POLARITON APPROACH TO UNDERSTANDING THE OPTICAL PROPERTIES OF CRYSTALS ( IN MEMORY OF KUN HUANG )

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


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80TH BIRTHDAY CELEBRATION OF K. HUANG IN BEIJING AT ICORS 2000

K. Huang and A. Rhys


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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


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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).


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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


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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 .


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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).


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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


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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);


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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


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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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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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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


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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


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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


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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


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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


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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


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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


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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.

POLARITON APPROACH TO UNDERSTANDING THE OPTICAL PROPERTIES OF CRYSTALS ( IN MEMORY OF KUN HUANG )

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