Transcript chapter 5-1---photons in semiconductors

CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Chapter 5
Photons in Semiconductors
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Semiconductors
• A semiconductor is a solid material that has electrical
conductivity in between a conductor and an insulator.
• Semiconductors can be used as optical detectors,
sources (light-emitting diodes and lasers), amplifiers,
waveguides, modulators, sensors, and nonlinear
optical elements.
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A. Energy bands and
charge carriers
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Energy bands in semiconductors
• The solution of the Schrödinger equation for the electron
energy in the periodic potential created by the atoms in a
crystal lattice, results in a splitting of the atomic energy levels
and the formation of energy bands.
E2’
Conduction band
E21
E1’
Valence band
ATOM
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Energy bands in semiconductors
• Each band contains a large number of finely separated discrete
energy levels that can be approximated as a continuum.
• The valence and conduction bands are separated by a
“forbidden” energy gap of width Eg
bandgap energy
Unfilled
bands
Conduction band
Bandgap
E
Filled
bands
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Energy bands in semiconductors
Materials with a large energy gap (>3eV) are insulators, those
for which the gap is small or nonexistent are conductors,
semiconductors have gaps roughly in the range 0.1 to 3 eV
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ev
Si
ev
GaAs
Conduction band
Conduction band
Eg
Eg
1.1ev
1.42ev
0
-5
Energy
Energy
Valence band
-15
0
Valence band
-5
-10
-10
(a)
5
5
(b)
-15
Figure 15.1-1 Energy bands: (a) in Si, and (b) in GaAs
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Electrons and holes
• In the absence of thermal excitations, the valence
band is completely filled and the conduction band
is completely empty. Thus, the material cannot
conduct electricity.
• As the temperature increases, some electrons will
be thermally exited into the empty conduction
band, result in the creation of a free electron in the
conduction band and a free hole in the valence
band.
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Electrons and holes
Electron energy E
Conduction band
Electron
Hole
Bandgap energy Eg
Valence band
Figure 15.1-2 Electrons in the conduction
band and holes in the valence band at T>0.
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Energy-momentum relations
2 2
p2
k
E

2m0 2m0
p is the magnitude of the momentum
k
p
k is the magnitude of the wavevector
associated with the electron’s wavefunction
m0 is the electron mass
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E
Eg=1.1ev
K
[111]
Si
[100]
E
Eg=1.42ev
K
[111]
GaAs
[100]
Figure 15.1-3 Cross section of the E-K function for Si and GaAs along
the crystal directions [111] and [100].
The energy of an electron in the conduction band depends not only on the magnitude
of its momentum, but also on the direction in which it is traveling in the crystal
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Effective mass
Near the bottom of the conduction band, the E-k relation may be
approximated by the parabola
2
k2
E  Ec 
2mc
2
k2
E  Ev 
2mv
Ec,Ev: the energy at the bottom of the conduction band and at the
top of the valence band
mc,mv: effective mass of the electron in the conduction band
and the hole in the valence band
The effective mass depends on the crystal orientation and the
particular band under consideration
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Effective mass
Typical ratios of the average effective masses
to the mass of the free electron mass
mc/m0
mv/m0
Si
0.33
0.5
GaAs
0.07
0.5
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Direct- and indirect-gap semiconductors
E
Eg=1.1ev
K
[111]
Si
[100]
E
Eg=1.42ev
K
[111]
GaAs
[100]
Figure 15.1-4 Approximating the E-K diagram at the bottom of the conduction band and at the top of
the valence band of Si and GaAs by parabols.
The direct-gap semiconductors such as GaAs are efficient photon
emitters, while the indirect-gap counterparts are not.
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B. Semiconducting
materials
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Semiconducting materials
• Si: widely used for making photon detectors but not useful
for fabricating photon emitters due to its indirect bandgap.
• GaAs, InP GaN etc.: used for making photon
detectors and sources.
• Ternary and quaternary semiconductors: AlxGa1-xAs,
InxGa1-xAsyP1-y etc.
tunable bandgap energy with variation of x and y.
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Semiconducting materials
AlxGa1-xAs is lattice matched
to GaAs, means it can be grown
on the GaAs without introducing
strains.
Solid and dashed curves
represent direct-gap and
indirect-gap compositions
respectively.
We can see that a material may
have direct bandgap for one
mixing ratio x and an indirect
bandgap for a different x.
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Doped semiconductors
• Dopants: alter the concentration of mobile charge
carriers by many orders of magnitude.
• n-type: predominance of mobile electrons
n  p
•
p-type: predominance of holes
p  n
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C. Electron and hole
concentrations
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Density of states
The density of states describes the number of states at
each energy level that are available to be occupied.
3/ 2
(2mc )
1/ 2
c ( E ) 
(
E

E
)
, E  Ec
c
2 3
2
Density of states near band edges
(2mv )3/ 2
1/ 2
v ( E ) 
(
E

E
)
, E  Ec
v
2 3
2
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E
E
Rc(E)
E
Ec Ec
Ec
Eg
Ef
Ef Ef
Rv(E)
K
Density of states
(a)
(b)
(c)
Figure 15.1-7 (a) Cross section of the E-K diagram (e.g., in the direction of the K1 component with K2
and K3 fixed). (b) Allowed energy levels (at all K). (c) Density of states near the edges of the conduction
and valence bands. Pc(E)dE is the number of quantum states of energy between E and E+dE, per unit
volume, in the conduction band. P(E) has an analogous interpretation for the valence band.
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Probability of occupancy
Under the condition of thermal equilibrium, the probability
that a given state of energy E is occupied by an electron is
determined by the Fermi function.
f (E) 
1
exp[( E  E f ) / kBT ]  1
Ef: Fermi level, the energy level for which the probability of
occupancy is 1/2.
f(E) is not itself a probability distribution, and it does not integrate
to unity; rather it is a sequence of occupation probabilities of
successive energy levels.
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E
E
T>0K
T=0K
f(E)
Ec
Ef
Eg
Ev
Ec
Ec
Ef
Ef
Ev
Ev
1-f(E)
0
0.5
1
f(E)
0
0.5 1
f(E)
Figure 15.1-8 The Fermi function f(E) is the probability that an energy level E is filled with
an electron; 1-f(E) is the probability that it is empty. In the valence band, 1-f(E) is the
probability that energy level E is occupied by a hole. At T=0K, f(E)=1 for E<Ef, and f(E)=0
for E>Ef, i.e., there are no elctrons in the conduction band and no holes in the valence band.
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E
E
n(E)
Ec
E
ED
Donor level
Ef
Ev
0
1
f(E)
p(E)
Carrier
concentration
Figure 15.1-10 Energy-band diagram, Fermi function f(E), and concentrations of
mobile electrons and holes n(E) and p(E) in an n-type semiconductor.
The Fermi level is above the middle of the bandgap
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E
E
Ec
p(E)
E
Acceptor level
EA
Ef
n(E)
Ev
0
1
f(E)
Carrier
concentration
Figure 15.1-11 Energy-band diagram, Fermi function f(E), and concentrations
of mobile electrons and holes n(E) and p(E) in an p-type semiconductor
The Fermi level is below the middle of the bandgap
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Exponential approximation of the Fermi function
n( E )  c ( E ) f ( E )

n   n( E )dE
Ec
p( E )  v ( E )[1  f ( E )]
Ev
p   p( E )dE

When the Fermi level lies within the bandgap, but away from its edges
by an energy of at least several times , the equations above gives:
n  N c exp(
Ec  E f
k BT
) p  N v exp(
E f  Ev
k BT
) np  N c N v exp(
Eg
k BT
)
As Nc and Nv are
N c  2(2 mc k BT / h 2 )3/ 2
N v  2(2 mv k BT / h 2 )3/ 2
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Law of mass action
Eg
2 kBT 3
3/ 2
np  N c N v exp(
)  4(
) (mc mv ) exp(
)
2
k BT
h
k BT
Eg
is independent of the location of the Fermi level
Thus
n  p  ni for intrinsic semiconductor
We have
ni  ( N c N v )
1/ 2
exp(
Eg
k BT
)
Therefore the law of mass action can be written as:
np  ni2
The law of mass action is useful for determining the
concentrations of electrons and holes in doped semiconductors.
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D. Generation,
recombination, and injection
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Generation and recombination in thermal equilibrium
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Thermal equilibrium requires
that the process of generation
of electron-hole pairs must be
accompanied by a
simultaneous reverse
process of deexcitation,.
The electron-hole
recombination, ooocurs
when an electron decays
from the conduction band to
fill a hole in the valence band.
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Generation and recombination in thermal equilibrium
Radiative recombination: the energy released take the form of an
emitted photon.
Nonradiative recombination: transfer of the energy to lattice
vibrations or to another free electron.
Recombination may also occur indirectly via traps or defect centers,
they can act as a recombination center if it is capable of trapping both
the electron and the hole, thereby increasing their probability of
recombining.
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rate of combination= np
ε(cm3/s) a parameter that depends on the characteristics of the material
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Electron-hole injection
Define G0 as the rate of recombination at a given temperature:
G0 = n0 p0
Now let additional electron-hole pairs be
generated at a steady rate R
A new steady state:
We get:
G0 +R= np
R=
n

As
n  n0  n, p  p0  p
1
With  
 [(n0  p0 )  n]
τ: the electron-hole recombination lifetime
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Electron-hole injection
For an injection rate such that
n
n0  p0
1

 (n0  p0 )
In a n-type material, where n0>>p0,the recombination lifetime
τ≈1/εno is inversely proportional to the electron
concentration.
Similarly, for p-type material, τ≈1/εp0.
This simple formulation is not applicable when traps play an
important role in the process.
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Internal quantum efficiency
The internal quantum efficiency ηi is defined as the
ratio of the radiative electron-hole recombination rate to
the total recombination.
It determines the efficiency of light generation in a
semiconductor material.
 nr
r
r

i  
 
  r   nr  r  r   nr
For low to moderate injection rates,
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1
 r (n0  p0 )
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Internal quantum efficiency
εr
τr
(cm3/s)
Si
10-15
10ms
GaAs 10-30 100ns
τnr
τ
ηi
100ns
100ns
100ns
50ns
10-5
0.5
The radiative lifetime for Si is orders of magnitude larger than
its overall lifetime, mainly due to its indirect bandgap, result in
a small internal quantum efficiency
On the other hand, direct bandgap material as GaAs, the decay is
largely via radiative transitions, consequently larger internal
quantum efficiency, useful for fabricating light-emitting devices.
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E. Junctions
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• Homojunctions: junctions between differently doped
reguions of a semiconductor material
The p-n junction
• Heterojunctions: junctions between different
semiconductor materials
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n-type
Ef
Ef
Carrier
concentration
Electron energy
P-type
P
n
n
P
position
Figure 15.1-15 Energy levels and carrier concentrations of a ptype and n-type semiconductor before contact.
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Figure 15.1-16 A p-n junction in thermal equilibrium at T>0K. The depletion-layer,
energy-band diagram, and concentrations (on a logarithmic scale) of mobile electrons
n(x) and holes p(x) are shown as functions of position x. The built-in potrntial difference
V0 corresponds to an energy eV0, where e is the magnitude of the electron charge.
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1. The depletion layer contains only the fixed charges, the
thickness of the depletion layer in each region is inversely
proportional to the concentration of dopants in the region.
2. The fixed charges created a built-in field obstructs the diffusion
of further mobile carriers.
3. A net built-in potential difference V0 is established.
4. In thermal equilibrium there is only a single Fermi function for
the entire structure so that the Fermi levels in the p- and nregions must align.
5. No net current flows across the junction.
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Energy-band and carrier concentrations in a forward-biased p-n junction.
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The biased junction
Forward biased
A misalignment of the Fermi levels in the p- and n-regions
Net current i=isexp(eV/kBT)-is
Reverse biased
Net current ≈-is as V is negative in exp(eV/kBT) and |V|>>kBT/e
i  is [exp(
eV
)  1]
k BT
Acts as a diode with a current-voltage characteristic
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The p-i-n junction diode
• Made by inserting a layer of intrinsic (or lightly doped)
semiconductor material between the p- and n-type region.
• The depletion layer penetrates deeply into the i-region.
• Large depletion layer: small junction capacitance thus fast
response
• Favored over p-n diodes as photodiodes with better
photodetection efficiency
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The p-i-n junction diode
A p-i-n diode has the depletion layer penetrates deeply into the i-region
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Heterojunctions
Junctions between different semiconductor materials
are called heterojunctions.
P
P
Electron energy
Eg1
n
Eg3
Eg2
They can provide substantial improvement in the
performance of electronic and optoelectronic devices
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Junctions between materials of different bandgap create
localized jumps in the energy-band diagram
Electron energy
Eg1
Eg3
Eg2
Electron energy
Ef
A potential energy discontinuity provides a barrier that can be useful in preventing
selected charge carriers from entering regions where they are undesired.
This property used in a p-n junction can reduce the proportion of current
carried by minority carriers, and thus to increase injection efficiency
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• Discontinuities in the energy-band diagram created by two
heterojunctions can be useful for confining charge carriers to a
desired region of space
• Heterojunctions are useful for creating energy-band
discontinuities that accelerate carriers at specific locations
• Semiconductors of different bandgap type can be used in the
same device to select regions of the structure where light is
emitted and where light is absorbed
• Heterojunctions of materials with different refractive indices
create optical waveguides that confine and direct photons.
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Quantum Wells and Super-lattices
• When the Heterostructures layer thickness is comparable to, or
smaller than, the de Broglie wavelength of thermalized electrons
(≈50 nm in GaAs), the energy-momentum relation for a bulk
semiconductor material no longer applies.
• A quantum well is a double heterojunction structure consisting of
an ultrathin (≤50 nm) layer of semiconductor material whose
bandgap is smaller than that of the surrounding material.
• The sandwich forms conduction- and valence band rectangular
potential wells within which electrons in the conduction-band well
and holes in the valence-band well are confined.
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(a) Geometry of the quantum-well structure. (b) Energy-level diagram for electrons
and holes in a quantum well. (c) Cross section of the E-k relation in the direction of
k2 or k3. The energy subbands are labeled by their quantum number q1 = 1,2,...
The E-k relation for bulk semiconductor is indicated by the dashed curves.
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The energy levels Eq in one-dimensional infinite rectangular
well are determined by Schrodinger equation:
Eq 
2
(q / d ) 2
, q  1, 2,....
2m
The energy-momentum relation:
2
k2
E  Ec  Eq1 
, q1  1, 2,3,...,
2mc
where k is the magnitude of a two-dimensional k = (k2, k3) vector in the y-z plane
For the quantum well, k1 takes on well-separated discrete values.
As a result, the density of states associated with a quantum-well
structure differs from that associated with bulk material.
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In a quantum-well structure the density of states is obtained
from the magnitude of the two-dimensional wavevector (k2, k3)
 mc
 2 d , ............E  Ec  Eq1
c ( E )  
1
0, ...................E  E  E
c
q1

q1  1,2,...,
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quantum-well structure exhibits a substantial density of
states at its lowest allowed conduction-band energy level
and at its highest allowed valence-band energy level
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Multiquantum Wells and Superlattices
Multiple-layered structures of different semiconductor materials that
alternate with each other are called multiquantum-well (MQW) structures
A multiquantum-well structure fabricated from
alternating layers of AlGaAs and GaAs
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Quantum Wires and Quantum Dots
The density of states in different confinement configurations: (a) bulk; (b)
quantum well; (c) quantum wire; (d) quantum dot. The conduction and
valence bands split into overlapping subbands that become successively
narrower as the electron motion is restricted in more dimensions.
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A semiconductor material that takes the form of a thin wire of
rectangular cross section, surrounded by a material of wider
bandgap, is called a quantum-wire structure
2
k2
E  Ec  Eq1  Eq 2 
2mc
Eq1 
2
(q1 / d1 ) 2
, Eq 2 
2mc
2
(q2 / d 2 ) 2
, q1 , q2  1, 2,...,
2mc
Density of states
 (1/ d1d 2 )(mc1/ 2 / 2 )
, .....E  Ec  Eq1  Eq 2

1/ 2
c ( E )   ( E  Ec  Eq1  Eq 2 )

0, .......................................otherwise
.q1 , q2  1, 2,...,
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In a quantum-dot structure, the electrons are narrowly
confined in all three directions within a box of volume d1d2d3.
E  Ec  Eq1  Eq 2  Eq 3
E q1 
Eq 2 
Eq 3 
2
( q1 / d1 ) 2
,
2mc
2
(q2 / d 2 ) 2
,
2mc
2
(q3 / d3 ) 2
2mc
q1 , q2 , q3  1, 2,...,
Quantum dots are often called artificial atoms.
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Chapter 15.2
INTERACTIONS OF PHOTONS WITH
ELECTRONS AND HOLES
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Mechanisms leading to absorption and emission
of photons in a semiconductor:
•
•
•
•
•
Band-to-Band (Inter-band) Transitions.
Impurity-to-Band Transitions.
Free-Carrier (Intraband) Transitions
Phonon Transitions
Excitonic Transitions.
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Examples of absorption and emission of photons in a semiconductor. (a) Bandto-band transitions in GaAs can result in the absorption or emission of photons
of wavelength  g  hc0 / Eg  0.87  m . (b) The absorption of a photon of
wavelength A  hc0 / EA  14 m results in a valence-band to acceptor-level
transition in Hg-doped Ge (Ge:Hg). (c) A free-carrier transition within the
conduction band.
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Examples of absorption and emission of photons in a semiconductor. (a) Bandto-band transitions in GaAs can result in the absorption or emission of photons
of wavelength  g  hc0 / Eg  0.87  m . (b) The absorption of a photon of
wavelength A  hc0 / EA  14 m results in a valence-band to acceptor-level
transition in Hg-doped Ge (Ge:Hg). (c) A free-carrier transition within the
conduction band.
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Absorption coefficient for some semiconductor materials
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Absorption coefficient for some semiconductor materials
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
• For photon energies greater than the bandgap energy
Eg, the absorption is dominated by band-to-band
transitions
• Absorption edge:
The spectral region where the material changes from being
relatively transparent to strongly absorbing
• Direct-gap semiconductors have a more abrupt
absorption edge than indirect-gap materials
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Band-to-Band Absorption and Emission
Bandgap Wavelength:
1.24
g ( m) 
Eg (eV )
The quantity g is also called the cutoff wavelength.
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Absorption and Emission
(a) The absorption of a photon results in the generation of an electron-hole pair. This
process is used in the photodetection of light. (b) The recombination of an
electron-hole pair results in the spontaneous emission of a photon. Light-emitting
diodes (LEDs) operate on this basis. (c) Electron-hole recombination can be
stimulated by a photon. The result is the induced emission of an identical photon.
This is the underlying process responsible for the operation of semiconductor
injection lasers.
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Conditions for Absorption and Emission
• Conservation of Energy
E2  E1  h
• Conservation of Momentum
p2  p1  hv / c  h / ....or
.....k2  k1  2 / 
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Conditions for Absorption and Emission
• Energies and Momenta of the Electron and
Hole with Which a Photon Interacts
mr
E2  Ec 
(h  Eg )
mc
mr
E1  Ev 
(h  Eg )  E2  h
mv
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Optical Joint Density of States
The density of states  ( ) with which a photon of energy hv interacts
under conditions of energy and momentum conservation in a direct-gap
semiconductor is determined by:
 ( ) 
(2mr )3/ 2

2
(h  Eg )1/ 2 , h  Eg
And illustrated as follow:
The density of states with
which a photon of energy hv
interacts increases with hv  E g
in accordance with a squareroot law
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Photon Emission Is Unlikely in an
Indirect-Gap Semiconductor ！
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Photon emission in
an indirect-gap
semiconductor
The recombination of an electron near the bottom of the conduction band with a
hole near the top of the valence band requires the exchange of energy and
momentum. The energy may be carried off by a photon, but one or more phonons
are required to conserve momentum. This type of multiparticle interaction is unlikely.
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Photon Absorption is Not Unlikely in an
Indirect-Gap Semiconductor！
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Photon absorption in
an indirect-gap
semiconductor
The photon generates an excited electron and a hole by a vertical transition;
the carriers then undergo fast transitions to the bottom of the conduction
band and top of the valence band, respectively, releasing their energy in the
form of phonons. Since the process is sequential it is not unlikely.
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Rates of Absorption and Emission
the probability densities of a photon of energy h being emitted
or absorbed by a semiconductor material in a direct band-toband transition are mainly determined by three factors:
• Occupancy probabilities
• Transition probabilities
• Density of states
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Occupancy Probabilities
• Emission condition: A conduction-band state of energy E2
is filled (with an electron) and a valence-band state of energy
E1 is empty (i.e., filled with a hole)
• Absorption condition: A conduction-band state of energy
E2 is empty and a valence-band state of energy E1 is filled.
The probabilities are determined from the appropriate
Fermi functions fc ( E ) and fv ( E ) associated with the
conduction and valence bands of a semiconductor in
quasi-equilibrium
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
The probability f e ( ) that the emission condition is
satisfied for a photon of energy hv is:
f e ( )  f e ( E2 )[1  f v ( E1 )]
The probability
satisfied is :
f a ( ) that
the absorption condition is
f a ( )  [1  f c ( E2 )] f v ( E1 )
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Transition Probabilities
• Satisfying the emission/absorption occupancy condition
does not assure that the emission/absorption actually
takes place.
• These processes are governed by the probabilistic laws
of interaction between photons and atomic systems
Transition Cross Section:

 ( ) 
g ( )
8 r
2
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Probability density for the spontaneous emission:
Psp ( )d 
1
r
g ( ) d
Probability density for the stimulated emission:
2
Wi ( )d  v ( )d  v
g ( )d
8 r
If the occupancy condition for absorption is satisfied, the
probability density for the absorption is also fit this formula
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Overall Emission and Absorption Transition
Rates of Spontaneous Emission、Stimulated Emission
and Absorption：
1
rsp ( )   ( ) f e ( )
r
2
rst ( )  v
 ( ) f e ( )
8 r
2
rab ( )  v
 ( ) f a ( )
8 r
the occupancy probabilities f e (v) and f a (v) is determined by the
quasi-Fermi levels Efc and Efv
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Spontaneous Emission Spectral Density in
Thermal Equilibrium
A semiconductor in thermal equilibrium has only a single Fermi function so
h
f e ( )  f ( E2 )[1  f ( E1 )]  exp(
)
k BT
then
rsp ( )  D0 (h  Eg )
1/ 2
where
D0 
Fundamentals of Photonics
exp(
(2mr )3/ 2
 r
2
h  Eg
exp(
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k BT
Eg
k BT
), h  Eg
)
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Spectral density of the direct band-to-band spontaneous emission rate rsp (v )
(photons per second per hertz per cm3) from a semiconductor in thermal
equilibrium as a function of hv. The spectrum has a low-frequency cutoff at
v  Eg / h and extends over a width of approximately 2kBT / h .
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Gain Coefficient in Quasi-Equilibrium
2
The net gain coefficient is  0 ( ) 
 ( ) f g ( )
8 r
Where the Fermi inversion factor is given by
f g ( )  f e ( )  f a ( )  f c ( E2 )  f v ( E1 )
The gain coefficient may be cast in the form：
 0 ( )  D1 (h  Eg )1/ 2 f g ( ), h  Eg
with
D1 
2mr3/ 2  2
h 2 r
The sign and spectral form of the Fermi inversion factor f g (v ) are governed by the
quasi-Fermi levels Efc and Efv, which, in turn, depend on the state of excitation of
the carriers in the semiconductor.
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Absorption Coefficient in Thermal Equilibrium
A semiconductor in thermal equilibrium
has only a single Fermi level, so:
fc ( E )  fv ( E )  f (E ) 
1
exp[( E  E f ) / kBT ]  1
Therefore the absorption coefficient:
 ( )  D1 (h  Eg )1/ 2[ f ( E1 )  f ( E2 )]
If Ef lies within the band gap but away from the band edges
 ( ) 
Fundamentals of Photonics
2c 2 mr3/ 2
r
1
1/ 2
(
h


E
)
g
(h ) 2
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
The absorption coefficient  (v)(cm1 ) resulting from direct band-to-band
transitions as a function of the photon energy hv (eV) and wavelength 0 (  m)
for GaAs
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Refractive Index
• The ability to control the refractive index of a semiconductor is important in the design of many photonic
devices
• Semiconductor materials are dispersive the refractive
index is dependent on the wavelength
• The refractive Index is related to the absorption coefficient  (v ) inasmuch as the real and imaginary parts
of the susceptibility must satisfy the Kramers-Kronig
relations
• The refractive index also depends on temperature
and on doping level
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Refractive index for high-purity, p-type, and n-type GaAs
at 300 K, as a function of photon energy (wavelength).
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Refractive Indices of Selected Semiconductor Materials at T = 300 K for
Photon Energies Near the Bandgap Energy of the Material hv  Eg
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Electron-hole generation
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Electron-hole Recombination via a trap
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
p-n junction
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
The Biased Junction
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Photon emission in an indirect-gap semicondutor
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CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS
Photon absorption in an indirect-gap semicondutor
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