Dynamical properties of polaritons in semiconductor microcavities

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(1)Universidad de Los Andes Physics Departament. DYNAMICAL PROPERTIES OF POLARITONS IN SEMICONDUCTOR MICROCAVITIES Presented to the Physics Departament of The Universidad de los Andes in Partial Fulfillment of the Requirements for the degree of Master in Physics. Ferney Chaves R. Physics Departament Universidad de los Andes e-mail: [email protected]. Ferney Rodriguéz, Ph.D Project Advisor Physics Departament Universidad de Los Andes. 2005.

(2) Contents 1 ABOUT MICRO-CAVITIES, EXCITONS AND POLARITONS 1.1 Excitonic states in semiconductors . . . . . . . . . . . . . . . 1.2 Semiconductor quantum wells . . . . . . . . . . . . . . . . . 1.3 Quantum well excitons . . . . . . . . . . . . . . . . . . . . . 1.3.1 Interaction with light . . . . . . . . . . . . . . . . . . 1.3.2 Semiconductor micro-cavities . . . . . . . . . . . . . 1.4 Microcavity Exciton Polaritons . . . . . . . . . . . . . . . . . 1.4.1 Exciton-Photon Coupling . . . . . . . . . . . . . . . 1.5 Strong-Coupling Theory . . . . . . . . . . . . . . . . . . . . 1.5.1 Quantum Theory . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 9 9 11 12 13 14 16 16 17 18. 2 Polariton-acoustic Phonon Interaction 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Phonons . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Acoustic and optical phonons . . . . . . . . . . . . 2.2 Scattering of excitons by acoustic phonons in a quantum well. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 23 23 23 23 24. . . . .. 3 Exciton-electron Interaction. 31. 4 Rate Equation Model. 37. 5 Relaxation kinetics of polaritons by phonons 5.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41 43. 6 Relaxation kinetics of polaritons by electrons 6.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51 53. 7 Conclusions and Perspectives. 61. Bibliography. 62. 3.

(3) 4. CONTENTS.

(4) INTRODUCTION The ability to manipulate electronic and photonic wave functions in semiconductors by controlled epitaxial growth to single mono-layer accuracy, has been the basis of enormous technological advance in recent years.Consequently, many of the basic physical properties are very well understood and even used in commercial devices such as semiconductor lasers, light emitting diodes (LED’s), vertical cavity surface emitting lasers (VCSEL’s) [1], microdisc lasers and displays. In VCSEL’s, for example, it is possible to suppress unwanted recombination modes by embedding the active layer in a wavelength size Fabry-Perot cavity formed by Distributed Bragg Reflectors (DBR) and thus enhancing the emission into the modes supported by the cavity [2]. Semiconductor hetero-structures, apart from their numerous applications in optoelectronics, have provided a playground for fundamental research. Confinement in quantum wells is a well-known example of quantization of energy levels in quantum mechanics textbooks. The main effect of the confinement is to increase the binding energy of excitons and modify their density of states, which has great influence on the quantum-well optical properties and allows the observation of exciton effects, even at room temperature for GaAs based quantum wells [3].Confinement also allows a coupling between excitons and radiation. Since no conservation of the momentum orthogonal to the quantum well plane is required, the excitons are thus coupled to a continuum of photon modes and undergo spontaneous emission. By confining the photon modes in a cavity to obtain a strong coupling with the excitons it is possible to enhance or inhibit the spontaneous emission rate. This idea has been taken from atomic physics. It was shown by Kleppner [4] that by placing the atom in an electromagnetic cavity, the spontaneous emission rate is either enhanced or inhibited, depending on the detuning between the atom and cavity resonance. Quantum well excitons are, in this sense, two dimensional analogs of atoms. This idea has led to the fabrication of the first semiconductor micro-cavities. In these structures, a very efficient exciton-photon coupling is achieved by placing a planar semiconductor quantum well inside a thin cavity (only a few wavelengths wide) which confines the electromagnetic field in the direction orthogonal to the quantum well plane. The first experiment on quantum wells embedded in planar Fabry-Perot micro-cavity were performed by Yokoyama [5]. In these experiments, modifications of the spontaneous emission rate was demonstrated in the weak coupling regime. On the other hand, the first evidence for strong coupling in micro-cavities showing Rabi splitting was presented by Weisbuch et al. [6]. Here, quantum well excitons were resonantly coupled to a single mode micro-cavity field to form new normal modes, called exciton polaritons. Their dispersion relation was measured in angle resolved luminescence measurements by Houdre et al. [7]. As a result of the strong mixing of excitons with photons, the polariton’s mass is reduced by four orders 5.

(5) CONTENTS. 6. of magnitude, with respect to the corresponding excitons’ mass, leading to a steep energy dispersion for polaritons, as opposed to excitons. The density of states at the polariton resonance is also dramatically reduced due to this new, modified dispersion relation. Novel properties of polaritons in bulk materials have been extensively studied experimentally in the past. In fact, polaritons are ideal for studying the non-linear dynamics of interacting particles at moderate excitation densities, because they can be directly manipulated by laser light with high precision and allow the observation of new collective excitation phenomena. Polariton luminescence has been the most debated property of bulk polaritons [8, 9]. More recently, effects such as stimulated scattering and the possibility of observing polariton condensation in a single quantum state, a phenomenon similar to Bose-Einstein condensation (BEC), was suggested due to the polaritons’ bosonic character [10] at excitation densities below exciton saturation, where a bosonic description of excitons is still appropriate. On the other hand, a recent and significant approach to a state condensate in a system of indirect excitons has been observed by Butov et al. revealing bosonic stimulation of exciton scattering, which is a signature of a degenerate Bose-gas of excitons. However, recent observations of parametric polariton scattering under stimulation of the final state gave a conclusive proof of the bosonic behavior of polaritons[11], and brought to the fore the inherently contradictory views of Bose condensation in excitonic systems. Among the many speculations on the advantages of polariton condensates is that of a new generation of optoelectronic devices based on a polariton laser or the so-called "phaser"[12]. However, Bose condensation of polaritons has not yet been observed from an initially incoherent population of excitons. The main obstacle has been a relaxation bottleneck which prevents excitons from rapidly relaxing into the strongly optically coupled polariton states at low energy[13]. The combination of phase space restricitions with the lifetime of the quasiparticles, gives rise the bottleneck effect, i.e., a dynamical barrier which blocks ( at the inflection point in the polariton dispersion relashionship) the polaritons enroute to the ground state. Surprisingly, strongly coupled semiconductor micro-cavities are found to be poor emitters allow carrier density because of this bottleneck. It is the purpose of the present work to study two of the main relaxation bottleneck mechanisms of polaritons such as: polariton-phonon and polariton-electron scattering, in order to elucidate which of those mechanisms are the most important to break the bottleneck and opening the way toward condensation in excitonic systems. The main aim of the present thesis work is to study the fundamental features of the different relaxation process for polaritons, when the semiconductor micro-cavity is excited by a non-resonant laser pump. Our numerical simulations of polariton relaxation, clearly demonstrate that, for the experimental situation, the electron-polariton relaxation mechanism is the most efficient, contrary to the recent published theoretical work and confirming the experimental results. In the present work, the polariton-polariton scattering is inefficient due to the low excitation densities. The present thesis is structured as follows: In the chapter 1 we show the basic concepts about what is an exciton state in a quantum well and how it does interact with the light to generate the new quasi-particle called polariton. In addition we give a brief description of which it is a cavity in semiconductors. In the chapter 2 is described theoretically the Hamiltonian of exciton-acoustic phonons interaction for excitons in , state. The chapter 3 is dedicated to the excitonelectron interaction taking into account both direct and exchange contributions. The deduction of . . .

(6) CONTENTS. 7. the rate equation from the Heisenberg equation is given in the chapter 4. This rate equation permits to get the distribution population of polariton in the k-space. In the chapter 5 we will resolve numerically the rate equations considering only scattering by acoustic phonon and the results are shown. Similarly the chapter 6 is dedicated to show the numerical results of rate equations considering, additionally, polariton relaxation by means of electron-polariton scatering process and we compared. Conclusions and perspectives are given in the chapter 7..

(7) 8. CONTENTS.

(8) Chapter 1 ABOUT MICRO-CAVITIES, EXCITONS AND POLARITONS 1.1 Excitonic states in semiconductors In semiconductors, when a photon with energy greater that the energy gap between valence and conduction bands is absorbed, an electron-hole pair is produced. Nevertheless, the Coulomb interaction between the electron and the hole reduces the energy of the pair and causes the appearance of lines with energies smaller than in the absorption spectra. These lines represent bound states of the two particles. Such a bound electron-hole state is called an exciton [14]. Excitonic states are known since 1931 (Frenkel). In bulk materials, the binding energy is very low and such states can be observed only in pure materials where their intrinsic nature, allows to characterize optical absorption or emission experiments. In poorer quality samples, these states merge with band-to-band transitions. In hetero-structures, due to spatial confinement, the binding energy increases. The oscillator strength is also increased. Thus one observes sharp excitonic transitions. There are two types of excitons Frenkel (small radius) and Wannier-Mott (large radius)(Fig. 1.1). We will discuss the latter ones because due to its oscillator strenght and the extension of the Bohr radius are present in semiconductors. For them we use the effective mass theory for the envelope . . . . Figure 1.1: schematic representation of the excitonic states. (Bulovic et. al. 2001). 9.

(9) CHAPTER 1. ABOUT MICRO-CAVITIES, EXCITONS AND POLARITONS. 10 function,. . . . . . . . . . . . . . (1.1) . . . . . . . r. . . . r. . . . . . . Then we introduce new coordinates r. . r. r k. K to get. r . . R . . r . . . . k . . . . k . . . . k . k. (1.2) . #. . %. . ). . . . . . . . . (1.3). '. . . +. r. . . . ,. . Here. is the reduced mass. Eigenvectors for the first term are of the form +. 7. 9. ;. =. (1.4) . . . .. 1. 3. 5. /. . The second term leads to the hydrogen atom problem, . ). . . r. '. . +. r. . . . ,. . (1.5). r . A A . . As a result P. P. P. (1.6) . . 1. 3. 5. G. I. K. M. N. O. A . C. D. C. . R. .. . R. U. . are central cell functions. The eigenvalues are. where R. .. V. U. % . + Y. . . X. . . . (1.7). . . . . . . . . . Z C. V. D. C C. . . . The kinetic term containing fective mass \. [. \. . . %. . . gives rise to a parabolic dispersion relation for the exciton, with ef. The ground state wave-function is the 1S hydrogenic wavefunction Y. . . r. . ^. _. a. 1 . 3. . 5. a. r. (1.8) b. g. A . b. . c. d. e. f. . a. where b. is the exciton Bohr radius: . . . a. (1.9). b. +. . . . a. For a typical semiconductor, such as GaAs, the exciton radius is approximately 150 and the meV. This binding energy is small compared corresponding binding energy is around to typical phonon energies at room temperatures, which are approximately 25 meV and suggests that different structures and materials allowing larger binding energies must be used to observe excitonic effects at room temperatures. b. Y. . m. . h. j. n. h.

(10) 1.2. SEMICONDUCTOR QUANTUM WELLS. 11. 1.2 Semiconductor quantum wells The energy dispersion relation of electrons (holes) in a bulk semiconductor around the minimum of the conduction band (maximum of the valence band)is usually described by a parabolic band with an effective mass: . . . (1.10). . . . . . . . . . . . is the minimum (maximum) of the conduction (valence) band, and is the effecwhere tive mass of electrons (holes) and the momentum. A semiconductor quantum well (QW) consists of a thin layer of one type of semiconductor material with a certain band-gap energy (GaAS or CdTe), lying between two thicker layers of another semiconductor material with a higher band-gap energy or [15]. The difference in band-gap energy leads to the formation a potential well along the growth direction in both the conduction and the valence band (Fig. 1.2a ), which leads to energy quantization, if the intermediate layer is thin enough. The quantized energy levels can be calculated in the enve. . . . . . . . . . . . . . . . $. . . . ,. -. %. ". '. (. *. ". .. /. Figure 1.2: a) Scheme of the energy band for a quantum well system. The difference in energy gap between the semiconductor in the well and in the barriers acts as an effective potential so as to create quantized energy levels for both electrons and holes in the quantum well. b)Energy dispersion relation for electrons and holes in the plane of the quantum well. lope function approximation [16] which allows one to write the Hamiltonian of the semiconductor quantum well as that of a standard 1-dimensional finite quantum well: 0. . . . 6. 7. . . . . . . . . . . 2. 4. 2. 4. (1.11).

(11) CHAPTER 1. ABOUT MICRO-CAVITIES, EXCITONS AND POLARITONS. 12. . Here, the z-axis has been chosen along the growth direction and minimum (maximum) of the conduction (valence) band. The resulting dispersion relation for electrons (holes) reads: . is the energy level of the . . . . . . . . . . . (1.12). . . . . . . . . where are the quantized energy levels of the 1-dimensional quantum well along the growth direction, and the second term represents the kinetic energy in the plane of the QW. The electron (hole) motion is therefore restricted to the 2D (x,y) plane of the QW. In two dimensions the density of states is given by: (1.13) . . . . . . . . . . . . and is therefore independent of the energy, and most importantly, is finite at the minimum (maximum) of the conduction (valence) band where all the relevant optical processes take place. This has fundamental consequences on the properties of two dimensional systems as it means that all dynamic phenomena remain finite at low kinetic energies, such as scattering processes In the approximation of infinite barriers is related to the effective mass and to the QW (L) width: . . . . . #. $. . #. . #. ). +. +. (1.14) +. . . . . . . . . ". This discussion of the band structure in terms of parabolic bands, characterized by a given effective mass, is often sufficient for electrons, but for holes the situation is more complicated. The valence band arises from the p-states (orbital angular momentum ) of the cation, which due to the spinorbit coupling results in three distinct hole energy bands, the so called heavy hole (total angular momentum , ), the light hole ( , ) and the split-off band , ). In bulk the heavy- and light-hole bands are degenerate and the split-off ( band is much lower in energy, which means that it can be neglected. Due to the difference in mass of the heavy- and light-hole bands, the confinement in a quantum well lifts their degeneracy. The result is an in-plane dispersion featuring two branches corresponding to the two different bands, in which the so called heavy-hole has an effective lighter in plane mass than the so-called light-hole (figure 1.2b). The dimensionality can be further reduced in more that one direction, creating quantum wires (QWRs) or quantum dots (QDs)[17]. .. $. ). $. /. . 1. . . . /. 3. $. 5. 3. ). 5. 1. . ). 1. . . 3. $. 5. /. 1. . 1. . 1. 1.3 Quantum well excitons In order to determine the energy spectrum of the exciton, the Hamiltonian can be written in terms of center of mass (R)and relative (r) coordinates: :. . . . . ;. . =. >. . C . ? . . . B >. r. r. (1.15).

(12) 1.3. QUANTUM WELL EXCITONS. 13. where is the total exciton mass, is the reduced exciton mass and is the dielectric constant of the semiconductor host material. The center of mass term describes the motion of the exciton as a whole with respect to the semiconductor lattice, while the term in relative coordinates is very similar to the problem of the Hydrogen atom. Excitons are therefore generally regarded as hydrogen-like complexes embedded in the semiconductor, resulting in modified energy and length scales expressed in terms of effective Rydberg and Bohr radius, given by . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.16). . . . . . . . . . . . . . where and are the atomic Hydrogen binding energy (13.6 eV) and Bohr radius (0.0529 nm) respectively. For typical values of effective masses and dielectric constants of semiconductor materials, exciton binding energies are approximately 1000 times smaller and Bohr radii are 100 times larger than those of atomic Hydrogen. With decreasing dimensionality the exciton binding increases and the limit of a purely two-dimensional case the effective Rydberg is given by [18]: . . . . . . (1.17) . . !. ". Since in a QW the heavy- and light-hole energies are non degenerate, the corresponding heavy- and light-hole excitons are also non-degenerate in energy, with a splitting that increases with decreasing well thickness. In a real QW the excitonic wave function penetrates the barriers layers and has a finite extent in the growth direction. Therefore, the system it not truly 2D; the excitonic binding energy therefore depends on the well width and depth, and has a value between that of the 3D and 2D limits. However, for simplicity, excitons in QWs are referred to as QW-excitons or 2D-excitons.. 1.3.1 Interaction with light Quantum well confinement has profound effects on the optical properties of excitons [19] in comparison with bulk excitons. The interband transition probability for particles confined in a well can be calculated by perturbation theory and is given by the product of an optical matrix element times the density of states. The transition rate is given by Fermi’s golden rule: 1. 2. %. $. 3. 5. 9. ' # . ,. .. /. ,. ,. 7 ). ;. (1.18). . ,. &. . 8 ' . 8. (. ). +. &. 1. 2. *. +. 3. are transition initial and final states with energies respectively, is the electric where dipole interaction Hamiltonian r E. The interband optical matrix element has the form <. <. ). /. 8. 8. 9. >. +. =. >. ,. .. /. ,. ?. E r. 3. k. 5. . ,. ,. F. r H. B. D. +. k. >. H. K. =. . I. F. L. N. rE r. r K. O. N. . r H. Q. (1.19) H. =. D. +. I. Q. and are, respectively, the electron’s and hole’s slowly varying envelope wave where and functions associated with the quantum well, E is the polarization vector of light, and . . E. D. E. D. K. K. F. E. L. E. N. O. N. Q.

(13) CHAPTER 1. ABOUT MICRO-CAVITIES, EXCITONS AND POLARITONS. 14. are usual Bloch functions (figure 1.3). The interaction of excitons with light, described by the expression in the integral, actually measures the electron and hole wave function overlap and is strongly dependent on the confinement. In QW’s the electron and hole overlap much more strongly. Figure 1.3: Electron and hole in a quantum well of width L which are the product of usual Bloch and with envelope wave functions and functions . . . . . . . . than in the case of delocalized free electron-hole pairs in bulk semiconductor. A useful quantity to characterize the strength of an optical transition is the oscillator strength defined as . . . . . . . . . . . . . E r. (1.20) !. . . . is the free electron mass and is the frequency. It can be shown that the oscillator where strength per unit crystal volume for a 2D exciton is 8 times larger than the corresponding 3D exciton oscillator strength due to the reduced Bohr radius, namely . . . . . . . . !. $. . . . $. &. . . . E r. !. &. (1.21) .. . *. ,. . . '. -. '. . E r where Bohr radius. . . . ,. is the standard valence band - conduction band matrix element and . -. is the. . 1.3.2 Semiconductor micro-cavities Spectacular advances in atomic physics in the last few years have allowed to investigate the interaction of single atoms with the cavity in which they are contained, leading to the creation of a.

(14) 1.3. QUANTUM WELL EXCITONS. 15. new field called cavity quantum electrodynamics. Recently it has been possible to mimic the atomcavity interaction using semiconductor hetero-structures. A quantum well formed by two different semiconductors is the analog of an atom, whereas a Fabry-Perot resonator, which takes the place of the cavity, can be created with two dielectric mirrors made out of alternating semiconductor films of different refraction indexes. The beauty of MBE is that it makes possible to build an almost perfect Fabry-Perot resonator with a quantum well in its middle. The mirrors of the Fabry-Perot resonator have reflectivity and , and are separated by a distance L containing a medium of refractive index n. For light incident at an angle , the total transmitted power T is given by [20] . . . . . . . (1.22). . . . . . . . . . . . . . where by. . . . is the phase difference between successive multiple reflections (figure 1.4a), and is given . . . . . . (1.23) . . . !. " . . where ". is the wavelength of the incident light. The transmission maxima occur when. Figure 1.4: a) Fabry-Perot cavity consisting of two plane mirrors separated by a distance L, with and respectively. b) Transmission spectrum showing the free spectral range reflectivity and linewidth of the transmission peak used for cavity finesse calculation. #. %. #. &. (. %. *. . +. . the phase after a round trip is a multiple of . The spacing between subsequent maxima in (figure transmission or the spectral range in frequency domain is then given by 1.4b), The width of transmission peak can be easily calculated to be equal to [20] . #. %. &. (. *. +. . .. . 0. 2. 3. 5. 6. . . #. %. . . . . . 7 . . +. : . . . . !. . . (1.24).

(15) CHAPTER 1. ABOUT MICRO-CAVITIES, EXCITONS AND POLARITONS. 16. It is very convenient to characterize the quality of the Fabry-Perot cavity in terms of the finesse which is defined as . . . ,. . . . . . . . . . (1.25). . . . . . . . . . . . . . The finesse indicates how much narrower the transmission peak is compared to the free spectral range, or how close is a Fabry-Perot cavity to the ideal case with delta-like resonances. Here, it has been assumed that the reflectivity of the mirrors is independent of the incident wavelength. Semiconductor micro-cavities are essentially Fabry-Perot resonators with a peculiar mirror structure. The mirrors, called Distributed Brag Reflectors (DBRs) are separated by a spacer of length and are stacks of semiconductor layers (with refractive indexes , ) of thickness for wavelengths centered at . The interference between light reflected at the boundary of subsequent layers results in a very useful property of the DBR to reflect light. The reflectivity coefficient is very close to unity in the region around center wavelength and has a phase which is linearly dependent on the frequency. Since the DBR has a certain penetration depth associated with it, the effective cavity length is extended by a characteristic quantity to become [21]. Similarly, the cavity mode linewidth can be rewritten for the case of stack mirrors. Assuming equal reflectivity for both mirrors we get . . . . . . . !. . . $. $. /. &. (. &. . . 1. +. ". ". #. . %. . #. . &. &. &. +. (. . . ,. . . .. -. . . 2. . . . (1.26) 8. 7. 6 . /. 3. . . 5. 6. 1.4 Microcavity Exciton Polaritons 1.4.1 Exciton-Photon Coupling Consider a GaAs quantum well (QW) located inside a planar DBR microcavity, as shown in Fig (1.5).The center-of-mass wave function of the two dimensional (2D) excitons confined in the QW can be written as . . . 3 J. . :. . =. ;. ;. L . >. ?. (1.27) ?. M. E. G. N. O. I. . . B. P 5. :. ; <. &. . . D. . D. @. @. where is the QW thickness, indicates the position of the QW inside the cavity, and n is an integer number. A longitudinal cavity photon mode of the cavity can be described by J. @. . D. @. . 3. . X. :. . =. ;. ;. L . >. ?. (1.28) ?. M. E. G. N. O. I. Q. B. /. . . 5. :. S. P. S ;. <. &. . . D. @. where is the effective cavity length. The coupling strength of the QW exciton to the cavity photon mode is embodied in the linear coupling constant : /. . . @. . Z. \. =. >. =. = >. >. . ? ?. ?. Q. L. B Z. a _. /. b. b. P. ]. : ;. : <. &. S. S ;. <. &. ^. . . D. @. @. @. (1.29).

(16) 1.5. STRONG-COUPLING THEORY. 17. Figure 1.5: A picture of a planar DBR microcavity. The Bragg mirrors consist of alternating and layers. The cavity spacer layer is The substrate is GaAs. . . . . . . . . . . . . . . . . . . . . . . . where is the excitonic oscillator strength per unit area. From 1.29, if the QW is located at a node of the intracavity field, is close to zero, and thus the QW excitons are decoupled from the cavity photon mode. However, if the QW is located at an antinode of the intracavity field, the exciton-photon coupling is significantly enhanced. QW excitons and cavity photons couple to the outside reservoirs; for example, the QW excitons can decay by emitting photons in the transverse direction, and the cavity photons can leak out of the cavity. Depending on the relative magnitude of the exciton-photon coupling constant compared with the exciton and photon decay rates , , there are two regimes. If , , the system is in the weak-coupling regime. In the weak-coupling regime, the spontaneous-emission rate and pattern can be significantly modified by a cavity. Nonetheless, the spontaneous emission remains an irreversible process. However, if , , the system is in the strong-coupling regime, where spontaneous emission becomes reversible. . . . . . . . . . . . . . . . . . . 1.5 Strong-Coupling Theory As semiconductor fabrication technology advanced, it became possible to fabricate high-Q semiconductor micro-cavities. Thus, the study of semiconductor cavity QED started in the weakcoupling regime and then entered the strong-coupling regime, where the exciton-photon coupling constant becomes larger than the exciton and cavity photon decay rates. In a high-Q semiconductor microcavity, the strong exciton-photon coupling leads to the formation of two new eigenstates of the exciton-photon coupled system, called "microcavity exciton polariton" states or simply "polaritons".The energy separation between the two polariton states increases as the exciton-photon.

(17) CHAPTER 1. ABOUT MICRO-CAVITIES, EXCITONS AND POLARITONS. 18. coupling increases. This exciton-polariton normal-mode splitting is the solid-state analog of the vacuum Rabi splitting in the atom-cavity case. In the time domain, the strong exciton-photon coupling makes the spontaneous-emission process reversible; namely, the emission from the microcavity shows an oscillation, instead of the usual exponential decay. This is because photons emitted by excitons are reabsorbed and reemitted a number of times before exiting the cavity. Hence, the excitation energy of the system is transferred back and forth between the QW exciton state and the cavity photon state, leading to a Rabi oscillation. This process can be modelled as a coupled pendulum system where the two pendulums correspond to the microcavity mode field at the frequency and the exciton at the frequency . The QW exciton state and the resonant-cavity photon state form a simple system of two harmonic oscillators coupled through the light-matter interaction. . . . 1.5.1 Quantum Theory The quantum theory of a microcavity exciton-photon coupled system is based on a Hamiltonian obtained through a microscopic theory of the coupling between the QW exciton and the quantized electromagnetic modes of the surrounding medium. The normal modes which diagonalize the Hamiltonian are the polariton modes. However, in the quantum theory, it is not easy to give a realistic description of the DBRs, in particular, to describe the frequency dependence of the phase of the DBR reflection coefficient. Consider a QW embedded in a planar DBR microcavity. The microscopic Hamiltonian of a Wannier exciton coupled to the radiation field is given by [22] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ". . . . . . . . . . . . . . !. . . . . . . . . . (1.30) %. . . . . . . . . !. . . . . . where and respectively and. are the creation (destruction) operators for the photons and excitons. . . . $. . . . . . *. ,. &. '. ). *. ,. & %. (1.31) 0. '. -. . . . . /. /. '. /. /. $. .. .. ,. Here is a dimensionless quantity characterizing the strength of the interaction between the oscillator and electromagnetic field. The first two terms represent the free Hamiltonian of transverse exciton and photon fields, whereas the last two terms represent the exciton-photon interaction. Owing to the in-plane translational invariance of the electronic and photonic system, the transverse momentum is conserved, namely, a QW exciton state with a well-defined transverse momentum is coupled to a single cavity photon mode with the same transverse momentum . This allows us to consider a single cavity mode and a single QW exciton mode with the same . In the following, we set: (1.32) . . . . . . . %. %. . . . . . !. . . !. .

(18) 1.5. STRONG-COUPLING THEORY. 19. . and omit the subscript from now on. Neglecting exciton disorder and knowing that only exciton and photon states with the same in-plane momentum are coupled, the total Hamiltonian of the system is . . . . . . . . . . . . . . . (1.33). . . . . . . . . . . . . . . . . . . . . . . . . . . . !. . . . where are exciton and photon creation operators with momentum and the term is the dipole interaction between exciton and the radiation field. This coupling depends on the oscillator strength of the exciton transition #. ". . . . . . . . . &. . (. *. . (1.34) ". / . . . % +. ,. . 3. 3. 0. 1. where. is the number of wells in the cavity and /. ,. .. . 5. 0. 4. (1.35). . (. *. ;. 7. 9 *. 8. 6. + +. is the radiative decay rate of the exciton, expressed in terms of the exciton oscillator strength defined in Eqn. 1.20. The above Hamiltonian can be diagonalized by the following unitary transformation: . . (1.36). =. . . . . . . >. . . <. =. . . . . . . . . where and are the upper and lower polariton annihilation operators similar to the 3D case. The resulting Hamiltonian in the new basis becomes <. . . . . . ?. (1.37) <. <. @. . . B. B. C. . . F. . . F F. The solution of the above Hamiltonian at each wave vector using exciton and cavity mode energies including damping (where is nonradiative exciton broadening) correspondingly yields #. E. E. . E. . ". . . . . . . . . . . . . . . . . F. F. E. >. . H. (1.38). . . . . . . E. >. . H. . . . . . producing energy eigenvalues:. . . . . F. . . . . . . F. . E. . . . . . . . . . . . . . . C. B. . &. &. . F. F. L. M. . E >. . . >. . (1.39) 5. 5. K. J. . . . . . . . !. . . . . . . . !. Q. 6. The new modes of the system are described by the real parts of the energy eigenvalue and can be expressed naturally as a function of detuning between exciton and cavity modes and are given by: >. R. . . . . . . . . . . . . L. > . . . 5 5. . . . . . . . K. J. . &. . . . 6. . . . . . . !. (1.40).

(19) CHAPTER 1. ABOUT MICRO-CAVITIES, EXCITONS AND POLARITONS. 20. Whereas the imaginary part is responsible for radiative decay or damping. The important difference between strong and weak coupling regimes lies in the result that, in the weak-coupling regime the eigenstates are a perturbation of the uncoupled system, whereas in the strong coupling regime coherent Rabi oscillations between the exciton and photon states can occur (in analogy with the atom-cavity coupling in atomic physics). In the strong coupling regime the relation: (1.41) . . . . . . . . . . . must be satisfied. The dispersion relations of the quantum well exciton and the empty cavity mode are given by . . k. . . . . . . . . . . (1.42). . . . . . . . . . #. . . . . . . . . (1.43). ' . . . % . . . . . & . . . . . . . . . . (. . . is the fundamental exciton energy, , is the electron where (hole) in-plane mass and is the refraction index of the intracavity. The last equation corresponds to strong in-plane dispersion which can be characterized by a very small in-plane mass of . Three different cases can be distinguished which are shown in Figure 1.6a) . . . ,. . ,. . . #. . &. . . . . ). . . . . . . *. %. &. . 0. . /. . . . 1. 3. 4. the exciton energy is higher than the cavity mode energy (negative detuning . ). 5. . 6. the exciton and cavity mode energies are equal at . '. . . (zero detuning) 7. 4. . the exciton energy is smaller than the cavity mode energy (positive detuning . ) 6. 4. In the case of negative detuning the resonance is achieved for higher in-plane wave vector which the relation is satisfied. For the crossing between exciton and cavity no resonance is achieved at any wave modes is achieved at normal incidence, and for vector. Note the changes in the dispersion relation as the exciton-cavity mode detuning is varied. Finally, the real coefficients and are given by the following expression '. 5. . 4. 8. 8. . . . . . 9. . . 9. 7. 6. 4. '. . . :. . 8. 8. . . . . <. . . . . 0. . . 8. 8 :. . . . . . . =. . . . . . (1.44). . These are plotted in Fig.[1.6b]. Later, they will be used in the calculation of the scattering processes. Figure 1.6a, shows the three typical dispersion energy curves of cavity polaritons for negative, zero and positive detunings. One can see that at small , the dispersion polariton relationship ( vs ) is essentially parabolic and can be characterized by an effective mass, while this mass varies dramatically as a function of the detuning, and it also depends on the well width. The possibility of tuning the polariton’s effective mass over a wide range is an important peculiarity of microcavities. One can see that, contrary to the bulk case, the dispersion relationship exhibits a well defined '. . '.

(20) 1.5. STRONG-COUPLING THEORY. 21. minimun located at . This makes cavity polaritons good candidates for Bose condensation. They have moreover an extremely small effective mass which provides a large critical temperature for condensation In order to understand the dynamical polaritons properties, let us describe the energy dispersion relationship. For positive detuning values ( ), the low branch polariton shows a plane dispersion which does not change with respect to the bare excitonic energy (for low values) and increases quadratically for higher momentum values. However, this relationship changes dramatically for . This situation makes it possible to define a point of crossing between the bare exciton and photon dispersion curves. Below this point, the dispersion of the polariton low energy is then characterized by a zone of strong curvature, the character of which is essentially photonic. For values larger than the crossing point, the polariton dispersion converges to the in plane dispersion excitonic curve. It should be noted that the density of states is directly related to the shapes of the energy relationship dispersion, vs , which plays a major role in the scattering process. For a two dimensional system, the density of states is given by: . . . . . . . . . (1.45). . . . . which, in the energy parabolic approximation is proportional to the confined quasi-particle mass . . (1.46). . . . . . . . By expanding the dispersion relation around ( to the confined photon in the microcavity . ), it is possible to assign an effective mass . . . . . (1.47). . . . . . . . . . . . . . . . . . . . . . . . . It is found that the effective mass associated to the photon in the microcavity is approximately time smaller than the exciton’s mass. This implies that polaritons themselves, even if they are of weak photonic nature, have a density of states which is enormously reduced compared with the bare exciton mass. Finally, it is important to remark that the density of states scales like the inverse of the slope ( ) which can be modulated through the variation. For a slope different from zero, the density of available states to be occupied for a quasiparticle is reduced and, therefore, it is possible to get large particle occupations for a given range of states. The evolution of the photonic weight of the polariton states located on the lower branch as a function of is depicted on Fig. 1.6b, for different detunings. In a general form, the photonic . For it increases remaining, however, smaller than the excitonic weight increases to weight, and the polaritons of lower energy take an excitonic character. Thus, with a very weak photonic weight in agreement with the previous discussion. On the other hand, for , polaritons on the lower branch, becomes strongly photonic and its photonic weight takes values close to 1 for small values. At the crossing point, both polaritons are exactly half photon and half exciton. . !. . ". $. . . . %. . . . . %. . .

(21) CHAPTER 1. ABOUT MICRO-CAVITIES, EXCITONS AND POLARITONS. 22. a) Dispersion relation. b) Hopfield coeficients Lower Polariton (LP). 1405. 1. E(meV). 1401. ∆=−3meV. 1397. Ecav Eex LP UP. 1393 1389 1385 1000. 10000. 1e+05 −1 k(cm ). 1e+06. 1405. E(meV). Ecav Eex LP UP. 1393 1389 10000. 1e+05 −1 k(cm ). 1e+06. 1405. 1e+05 −1 k(cm ). 1e+06. 1e+07. Ck Xk. 0.5. 0 1e+07 1000. 10000. 1e+05 −1 k(cm ). 1e+06. 1e+07. 1. 1401 E(meV). 10000. ∆=0. 1397. ∆=3meV. 1397. Ecav Eex LP UP. 1393 1389 1385 1000. 0 1e+07 1000 1. 1401. 1385 1000. Ck Xk. 0.5. 10000. 1e+05 −1 k(cm ). 1e+06. Ck Xk. 0.5. 0 1e+07 1000. 10000. 1e+05 −1 k(cm ). 1e+06. 1e+07. Figure 1.6: a)Exciton-polariton dispersion relation for negative, 0 and positive detuning . Dot-line correspond to the photon, dashed line to the exciton whereas solid lines correspond to the upper and lower polariton dispersion. b) Corresponding exciton and photon components of lower polariton branch . . . . . . .

(22) Chapter 2 Polariton-acoustic Phonon Interaction 2.1 Definitions 2.1.1 Phonons A phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because phonons play an important role in many of the physical properties of solids, such as the thermal conductivity and the electrical conductivity. In particular, the properties of long-wavelength phonons give rise to sound in solids – hence the name phonon. In insulating solids, phonons are also the primary mechanism by which heat conduction takes place. Phonons are a quantum mechanical version of a special type of vibrational motion, known as normal modes in classical mechanics, in which each part of a lattice oscillates with the same frequency. These normal modes are important because, according to a well-known result in classical mechanics, any arbitrary vibrational motion of a lattice can be considered as a superposition of normal modes with various frequencies; in this sense, the normal modes are the elementary vibrations of the lattice. Although normal modes are wave-like phenomena in classical mechanics, they acquire certain particle-like properties when the lattice is analyzed using quantum mechanics. They are then known as phonons. Phonons are bosons possessing spin zero.. 2.1.2 Acoustic and optical phonons In real solids, there are two types of phonons: "acoustic" phonons and "optical" phonons. "Acoustic phonons" have frequencies that become small at long wavelengths, and correspond to sound waves in the lattice. Longitudinal and transverse acoustic phonons are often abbreviated as LA and TA phonons, respectively. "Optical phonons," which arise in crystals that have more than one atom in the unit cell, always have some minimum frequency of vibration, even when their wavelength is large. They are called "optical" because in ionic crystals (like sodium chloride) they are very easily excited by light (in fact, infrared radiation). This is because they correspond to a mode of vibration where positive and 23.

(23) 24. CHAPTER 2. POLARITON-ACOUSTIC PHONON INTERACTION. negative ions at adjacent lattice sites swing against each other, creating a time-varying electrical dipole moment. Optical phonons that interact in this way with light are called infrared active. Optical phonons which are Raman active can also interact indirectly with light, through Raman scattering. Optical phonons are often abbreviated as LO and TO phonons, for the longitudinal and transverse varieties respectively.. 2.2 Scattering of excitons by acoustic phonons in a quantum well The information about the interaction between excitons and their environment in a semiconductor is contained in the homogeneous line-width of exciton luminescence. During the past two decades, the homogeneous line-width of exciton in several kinds of quantum well and super-lattice systems has been investigated extensively in both time and frequency domains. In the time domain, the exciton dephasing time was measured from four-wave mixing (FWM), and then the homogeneous linewidth could be deduced [23, 24]. In the frequency domain, the line-width was measured directly from photoluminescence, transmission, reflection or absorption and Raman spectroscopy[25, 26]. By modeling of experimental data, extensive information about the interaction among excitons and acoustic phonons has been deduced. In those investigations, excitons are quasi-two-dimensional (Q2D). That is, they can move freely in the wells or are localized weakly with a localization energy of several meV. On figure 2.1, we schematically present the principal steps of polaritons relaxation in microcavities at strong coupling before the annihilation of polaritons by emission of photons. In the general case, as in experiments, the excitation is carried out with an energy above the radiative level in the continuum of the electron-hole pairs. The electron-hole pairs created in this way redistribute their energy and momentum, in a such way that excitons with a large in-plane momentum can be created. Two processes are distinguished: The first consists of the direct formation of an exciton by coupling of the electron-hole pair with an LO phonon which absorbs the excess of energy without the pair losing its correlation. The second process, considered as most probable, requires a preliminary thermalisation of the electrons and holes by various physical processes. The excitons are formed finally by the overlapping of the functions of a hole and an electron. It was shown in experiments that the processes of formation of excitons are very effective: They are held, typically, on a scale of time of T < 20 Ps. In both cases, the loss of energy excess implies a redistribution of the states excited in reciprocal space. The excitons thus created are characterized by large wave vectors. In general, coherence is destroyed with the excitation during this first stage of relaxation. In the case of a non-resonant excitation, it is then convenient to consider excitons in terms of population and not in terms of polarization. Once created, polaritons release then along the curve of dispersion towards K = 0. In the majority of cases, this second stage is controlled by the interactions with the phonons. It is characterized.

(24) 2.2. SCATTERING OF EXCITONS BY ACOUSTIC PHONONS IN A QUANTUM WELL. 25. Figure 2.1: Relaxation processes of polaritons in a microcavity.. by multiple steps utilizing small exchanged quantities of energy and is held on a scale of time larger than 20 ps, until the excitons return in the radiative zone. It is only then that one significant difference between dynamics in a quantum well and a microcavity occurs: In a microcavity, the excitons of spin couple themselves strongly with light, thus forming polaritonic states, whose dispersion strongly differs from bare excitons. This modification of the dispersion relation means, among other things, an enormous reduction of the density of states around = 0, which involves, as we will see, a shake up of the relaxation processes. With regard to the probability of emission through a surface by the polaritonic states, it is proportional to their photonic weight and is not constant, as it is the case in a quantum well without cavity. This last step is then characterized by the competition between annihilation by emission of photons and the later relaxation which prevents the excited states to reach the complete thermalisation . 0. In order to describe all of relaxation process, we first study the interaction Hamiltonian of the quasi-two-dimensional exciton with acoustic phonons derived from the deformation-potential which is of interest in this work. In the case of GaAs-AlAs structures, the electron and hole of excitons are considered to be well confined within a QW since the band-gap discontinuity is quite large. On the other hand, the lattice properties of GaAs and AsAl, for example, the lattice constant and elastic moduli, are in close proximity. Thus, the acoustic phonon which interacts with the quasitwo-dimensional exciton in a GaAs layer can be considered to have three-dimensional character. The interaction Hamiltonian of the quasi-two-dimensional with acoustic phonons can be derived starting from the three-dimensional exciton-phonon interaction Hamiltonian [27]..

(25) CHAPTER 2. POLARITON-ACOUSTIC PHONON INTERACTION. 26. The quasi-two-dimensional exciton state with total wave vector K can be represented by v . . K . . . K F r . . . . . . r . z z . . . . r r. (2.1). . r . . r. . . . . . . . where v y L are, respectively, the volume of the unit cell and the linear dimension of the quantization volume , r r the creation (annihilation) operator of the th (conduction or valence) the crystal ground state, and R the coordinate of band electron in the Wannier representation, the exciton’s center of mass, defined by . . . . . . . . . . . . . R. . . r . . . . . r . . . . . . (2.2) . . . . . .The envelope function for the th electron-hole inwith the electron (hole) effective mass ternal motion is denoted by F . In the following, any position vector or wave vector will be decomr or k k . posed into components parallel and perpendicular to the QW interface r Then, rewriting the operators in the Wannier representation with those in the Bloch representation by a well-know relation kr (2.3) k N k , and transforming then the discrete where N is the number of unit cells related to L by sum over the lattice sites into a spatial integral by . . . . . . . . . . . . . . . !. !. #. . . . . . . . $. . ". . . &. . %. *. . '. (2.4) . ). . &. r . one obtains . . . . K . . f k k’ K. . k k’ K. ,. . . k k’. (2.5). -. . . k k’ . . . . . .. with #. . /. /. f. * 1. . 2 . . ,. . ) '. ) '. ) '. 3. F . 0. 2. . . . . . . . . . 4. 5. . 1. . /. 7. 3. . 5 5. 9. (2.6). 9 . . . . . . 0. . . !. ! !. !. : . . . where and . The three-dimensional electron-phonon (e-ph) interaction for the deformation-potential (DF) coupling is written as . . . . . . . . . . . . . . . . !. . . . . . !. . . . ?. 2 . D. q . F C. @. . A. . . . . I. . . . . . K. . (2.7) . E . H. B . . . B . . . . O. . O. . G . B. B. . . . . . . .. .. . B L . . L. . . q. . . . . q . . B . B. . . . . . (2.8). O. . O. . . B . B . . . . . .. B . . . F. . and are the deformation potentials for the conduction (valence) bands, the mass where density, and the sound velocity of the longitudinal acoustic (LA) phonon mode, respectively, and the . G. L. . L. . .

(26) 2.2. SCATTERING OF EXCITONS BY ACOUSTIC PHONONS IN A QUANTUM WELL. 27. coupling functions and are introduced for simplify the notation. The interaction Hamiltonian of the quasi-two-dimensional exciton with acoustic phonons for the deformation-potential coupling between two exciton states and is obtained by calculating the matrix element where, for simplicity, the change of the electron-hole internal motion is not taken into account. The result is given by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.9) . . . . . . . . . . . Substituting expression (2.6) and converting the discrete sum over. into an integral by (2.10). ). #. %. ' &. $ . (. . one can reduce the first term as . . . . . -. -. . 0. . 0. , . ,. . ' &. ' &. . . . . . ' &. ' &. ' &. . . ' &. . /. /. . . . . . *. . . 0. . 0. . . . :. ;. . ;. . . . . <. 0. . ,. . . . . .. 2. .. . . . . .. 5. . . . . .. 3. . . . 7. .. .. 9. .. .. . . > . 0 :. ;. . . ;. . . . . <. 0. ,. . . . , . . & . '. &. & '. '. . . . C. . .. .. . .. 5. 7. . 9. . . / . . >. . *. . . .. .. .. (2.11) .. Similarly, the second term in 2.9 can be reduced as . . . . . -. . . 0 :. , . . ;. '. &. '. &. . . . . <. 0. . . . &. ;. ,. . . F. . C. . ' 5. 7. 9. . . (2.12). / . . . >. . . *. .. . .. . .. . Thus, the quasi-two-dimensional exciton-phonon interaction Hamiltonian mation potential coupling is given by . . ,. . for the defor-. . . I. 2. . 3. . . . . ,. :. . . . . . . . . . . . . . . I. . 2. . 3. . . . . . . . . *. . . . . . . . . . . . . . . . . (2.13). . . . F . . . . . . . L >. . . L. M. . . . *. . . *. . . . . with . . . -. : 0. ;. . ;. . <. 0. , . . , . ' &. &. ' &. . C F. . . 5 '. 7. 9. (2.14). / > . . . 0. 0. is supposed to hold. This is a quite general exwhere the symmetry pression for the deformation-potential coupling. To obtain a more explicit expression, the envelope . . . .. . .. .. . /. /. . . . . .. .. .. .. .. ..

(27) CHAPTER 2. POLARITON-ACOUSTIC PHONON INTERACTION. 28 . . function function is assumed . . must be specified. For the lowest (1s) exciton state, a variational envelope. . . . . . . . . (2.15). . . . . . . . . . . . . . . where is the normalization constant, is the variational parameter to minimize the energy and ( ) is the wave function of the electron(hole) orthogonal to the QW plane. These functions depend on the geometry of the QW, which allows to extend this formulation to parabolic quantum wells and indirect quantum wells. With the use of this envelope function, the function is given by . . . . . . . . . . . . . . . . . . . . !. . ". . ". . %. . . . '. . . . . . . . . . $. ( . . . . . . . . . . (2.16). . . +. *. . *. *. *. . ). . . . . . . . . The last equation can be expressed as . . . . . . . . . (2.17). . . . . 1. . . . . . . . -. .. . *. . *. *. *. . /. . . . . -. . . . . . . . . . and making use of the decomposition. Introducing the two-dimensional polar coordinate for formula [28] . . !. ". . <. $. (. (2.18) . :. 3. 4. 6. 7. 2. 9 . . . . ;. =. 8. . . ). . . . so that . < . . . (. . ?. *. . . ?. . . . . -. .. /. 2. . 9. . where. ?. 9. (2.19) . *. . . is the zeroth-order Bessel function. By use of the formula [28] C. B. . (2.20) . ?. A. ? . . . 2 . . 9. C. $. . . . @. @. D. E. the integral over in 2.17 can be performed as . <. . . . . . . . (2.21). F . . . . . 1 *. * *. *. . !. . . . . . . . . . . . $. . *. . * . F =. . . ). <. . . . . . . F G. . !. . . . . . $. . . *. . *. F. =. . ). . where the function G. H. is defined by . . G. . H . . (2.22). . +. 1 *. . . . . . . -. *. . *. . *. . . . . . M . M . In order to determine the normalization constant and thus finally to obtain for square wells:. , it is necessary to make . . L. L. K. . N *. . .

(28) 2.2. SCATTERING OF EXCITONS BY ACOUSTIC PHONONS IN A QUANTUM WELL. 29. . . . . . . . . . . . . . . . . % . . . !. (2.23). #. (. . . . !. #. . . . . $. '. . &. . . . . . . . . . 0 *. . . +. . . $. . +. . . . $. . . -. -. ,. 2. . . ) ). . 2 . /. . #. ,. . /. #. . . +. 3 +. 3. . 4. ,. . 5 ,. . . . . . . . . . . with . . . . 3. . . . . . 7. . 4. . ,. . 4. 4. -. . . 4. -. . (2.24). ,. So far, the deformation-potential coupling has been discussed, which will be used later to determine the dynamics of polaritons interacting with acoustic phonons..

(29) 30. CHAPTER 2. POLARITON-ACOUSTIC PHONON INTERACTION.

(30) Chapter 3 Exciton-electron Interaction In semiconductor microcavities, electron-polariton scattering has been proposed as an efficient process that can drive polaritons from the bottleneck region to the ground state, achieving Bose amplification of the optical emission[29]. Recently, observations of parametric polariton scattering under stimulation of final state gave conclusive proof of bosonic behavior of polaritons [11] and brought to the fore the inherently contradictory views of Bose condensation phenomena in exciton systems [30]. However, Bose condensation of polaritons has not yet been observed from an initially incoherent population of excitons. The main obstacle has been a relaxation bottleneck which prevents excitons from rapidly relaxing into the strongly optically coupled polariton states at low energy[12, 13]. Surprisingly, strongly coupled polariton semiconductor microcavities (MCs) are found to be poor emitters at low carrier densities, because of this bottleneck. Polariton-polariton scattering has been found to be insufficiently effective at bypassing this bottleneck for carrier densities below the limit which retains strong coupling[31, 32]. In this work it is shown that the electron-polariton scattering relaxation mechanism is an excellent candidate for preventing the bottleneck. In this chapter, the electron scattering on excitons in quantum wells is presented theoretically by means of a microscopic model, taking into account elastic scattering. This model is based on calculating the exciton-electron direct and exchange interaction matrix elements, from which the exciton scattering rates can be derived. . . Separating the coordinates in the QW from the perpendicular coordinate , and deand , respectively, we write the in-plane noting the electron and hole in-plane momenta by , Fourier transform of the exciton wave function . . . . . . . . . . . . . . . . . . . . . (3.1). . . . . '. . (. (. . . ). . . . . . . " . $. %. &. %. &. *. !. . . . . . . . . . . . where denotes the QW surface area. Transforming to center of mass (CM) and relative coordinates in the QW plane ( ) where , are the electron and exciton in-plane effective masses, respectively) it is possible to and , and a free motion decompose the exciton wave function into an envelope function, +. -. ,. . 1. . . -. . . 1. . . 3. 6. 5. . . :. 31. . . . . 6. . 5. . . .

(31) CHAPTER 3. EXCITON-ELECTRON INTERACTION. 32. part related to the in-plane CM coordinate. Denoting. as the in-plane CM momentum then . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . !. . . . . . . #. ". #. . . . (3.2) &. . . . . . . $. . . . . is the simplest exciton wave function for the exciton ground state: , where is a normalization factor and is a variational parameter associated with the exciton Bohr radius in the QW, which is fixed by maximizing the binding energy of the exciton [33]. It is remarkable that the use of a wave function separable en and , facilitates considerably the calculation of the scattering matrix element, although it is strictly justifiable only for narrow well structure. Assuming perfect confinement of electrons and holes in QW, and taking the axis origin in the center of the QW, the confinement functions are . . . . +. . . . . . . . . .. . . . . *. *. -. . ,. ). ). (. (. . . . . . . . . . . . 6. :. 4. 9. . 9. . . . . (3.3). 0. 7. 7. : 1. *. 3. 8 8. *. 9. 9. . 7 . 8. . . . We now consider a state of a single exciton with an in-plane CM momentum in the fermionic Hilbert space of electron-hole pairs, using the notations of Tassone and Yamamoto [34]. It is a superposition of wave functions with different electron momenta and electron and hole coordinates, given by . . $. . =. . . (3.4). . . . 9 9. . . . . ;. @. ?. . . ?. . . . . . . . A . . >. . . . . . . where is the electron (hole) creation operator with in-plane momentum and coordinate, and is the in-plane Fourier transform of the exciton wave function. A state comprising an exciton and unbound electron having and will be written as . . . . . @. ?. ?. . . . . . . A. . C. . . .. . >. . . F. E. . . . . *. . I. *. ,. . . ). G. . . . . (. . . . . = =. . . . . . . . (3.5). . . . 9 9. K. . . . . ;. . @. L. . @. N. . L. . @. ?. ?. . . . ?. . . . . . A. . . >. >. . . . . . . . . is the electron wave function. Applying the Coulomb interaction operator where to this state, the electron-hole and electron-electron interactions are O. . . O. . @. . N. L. . . #. . O. . . . . O . 9. K. . . . ;. L. . = =. . . . . . . . . . . . . . =. =. @. . @. N. . . L. . ?. ?. . . . . . . . . . . . . A. >. . . E. . . . . @ @. ?. @. N . . . L. . ?. . . . . . . . . > >. . . . (3.6). . . 9. . . . @. ? ?. . . . . . . . . A. . >. . . . .

(32) 33. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.7). . . . . . . . . . . . . . . . . . . The first term in Eq. (3.6) represents the Coulomb interaction between the exciton’s constituents, thus contributing its self-energy, and can be discarded in the calculation of the scattering matrix element. In Eqs. (3.6) and (3.7), is the two-dimensional Fourier transform of the Coulomb interaction, where A is the QW area, and the plus (minus) sign is used . in Using the anti-commutation relations for the fermion operators, one can easily find the scattering matrix elements to be . . . #. '. ". ). %. (. $. . $. !. &. +. . . . . *. . . . . . . . . . .. . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.8) . . . . . . . . . . . . . . . 8. . . . . . . . . . . . .. . . . . . . . . . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.9) . . . . . . . . . . . . . . 8. . . . . . . The first term in each of Eqs. (3.8,3.9) contributes to the direct (classical) Coulomb interaction and the second contributes to the exchange matrix element. The direct term reads . . . . >. <. ". %. %. . . %. A. $. C ;. % . . . B. $ '. . . . . ). +. . . . . (. %. :. *. A. @. 0. . E. . . . . . ? >. . > >. B. . C. .. . . . . . . . D D. . D. . . . * #. %. . * #. % . I. . 8. . * #. % . I. . I. . &. & &. . G. H. ?. where. . G. H. . G. . 0 . ? H. . . > >. K. %. %. .. . . . (3.10). . ?. . @. B. (3.11). 8. is a dimensionless function. It is remarkable that the electron wave functions in the QW plane are absent, as they contribute, together with the in-plane center of mass part of the exciton wave.

(33) CHAPTER 3. EXCITON-ELECTRON INTERACTION. 34. function, a fixed phase factor which is unimportant for our purpose. The momentum integral is a simple convolution and can be readily evaluated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.12) !. . . . . where . . . . . . . (3.13). . . . % . . ". $. . integrals, it is. is the exciton spatial wave function in the QW plane. In order to evaluate the convenient to change to the coordinates &. . . '. '. (3.14) . *. ). . . (. (. which implies the following change in the integration limits: &. &. &. &. . &. &. .. . . /. -. -. '. *. $ $. . . . . (. . . . / -. -. 0. ". " $. " $. $. 1. ". . . /. -. & &. &. &. (3.15). . *. $. ". .. . . / -. 0. 1. " ". & &. and performing the integrations results in the direct term . . . 4. . . . 9 . 2. . . . . . (3.16) . 5. 7. . . . . . .. . 6 . . . . . 3. 8. where is the QW width, the QW, and. is a variational parameter associated with the exciton Bohr radius in. 8. . . . . . <. . . -. . -. -. 0. 1. 0. 1. . . 9. (3.17) :. : . ;. 5. . : = ". 8. . . . . . . . . . . 5. . . . 5. . . 8. Similarly, the exchange term is . . . . A. '. A. @. 9 . B 2. . . !. . ;. 7. . . . .. . (. 6 . . . . . . . A. 8. A. . !. . . . (3.18). . . A. . The exchange term (3.18) does not have analytic solution and must be where computed numerically. It is convenient to transform to dimensionless direct and exchange integrals given by '. . @. . . . 2. 5. . 7. .. . 6. 8. . (3.19).

(34) 35. Figure 3.1: Calculated direct and indirect integrals vs transferred momentum. The inset shows on an expanded scale. Taken from Ref. [36] . . The direct and exchange integrals are plotted in Fig. 3.1 as a function of the transferred momen, where the angular dependence of disappears. As the tum , for the case direct integral approaches zero, while the exchange integral its maximum (this is also the case for exciton-exciton interaction [35]). In the general case, the exchange integral is a function of the transferred momentum , the momentum difference (which can be regarded, for convenience, as the in-plane momentum of the colliding electron in the rest frame of the exciton), and of the angle . The exchange interaction term has the following features: (i) The interaction . Physically, this means that the electron is inclined to transfer as much favors the case momentum as possible to the exciton, preferably in the same direction. (ii) The interaction retains its strength for quite large values of (or ) even though the excitonic wave function vanishes much more rapidly with momentum. Above, the electron spin degree of freedom has been disregarded. Considering parallel spins (triplet configuration) will retain the sign of the exchange term with respect to the direct term, whereas for anti-parallel spins (singlet configuration) the sign would be reversed. The spin configuration is irrelevant in this case, since the direct term is much smaller than the exchange term. So far, the exciton-electron interaction has been discussed, which will be used later to determine the dynamics of polaritons interacting with acoustic phonons and with a cold electron gas. . . . . . . . . . . . . . . . . . . . . . . . . .

(35) 36. CHAPTER 3. EXCITON-ELECTRON INTERACTION.

(36) Chapter 4 Rate Equation Model In this chapter, we present a derivation of a rate equation performed by Stanley et. al. [22], which describes the polariton dynamics inside a microcavity. The starting point is the Hamiltonian of the coupled exciton-photon-phonon system which, to first order in coupling, is given by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ . !. (4.1). . . . . ! . . . . . . . . . . . . . . . . . similarly to Eq. 1.30, where we have changed the notation for simplicity. and are the annihilation operatos of the microcavity photon, the quantum well exciton, and the phonon, respectively, and any direct coupling between phonon and photon has been neglected. The phonon-exciton coupling constant is that for a three-dimensional phonon interacting with a two-dimensional exciton and includes the deformation potential. The expression for was given in Eq. (2.23). Eq. 4.1 can be reexpressed in terms of the polariton operator, using the Hopfield transformation [37], . . . . . . . . . !. . . . . . . !. . . . . . . . . (4.2). . ). ). !. . . . . . . ). . . . . . . . . . . . . . . . . . . % %. %. % . % . . . . '. . . . . . . . . % %. &. . where is the polariton operator of branch and vector , and we keep only the terms in the Hamiltonian which describe the scattering of polaritons with absorption and emission of phonons. Thus, (4.3) . . . . %. .. . . . ). ,. .. . . . . . . . . . . . . ). 1. 3. ). !. !. !. !. -. -. '. . . . . *. *. . . . . . . 0. . + %. %. + . % . &. . in which is the unit polarization vector, and corresponds to the upper and lower polariton branch, respectively. The Hamiltonian (4.2) has the form of coupled harmonic oscillator system with the polariton being the harmonic oscillator with a coupling constant dependent on the phonon population. The expressions for polariton dispersion and the coefficients and are not relevant in this chapter, however they are given in Ref.[22]. 1. 3. . *. . . . . . -. . . %. %. . 37.

(37) CHAPTER 4. RATE EQUATION MODEL. 38. The motion of polaritons and phonons is described, through Heisenberg’s equation of motion, making use of the Hamiltonian 4.2. In a real experiment, the microcavity is not a closed system and is subject to dissipation, which can be modeled by coupling of the microacavity photon field to a photon bath, or equivalently, the microcavity polariton field to a polariton bath. Coupling to a bath introduces well known damping and fluctuation terms under the Markoff approximation [38]. Under the presence of a polariton bath, the Heisenberg equations of motion for the amplitude of polariton and phonon fields take the form . . . . . . . . . . (4.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.5) . . . . . . . . . . . . . . . . . . . . . . . . . is the net amplitude damping for polariton in branch and of wave vector , and is the where net fluctuation operator due to the polariton bath. Since we are not interested in the noise properties of our system, we neglect from now on. We define the number operators and . Using , one can derive the equations for the occupation number, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $. . . . . . . . . . . #. . %. %. (4.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . $. . . . . . . . . . . . . . % . %. (4.7) . . . . . . . . . . . . . . . . . . . Having obtained the rate equations, the adiabatic approximation [39] is realized on the transition and next. For example operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *. . . . . -. . . . . . . . . . . . . . . /. . . . . . (4.8) and substitute the steady state values for the transition operators into 4.6 to get . # #. . . 1. . . . . 0 0. . . . . . . . . . . . . . . . . . . . . . . . -. . . *. . . . *. . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1. . 1. . . . . . . . /. . . . . . . . . . . . . *. . . . . . . . . . . . . . . . . . . . . . 1 . . . 1. 1. /. . . . . . . . /. . . . . (4.9).

(38) 39 Note that from now on, we treat identity. as an expectation value and not as an operator. Using the . . . . (4.10) . . . . . . . . . . . . . and setting . , we arrive at final result,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . !. . . . . . . . . ". . ". ". . . . . . . . . . !. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . !. . (4.11). . . %. . ". . . . . . . ". . ". . . . . . . !. . . Here, a phenomenological pump term has also been added. In setting , it is assumed that the polariton damping rate is long in comparison with scattering processes. Once we have the rate equation, the exciton-phonon and exciton-electron matrix elements, we are ready to solve it and thus to find the dynamical properties of polaritons on microcavities. . . . . . . . . . . . . .

(39) 40. CHAPTER 4. RATE EQUATION MODEL.

(40) Chapter 5 Relaxation kinetics of polaritons by scattering with acoustic phonons In this chapter the rate equation is solved numerically considering polariton-acoustic phonons interaction as the only relaxation mechanism and we show that a suppression of the bottleneck of the lower polariton states occurs at sufficiently high polariton densities. Besides, we show that small values of exciton-cavity detuning are favorable for the suppression of the bottleneck. We follow the two-coupled band model of chapter two in the strong coupling regime and we change slightly the notation. Thus, the state of the heavy-hole quantum well exciton with an in-plane dispersion relation is given by . . . . . . . . . . k. . . (5.1) . . . . . . . . . . is the fundamental exciton energy and is the effective mass describing the in-plane motion of the exciton center of mass. The empty cavity mode has a dispersion given by . . . . . . . #. . . $. (5.2). $. . . . . . . . . . ". is the cavity’s effective refraction index, the speed of light, and and the wave where vectors along and in the layer plane. Since a planar microcavity has an in-plane translation invariance, each exciton with an in-plane wave vector is coupled to a photon with the same inplane wave vector. Therefore, the upper and lower branch polaritons with the in-plane wave vector are approximated as the eingenstates of the 2 2 Hamiltonian $. . . ". . &. '. . . ). . (5.3). (. . . . . (. . . .. . is the radiative coupling between exciton and cavity modes. If for the upper an lower polariton branches, these eigenstates are . . (. . +. ,. -. and superscripts hold. . . . . . . . . . . -. . 1 . 2. 1 . . 2. 4. 3. 3. 0. 3. 3. . 41. (5.4).

(41) CHAPTER 5. RELAXATION KINETICS OF POLARITONS BY PHONONS. 42 with. (5.5) . . . . . . . . . . . . . . . . . . where where. . and. . . are the photon and exciton eigenstates at wave vector . respectively, and. . . . . . . . . . . . (5.6) . . . . . . The eigenenergies are given by . . . . . . . . . . (5.7). . . . . . . . . . . . . can scatter a polariton from Racall from chapter 2 that acoustic phonons of wave vector a state to state , through the deformation potential Hamiltonian , generating a coupling between the corresponding excitons and . The transition rates between and is the sum of two contributions arising from phonon the two polariton states, of energy emission and absorption and are given by . . !. . ". . ". . . . '. $. . . . (. . . . . . . . . . ). ,. . . &. . . . +. . -. +. . *. . . . -. . !. 4. /. . " . . . ". . . ) . 9. '. , ,. $. . 5 . ( . . 5 . . . . . . 1 . 2. . +. . -. + +. . +. +. +. *. 0. * 3. 3. &. '. (. ,. '. (. 3. . 1. 1. . 9. . 9. . (5.8) . . 5. 0. . 0. . . *. :. . . . =. . . . . :. =. . . +. 3. . . . Here is the exchanged in-plane momentum, is a phonon number state, is the polariton energy of the branch and wave vector , is the Bose occupation number for is the phonon energy. are the Hopfield coefficients, whose square phonons, and modulus give the exciton content in the polariton state of branch and wave vector . The explicit as a function of the conduction and valence band deformation potential expression of and can be expressed as . 5. . . . '. *. (. . +. . . . . 3. 5. . . '. (. . 1. . >. 0. . . . . . +. . . . *. ). ,. . +. -. . . +. . *. ?. @. ,. ?. . , ,. B. . . . . . . . . ) ,. . . E. . *. . . . . . + +. -. A. * *. 1 C. +. . . (5.9). -. + . . . -. + +. H *. 0 H. G. -. &. ?. where ,. ,. . . . . (5.10). . *. + -. *. . 1. >. +. 0. ,. . . . . (5.11). . . . . . +. *. . +. . . . I. . J. @. ". I. . J. ". 9. . (. . 9. . $. . . . ?. =. =. K. ?. . . ?. L. M. K L. M. (5.12).