Lecture 8 – Light-Matter Interaction
Part 2
Basic excitation and coupling
EECS 598-002 Winter 2006
Nanophotonics and Nano-scale Fabrication
P.C.Ku
Schedule for the rest of the semester
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Introduction to light-matter interaction (1/26):
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Basic excitations and measurement of ε(r). (1/31)
Structure dependence of ε(r) overview (2/2)
Surface effects (2/7 & 2/9):
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Surface EM wave
Surface polaritons
Size dependence
Case studies (2/14 – 2/21):
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How to determine ε(r)?
The relationship to basic excitations.
Quantum wells, wires, and dots
Nanophotonics in microscopy
Nanophotonics in plasmonics
Dispersion engineering (2/23 – 3/9):
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Material dispersion
Waveguide dispersion (photonic crystals)
EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku
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Last time
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We learned:
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To determine ε(r), we need to study how the microscopic
interaction between atoms/electrons with the light.
This interaction is similar to coupling of two SHO’s.
The only details we need to know are the interaction near
resonances of basic excitations.
The rest of the information needed to complete the calculation of
ε(r) is through the Kramers-Kronig relation.
EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku
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Today
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Basic excitations by photons Æ polaritons
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Plasmons (last time)
Phonons
Excitons, biexcitons, etc.
Measurement of ε(r)
Ref: D. L. Mills and E. Burstein, “Polaritons,” Rep. Prog. Phys., 37 (1974) 817.
P. Y. Yu and M. Cardona, Fundamentals of Semiconductors, 2nd ed.,
Springer-Verlag (1999) chapters 6 and 7.
EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku
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Review of the concept of polaritons
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In a dielectric medium:
⎡
⎤
2
⎢
⎥
ω pn
ε (ω ) = ε 0 ⎢1 − ∑
⎥
2
2
⎢ n basic (ω − ω0 n ) + iγ nω ⎥
⎢⎣ excitations
⎥⎦
or oscillators
N n qn2
where ω ≡
mnε 0
2
pn
ω p2
=ε ∞ − ε 0 2
(ω − ω02 ) + iγω
„
where the index denotes the n-th kind of basic excitation
or SHO.
QM analogue:
2
exˆ
nq
ω =
→
ε 0m
ε0
2
2
p
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Transverse and longitudinal vibrations
Longitudinally vibrated SHO’s
Vertically vibrated SHO’s
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Transverse and longitudinal polaritons
For homogeneous media:
K
∇⋅ D = 0
K K
K
⇒ ∇ ⋅ (ε E ) = 0 ⇒ ε k ⋅ E = 0
K K
⇒ k ⋅ E = 0 or ε = 0
Normally EM wave couples only to transverse SHO’s unless
the dielectric constant vanishes.
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Longitudinal polaritons
ε=
k2
ω µ0
ω
2
=0
EM wave can couple to
the longitudinal vibration.
Photons
In free space
ω02 + ω p2
ω0
k
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Phonons
M
m
a
vn
un
⎧ mun = −α (un − vn −1 ) − α (un − vn +1 )
⎨
⎩ Mvn = −α ( vn − un −1 ) − α ( vn − un +1 )
optical
ω
By periodicity:
⎧⎪un = u exp [i (2nka − ωt )]
⎨
⎪⎩ vn = v exp [i (2( n + 1)ka − ωt ) ]
acoustic
⎧ −mω 2 u = α ⎡ v ( eika + e − ika ) − 2u ⎤
⎪
⎣
⎦
⇒⎨
ika
2
− ika
⎪⎩ − M ω v = α ⎡⎣ u ( e + e ) − 2v ⎤⎦
⎛m+M
⇒ ω2 = α ⎜
⎝ mM
k
π /a
1/ 2
⎡⎛ m + M ⎞ 2(1 − cos ka ) ⎤
⎞
±
α
⎢⎜
⎥
⎟
⎟ −
mM
mM
⎠
⎝
⎠
⎢⎣
⎥⎦
2
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Orders of magnitude
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At optical frequencies, k=ω/c~107.
For typical crystal lattice, π/a~1010.
Only optical phonons couple to the light.
k ≈0
For optical branch:
M
u
Can generate the dipole moment
≈−
m
v
For acoustic branch:
u
≈1
v
EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku
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Examples of phonon dispersion curves
silicon
GaAs
Taken from P. Yu and M. Cardona.
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Dispersion curve for phonon polaritons
Free-space-photon-like
Photons
In free space
ω
ω02 + ω p2 = ωLO
ω0 = ωTO
k
Photon-like but with
strong phonon influence
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Raman processes
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When light is not at the infrared frequency, phonons can
still participate in the inelastic processes with light Æ
Raman processes.
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The scattered light has a frequency shift w.r.t to the
incident light due to its energy lost (or gain) to phonons.
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Energy lost: Stokes process
Energy gain: Anti-Stokes process
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Excitons (two-level system)
el
The Coulomb interaction b/w electron and
hole makes the exciton. Exciton is like a
hydrogen atom.
hole
Exciton absorption
Coulomb
enhancement
Eg
E
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Hot carriers relaxation processes
Carrier capture
Phase relaxation
T2~100fs - ps
k
Thermalization
Recombination (T1~ns)
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Size dependence (quantum confinement)
g(E) = Density of states
3D
g(E)
2D
g(E)
Eg
wire
sheet
bulk
E
1D
g(E)
Eg
dot
E
0D
g(E)
Eg
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Eg
E
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Exciton absorption in low-dim structures
Exciton binding energy:
2D:
EB2 d,n =1 = 4 EB3d,n =1
Exciton absorption:
2D:
1D:
1D:
a0=exciton Bohr radius~100A
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Excitons (three-level system)
If we consider the polariton
corresponding to exciton 1:
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Electromagnetically induced transparency
α
1
ωp= ω31
3
ωs= ω21
n
2
EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku
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Measurement of ε(r) - ellipsometry
ellipsometry
Sensitive to:
1. Film thickness
2. Surface roughness
3. Anisotropy
Need to know underlying
composition of materials.
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