UNIT - III Light-Semiconductor Interaction · UNIT - III Light-Semiconductor Interaction. Table of...
Transcript of UNIT - III Light-Semiconductor Interaction · UNIT - III Light-Semiconductor Interaction. Table of...
UNIT - III Light-Semiconductor Interaction
Table of Contents
• Optical transitions in bulk semiconductors: absorption, spontaneous emission, and stimulated emission
• Joint density of states, Density of states for photons
• Transition rates (Fermi's golden rule)
• Optical loss and gain
• Photovoltaic effect, Exciton
• Drude model
Optical transitions in bulk semiconductors
Absorption • Spontaneous event in which an atom or molecule absorbs a photon from an incident optical field
• The asborption of the photon causes the atom or molecule to transition to an excited state
Spontaneous Emission
• Statistical process (random phase) – emission by an isolated atom or molecule
• Emission into 4π steradians
Stimulated Emission
• Same phase as “stimulating” optical field
• Same polarization
• Same direction of propagation
E2
hn
E1
2hn
Putting it all together…
• Assume that we have a two state system in equilibrium with a blackbody radiation field.
E2
E1
Stimulated emission
AbsorptionSpontaneous
emission
Joint Density of States
The probability of absorption or emission will depend on the overlap
and energy difference of the initial and final state, and the density of
these states.
In order to determine the probability or amplitude of the absorption we
must find the overlap of the initial and final wavefunctions.
Instead of single initial and final states in single-particle picture, we
have in principle a large density of final states -(k)
In quantum structures case
Eg
Probe
EgHHx
Pump
E2
E1
HH1
LH1
K||
Eg
Choice of the wavefunctions for the initial and final states
Two different kinds of possibilities in quantum structure
Transitions between the valence and conduction bands
Transitions between the quantum-confined states within a given band,
so-called "intersubband“ transitions
E
V.B
C.B
Probe
HH1
LH1
E2
E1
Eg
Barrier QW Barrier
Emission
LHx HHx
e1-e2 ISBT
Resonant optical transition
Photovoltaic effectThe photovoltaic effect is the creation of voltage and electric currentin a material upon exposure to light and is a physical and chemicalphenomenon.
The photovoltaic effect is closely related to the photoelectric effect.In either case, light is absorbed, causing excitation of an electron orother charge carrier to a higher-energy state. The main distinction isthat the term photoelectric effect is now usually used when theelectron is ejected out of the material (usually into a vacuum) andphotovoltaic effect used when the excited charge carrier is stillcontained within the material. In either case, an electric potential (orvoltage) is produced by the separation of charges, and the light has tohave a sufficient energy to overcome the potential barrier forexcitation. The physical essence of the difference is usually thatphotoelectric emission separates the charges by ballistic conductionand photovoltaic emission separates them by diffusion, but some"hot carrier" photovoltaic device concepts blur this distinction.
55Excitonic Effect : Two particle (e-h) interaction
Absorption via Excitons
Electron-Hole interaction: Excitons
Let the electric field of optical wave in an atom be
E=E0e-iwt
the electron obeys the following equation of motion
EXXX emdtdm
dtdm
2
02
2
w
X is the position of the electron relative to the atom
m is the mass of the electron
w0 is the resonant frequency of the electron motion
is the damping coefficient
Classical Electron Model ( Lorentz Model)
+ -
w0
X
E
The solution is tieim
e w
www
)( 22
0
0EX
The induced dipole moment is
EEXp www
)( 22
0
2
im
ee
)( 22
0
2
www
im
e
is atomic polarizability
The dielectric constant of a medium depends on the manner in which the atoms are assembled. Let N be the number of atoms per unit volume. Then the polarization can be written approximately as
P = N p = N a E = e0 c E
Classical Electron Model ( Lorentz Model)
)(1
22
00
2
0
2
www
im
Nen
)(21
22
00
2
www im
Nen
If the second term is small enough then
The dielectric constant of the medium is given by
= 0 (1+c) = 0 (1+N/ 0)
If the medium is nonmagnetic, the index of refraction is
n= ( /0)1/2 = (1+N/ 0 )1/2
Classical Electron Model ( Lorentz Model)
])[(2])[(2
)(1
22222
00
2
22222
00
22
0
2
www
w
www
ww
m
Nei
m
Neinnn ir
The complex refractive index is
-6 -4 -2 0 2 4 6
(w-w0)/
nr
ni
Normalized plot of n-1 and k versus ww0
])2/()[(8])2/()[(4
)(1
22
000
2
22
000
0
2
www
www
ww
m
Nei
m
Neinn ir
at w ~w0 ,
Classical Electron Model ( Lorentz Model)
j jj
j
i
f
mNen
)(1
220
22
www
For more than one resonance frequencies for each atom,
Zfj
j
Classical Electron Model ( Drude model)
If we set w0=0, the Lorentz model become Drude model. This model can be used in free electron metals
)(1
2
0
22
ww im
Nen
0
2
n
ir innn
21 i
0
2/1
1
2/12
2
2
1 /]})[(21{ rn
0
2/1
1
2/12
2
2
1 /]})[(21{ in
By definition,
We can easily get:
Relation Between Dielectric Constant and Refractive Index
Real and imaginary part of the index of refraction of GaN vs. energy;
An Example to Calculate Optical Constants
The real part and imaginary part of the complex dielectric function w) are not independent. they can connected by Kramers-Kronigrelations:
P indicates that the integral is a principal value integral.
K-K relation can also be written in other form, like
www
ww
w
dP0 22
201
)(2)(
www
w
ww
dP0 22
012
)(2)(
Kramers-Kronig Relation
dPn0 2)/(1
)(1)(
Typical experimental setup
( 1) halogen lamp;
(2) mono-chromator; (3) chopper; (4) filter;
(5) polarizer (get p-polarized light); (6) hole diaphragm;
(7) sample on rotating support (); (8) PbS detector(2)
A Method Based on Reflection
xirx
xirx
xx
xxp
kinnk
kinnk
knkn
knknr
1
2
2
1
2
2
1
2
22
2
1
1
2
22
2
1
)(
)(
Snell Law become:
Reflection of p-polarized light
sin2
21 zz kk
2/122
1 ])2[( xk
2/1222
2 ])()2[( irx innk
Reflection coefficient:
In this case, n1=1, and n2=nr+i n i
Reflectance:R(1, , nr, n i)=|r p|2
From this measurement, they got R, for each wavelength ,Fitting the experimental curve, they can get nr and n i .
Calculation
E1
E1'
n1=1
z
x
n2=nr+i n i
Results Based on Reflection Measurement
22
0
021
EE
EEn d
r
)ln(122
24
3
0
2
0
2
EE
EEE
E
EE
E
En
fddr
FIG. 2. Measured refractive indices at 300 K vs.
photon energy of AlSb and AlxGa1-xAsySb1-y
layers lattice matched to GaSb (y~0.085 x).
Dashed lines: calculated curves from Eq. ( 1);
Dotted lines: calculated curves from Eq. (2)
E0: oscillator energyEd: dispersion energyE: lowest direct band gap energy
22
0
22 EEE f
)(222
0
3
0
EEE
Ed
Single effective oscillator model
(Eq. 1)
(Eq. 2)