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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 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.

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

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  1. 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

  2. DEDICATED TO THE MEMORY OF KUN HUANG 1971 Beijing Kun HUANG (1919-2005) 1991 An Arbor, Michigan Chinese University of Hong Kong, November 2012

  3. 80TH BIRTHDAY CELEBRATION OF K. HUANG IN BEIJING AT ICORS 2000 K. Huang and A. Rhys Chinese University of Hong Kong, November 2012

  4. 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

  5. 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(2px/lo)-wt)] • w=angular frequency and lo=wavelength of EM oscillation wave in vacuum and c=speed of EM wave with w=c(2p/lo). Chinese University of Hong Kong, November 2012

  6. 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 l ~mm 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)=Spi Chinese University of Hong Kong, November 2012

  7. MACROSCOPIC MAXWELL EQUATIONS • For small E(r,t) we can assume thatP(r,t) ~ E: • P(r’,t’)=(r’,r,t’,t)E(r,t)drdt •  =linearelectric susceptibility • The field inside the medium = the external field produced by some charge density r+ field produced by P so Gauss’s Law is replaced by:  (E+4pP)= 4pr. • Introduce the electric displacement vector D: • D =E+4pP= E(1+4p)=eE. • e is the dielectric function and determines entirely the linear response of the medium to the external field e. Chinese University of Hong Kong, November 2012

  8. 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 permeabilitym. • From the Macroscopic Maxwell’s Equations one obtains the EM wave equation: • 2E-(me/c2)(2E/t2) =0. • For non-magnetic medium m=1. The velocity of the EM wave is now v: v=c/(e1/2). Chinese University of Hong Kong, November 2012

  9. DISPERSION OF EM WAVE • When we study optics we define a refractive index: n=c/v so e and n are related by: e=n2. • To define both the direction of propagation and wavelength of EM wave we introduce the wave vector k whose magnitude |k|=2p/l • w=vk=ck/(e1/2) is known as the dispersion relation for the EM wave inside the medium w Photon Dispersion Slope=c/(e1/2) k Chinese University of Hong Kong, November 2012

  10. 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. + + k + + u _ _ _ Transverse Optical Phonon u _ _ _ _ + + + + k Longitudinal Optical Phonon Chinese University of Hong Kong, November 2012

  11. 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 w and k they can couple to form a mixed wave just like two coupled Simple Harmonic Oscillators. w LO Phonon wTO TO Phonon Formation of Coupled Mode Photon k Chinese University of Hong Kong, November 2012

  12. 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

  13. 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

  14. POLARITON DISPERSION Phonon-Polariton Dispersion Curve of K. Huang (1951) Uncoupled Phonon and Photon Dispersion Curves Coupling w Photon Dispersion at w~ TO Phonon wTO LO Phonon Photon Photon Dispersion at w~0 Chinese University of Hong Kong, November 2012

  15. POLARITON DISPERSION • One way to obtain the phonon-polariton dispersion is to calculate e by classical EM theory by treating the TO phonon wave as a collection of identical charged SHO with frequency wTO. The displacement of the ions u induced by an EM field E(w) is • u~1/M[wTO2-w2] where M = reduced mass of the two oscillating ions Chinese University of Hong Kong, November 2012

  16. 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: e=1+{4pN(e*)2/MwTO2[1+(wTO2-w2)]} • The valence electrons produce a background contribution to e: e. This e can be measured by choosing w to be>> wTO so that the optical phonons are not able to follow the EM oscillation but w is too low to excite electrons by interband transitions. e includint electronic contribution is: Chinese University of Hong Kong, November 2012

  17. POLARITON DISPERSION 4pN(e*2)/M can be replaced by measurable quantities such as e0=e(w=0). or Or by the LO frequency wLO defined by setting e(wLO)=0 or Chinese University of Hong Kong, November 2012

  18. POLARITON DISPERSION (Curvea is for light in vacuo) • The polariton dispersion is obtained by writing: • For k=>0 and w=>0 the dispersion is given by: k2~(w/c)2[e+(e0-e)] or w=ck/(e0)1/2 • when w=>, k2=>(w/c)2[e] and w=ck/(e)1/2 Curve c= LO Chinese University of Hong Kong, November 2012

  19. 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 Chinese University of Hong Kong, November 2012

  20. 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. SAMPLE Reflected Polaritons Incident Photon Transmitted PolaritonS Transmitted Photon Reflected Photon Polaritons are: Scattered, Trapped or Annihilated Chinese University of Hong Kong, November 2012

  21. PHONON-POLARITON REFLECTIVITY • For wTO<w<wLO k2 and e<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 w2 by w2-iwg. Chinese University of Hong Kong, November 2012

  22. 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 Chinese University of Hong Kong, November 2012

  23. 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). Chinese University of Hong Kong, November 2012

  24. 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 Chinese University of Hong Kong, November 2012

  25. 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 Chinese University of Hong Kong, November 2012

  26. 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. Chinese University of Hong Kong, November 2012

  27. EXCITON-POLARITON DISPERSION The exciton-polariton dispersion can be obtained the same way as for TO phonon by replacing wTO with the exciton frequency: wX = wx(0)+[k2/(2M)] Upper Branch Lower Branch Exciton-like Exciton Mixed with Photon Photon-like Chinese University of Hong Kong, November 2012

  28. EXCITON-POLARITON DISPERSION Differences between exciton-polariton and phonon-polariton di: • There is no Restrahlen Band in exciton-polaritons. • For w≥wLthe 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=w/k while vgr=dw/dk). Chinese University of Hong Kong, November 2012

  29. TRANSMISSION SPECTRUM OF CdS (1.2 mm 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: A Exciton B Exciton Experiment: Theory: - - - - assumes a damping linearly dependent on k. Chinese University of Hong Kong, November 2012

  30. 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 d-function.Defects and scattering with phonons will broaden this d-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] High T Low T Chinese University of Hong Kong, November 2012

  31. 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). A B C D Bottleneck E F G Chinese University of Hong Kong, November 2012

  32. POLARITON EMISSION SPECTRA (F. Askary and P.Y. Yu, Solid State Comm. 47, 241 (1983)) THEORY (crystal:CdS) Lower Branch Population background - - - - Pekar ABC; Solid curve: Experiment Chinese University of Hong Kong, November 2012

  33. 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

  34. RAMAN SCATTERING MEDIATED BY EXCITONS 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).) EXCITON PICTURE Non-resonant Regime Resonant Regime Chinese University of Hong Kong, November 2012

  35. 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

  36. 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 ascavity 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. Chinese University of Hong Kong, November 2012

  37. 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. Chinese University of Hong Kong, November 2012

  38. Reflectance curves of microcavity as a function of photon energy for various values of q. REFLECTION FROM MICROCAVITY (Dimitri Dini, Rüdeger Köhler,Alessandro Tredicucci, Giorgio Biasiol, and Lucia Sorba. Phys. Rev. Lett. 90, 116401 (2003)) The insert shows the cavity polariton dispersion curve. sinq is proportional to the in-plane wave vector of the EM wave. Microcavity Substrate= undoped [100]GaAs Experimental arrangement to access the polariton black arrows = optical path of the incident light The shape of the substrate allows the angle of incidence (q) to be varied. Chinese University of Hong Kong, November 2012

  39. 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 Chinese University of Hong Kong, November 2012

  40. 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. Chinese University of Hong Kong, November 2012

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