hv>Eg : (a) an EHP is created during photon absorption; (b) the excited electron gives up energy to the lattice by scattering events; (c) the electron recombines with a hole in the valence band.
ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)
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Description of Optical Absorption
Monochromatic light is directed towards a semiconductor, a fraction of the light will be absorbed, described by differential equation:
negligible absorption at long wavelength (small energy) and considerable absorption with energies larger than Eg
dI ( x )

 I ( x )
dx
where I(x) = light intensity at the position x and  is called the absorption coefficient with unit of cm‐1. The solution to this equation is :
 x
I ( x)  I 0e
I0 is the flux of photon entering the sample at x=0.
Sample with thickness l, amount of light not absorbed (hence transmitted) :
Fig 4‐2
I (l )  I 0 e  l
Dependence of optical absorption coefficient  for a semiconductor on the wavelength of incident light.
ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)
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Experiment Setup Fig. 4‐3
Optical absorption experiment.
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4.2 Luminescence
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Energy must be dissipated when electrons and holes recombine.
Light is given off if the dissipated energy appears in the form of photons.
Recombination via the emission of light is particularly strong in semiconductors with direct band gaps (such as GaAs), but much less likely in semiconductors with indirect band gaps (such as Si). (Why?)
Depending on how the excess carriers are generated, luminescence can be classified as :
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Photoluminescence (PL) : excess carriers generated from absorption of photons.
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Cathodoluminescence (CL) : excess carriers generated by high energy electron bombardment on the semiconductor.
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Electroluminescence (EL) :excess carriers generated by carrier injection.
Fig. 4‐4
Band gaps of some common semiconductors relative to the optical spectrum.
ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)
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Excitation and recombination mechanisms
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Among all the mechanisms of light emission
from a semiconductor, the simplest one is
direct recombination of EHPs.
Direct recombination is a fast process,
typically occurring with 10ns (or 10 8 s). Fast
luminescence is called fluorescence.
In some materials, emission continues for
durations up to seconds or even minutes after
the removal of the excitation source. This
slow luminescence is called phosphorescence.
Defect levels (perhaps due to impurities)
within the energy band gap with a strong
tendency to trap electrons from the CB are
involved in the process.
The delay time between the excitation and
the recombination can be relatively long if the
electrons trapped in these levels are only
slowly released to the CB for radiative
recombination.
Fig. 4‐5
Excitation and recombination mechanisms in photoluminescence with a trapping level for electrons.
ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)
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Cathodoluminescence and Electroluminescence
CRT display:
• Electrons emitted from a heated cathode are
accelerated and directed towards the anode.
• When the energetic electrons strike on a
phosphorescent screen, the resident electrons
in the phosphor are excited to higher energy
states.
• When these electrons recombine, light is
emitted.
• Basis of old TV, computer, workstation
monitors.
Fig. 4



Electric current results in the injection of
carriers
and
causes
the
carrier
concentrations to deviate from the
equilibrium values.
The excess carriers recombine radiatively.
Basis of light emitting diodes (LEDs) and
lasers.
Simplified view of a cathode‐ray tube.
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4.3 Photoconductivity
At thermal equilibrium, the thermal generation rate (gth) is exactly balanced by the recombination rate (which is 0 0
), we have
g th   r n
2
i
Where ar is the recombination constant
Steady flux of light is incident on the semiconductor, an additional carrier generation giving rise to an optical generation rate, gop.
dn(t )
where n(t) and p(t) electron and hole concentrations under the illumination. The last term on the RHS is the recombination rate according to the Law of Mass Action.
dt
2
= g op   r n   r n(t ) p(t )
i
δn(t) and δp(t) denote respectively the excess instantaneous electron and hole concentrations. i.e.
dn(t ) dn(t )

dt
dt
Assume δn(t)=δp(t)  g op  g th   r n(t ) p(t )
(1)
n(t )  n 0  n(t )
2
d  n (t )
 g op   r n   r [ n0  n (t )][ p 0   p (t )]
i
dt
= g op   r [( n0  p 0 )n (t )   n 2 (t )]
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Kinetic of Excess Carrier Generation and Recombination
d n ( t )
 g op   r p 0 n(t )
dt
low level injection (LLI) condition: n(t )  (n0  p0 )
assumed material is p‐type so that n0  p0  p0
1
n
 r p0
We consider two limiting cases in this LLI condition :
d n ( t )
1. Steady state (SS): By definition
0
dt
n ss  p ss  0 , it must be true that
Since •
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•
dn(t )
n(t )
 g op 
n
dt
(2)
 n ss   n g op
2

np   n0 p0  n 
i

The derivations so far assume no carrier trapping in localized energy states.
When there is trapping, excited electrons and holes could reside in traps so that they are no longer created and annihilated in pairs in the CB and VB.
In this case, the majority carrier lifetime and the minority carrier lifetime may no longer be identical.
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Two limiting cases—Transient
light source is removed after a long illumination, we would expect the excess carriers to recombine so that equilibrium is recovered.
How long does it take?
Eqt. 2 again govern this case, but with gop  0
:
dn(t )
n(t )

dt
n
The solution to this first order differential equation is given by
n(t )  ( n g op )e

t
n
is called the recombination lifetime.
For direct recombination, since EHPs recombine in pairs, the excess majority carriers decays at the same rate as the excess minority carriers.
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Example
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Assume p‐type GaAs with
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For
GaAs,
10
.
ni=1016cm‐3,
10
0
hence
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0
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This is clearly a p‐type material since p0>>n0.
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Suppose 1014/cm3 EHPs are generated at t = 0.
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LLI condition is valid since p0>> δn. (Note : even
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at
t = 0, p0 = 10n)
If
=10ns, the time dependence δn of δp and
is shown in the following graph.
Fig. 4‐7
Decay of excess electrons and holes by recombination, for n= p=0.1p0, with n0 negligible, and =10ns. The exponential decay of δn(t) is linear on this semilogarithmic graph.
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Quasi‐Fermi Levels‐I
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When the CB and the VB are in equilibrium with each other, a Fermi Level (Ef) can be defined for both bands such that :
E f  Ei
Ei  E f
n0  ni e
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, p0  ni e
kT
When there are excess carriers, the system is no longer in equilibrium and ONE Fermi Level for both the CB and VB cannot be defined.
It happens that the electrons in the CB, though not in equilibrium with the holes in the VB, can interact with the lattice to achieve a quasi‐Fermi Level (Efn) which can be defined for the quasi‐equilibrium electron concentration in the CB such E fn  Ei
that
n  ni e
kT
quasi‐Fermi Level (Efp) can be defined for the quasi‐equilibrium hole Ei Efp
concentration in the VB.
p  ni e
•
kT
kT
Efn  Efp implies a “splitting” of the Fermi Level when the system deviates from equilibrium. In fact, the difference Efn ‐ Efp can serve as a quantitative measure of the degree of deviation from equilibrium
quasi‐Fermi Level is also given the name IMREF. (Can you see why?)
ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)
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Quasi‐Fermi Levels‐III
considering an n‐type semiconductor under LLI (?)
• LLI  n≅ 0 , hence ≅ . I.e. the quasi‐Fermi Level for the majority carriers (electrons in this case) does not deviate too much from the equilibrium Fermi Level.
• However, even in LLI, the excess carrier concentration can easily overwhelm the minority carrier concentration. I.e.
p  p0
This implies
Ei  E fp  Ei  E f
or
E fp  Ef
Therefore, the quasi‐Fermi Level for holes is below the equilibrium Fermi Level. ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)
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Photoconductivity
optical generation rate is gop, steady state excess carrier concentrations (δn, δp) are
n   n gop
 p   p g op
The change in conductivity (δ ) is therefore given by
  qg op ( n  n   p  p )
µn and µp are the electron and hole mobilities, respectively. To maximize δ , we need long carrier lifetimes and high mobilities.
The effect of photoconductivity can be used to
• make light detectors for automatic control of night lights,
• exposure meters in cameras,
• moving objects counters,
• burglar alarms, etc.
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