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POLARITON APPROACH TO UNDERSTANDING THE OPTICAL PROPERTIES OF CRYSTALS ( IN MEMORY OF KUN HUANG ). PETER Y YU Physics Department University of California Berkeley, CA, USA. DEDICATED TO THE MEMORY OF KUN HUANG. 1971 Beijing. Kun HUANG (1919-2005). 1991 An Arbor, Michigan. - PowerPoint PPT Presentation
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POLARITON APPROACH TO UNDERSTANDING THE OPTICAL PROPERTIES OF CRYSTALS(IN MEMORY OF KUN HUANG)
PETER Y YUPhysics DepartmentUniversity of CaliforniaBerkeley, CA, USA
Chinese University of Hong Kong, November 2012
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DEDICATED TO THE MEMORY OF KUN HUANG
Kun HUANG (1919-2005)
1971 Beijing
1991 An Arbor, Michigan
Chinese University of Hong Kong, November 2012
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80TH BIRTHDAY CELEBRATION OF K. HUANG IN BEIJING AT ICORS 2000
K. Huang and A. Rhys
Chinese University of Hong Kong, November 2012
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OUTLINE
Introduction Classical Electromagnetic (EM) Theory Polarization Waves in Ionic Crystals: Optical Phonons
Coupled EM-Optical Phonon Waves (Phonon-Polaritons)
Exciton as Polarization Waves & Exciton-Polariton Optical Properties (Transmission, Reflection and
Absorption, Emission & Scattering) described using the Polariton Approach
Confined Polariton Modes (Cavity-Polaritons) Conclusions
Chinese University of Hong Kong, November 2012
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INTRODUCTION
Microscopic Maxwell’s Equations in vacuum (in cgs):
E=B=0, xB=(1/c)(E/t), and
xE= - (1/c)(B/t) Combining these equations to eliminate B one obtains
a wave equation for E of the form:
2E-(1/c2)(2E/t2) =0 One solution of this equation is that of a traveling
plane wave:E(x,t)=Eoexp[i(2x/o)-t)] =angular frequency and o=wavelength of EM
oscillation wave in vacuum and c=speed of EM wave with =c(2/o).
Chinese University of Hong Kong, November 2012
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MAXWELL EQUATIONS IN MACROSCOPIC MEDIUM
A macroscopic medium contains many charges in the form of electrons and ions.
An applied E field can induce microscopic dipoles of moment pi.
The wavelength of the EM field ~m but typical separation of pi is ~nm so we can assume that the EM wave “sees” an “averaged dipole moment per unit volume”
P (polarization) =pi
Chinese University of Hong Kong, November 2012
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MACROSCOPIC MAXWELL EQUATIONS
For small E(r,t) we can assume that P(r,t) ~ E: P(r’,t’)=(r’,r,t’,t)E(r,t)drdt = linear electric susceptibility The field inside the medium = the external field
produced by some charge density field produced by P so Gauss’s Law is replaced by: (E+4P)= 4.
Introduce the electric displacement vector D: D =E+4P= E(1+4)=E. is the dielectric function and determines entirely the
linear response of the medium to the external field .
Chinese University of Hong Kong, November 2012
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MACROSCOPICMAXWELL EQUATIONS
The same assumptions and arguments can be applied to the magnetic response of the medium to an applied H field leading to the definition of the magnetic susceptibility and permeability .
From the Macroscopic Maxwell’s Equations one obtains the EM wave equation:
2E-(/c2)(2E/t2) =0. For non-magnetic medium =1. The velocity of
the EM wave is now v: v=c/(1/2).
Chinese University of Hong Kong, November 2012
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DISPERSION OF EM WAVE
When we study optics we define a refractive index: n=c/v so and n are related by: =n2.
To define both the direction of propagation and wavelength of EM wave we introduce the wave vector k whose magnitude |k|=2
=vk=ck/(1/2) is known as the dispersion relation for the EM wave inside the medium
k
Slope=c/(1/2)
Photon Dispersion
Chinese University of Hong Kong, November 2012
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POLARIZATION WAVES IN CRYSTALS
In ionic crystals, like NaCl or partially ionic semiconductors like GaAs, atoms are charged.
The oscillatory waves in crystals are quantized like SHO into phonons.
Some phonons, like sound waves, do not interact with EM wave.
Some phonons interact strongly with EM wave and are known as optical phonons.
Optical phonons are classified as transverse or longitudinal depending on the direction of the relative atomic displacement vector (u) relative to the wave vector k.
+ +++
_ _ _
ku
Transverse Optical Phonon
+ +++_ _ _
k
u
Longitudinal Optical Phonon
_
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POLARIZATION WAVES FORMED BY OPTICAL PHONON
EM wave is a transverse wave so it cannot couple to the Longitudinal Optical (LO) Phonon but it can couple strongly to the Transverse Optical (TO) Phonon.
When the EM and polarization waves have the same and k they can couple to form a mixed wave just like two coupled Simple Harmonic Oscillators.
k
Photon
TO PhononTO
Formation of Coupled Mode
LO Phonon
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COUPLED POLARIZATION-EM WAVE: POLARITON
Such coupled EM and Polarization waves were first named POLARITONS by Prof. Kun HUANG in 1951. Lattice Vibrations and Optical Waves in Ionic
Crystals. Nature, 167, 779 (1951); On the Interaction between the Radiation Field
and Ionic Crystals. Proc. Roy. Soc. Lond. A208, 352 (1951);
Chinese University of Hong Kong, November 2012
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POLARITON DISPERSION
When two SHO with same frequency are coupled by a spring, the resultant coupled oscillators will oscillate with two different normal mode frequencies.
In quantum mechanics when two degenerate states are coupled the degeneracy is split.
TWO COUPLED SIMPLE PENDULUM
TWO DEGENERATE STATES SPLIT BY INTERACTION
Chinese University of Hong Kong, November 2012
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POLARITON DISPERSION
Photon
TO PhononTO
Uncoupled Phonon and Photon Dispersion Curves
Phonon-Polariton Dispersion Curve of K. Huang (1951)Coupling
LO Phonon
Photon Dispersion at ~
Photon Dispersion at ~0
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POLARITON DISPERSION
One way to obtain the phonon-polariton dispersion is to calculate by classical EM theory by treating the TO phonon wave as a collection of identical charged SHO with frequency TO. The displacement of the ions u induced by an EM field E() is
u~1/M[TO2] where M = reduced
mass of the two oscillating ions
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POLARITON DISPERSION
The polarization P=Ne*u where N is the number of ion pairs per unit volume and e* is an “effective” charge of each ion.
The dielectric constant is given by: =1+{4N(e*)2/MTO
2[1+(TO2)]}
The valence electrons produce a background contribution to : . This can be measured by choosing to be>> TO so that the optical phonons are not able to follow the EM oscillation but is too low to excite electrons by interband transitions. includint electronic contribution is:
22
2*)(4)(
TOM
eN
Chinese University of Hong Kong, November 2012
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POLARITON DISPERSION
4N(e*2)/M can be replaced by measurable quantities such as =0).
2
2
0
*)(4
TOM
eN
or 2
0
2*)(4TOM
eN
Or by the LO frequency LO defined by setting LO)=0
22
2*)(40
LOTOM
eN
M
eNTOLO
222 *)(4
or
Chinese University of Hong Kong, November 2012
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POLARITON DISPERSION
The polariton dispersion is obtained by writing:
For k=>0 and =>0 the dispersion is given by: k2~(/c)2[or ck
when =>, k2=>(/c)2[] and =ck/(
(Curve a is for light in vacuo)
Curve c= LO
Chinese University of Hong Kong, November 2012
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QUANTUM MECHANICAL INTERPRETATION OF POLARITON
Without coupling: Ψphoton =photon wavefunction
Ψphonon =phonon wavefunction
With coupling:Ψpolariton=a Ψphoton +b Ψphonon
In quantum computation terminology a polariton is result of entanglement between photon and phonon.
b~1
a~0.5, b~0.5
a~1
a~0.5, b~0.5
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REFLECTION, ABSORPTION & TRANSMISSION OF POLARITONS
At sample surface external photon is reflected and converted into polaritons inside the medium.
Inside the sample polaritons will propagate through the crystal and emerges from the other side of the crystal as transmitted photon.
Absorption of light occurs inside the crystal when polaritons are scattered or dissipated inside the medium.
Dissipation of polaritons is dominated by the polarization component of the polariton.
Incident Photon
Reflected Photon
Transmitted PolaritonS
Reflected Polaritons
Polaritons are: Scattered, Trapped or Annihilated
Transmitted Photon
SAMPLE
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PHONON-POLARITON REFLECTIVITY
For TO<<LO
k2 and <0 so n is purely imaginary. Reflectivity R =1 since
This region is known as the Reststrahlen (German for residual rays) region.
In reality TO phonons are damped SHO. They decay into acoustic phonons in time scales of ~10-12 sec. This damping effect is included by replacing by i.
2
1
1
n
nR
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COMPARSION WITH EXPERIMENTS
The polariton concept is needed to explain why the LO and TO phonons become degenerate at k=0 (when k=0 there is no way to distinguish between TO and LO modes).
1 meV<=>12.396 cm-1
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EXPERIMENTAL DETERMINATION OF PHONON-POLARITON DISPERSION
k cannot be varied in absorption or reflection experiment so the phonon-polariton dispersion has to be determined by inelastic light scattering (or Raman scattering) using a laser. This experiment was in 1965 after the invention of the He-Ne laser by C. H. Henry & J. J. Hopfield: Raman scattering by Polaritons. Phys. Rev. Lett. 15, 964 (1965).
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POLARIZATION WAVES FORMED BY ELECTRONS : EXCIONS
In an insulator the filled valence bands (VB) are separated by a band gap from the empty conduction bands (CB).
Photons can excite electron from VB to CB leaving a positively charged hole in the VB.
This process can be expressed by: ω→e+h.
Mass of e and h: me and mh
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EXCITON AS POLARIZATION WAVE The e and h have opposite
charge so they are attracted to each other by Coulomb force to form a bound sate: exciton.
The exciton is like a “positronium” inside the crystal. Its bound states can be classified as 1s, 2s, 2p etc by their angular momentum.
Each exciton has a dipole moment and it moves like a particle with mass: M=me+mh.
Excitons form a polarization wave with wave vector K: K=ke+kh
Dispersion Curve of Exciton in the 1s bound state
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EXCITON-POLARITON
By analogy with the TO Phonon, the exciton wave can be longitudinal or transverse.
The coupling between the transverse exciton wave with the EM wave results in an exciton-polariton.
References: J. J. Hopfield, Theory of the
contribution of excitons to the complex dielectric constant of crystal. Phys. Rev. 112, 1555 (1958).
Strong Mixing of the Exciton with the Photon occurs at the point where their disperision curves intersect.
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EXCITON-POLARITON DISPERSION
The exciton-polariton dispersion can be obtained the same way as for TO phonon by replacing TO with the exciton frequency: X =
x(0)+[k2/(2M)]
Photon
Exciton
0
I
I
T
L
WAVEVECTOR
22
2
2
2
22
2
2
22
)0()0(
)/(41
2)0(
)/(41
Mk
MeN
Mk
MeNkc
XX
b
X
b
b Photon-like
Exciton-like
Exciton Mixed with Photon
Upper Branch
Lower Branch
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EXCITON-POLARITON DISPERSION
Differences between exciton-polariton and phonon-polariton di:
There is no Restrahlen Band in exciton-polaritons.
For ≥L the exciton-polariton dispersion has two polariton waves co-existing for the same frequency.
The existence of two polariton branches means that when EM wave crosses the interface between vacuum and the crystal, the continuity of E, D, B and H from the Maxwell Equations does not generate enough equations to determine uniquely all the fields inside the crystal. Typically an Additional Boundary Condition (ABC) is required. This ABC specifies what happens to the exciton at the interface.
The upper and lower polariton branches have quite different phase and group velocities (vph=/k while vgr=d/dk).
Photon
Exciton
0
I
I
T
L
WAVEVECTOR
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TRANSMISSION SPECTRUM OF CdS (1.2 m thick) from M. Dagenais, and W. Sharfin, Phys. Rev. Lett. 58, 1776-1779 (1987).
Oscillations at frequencies below each exciton are due to interference between the two polariton branches associated with the respective excitons. Exciton parameters obtained by fitting experiment:
Wavenumber (cm )-120500 20600 20700 20800
0
1
2
3
4 A Exciton
B Exciton
Experiment:Theory: - - - - assumes a damping linearly dependent on k.
Exciton A B
Transverse
Frequency (cm-1)
20583.8 20706.0
L-T splitting (cm-1)
15.8 14.5
Chinese University of Hong Kong, November 2012
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EMISSION LINESHAPE WHEN POLARITON EFFECT IS NEGLECTED
Emission from a crystal usually originates from a thermalized population of excitons.
Exciton emission have a sharp cutoff at the energy minimum ET (the k=0 exciton energy).
At low T the emission is a sharp -function.Defects and scattering with phonons will broaden this -function into a Gaussian (inhomogeneous broadening) or a Lorentzian (homogeneous broadening)
If T is large then the lineshape will be asymmetric since the high energy tail should fall off according to the Boltzmann distribution: exp[-(E-Eo)/kBT]
Low T
High T
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EXCITON-POLARITON “BOTTLENECK” & EMISSION
Toyozawa [Y. Toyoawa, Prog. Theor. Phys. Suppl. 12,111 (1959)] noted that
(1) the lower branch polariton has no energy minimum where polaritons can thermalize
(2) the lower branch has a “bottleneck” where the polariton lifetime reaches a maximum.
Above the region:C-D-E polariton decays very fast via exciton-phonon scattering. Around C-D-E exciton content decreases leading to a slow down in exciton-phonon scattering
Below C-D-E photon content increases and polariton lifetime become short again now as they escape very fast from the crystal as photons (radiative decay).
Photon
Exciton
0
I
I
T
L
WAVEVECTOR
AB
CD
EF
G
Bottleneck
Chinese University of Hong Kong, November 2012
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POLARITON EMISSION SPECTRA (F. Askary and P.Y. Yu, Solid State Comm. 47, 241 (1983))
THEORY (crystal:CdS)
Lower Branch
Population
- - - - Pekar ABC; Solid curve: Experiment
background
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INELASTIC SCATTERING OF POLARITONS BY PHONONS
Light scattering is responsible for the sky being blue (Rayleigh Scattering).Light can also be scattered inelastically by acoustic waves (Brillouin Scattering) or by optical phonons (Raman Scattering).
Since visible light does not couple strongly to low frequency optical phonons, the scattering is mediated by excitons.
Two possible regimes for exciton-mediated Raman scattering: Non-resonant regime when the photon energy is below the
bandgap so that excitons are excited only virtually Resonant regime when photon energy is above the band gap so
that exciton are excited by absorption of photons
Chinese University of Hong Kong, November 2012
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RAMAN SCATTERING MEDIATED BY EXCITONS
EXCITON PICTURE POLARITON PICTURE (J. J. Hopfield, Phys. Rev. 182,945 (1969); B. Bendow, J. L. Birman, Phys. Rev. B1, 1678 (1970);W. Brenig, R. Zeyher and J. L. Birman, Phys. Rev. B6, 4617 (1972).)
Non-resonant
Regime
Resonant
Regime
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BRILLOUIN SCATTERING OF POLARITONS BY ACOUSTIC PHONONS IN GaAs (R. G. Ulbrich & C. Weisbuch, Phys. Rev. Lett.38,865 (1977)
FOUR BRILLOUIN MODES ARE POSSIBLE SINCE THERE ARE 2 POLARITON BRANCHES TO SERVE AS INITIAL AND FINAL STATES
Chinese University of Hong Kong, November 2012
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CAVITY POLARITONS
In cavities the EM modes are confined in one or more directions but can propagate as a wave in the unconfined direction. Polaritons in microcavities are known as cavity polaritons,
Cavity polaritons are important for understand a class of lasers known as vertically integrated cavity surface emitting lasers (VICSEL) which contain microcavities formed by Bragg reflectors.
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A 2D MICROCAVITY FORMED BY BRAGG REFLECTORS
The vertical cavity is formed by multiple reflections in a 1D grating consisting of a periodic array of multiple layers whose thickness (given in nm) are chosen to satisfy the Bragg reflection condition
The GaAs Quantum Well is doped with electrons by Si donors. These electrons will occupy the lowest subband and can be excited into the next subband by photons with 136 meV energy. These excited electrons produce the polarization wave which can propagate within the plane while being confined vertically.
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REFLECTION FROM MICROCAVITY (Dimitri Dini, Rüdeger Köhler,Alessandro Tredicucci, Giorgio Biasiol, and Lucia Sorba. Phys. Rev. Lett. 90, 116401 (2003))
Reflectance curves of microcavity as a function of photon energy for various values of .
Experimental arrangement to access the polariton black arrows = optical path of the incident light The shape of the substrate allows the angle of incidence () to be varied.
Substrate=
undoped [100] GaAs
Microcavity The insert shows the cavity polariton dispersion curve. sin is proportional to the in-plane wave vector of the EM wave.
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RECENT DEVELOPMENTS: “POLARITON BULLETS”
“Motion of Spin Polariton Bullets in Semiconductor Microcavities”C. Adrados, et. al. Phys. Rev. Lett. 107, 146402 (2011). The dynamics of optical switching in semiconductor microcavities in the strong
coupling regime is studied by using time- and spatially resolved spectroscopy. The switching is triggered by polarized short pulses which create spin bullets of high polariton density. The spin packets travel with speeds of the order of 106 m/s due to the ballistic propagation and drift of exciton polaritons from high to low density areas.The speed is controlled by the angle of incidence of the excitation beams, which changes the polariton group velocity.
Experiment Theory
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CONCLUSIONS
More than 60 years after K. Huang introduced the “polariton” as a coupled photon-polarization wave, this concept remains the most fundamental one in our understanding of the interaction between EM wave and elementary excitations in crystals.