Optical Properties of Lattice Vibrations
z For a collection of classical charged Simple
Harmonic Oscillators, the dielectric function is
given by:
z Where Ni is the number of oscillators with
frequency ωi and e is the charge of the
oscillators
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Optical Properties of Lattice Vibrations
z For optical phonons it is necessary to consider that
the polarization is now in the form of a wave and also
the effect of retardation.
z A wave has a wave vector k which determines
whether the wave is transverse ( E┴k) or longitudinal
(E||k)
z In the absence of free charges the medium has to
satisfy the Gauss theorem: ∇•D=0 or ε(k•Eo)=0.
z If the EM wave is a plane wave described by:
E=Eoexp[i(k•r-ωt)] then the equation ε(k•Eo)=0 can
be satisfied either by:
ε=0 or (k•Eo)=0
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Optical Properties of Lattice Vibrations
zWhen (k•Eo)=0 the EM wave is
transverse and since P~E the
polarization wave induced is also
transverse.
zFor a longitudinal wave (k•Eo)≠0 so
ε=0. The frequency of this longitudinal
wave will be denoted by ωL and it is
given by the frequency when ε(ωL)=0
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Optical Properties of Lattice Vibrations
z The classical expression for the dielectric function due
to optical phonon with oscillation frequency ωT is:
4πNQ 2
ε (ω ) = 1 +
M ωT 2 − ω 2
z To include the contribution to ε due to the valence
electrons we will add a constant ε∞ to ε:
(
ε (ω ) = ε ∞ +
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)
4πNQ 2
(
2
M ωT − ω 2
2009 Lecture 16
)
4
Optical Properties of Lattice Vibrations
4πNQ 2
z ε(ωL)=0 => ε (ω L ) = ε ∞ + M (ω 2 − ω 2 ) = 0
T
L
z Solving this equation=>
z Using this equation we can express ε in terms of
ωT and ωL:
(LST Relation)
z The longitudinal electric field of the longitudinal phonon is
given by:
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2009 Lecture 16
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Optical Properties of Lattice Vibrations
z In order to include retardation effect we will take
the classical approach by going back to the
Maxwell’s Equations and derive the wave
equation in the medium:
∇2E-(ε/c2)(∂E/∂t)2=0
Next substitute in the plane wave solution for E:
E=Eoexp[i(k•r-ωt)], we obtain the photon
dispersion:
2
⎛ω ⎞
2
k =⎜ ⎟ ε
⎝c⎠
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2009 Lecture 16
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Optical Phonon Polaritons
zSubstituting into this
expression the
optical phonons
contribution to ε we
obtain the polariton
dispersion :
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2009 Lecture 16
7
Coupled EM-Polarization Waves (Polariton)
I
Upper
Branch
Photon
ωL
Lower Branch
I
Two degenerate waves: photon
and exciton
Exciton
ωT
0
WAVEVECTOR
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Any Interaction due to Her will split
this degeneracy. The results are
two “mixed waves” or polariton.
There are two branches to the
polariton dispersion (upper branch
and lower branch)
2009 Lecture 16
8
Exciton-Polariton
Exciton-Polariton Dispersion
ε =Aεb +
Exciton
4πNX (e)2
(
mX ωX2 − ω2
CdS
)
Combine
with
ωX = ωx(0)+[hk2/(2mx)]
c2k 2
2
εbω
= 1+
4πNX e2 /(εbmx )
2
⎡
⎛ hk 2 ⎞⎤
⎟⎥ − ω2
⎢ωX (0) + ⎜⎜
⎟
⎢⎣
⎝ 2mX ⎠⎥⎦
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≈ 1+
4πNX e2 /(εbmx )
⎡ 2
⎛ hk 2 ⎞⎤
⎟⎥ − ω2
⎢ωX (0) + ωX (0)⎜⎜
⎟
⎢⎣
⎝ mX ⎠⎥⎦
2009 Lecture 16
9
Exciton-Polaritons Transmission in CdS
4
3
B
Exciton
A
Exciton
2
Experiment
Theory
1
0
20500
20600
Wavenumber
20700
(cm-1 )
20800
Experimental transmission Spectrum of CdS from Dagenais, M. and Sharfin,
W. Phys. Rev. Lett. 58, 1776-1779 (1987). Oscillations due to interference
between the two polariton branches
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10
Absorption in the Polariton Picture
z Polariton is a propagating wave in a medium.
External wave is converted into a polariton inside the
medium with a reflection and transmission
coefficients.
z Absorption occurs when polaritons are scattered or
disappear inside the medium (note the similarity
between this case and the Landauer-Büttiker
formalism for transport of charges)
z Since excitons and phonons are more strongly
scattered, dissipation of polaritons is usually
dominated by the polarization component of the
polariton
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Cavity Polaritons
z Polaritons (as a form of coupled mode) can also exist
in micro-cavities
z In cavities the EM modes are confined in one or more
directions but in most cases can propagate as a
wave in at least one direction. When these cavity
modes (either standing or guided waves) resonate
with excitons in the medium, coupled EM-polarization
modes, known as cavity polaritons, are formed.
z Cavity polaritons are important for understand the
properties of a class of lasers known as vertically
integrated cavity surface emitting lasers (VICSEL)
which contain micro-cavities formed by Bragg
reflectors.
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2009 Lecture 16
12
Polariton
Reflection
from a Microcavity
Reference: Phys. Rev. Lett. 90, 116401 (2003)
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2009 Lecture 16
13
Polariton
Reflection from
a Micro-cavity
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14
Theory of Emission
z Classically emission of light is a common everyday
experience
z Classical theory: an oscillating dipole will radiate EM
wave so the medium must be excited first
z Emission excited by
{ lightÙphotoluminescence
{ ElectronsÙelectroluminescenc
{ HeatingÙthermoluminescence
{ Sound waveÙsonoluminescence
z The semiclassical approach we have adopted
cannot explain spontaneous emission since there is
no EM field before emission making the interaction
Hamiltonian between electron and EM field=0
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Theory of Emission
z This problem is solved when we quantize the EM
wave into photons.
z Probability of creating a photon is proportional to
(1+N) where N is the photon occupancy:
N=1/[exp(hω)/KbT-1]. The constant of proportionality
is same as that for annihilation of a photon or
absorption. Notice that even if N=0 there is still a nonzero probability of emitting a photon.
z Thus if the probability of absorbing a photon is given
by BN, the probability for spontaneous emission is A
while the probability for stimulated emission is AN. A
and B are known as the Einstein’s A and B
coefficients and are related to each other by the
photon energy density. The reason is because the
incident EM wave usually has a well-defined k but in
emission the EM wave is emitted in all directions.
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16
Einstein Theory of Stimulated Emission
z Einstein did not believe in Quantum Mechanics but
he was able to derive the QM result before the
development of QM.
z Einstein used Boltzmann’s theory of statistical
mechanics and Planck’s radiation laws to argue that
an electron with two levels cannot be in thermal
equilibrium with a radiation field without stimulated
emission
z The reason is: rate of absorption is proportional to
intensity of light. If emission is entirely due to
spontaneous emission, its rate is independent of
intensity. By increasing the intensity one can make
the excited state population larger than the ground
state populationÙ violation of Boltzmann’s result that
the excited state population is smaller than the
ground state population by the factor : exp[-ΔE/KbT].
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Einstein’s A & B coefficients
z Let |n> and |m> represent 2 non-degenerate
levels with En>Em. The rate for absorption
(transition from |m> to |n>) for unit of incident
EM energy density is Bmn and is equal to the
rate for the reverse process (stimulated
emission) Bnm.
z The rate for spontaneous emission is given
by Anm and since spontaneous emission is
spread over all directions the emission rate
per unit EM energy density=Anm/ρ(ν) where
ρ(ν) is energy density of the EM wave with
frequency between ν and ν+Δν and is given
by: Np(hν) (8πν2) (n/c)3.
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Einstein’s A & B coefficients
z Using the Principle of Detailed Balance
Einstein obtained:
Bmn = Bnm and Anm =(8πhν3) (n/c)3 Bnm
z From Einstein’s result the total rate of
emission is:
Anm + Bnm ρ= Anm(1+ρBnm /Anm)
= Anm(1+Np)
The rate of absorption is Bnm ρ= Anm Νp
Τhis is exactly the same result as obtained by
QM!
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2009 Lecture 16
19
Emission Processes in Semiconductors
z Based on Einstein’s result we
expect the emission probability R
can be determined from the
absorption coefficient α
(Roosbroek-Shockley relation):
z Note that the indirect edge is
almost as strong as the
direct edge in emission
because of the Boltzmann
factor
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2009 Lecture 16
Ge
300 K
10 -2
10 12
P (v )ρ (v )
k
1011
10 -3
10
10
10 -4
10 9
10 -5
0.62
0.70
0.80
0.90
1.00
hν (eV)
20
Photoluminescence Processes in
Semiconductors
z PL involves 3 distinct
steps:
{Real Excitation of e-h pair
via absorption
{Relaxation of e-h to
lowest energy states
(favored by the
Boltzmann factor) and
equilibrium with phonons
{Emission via radiative
recombination of e-h
pairs
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21
Emission Processes in Semiconductors
z In pure semiconductors emission is intrinsic and
dominated by recombination of free excitons at low T
and conduction band-to-valence band transition at
high T
z In extrinsic semiconductors emission is dominated by
defects and impurities:
{ Excitons bound to donors, acceptors or neutral centers
{ Free-to-bound transitions such as donor=>valence band
{ Donor-acceptor pair (DAP) transitions
z Since the Boltzmann factor tends to favor low energy
states emission is a very sensitive probe of defect
and impurity states
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22
Free-to-bound transitions in Doped Semiconductors
and Mott transition
z Electrons in Shallow Donors have Bohr radii
typically of the order of tens of lattice
constants (or ~ several nm to 10 nm).
z When their concentration is high enough
these electron wave function will overlap and
electrons from one donor can hop to another.
In another word the semiconductor becomes
metallic and the discrete impurity levels will
form bands, known as impurity bands. This
transition from an insulating to a metallic state
is known as Mott Transition. It is a classic
example of a many-body effect and quantum
phase transition (a transition which occurs
even at T=0).
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Free-to-bound transitions in Doped Semiconductors
z The emission spectra
of Zn-doped p-type
GaAs as a function
of doping
concentration shows
2 many-body effects:
z Band gap shrinkage
(or renormalization)
z Formation of a Fermi
sea of electrons
1.9 X 10 18
1.25 X 10
3.7 X 10 17
20
1.3 X 10 19
1.40
1.50
Photon energy
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2009 Lecture 16
Fermi
Level
(eV)
24
DAP transitions
z In a compensated semiconductor there are
both donors and acceptors
z Recombination of electrons at a donor with a
hole at an acceptor exhibits also a final state
interaction:
Do+Ao=>hω+ D++Az Since the donor and acceptor becomes
charged in the final state there is a Coulomb
attraction between them. As a result the
emitted photon energy hω is given by:
z hω =Eg-ED-EA+e2/(εoR)
z Where R= distance between donor and
acceptor
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DAP transitions
zSince the donors and acceptors can
only sit on specific lattice sites R is
discrete leading to sharp DAP lines
zThe position of the DAP lines depends
on the lattice constant and on whether
the donor and acceptor sit on the same
sublattice (Type 1) or different
sublattices (Type 2)
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Type I DAP transitions
Numbers label
the Shells
counting from
either D or A
Notice how the
series converge
towards lower
energy!
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Type II DAP transitions
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DAP transitions
z Curve CÙCoulomb
interaction alone
z Curve C+vdWÙ Coulomb
interaction plus van der
Waals interaction between
D and A pair in the initial
state
z Parameters obtained from
these fits: ε and (ED+EA)
z DAP transitions provide
the most accurate
determination of defect
energies and separation
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29