氧化鋅微共振腔結構下之激子-極激子雷射之光學特性

(1)

國立交通大學

顯示科技研究所

碩士論文

氧化鋅微共振腔結構下之激子-極激子雷射

之光學特性

Optical Characteristics of Exciton-Polariton

Lasing in ZnO-Based Microcavity

研

究

生

：

吳

永

吉

..

指

導

教

授

：

盧

廷

昌

教

授

郭

浩

中

教

授

中

華

民

國

九

十

九

年

七

月

(2)

氧化鋅微共振腔結構下之激子-極激子雷射之光學特性

Optical Characteristics of Exciton-Polariton Lasing in

ZnO-Based Microcavity

研究生:吳永吉 Student: Yung-chi Wu

指導教授: 盧廷昌教授 Advisor: Prof. Tien-chang Lu

郭浩中教授 Prof. Hao-chung Kuo

國立交通大學電機資訊學院

顯示科技研究所

碩士論文

A Thesis

Submitted to Display. Institute

College of Electrical Engineering and Computer Science National Chiao Tung University

in Partial Fulfillment of the Requirements for the Degree of

Master In

Display. Institute

July 2010

Hsinchu, Taiwan, Republic of China

中華民國九十九年七月

(3)

氧化鋅微共振腔結構下激子-極激子雷射之光學特性

研究生:吳永吉指導教授: 盧廷昌教授郭浩中教授

國立交通大學顯示科技研究所

摘要

在本篇論文中，我們利用氧化鋅的微共振腔結構作為研究對象。在特性分析中，主要是光學特性上的相關研究。在發光特性方面，利用光激發螢光光譜(PL)以及反射頻譜，量測不同角度的發光特性，藉由變角度的譜譜可以定義出極激子的能量對波向量的關係圖，藉此了解光與激子之間的耦合情形，並且確認兩者之間耦合現象確實存在。接著我們利用調變共振腔長度來調整光存在微共振腔內的能量，使激子與光之間的耦合情形發生改變。觀察此時的極激子發光情形。在波向量等於零，當光子能量較激子能量來的低時，能量對波向量的曲線可以觀察到明顯的轉折處，使得當極激子由較大的能量向低能處掉落時，會容易阻塞再轉折處，這種瓶頸對於我們的目標-激子極激子雷射-是一項需要克服的問題，要克服這種瓶頸現象，便進行了不同的實驗來探討此一現象，我們藉由改變共振腔長度的方法，了解在光與激子對瓶頸現象的關係，接著我們利用變溫的變角度光激發螢光光譜(ARPL)，確認了極激子與聲子的散射可以幫助克服瓶頸現象，最後我們利用變功率的變角度光激發螢光光譜 (ARPL)，使瓶頸現象可以藉由極激子對極激子本身的散射現象來克服。為了在室溫的條件下觀測到玻色-愛因斯坦凝聚的現象，我們使用摻釹釩酸釔晶體脈衝雷射來進行激發並且順利在室溫下觀測到同調性的發光現象，與現有類似規格的面射型雷射相比所需的功率只有十分之一，因此我們認為這個是因為玻色-愛因斯坦凝聚的現象所形成的激子-極激子雷射。我們初步的證明，一樣是利用變角度的光激發螢光光譜(ARPL)，可以觀察到極激子克服平井現象後，集中於底部形成玻色-愛因斯坦凝聚的情形。這是在氧化鋅材料下觀測到玻色-愛因斯坦凝聚的例子。

(4)

Optical Characteristics of Exciton-Polariton Lasing in ZnO-Based Microcavity Student: Yung-chi Wu Advisors: Prof. Tien-chang Lu

Prof. Hao-chung Kuo

Display. Institute, National Chiao Tung University

Abstract

In this paper, we use zinc oxide micro-cavity structure to study the characteristics of

exciton-polaritons. By performing angle-resolved photoluminescence and angle-resolved reflection

measurements, we can probe the energy-wavevector dispersion curves of cavity polaritons. This

dispersion relationship can be used to understand the coupling strength between photons and excitons,

and confirm the existence of cavity polaritons. Furthermore, we observed the coupling between

different exciton-photon detunings by varying the length of microcavity. When the detuning is

negative, photon energy lower than the exciton energy, the anticrossing can be observed in the

dispersion curves, which causes a significant change in density of states. Under the condition, the

bottleneck behavior should be observed during the process of polariton relaxation. This consequence

may originate from the polariton states with very high photon fractions in the low angle region.

The bottleneck effect is an important obstacle to the realization of Bose-Einstein condensation in

microcavity. Several systematical experiments are performed to understand the possible physical

mechanisms inducing the polariton bottleneck effect. First, we change the cavity length in order to get

different exciton-photon detunings, which gives rise to different photon and exciton fractions, and the

(5)

significantly suppressed by the mechanism of polariton-phonon scattering at high temperature.

Consequently, we use the temperature-dependent and angle-resolved photoluminescence to confirm

the effect of polariton-phonon scattering. Finally, the polariton-polariton interaction is an important

factor under high pumping power condition, and the power-dependent angle-resolved

photoluminescence can help us to understand the factor.

In order to observe Bose-Einstein condensates at room temperature, we use Nd:YVO4 pulsed

laser as excitation source and observe a coherence light at room temperature with a low-threshold

pumping power, 1 order of magnitude smaller than in previously reported nitride-based vertical-cavity

surface-emitting lasers. This is an important evidence of an exciton-polariton laser induced by

Bose-Einstein condensate. In addition, based on the experimental results of angle-resolved

photoluminescence, we can observe the phenomenon that the polaritons could overcome the relazation

bottleneck, and then approach to the bottom of low polariton branch. This result demonstrates the

experimental observation of Bose-Einstein condensation in ZnO-based microcavity at room

(6)

Acknowledgement

去年的此時，我才剛成為碩二生，一點都沒有身為學長姐的自覺，所以有許多的事情都需要借助他人的幫助。首先要感謝實驗室的三位老師：王老師，盧老師，郭老師。三位老師不只在研究上的幫助，給予我們莫大的幫助，在基本的學問上，也幫助我釐清了許多疑惑，填補大學課業的大爛帳。在這點上，我特別感謝盧老師所給予的教導。也感謝實驗室給予的充足資源，良好的實驗地點，豐富且精良的實驗儀器。感激老師對於實驗結果的嚴格態度，使我們學到足夠嚴謹的實驗方法。對老師的感謝，並非三言兩語可以道盡。此外也感激謝文峰老師替我們成長了如此高品質的氧化鋅，以及李正中老師替我們鍍上了高反射率的上ＤＢＲ。感激俊榮學長給予的幫助，從我大三做專題開始，就發揮相當大的耐心再教導我。從半導體雷射的模擬，到微共振腔的教學，總是一步一步耐心的教導。在實驗室內老是看到他本身就有許許多多的計畫以及論文要趕工，在如此忙碌的情形下仍然要花時間在一位懵懵懂懂的大三專題生身上。這些事情的可貴在自己要帶學弟時更加清楚的感受到。十分感激他願意浪費時間再如此駑鈍的我身上，也祝福學長在接下來的道路上一帆風順。感謝我的實驗同伴詳淇所給予的巨大幫忙，沒有他的協助以及精準的調整光路技巧，這些實驗的結果一定無法如此順利，要他在大四時一邊忙於課業還要一邊兼顧實驗跟模擬，感到相當愧疚，甚至在畢業前的一段時間，即便自己的論文內容已經足夠，仍然花費不少時間協助我的實驗，真的萬分感激。也了解你在畢業前所面對的巨大的壓力，但你仍然可以順利克服，祝福你在接下來的博士生涯一切順利。感謝思維的耐心，很抱歉讓你在碩一生涯就經歷更換題目的窘境。也希望你能原諒學長在學問上的不成熟。祝福你在接下來的碩二生涯可以一帆風順。感謝同學們給予在學業上和生活上的幫助，阿伯，祝你畢業順利以及早日達陣。獸皇，我覺得魔術方塊比魔獸來的帥氣多了，考慮一下吧。二六，我知道你在畢業前夕並不順遂，我還是祝你早早畢業，一切都會過去的。啾博，你算是９７級的博班代表吧，祝你接下來的學業順利，別把身體搞壞了。要感謝的人事物實在是太多了，有太多的人給予我適時的協助，但是篇幅實在是不夠了，原諒我無法一一列出。祝同學跟朋友未來都一路順風

(7)

Abstract (Chinese)………....…i

Abstract (English) ………...ii

Acknowledgement………...iv

Content………..……...iv

List of Tables………vii

List of Figures………vii

Chapter1. Introduction……….1

1.1 Polaritons For Bose-Einstein Condensation(BEC) Study………1

1.1.1 Atomic System BEC……….1

1.1.2 Exciton-Polariton System BEC……….2

1.2 Properties Of Microcavity………..4

1.2.1 Q-factor and finesse………5

1.3 Introduction to Strong Coupling Between Photon And Exciton………..7

1.3.1 Strong Coupling………..7

1.3.1.1 Strong coupling between photon and exciton………11

1.3.2 Weak Coupling………..13

1.4 Difference of Material in Semiconductor Microcavity……….13

1.4.1 GaAs-Based Microcavity………15

1.4.2 CdTe-besed Microcavity………15

1.4.3 GaN-Based Microcavity………16

1.4.4 ZnO-Based Microcavity………20

1.4.4.1 Material Characteristics of Zinc-Oxide………21

Chapter2. The Coupling Between Exiton With Photon………..32

2.1 Quasi-Particle Model……….32

2.1.1 Wannier-Mott Exciton………..32

(8)

2.1.3 Quasi-Particle Model Simulation in MatLAB………..43

2.2 Transfer Matrix Method………..46

2.3 The Scattering Mechanisms Between Polariton With Other Particle………..50

2.3.1 Polariton dynametic rate equation……….51

2.3.2 The scattering mechanisms between polariton with other particle………52

2.3.2.1 The polariton-phonon scattering………..52

2.3.2.2 The polariton-polariton scattering………53

Chapter3. Experimental Systems………58

3.1 Sample Materials And Structures………..58

3.2 Photoluminescence Measurement………..59

3.2.1 Micro-Photoluminescence Measurement……….60

3.2.2 Angle-Resolved Photoluminescence Measurement………..61

3.3 Reflection Measurement……….61

Chapter4. Experiment Result And Discussion………..66

4.1 Polariton dispersion………66

4.2 Micro-Photoluminescence experiment results………67

4.3 Angle-resolved reflective experiment results……….68

4.3.1 Angle-resolved Photoluminescence experiment results………..70

4.3.2 Temperature dependent Angle-resolved Photoluminescence………71

4.3.3 Power-dependent Angle-resolved Photoluminescence……….74

4.4 Nonlinear emission ………75

4.4.1 Nonlinear emission from ZnO-based microcavity……….75

Chapter5. Conclusions………..90

(9)

List of Tables

Table 1.1: Parameter Comparison of BEC Systems………(31)

Table 1.2: Comparison of material use in exciton-polariton BEC Systems……….(31)

List of Figures

Fig 1.1 (a) Before condensation, the atoms look like fuzzy balls. (b) After condensation, the atoms lie exactly on top of each other . (c) Schematic of the apparatus. Six laser beams intersect in a glass cell, creating a magneto-optical trap (MOT). (d) Bose-Einstein Condensation at 400, 200, and 50 nK…(25) Fig 1.2 (a) the dispersion of bulk polariton (b) the dispersion of microcavity polariton………(26)

Fig 1.3 The different types of microcavity (a) planar microcavities (b) pillar microcavities (c) Whispering-gallery microdisk resonator (d) photonic-crystal cavity ……….………(26)

Fig 1.4 DBR reflectivity spectrum with different wavelength……….(27)

Fig 1.5 Microcavity reflectivity spectrum with different wavelength……….(27)

Fig 1.6 Schematic of two level system……….(27)

Fig 1.7. Probability for finding the atom in either the upper or lower level in the strong-field limit in the absence of damping. ………...……….(28)

Fig1.8 Polariton lasing and photon lasing. (a) The emission energy vs pump power for a N=12

multi-QW planar microcavity. (b) The dispersion characteristics of polariton BEC (green diamond) and photon lasing (pink triangle) as well as the linear dispersion of UP (red square) and LP (blue

(10)

circle) at low pump power. ………...……….(28)

Fig 1.9 A negative conductance polariton amplifier. (a) The LP emission intensity taken as a function

of energy and in-plane wave vector. The system is above the quantum degeneracy threshold at the

bottleneck. The solid line and the dashed line indicate the theoretical dispersion (b) Observed probe

emission for pump only, probe only, and simultaneous pump and probe excitation. ……….(29)

Fig 1.10(a) Semi-logarithmic plot displaying RT emission spectra at average pump power densities

ranging from 0.16 to 28.8 W/cm2at k=0, shifted for clarity. C and X are also reported (arrows). (b) Three-dimensional representation of the far-field emission with emission intensity displayed on the vertical axis linear vertical scale above threshold. C and X are also reported(white lines). …..….(29) Fig. 1.11 (a) Angle-resolved PL spectra at RT in the range of 0 º to 40 º for a ZnO hybrid MC. The

dotted line is the exciton mode. The solid lines are guides to the eye. (b) Experimental cavity olariton

dispersion curve. The dashed lines represent the cavity and exciton modes. …..………...(30)

Fig 1.12 (a) Structure of wurtzite, which is a member of the hexagonal crystal system and consists of

tetrahedrally coordinated zinc and sulfur atoms that are stacked in an ABABAB pattern. (b) the bangap

structure of wurtzite structure…..……….……...(30)

Fig 2.1 Excitons may be treated in two limiting cases, depending on the properties of the binding energyE . (a) Wannier-Mott exciton (b)Frenkel exciton …..………..….……...(55) _b

Fig. 2.2 polariton dispersion and corresponding Hopfield coefficients at (a) positive detuning (b) zero

(11)

Fig 2.3the simulation result of the reflectivity of microcavity in transfer matrix method………....(56)

Fig 2.4 Schematic of the structure for transfer matrix model ………....(56)

Fig 2.5 schematic of the polariton scattering processes (a) polariton-phonon scattering and polariton-electron scattering (b) polariton-polariton scattering………...(57)

Fig3.1 The schematic sketch of the ZnO-based microcavity structure………...(62)

Fig 3.2 The interface between the AlN/AlGaN DBR and the ZnO cavity is smooth as seen from the cross-sectional scanning electron microscope (SEM) image………...(62)

Fig 3.3 Refractive index profile and electric-field intensity in the growth direction for normal incidence at photon energy of 3.23 eV. ………...(63)

Fig 3.4The RT reflectivity spectra of a 30-pair AlN/Al0.23Ga0.77N DBR (dashed line) and a nine-pair SiO2 /HfO2 DBR(solid line). RT PL spectrum from a half cavity is located within the stop band of the bottom and top DBRs. . ………..…………...(63)

Fig3.5 The schematic diagram of photoluminescence setup ………..…………...(64)

Fig3.6 The schematic diagram of micro-photoluminescence setup……….……...(64)

Fig3.7 The schematic diagram of angle-resolved photoluminescence setup………(65)

Fig3.8 The spectrum of Xenon lamp………(65)

Fig 4.1 the micro-PL measurement result on different detuning case………(79)

Fig 4.2Experimental (open blue circle) and simulated (solid line) absorption spectra of a bulk ZnO at RT. ……….…………(79)

(12)

Fig 4.3 (a) Color map of the angular dispersion of measured reflectivity spectra from 8 to 38° at RT. (b) Color maps of the calculated angle-resolved reflectivity spectra with taking the resonant exciton into

account. (c) Simulation of angle-resolved reflectivity spectra for the bulk ZnO MCs after taking the

absorption of scattering states into account. ……….………...(80)

Fig 4.4 he experimental angle-resolved PL spectra of the ZnO MCs with approximate exciton-photon detunings of: (a) δ = −78 meV, and (b) δ = −26 meV at RT. The dashed line corresponds to the uncoupled exciton energy. The curve red line is a guide for the eyes, showing the dispersion of lower

polariton branch. ……….……….…...(81)

Fig 4.5 The color maps of the experimental angular dispersions of measured PL spectra at (a) 100 K, (b) 200 K, and (c) 300 K for the case of δ = −78 meV at RT. The curved dashed lines represent the calculated LPBs and the curved dot line and horizontal dot line show the pure cavity and exciton

modes, respectively. (d)~(f) show the calculated angle-dependent composition of the cavity photon and

exciton modes for the three detunings induced by different temperatures. ……….…(82)

Fig 4.6 The color maps of the experimental angular dispersions of measured PL spectra at (a) 150 K, (b) 250 K, and (c) 300 K for the case of δ = −26 meV at RT. (d)~(f) show the calculated angle-dependent composition of the cavity photon and exciton modes for the three detunings induced

by different temperatures. ………..……….…(83)

Fig 4.7 The color maps of the experimental angular dispersions of measured PL spectra at (a) 150 K, (b) 200 K, and (c) 250 K for the case of δ = −8 meV at 250 K. (d)~(f) show the calculated

(13)

angle-dependent composition of the cavity photon and exciton modes for the three detunings induced

by different temperatures. ………..……….…(84)

Fig 4.8 Experimental LP-PL intensities as a function of the external detection angle for different

excitation power densities at room temperature. The detuning between the uncoupled photon and

exciton modes at k=0 is − 68 meV. The intensities are normalized to the excitation power density.

Fig 4.9 The PL spectra below threshold and above threshold with different detuning condition…..(85)

Fig 4.10 The color maps of the experimental angular dispersions of measured PL spectra below

threshold and above threshold at 300 K. ………..……….……….….(86)

Fig 4.11 Integrated intensity vs pump power (solid points), with a guide line for the eyes(red dash

line). ………..……….……….(87)

Fig4.12 Experimental LP-PL intensities as a function of the external detection angle for different

excitation power at a room temperature. ……….…………...…….(88)

Fig 4.13 The color maps of the experimental angular dispersions of measured PL spectra below

(14)

Chapter1. Introduction and Motivation

1.1 Polaritons for Bose-Einstein condensation(BEC) Study

Bose-Einstein condensation (BEC) has been a source of imagination and innovation of physicists

ever since its first proposal by Einstein in 1925. Bose–Einstein condensation (BEC), also simply

“Bose condensation”, is a phase transition for bosons leading to the formation of a coherent

multiparticle quantum state characterized by a wavefunction, and the BEC occupies the lowest energy

level of the system that coincides with the chemical potential. In particle physics, bosons are

subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum

state, include photon, meson, some of atom with integer spin, exciton, etc. In 1995, the first

unambiguous realization of BEC was achieved in dilute atomic gasses. The effort devoted to atomic

systems has harvested a modern branch of physics, ( ultra- )cold atom physics, which continues to be a

test-ground of theories and a cradle of novel applications. In recent decades, the well developed

fabrication techniques in semiconductor make it possible to observe the BEC phenomenon at

laboratory through the microcavity-polariton BEC system. Below we briefly review the history of

BEC research.

1.1.1 Atomic system BEC

The BEC phenomenon is predicted in 1924 by Satyendra Nath Bose and Albert Einstein, and is

(15)

metastable helium gases. In atomic system BEC, because the condensation at low densities is

achievable only at very low temperatures (~nK), the laser-cooling technique is necessary. The primary

force used in laser cooling and trapping is the recoil when momentum is transferred from photons

scattering off an atom, and the cooling is achieved by making the photon scattering rate

velocity-dependent using the Doppler effect. Moreover, a magnetic quadruple field generated by two

coils carrying equal currents flowing in opposite directions traps those atoms, as shown in fig 1.1(c).

because of the laser cooling technique, the Bose-Einstein condensation can be observed at 200nK, as

shown in fig 1.1(d), and Cornell, Wieman and Ketterle won the 2001 Nobel Prize in Physics for the

achievements [18]. However, the application of atomic system BEC in room temperature is extremely

different.

1.1.2 Exciton-polariton system BEC

Exciton BEC was first proposed in 1962 by Moskalenko et al. [4] and Blatt et al. (18). A most

well-known experimental system is the ortho-excitons in bulk Cu2O. This system was considered to

have shown, in the first conference on BEC held in 1995, the most convincing evidence of BEC [5].

Yet it was found out later that the auger-recombination of excitons prevented the system from reaching

the critical density of BEC. In 2002, a few macroscopic phenomena observed in quantum-well exciton

systems were again proposed to be related to BEC. Yet more careful analysis later concluded otherwise.

(16)

of a phase transition was inferred from these experiments, e.g. the coherenceproperties and momentum

distribution functions of the excitons. The search continuesfor exciton BEC and the question remains

open if BEC is ever possible in a solid state system.

In 1968, BEC was proposed to be also possible with bulk polaritons [7]. However, these

polaritons are outside of the optical cone and do not directly couple to light, as shown in Fig 1.2(a). It

is very difficult to study the bulk polaritons experimentally. Moreover, the minimum energy of the

bulk polariton bands are the crystal ground state with zero excitation energy. BEC is possible only

with states at an energy-relaxation bottleneck. These states have a large degeneracy in momentum,

adding much complication to the physics. There has been no successful experimental effort toward

bulk polariton condensation.

In decades, a more experimentally accessible solid-state system becomes available when the

strong-coupling regime is reached in an epitaxially grown quantum-well microcavity [8]. Due to

confinement of both the cavity photon field and the quantum excitons along the growth direction, translational symmetry is broken in the longitudinal direction, only the transverse wavenumber k_ is

a good quantum number for microcavity polaritons. Hence for the relevant polariton states, there exists a one-to-one coupling between each internal polariton mode with certain k_ at energy E_LP

( )

k_ and each external photon mode with the same k_ and E_LP

( )

k_ emitted into certain angle θ relative to the

growth direction, as shown in Fig 1.2(b). The coupling rate is determined by the fixed cavity photon

(17)

the external photon emission by well developed optical techniques. Within a decade after the first

observation of microcavity polaritons, stimulated scattering threshold of polaritons were reported by

various groups [9, 10, 11, 12].

Table 1.1 compares the basic parameters of atomic gases to excitons and polaritons in

semiconductors. The parameter scales of these systems differ by many orders of magnitude. Even in a

common quantum phase transition, each system is expected to have its own characteristics, to reveal

particular pieces of unexplored fundamental physics, and to have unique applications. Most notable for

the polariton system is its very light effective mass and very short time scale. The former leads to a

critical temperature of phase transitions ranging from 1 K up to room temperature. The latter dictates

the dynamic nature of polariton phase transitions.

1.2 Properties of Microcavity

A microcavity is an optical resonator close to, or below the dimension of the wavelength of light.

Micrometre- and submicrometre-sized resonators use two different schemes to confine light. In the

first, reflection off a single interface is used, for instance from a metallic surface, or from total internal

reflection at the boundary between two dielectrics. The second scheme is to use microstructures

periodically patterned on the scale of the resonant optical wavelength, for instance a planar multilayer

Bragg reflector with high reflectivity, or a photonic crystal. Since confinement by reflection is

(18)

the same microcavity. The resonant optical modes within a microcavity have characteristic lineshapes,

wavelength spacings and other properties that control their use. A longitudinal resonant mode has an

integral number of halfwavelengths that fit into the microcavity, while transverse modes have different

spatial shape. The common designs of microcavity are listed in Fig 1.3

The most common microcavity is the planar microcavity in which two flat mirrors are brought

into close proximity so that only a few wavelengths of light can fit in between them. To confine light

laterally within these layers, a curved mirror or lens can be incorporated to focus the light, or they can

be patterned into mesas.

Due to well developed fabrication techniques, Distributed Bragg reflector (DBR) can achieve the

request of high reflectance mirror. A DBR is made of layers of alternating high and low refraction indices, each layer with an optical thickness of λ/4. Light reflected from each interface destructively interfere, creating a stop-band for transmission. As shown in Fig 1.4, the DBR stop-band width is

considered the number of pairs and the refraction index difference between two materials.

When two such high-reflectance DBRs are attached to a layer with an optical thickness integer

times of λ/2, a cavity resonance is formed at λ, leading to a sharp increase of the transmission T at λ, as

shown in Fig1.5.

1.2.1 Q-factor and finesse

The quality-factor (or Q-factor) has the same role in an optical cavity as in an LCR electrical

(19)

characteristic parameter, the cavity quality factor Q of the cavity quality. c c Q ω λ δω λ = ≈ ∆ Eq(1.1)

The finesse of the cavity is defined as the ratio of free spectral range (the frequency separation

between successive longitudinal cavity modes) to the linewidth (FWHM) of a cavity mode :

1 c c R F R ω π δω ∆ = = − Eq(1.2)

Q is the average number of round trips a photon travels inside the cavity before it escapes. That is,

the higher Q value means the higher ability to confirm a photon. Equivalently, the exponentially decaying photon number has a lifetime given by

c

Q

τ = _ω . Because the mode frequency separation 2 c c L π ω

∆ = is similar to the cavity mode frequency in a wavelength-scale microcavity, the finesse and the Q-factor are not very different. This is not the situation for a large cavity, in which case the

Q-factor becomes much greater than the finesse because of the long round-trip propagation time.

Instead, the finesse parameterizes the resolving power or spectral resolution of the cavity. 2n_neffL_cav

Q F

λ

= Eq(1.3)

In order to achieve the higher quality factor (Q) and higher finesse, we have to use the higher

(20)

1.3 Introduction to coupling between photon and exciton

1.3.1 The time-dependent Schrödinger equation

The quantum treatment of the interaction between light and atoms is usually developed in terms

of the two-level atom approximation. This approximation is applicable when the frequency of the light

coincides with one of the optical transitions of the atom. The condition is depicted schematically in Fig

1.6.

The time-dependent Schrödinger equation for a two-level system in the presence of the light is a

great method to understand the coupling between photon and exciton. In other words, we must solve: ˆ H i t ∂Ψ Ψ = ∂  Eq(1.4)

for an atom with two energy levels E1 and E2 in the presence of a light wave of angular frequency

ω. We shall assume that the light is very close to resonance with the transition, so that

0 ω ω δω= + Eq(1.5) Where

(

2 1

)

0 E E ω = − _ , and δω<<ω₀ Eq(1.6)

Exact resonance thus corresponds to δω =0, We start by splitting the Hamiltonian into a time-independent partH which describes the atom in the dark, and a perturbation term ˆ₀ V t which ˆ

( )

accounts for the light–atom interaction:

( )

0

ˆˆˆ

H =H r +V t Eq(1.7)

(21)

system: 0 ˆ i i H i t ∂Ψ Ψ = ∂  Eq(1.8)

with Ψ_i

( )

i E t2

1ˆˆ e1 2 2e 1 1e 2 2e

i E t i E t i E t i E t

c Vψ −  +c Vψ −  =i cψ −  +i cψ −  Eq(1.15) On multiplying by ψ , integrating over space, and making use of the orthonormality of the ₁*

(22)

* 3

i jd r ij

ψ ψ =δ

∫

Eq(1.16)

where δ is the Kronecker delta function, we find that: _ij

( )

(

( )

)

2 1 21 2 22 i t i c t = − c t V eω +c t V  Eq(1.19)

To proceed further we must consider the explicit form of the perturbation ˆV . In the

semi-classical approach, the light–atom interaction is given by the energy shift of the atomic dipole in

the electric field of the light:

( )

ˆ

V t = ⋅er E t Eq(1.20)

We arbitrarily choose the x-axis as the direction of the polarization so that we can write:

( ) (

0, 0, 0 cos

) ( )

E t = E ωt Eq(1.21)

where E is the amplitude of the light wave. The perturbation then simplifies to: ₀

( )

(

)

0 0 ˆ _cos 2 i t i t V t exE t exE eω e ω ω − = = + Eq(1.22)

and the perturbation matrix elements are given by:

( )

0

(

)

* 3 2 i t i t ij i j eE V t = eω +e−ω

∫

ψ ψx d r Eq(1.23)

Now the dipole matrix element µ is given by: _ij µ = −

∫

ψ ψ ≡ −

(23)

so that we can write:

( )

0

(

)

2 i t i t ij ij E V t = − eω +e−ω µ Eq(1.25)

Since x is an odd parity operator and atomic states have either even or odd parities, it must be the case that µ₁₁=µ₂₂ = Moreover, the dipole matrix element represents a measurable quantity and 0

must be real, which implies thatµ₂₁=µ₁₂, because µ₂₁ =µ₁₂*

With these simplifications, equation 1.17, 1.19 reduce to:

( )

(

( ) ( )

)

_{( )}

( )

(

( ) ( )

)

_{( )}

0 0 0 0 0 12 1 2 0 12 2 1 2 2 i t i t i t i t E c t i e e c t E c t i e e c t ω ω ω ω ω ω ω ω µ µ − − + − − + = + = +     Eq(1.26)

We now introduce the Rabi frequency defined by:

12 0

R µ E

Ω =  Eq(1.27)

We then finally obtain:

( )

(

( ) ( )

)

_{( )}

( )

(

( ) ( )

)

_{( )}

0 0 0 0 1 2 2 1 2 2 i t i t R i t i t R i c t e e c t i c t e e c t ω ω ω ω ω ω ω ω − − + − − + = Ω + = Ω +   Eq(1.28)

These are the equations that we must solve to understand the behavior of the atom in the light

field. It turns out that there are two distinct types of solution that can be found, which correspond to

(24)

1.3.1.1 Strong coupling between photon and exciton

In order to find a solution to equation 1.28 in the strong-field limit we make two simplifications. First, we apply the rotating wave approximation to neglect the terms that oscillate at ±

(

ω ω+ ₀

)

, as in the previous section. Second, we only consider the case of exact resonance with δω=0. With these simplifications, equation 1.28 reduces to:

( )

1 2 2 1 2 2 R R i c t c t i c t c t = Ω = Ω   Eq(1.29)

We differentiate the first line and substitute from the second to find:

2 1 2 1 2 R 2 R i i c = Ω c =_ Ω _ c     Eq(1.30) We thus obtain 2 1 1 0 2 R c +_Ω _ c =    Eq(1.31)

which describes oscillatory motion at angular frequency 2

R

Ω

. If the

particle is in the lower level at t = 0 so that c₁

( )

0 = and 1 c₂

( )

0 = , the solution is: 0

( )

(

)

( )

(

)

1 2 cos 2 sin 2 R R c t t c t i t = Ω = Ω Eq(1.32)

The probabilities for finding the electron in the upper or lower levels are then given by:

( )

(

)

( )

(

)

2 ₂ 1 2 ₂ 2 cos 2 sin 2 R R c t t c t t = Ω = Ω Eq(1.33)

(25)

the upper level, whereas at t =2π Ω it is back in the lower level. The process then repeats itself _R with a period equal to 2π Ω . The electron thus oscillates back and forth between the lower and _R upper levels at a frequency equal to Ω_R 2π . This oscillatory behaviour in response to the strong-field

is called Rabi oscillation or Rabi flopping. When the light is not exactly resonant with the transition, it

can be shown that the second line of equation 1.33 is modified to:

( )

2 2 ₂

(

)

2 2 sin 2 R c t =Ω Ωt Ω Eq(1.34) where 2 2 2 R δω Ω = Ω + Eq(1.35)

δω being the detuning. This shows that the frequency of the Rabi oscillations increases but their amplitude decreases as the light is tuned away from resonance. For transitions in the visible-frequency

range, the experimental observation of Rabi flopping requires powerful laser beams. In many cases,

these lasers will be pulsed, so that the electric field amplitude E0 varies with time. Equation 1.27

then tells us that the Rabi frequency Ω_R 2π

1.3.2 Weak coupling between photon and exciton

Weak coupling between two systems refers to the regime opposed to strong coupling where

dissipation dominates over the system interaction so that the coupling between the modes can be dealt

with pertubatively and both modes retain essentially their uncoupled properties.

With a low-intensity source, the electric field amplitude will be small and the perturbation weak.

The number of transitions expected is therefore small, and it will always be the case that

( )

1 2

c t c t . In these condition, we can get the solution of equation 1.28:

( )

2 1 2 2 ₂ 2 1 2 R c t c t t = Ω   = _ _ Eq(1.37)

The weak coupling between exciton and light manifests itself in the appearance of the splitting

between the imaginary parts of the eigenfrequencies of exciton-polariton modes at the resonance

between bare exciton and photon modes. In this regime the real parts of two polariton

eigenfrequencies coincide at the resonance, and two polariton resonances in the reflection or

transmission spectra usually coincide, [17].

1.4 Difference Of Material In Semiconductor Microcavity

Semiconductor microcavities have recently attracted much attention because of the control that

they provide on the light-matter interaction in solid-state systems. In the strong coupling regime,

(27)

which open the way to a broad area of fundamental and applied investigations.

The first observations of exciton–polaritons in microresonators were reported for GaAs and

CdTe-based resonators at low temperatures (T < 30 K) [19, 20, 21]. In these materials, the exciton

binding energy is smaller than the thermal energy at RT and excitons are not stable at RT. Therefore,

many efforts have been made to obtain microresonators with gain media that reveal an exciton binding

energy larger than the thermal energy at RT, such as organic semiconductors [22, 23], or GaN [24, 25,

26] and ZnO [27, 28, 29]. The advantage of organic semiconductors is their huge exciton oscillator

strength resulting in a large coupling strength between the excitons and the cavity photons. However,

often the low crystal quality of such materials leads to emission spectra that are superposed from the

emission from localized and delocalized states. In contrast, inorganic semiconductors reveal high

crystal qualities with an emission from well-defined exciton states. Here, GaN and ZnO are the most

prominent candidates that are considered for high temperature applications. ZnO offers some

advantages, since it reveals the largest exciton oscillator strength of the technologically relevant

semiconductors, about three times larger than that of GaN [30 ,31], and its exciton binding energy is

about twice the thermal energy at RT.

The critical temperature is determined by the Rydberg energy of exciton, oscillator strength, etc.

Most of those factors are depended on the material properties. Table 1.2 shows several common

semiconductor materials, including GaAs, GaN, and ZnO. GaN and ZnO are the well-known

(28)

60meV, respectively. That is to say, at room temperature, more excitons exist in the room temperature

in GaN and ZnO microcavity.

1.4.1 GaAs-Based Microcavity

The choice of a direct band-gap semiconductor depends first on the fabrication technology. The

best fabrication quality of both quantum well and microcavity has been achieved via molecular-beam epitaxy growth of AlxGa1−xAs-based samples

(

0≤ ≤ , thanks to the close match of the lattice x 1

)

constants a of AlAs and GaAs and a relatively large difference between their band-gap energies_lat E . g

At 4K, GaAs has a =5.64 Å and _lat E =1.519 eV and AlAs has g a =5.65 Å and lat E =3.099 eV. g

Nearly strain and defect-free GaAs QWs are now conventionally grown between AlxGa1−xAs.

Inhomogeneous broadening of exciton energy is limited mainly by monolayer QW thickness

fluctuation. Nearly defect-free microcavity structures can be grown with more than 30 pairs of

AlAs/GaAs layers in the DBRs and with a cavity quality factor Q exceeding 105 [32]. Many signatures

of polariton condensation were first obtained in GaAs-based systems , [33, 34, 35], as shown in Fig

1.8 .

1.4.2 CdTe-based Microcavity

Another popular choice is the CdTe-based II-IV system, with CdTe QWs and MgxCd1−xTe and

(29)

energy and larger oscillator strength, as well as larger refractive index contrast (hence less layers

needed in the DBRs). The smaller Bohr radius of CdTe excitons, on the one hand, allows a larger

saturation density and, on the other hand, reduces the polariton and acoustic phonon scattering. Hence

the energy relaxation bottleneck is more persistent in this system, which prevented condensation in the

LP ground state in early experiments, [37, 38]. By adjusting the detuning to facilitate thermalization,

partially localized polariton condensation into the ground state was finally observed in 2006, [39] , as

shown in fig 1.9.

1.4.3 GaN-Based Microcavity

GaN-based MCs are beginning to receive interest in the research community. A realistic model

for room temperature polariton laser has been proposed for a GaN MC by Malpuech [40]. In the preceding report, the model structure was a 3λ/2 MC which consisted of a cavity layer with 4

monolayers thick 9 QWs between Al0.2Ga0.8N/Al0.9Ga0.1N DBRs, 11 pairs on the top and 14 pairs at

the bottom. The critical temperature of BEC of cavity polaritons was predicted to be 460 K with a

room temperature polariton lasing threshold as small as 100mW. Several groups have already reported

polariton luminescence at room temperature from bulk [41, 42, 43, 44] and QW MCs. The first

experimental results of the strong coupling regime in GaN-based MCs were reported by

Antoine-Vincent et al, [45]. The MCs fabricated by a wafer-bonding technique was composed of

(30)

SiO2/ZrO2 DBRs. The anticrossing behavior was observed by angle-resolved reflectivity

measurements showing a vacuum Rabi splitting of 6 meV. By increasing the number of QWs from 3 to

10, the vacuum Rabi splitting was increased to 17 meV. An impediment for strong coupling regime in

this particular InGaN QW-MC was a low finesse cavity and/or large inhomogeneous broadening of the

QW emission, [46]. Bulk GaN-based MCs were further studied for polariton emission in the strong

coupling regime [41, 42, 43, 44]. In a bulk GaN MC with lattice matched AlInN/(Al)GaN DBRs, a

strong bottleneck effect was observed at room temperature by photoluminescence (PL) measurements

[43]. In an attempt to use ubiquitous Si substrates, bulk GaN MCs with a 10 pair AlN/Al0.2Ga0.8N DBR have been grown directly on Si (111),[41, 42]. A vacuum Rabi splitting of approximately 50

meV was observed up to room temperature by angle-resolved reflectivity and PL measurements. A

vacuum Rabi splitting of 43 meV in GaN hybrid MCs in the strong coupling regime was reported by

Alyamani et al, [44]. despite a cavity Q factor of about 160 or less. A GaN/Al0.2Ga0.8N QW-MC with a

sharper linewidth enabled observation of cavity polaritons at room temperature using angle-resolved

PL [47]. A vacuum Rabi splitting of 30 meV was observed and the exciton oscillator strength was

estimated to be ~ 3 × 1013 cm-2 per QW.

Room temperature polariton lasing in a bulk GaN MC under nonresonant pulsed optical pumping

has been demonstrated by Christopoulos et al, [48]. The 3λ/2 bulk GaN cavity was sandwiched

between a bottom 34 pair Al0.85In0.15N/Al0.2Ga0.8N DBR and a top 10 pair SiO2/Si3N4 DBR. The Q

(31)

above the upper DBR stop band, and the system in the strong coupling regime was confirmed by the

observed anti-crossing behavior from angleresolved PL. A clear nonlinear behavior is seen for the emission at λ ≈ 365 nm above the critical threshold of Ith = 1.0 mW. This corresponds to a carrier

density of N3D ~ 2.2 × 1018 cm-3, which is an order of magnitude below the Mott density ≈ 1 – 2 × 1019

cm-3 in GaN at 300 K. Additionally, the emission line was observed to blueshift with increasing pump

power and lock at threshold due to polariton-polariton interactions.

Further challenging, room temperature strong coupling regime and nonlinear effects in

GaN-based QWs MCs were studied. Christmann et al [49, 50, 51]. employed GaN-based hybrid MCs which consist of a 3λ cavity layer with 67 period of GaN/Al0.2Ga0.8N MQWs sandwiched between a

35 pair of lattice-matched Al0.85In0.15N/Al0.2Ga0.8N DBR and a 10 pair SiO2/Si3N4 DBR. Due to high

quantum efficiency, InGaN-based emitting devices are commonly used. However, large

inhomogeneous broadening of QWs at room temperature is a serious problem in efforts to attain the

strong coupling regime, [52]. By comparison, GaN/AlGaN QWs have a narrower emission linewidth

with a broadening of ~ 38 meV which is capable of paving the way for achieving the strong coupling

regime. In the lattice-matched GaN/AlGaN QWs MC system, the strong coupling regime at room

temperature was demonstrated using angle-resolved reflectivity measurements observed at small

angles followed by an asymptotic trend towards the uncoupled exciton energy (X). Anticrossing

between the lower polariton branch (LPB) and the upper polariton branch (UPB) was observed at 17°,

(32)

observed at room temperature. In order to observe the nonlinear optical properties, MCs were

nonresonantly excited by a pulsed laser (λpump = 266 nm). Fig. 1.10 (a) shows a series of emission

spectra at average pump power densities ranging from 0.16 to 28.8 W/cm2 at k// = 0. The nonlinear

behavior is clearly observed at a relatively low threshold pump power density ~18 W/cm2,

corresponding to a calculated density of 8 × 109 cm-2 per QW. This threshold pump power density is ~

1/3 smaller than that in GaN-based VCSELs [53]. It should note that a further increase of the pump

power results in broadening of the emission peak due to increasing polariton-polariton interactions

occurring in the condensates. Above threshold, the linewidth reduced from ~15 meV to ~ 0.46 meV.

In a perfect polariton laser, polarization should randomly change for each realization of

condensate. Baumberg et al. observed the spontaneous polarization build up in room temperature

GaN-based polariton lasers excited by short optical pulses [54]. The Stokes vector of the emitted light

changes its orientation randomly from one excitation pulse to the other. Although it was unpolarized

below threshold, the polartization of polariton emission above threshold are linearly polarized, but

with no preferential orientation. This behavior is completely different from any conventional laser

including VCSELs. A spontaneous build up of polarization could be interpreted as spontaneous

symmetry breaking in a Bose-Einstein condensate of exciton-polaritons.

Interest is now brought to wide band-gap materials because the strong coupling regime is stable

up to room temperature and the exciton binding energy is much larger, leading to stronger

(33)

achieved.

1.4.4 ZnO-Based Microcavity

Another wide bandgap semiconductor, ZnO is an attractive candidate for ultraviolet (UV)

optoelectronics devices. ZnO has an exciton binding energy (60 meV) that is more than twice that of

GaN (~26 meV). Zamfirescu et al [55]. predicted a large Vacuum Rabi splitting ~120 meV for cavity

polaritons in a model ZnO MC sandwiched between Mg0.3Zn0.7O/ZnO DBRs, which projects to an

extremely low threshold polariton laser (~2 mW) at room temperature. A record of ~191 meV has been

predicted but not yet experimentally observed [56]. On the reflector side, Chichibu et al. reported high

reflectivity SiO2/ZrO2 DBRs for ZnO based MCs owing to the large refractive index contrast between

SiO2 and ZrO2, giving rise to a high reflectivity (> 99%) and a wide stop band even for an 8 pair

SiO2/ZrO2 DBR. Recently, ZnO-based MCs were grown by different growth techniques and tested

under optical pumping. Shimada et al, [58, 59, 60]. observed a vacuum Rabi splitting of 50 meV in

ZnO-based hybrid MCs grown by molecular beam epitaxy (MBE) sandwiched between a 29 pair of

AlGaN/GaN bottom DBR and an 8 pair dielectric (SiO2/SiNx) top DBR [58]. Fig. 1.11 (a) shows the

angle-resolved PL spectra at room temperature up to 40°. It is clear that the lower polariton mode gets

closer to the uncoupled exciton mode, and the upper polariton mode is dispersed from the exciton

mode to the cavity mode. The experimental cavity polariton dispersion curve shown in Fig. 1.11 (b)

(34)

mode energy crosses the exciton mode. Schmidt-Grund et al. grew λ/2-thick ZnO-based planar MCs

which consist of a ZnO cavity layer surrounded by a 10.5 pair ZrO2/MgO DBR prepared by

pulsed-laser deposition (PLD). A large vacuum Rabi splitting of ~ 78 meV was obtained from

angle-resolved reflectivity and PL measurements [59]. Using a dielectric MC consisting of a λ-thick

ZnO cavity layer and two HfO2/SiO2 DBRs by PLD and RF magnetron sputtering, respectively, cavity

polariton formation was demonstrated by Nakayama et al [60]. The vacuum Rabi splitting energy was

estimated ~ 80 meV. However, no polariton lasing was reported in any ZnO-based MCs as of yet.

Nevertheless, the above mentioned results are promising towards the realization of room temperature

ZnO-based polariton devices [61]. Moreover, ZnO-based electrical injection polariton lasers may also

be realizable in the future when reproducible and reliable p-type conductivity is achieved in ZnO. [62,

63]

1.4.4.1 Material Characteristics Of Zinc-Oxide

In recent years, the desire for blue and UV diode lasers and light emitting diodes has prompted

enormous research efforts into II–VI and III–V wideband gap semiconductors. Among the well-known

semiconductor materials employed in various technical applications, two unique positions are held by

gallium nitride (GaN) and zinc oxide (ZnO) in the wide direct band gap semiconductor. In the material

property, both GaN and ZnO have many similar aspects, such as material structure, lattice constant,

(35)

exciton binding energy of 60 meV, which is only 30 meV for GaN. Owing to the larger exciton

binding energy, more excitons exist in the room temperature, resulting in higher luminescence than

GaN. Furthermore, ZnO can be grown at lower temperature on the cheaper substrate and lead to low

cost of growth. However, because of more intrinsic defects, the hard growth of p-type ZnO to achieve

the p-i-n junctions, and the degradation of material quality, the current commercial blue and UV LEDs

are primitively composed of GaN. However, GaN-based LEDs still face some problems of the

luminescence, such as more defects in the material and low electron-hole recombination of c-direction

growth. Therefore, it is worth making the further researches on the material of ZnO and GaN on

purpose of possessing well-performed LEDs and LDs.

In materials science, ZnO is often called a II-VI semiconductor because zinc and oxygen belong

to the 2nd and 6th groups of the periodic table, respectively. This semiconductor has several favorable

properties: good transparency, high electron mobility, wide band-gap, strong room-temperature

luminescence, etc.

Zinc oxide crystallizes is hexagonal wurtzite, as shown in Fig1.12(a). The hexagonal structure has

a point group 6°mm or C6v, and the space group is P63mc or C6v4. The lattice constants are a = 3.25 Å

and c = 5.2 Å; their ratio c/a ~ 1.60 is close to the ideal value for hexagonal cell c/a = 1.633. As in

most II-VI materials, the bonding in ZnO is largely ionic, which explains its strong piezoelectricity.

Due to this ionicity, zinc and oxygen planes bear electric charge (positive and negative, respectively).

(36)

colorless and transparent. Advantages associated with a large band gap include higher breakdown

voltages, ability to sustain large electric fields, lower electronic noise, and high-temperature and

high-power operation. The bandgap of ZnO can further be tuned from ~3–4 eV by its alloying with

magnesium oxide or cadmium oxide.

Most ZnO has n-type character, even in the absence of intentional doping. Native defects such as

oxygen vacancies or zinc interstitials are often assumed to be the origin of this, but the subject remains

controversial. An alternative explanation has been proposed, based on theoretical calculations, that

unintentional substitutional hydrogen impurities are responsible. Controllable n-type doping is easily

achieved by substituting Zn with group-III elements Al, Ga, In or by substituting oxygen with

group-VII elements chlorine or iodine. Reliable p-type doping of ZnO remains difficult. This problem

originates from low solubility of p-type dopants and their compensation by abundant n-type impurities,

and it is pertinent not only to ZnO, but also to similar compounds GaN and ZnSe. Measurement of

p-type in "intrinsically" n-type material is also not easy because in-homogeneity results in spurious

signals.

Current absence of p-type ZnO does limit its electronic and optoelectronic applications which

usually require junctions of n-type and p-type material. Known p-type dopants include group-I

elements Li, Na, K; group-V elements N, P and As well as copper and silver. However, many of these

form deep acceptors and do not produce significant p-type conduction at room temperature.

(37)

common potential applications are in laser diodes and light emitting diodes (LEDs). Some

optoelectronic applications of ZnO overlap with that of GaN, which has a similar bandgap (~3.4 eV at

room temperature). Compared to GaN, ZnO has a larger exciton binding energy (~60 meV, 2.4 times

of the room-temperature thermal energy), which results in bright room-temperature emission from

ZnO. Recent studies of ZnO epilayers have observed spontaneous emission from free-exciton (FE)

radiative recombination as well as stimulated emission from exciton-exciton scattering (EES) and

electron-hole-plasma (EHP) radiative recombination at temperature up to ~550K.[3] Other properties

of ZnO favorable for electronic applications include its stability to high-energy radiation and to wet

chemical etching. The pointed tips of ZnO nanorods result in a strong enhancement of an electric field.

Therefore, they can be used as field emitters. Transparent thin-film transistors (TTFT) can be produced

with ZnO. As field-effect transistors, they even may not need a p–n junction, thus avoiding the p-type

doping problem of ZnO. Some of the field-effect transistors even use ZnO nanorods as conducting

(38)

Fig 1.1 (a) Before condensation, the atoms look like fuzzy balls. (b) After condensation, the

atoms lie exactly on top of each other . (c) Schematic of the apparatus. Six laser beams intersect in a

glass cell, creating a magneto-optical trap (MOT). (d) Bose-Einstein Condensation at 400, 200, and 50

(39)

z k k E exc E 0 E=

LP

can’t couple to light

optical cone

UP

couples to light

exc E k_ E

LP

can’t couple to light

(a)

(b)

Fig 1.2 (a) the dispersion of bulk polariton (b) the dispersion of microcavity polariton

Fig 1.3 The different types of microcavity (a) planar microcavities (b) pillar microcavities [13]

(40)

250 300 350 400 450 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength(nm) Reflectivity

Fig 1.4 DBR reflectivity spectrum with different wavelength

300 350 400 450 500 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

R

efe

cti

v

ity

(a

.u

.)

Wavelength (nm)

Fig 1.5 Microcavity reflectivity spectrum with different wavelength

2 1 E −E = ω 2 E 1 E ω 

(41)

Probability

1.0 0.8 0.6 0.4 0.2 0.0 0

2

R

π

Ω

4

R

π

Ω

6

R

π

Ω

8

R

π

Ω

Time(t)

( )

2 1

c t

( )

2 1

c t

Fig 1.7. Probability for finding the atom in either the upper or lower level in the strong-field limit

in the absence of damping. The electron oscillates back and forth between the two levels at the Rabi angular frequency, ΩR. This phenomenon is either called Rabi flopping or Rabi oscillation, [16].

Fig1.8 Polariton lasing and photon lasing. (a) The emission energy vs pump power for a N=12 multi-QW planar microcavity. (b) The dispersion characteristics of polariton BEC (green diamond)

and photon lasing (pink triangle) as well as the linear dispersion of UP (red square) and LP (blue circle) at low pump power, [36].

(42)

Fig 1.9 A negative conductance polariton amplifier. (a) The LP emission intensity taken as a

function of energy and in-plane wave vector. The system is above the quantum degeneracy threshold at

the bottleneck. The solid line and the dashed line indicate the theoretical dispersion (b) Observed

probe emission for pump only, probe only, and simultaneous pump and probe excitation, [37].

Fig 1.10(a) Semi-logarithmic plot displaying RT emission spectra at average pump power

densities ranging from 0.16 to 28.8 W/cm2, shifted for clarity. C and X are also reported (arrows). (b)

Three-dimensional representation of the far-field emission with emission intensity displayed on the vertical axis linear vertical scale above threshold. C and X are also reported(white lines).

(43)

Fig. 1.11 (a) Angle-resolved PL spectra at RT in the range of 0 º to 40 º for a ZnO hybrid MC. The

dotted line is the exciton mode. The solid lines are guides to the eye. (b) Experimental cavity olariton

dispersion curve. The dashed lines represent the cavity and exciton modes. (Courtesy of R. Shimada).

Fig 1.12 (a) Structure of wurtzite, which is a member of the hexagonal crystal system and consists

of tetrahedrally coordinated zinc and sulfur atoms that are stacked in an ABABAB pattern. (b) the

(44)

systems atomic gases excitons polaritons effective mass * c m m 103 10-1 10-5 Bohr radius a B 10 -1 Å 102 Å 102 Å particle spacing: 1 d n− 103 Å 102 Å 1μm critical temperature T _c 1nK~1μK 1mK~1K 1K~>300K thermalization time lifetime 1ms/1s~10 -3 1ps/1ns~10-2 (1~10ps)/(1~10ps) =0.1~10

Table 1.1: Parameter Comparison of BEC Systems

Material Bandgap Exciton

binding energy

Rabi

splitting

Advantages Drawbacks

GaAs 1.519eV ~10meV 4meV lattice-match DBR

crystal quality

small exciton binding

energy

GaN 3.507eV ~26meV(Bulk)

~40meV(QW)

26meV large exciton

binding energy

QCSE in QW

lattice-mismatch DBR

crystal quality

ZnO 3.289eV ~60meV 120meV large exciton

binding energy

lattice-mismatch DBR

crystal quality

(45)

Chapter2.

THE COUPLING BETWEEN EXITON WITH PHOTON 2.1 Quasi-particle model

2.1.1 Properties of Wannier-Mott Exciton

A solid consists of 1023 atoms. Instead of describing the 1023 atoms and their constituents in full

detail, the common approach is to treat the stable ground state of an isolated system as a quasi-vacuum

( the state with filled valence band and empty conduction band for a semiconductor ) and to introduce

quasi-particles as a unit of elementary excitation, which only weakly interact with each other. An

exciton is a typical example of such a quasi-particle, consisting of an electron and a hole bound by the

Coulomb interaction. The quasi-vacuum of a semiconductor is the state with filled valence band and

empty conduction band. When an electron with charge − is excited from the valence band into the e

conduction band, the vacancy it leaves in the valence band can be described as a quasi-particle call a

‘hold ’. A hole in the valence band has charge + , and an effective mass defined by e

1 2 2 E p − ∂  −_∂    . A

hole and an electron at p~ 0 interacts with each other via Coulomb interaction and form a bound pair (an exciton) analogous to a hydrogen atom where an electron is bound to a proton. The envelope

wavefunction of an exciton is also analogous to that of a hydrogen atom. However, due to the strong

dielectric screening in solids and a small effective mass ratio of the hole to the electron, the binding

energy of an exciton in GaAs, GaN, and ZnO is on the order of 10 meV, 26meV, and 60meV,

respectively, three orders of magnitude smaller than that of hydrogen atoms, and the radius of an

(46)

An exciton can be classified into Wannier-Mott exciton and Frenkel exciton, depending on the

properties of the material in question, as shown in Fig 2.1. In Wannier excitons, typically observed in

covalent semiconductors and insulators, the electron and hole are separated by a distance much larger

than the atomic spacing, so that the effect of the crystal lattice on the exciton can be taken into account

primarily via an average permittivity.

An exciton is a typical example of such a quasi-particle, consisting of an electron and a hole

bound by the Coulomb interaction. Therefore, an effective is that of a hydrogen-like atom formed by

an electron and a hole interacting, though simplified picture of the exciton state. The energy of exciton

* 2 1 n y E R n

= ⋅ Ry is the Rydberg energy.Following Hanamura and Haug [64], the Hamiltonian of the

electronic system of a direct two-band semiconductor is:

( ) ( ) ( )

( ) ( ) (

) ( ) ( )

†3 0 ††3 3 ˆˆ _ˆˆ 1 _ˆ ˆˆˆˆ 2 H x H x x d x x y V x y x y d xd y ψ ψ ψ ψ ψ ψ = + −

∫

Eq(2.1)

exp

(

)

kj x kj x ik x N ψ =µ ⋅ Eq(2.3)

Here j=c v, denotes the conduction or valence band, µ_kj

( )

x is the Block wavefunction and N is the number of unit cells of the lattice. aˆ_kj is the fermionic annihilation operator for an electron. It

氧化鋅微共振腔結構下之激子-極激子雷射之光學特性

國 立 交 通 大 學

顯 示 科 技 研 究 所

碩士論文

氧化鋅微共振腔結構下之激子-極激子雷射

之光學特性

Optical Characteristics of Exciton-Polariton

Lasing in ZnO-Based Microcavity

研

研

究

究

生

生

：

：

吳

吳

永

永

吉

吉

指

指

導

導

教

教

授

授

：

：

盧

盧

廷

廷

昌

昌

教

教

授

授

郭

郭

浩

浩

中

中

教

教

授

授

中

中

華

華

民

民

國

國

九

九

十

十

九

九

年

年

七

七

月

月

氧化鋅微共振腔結構下之激子-極激子雷射之光學特性

Optical Characteristics of Exciton-Polariton Lasing in

ZnO-Based Microcavity

研究生:吳永吉 Student: Yung-chi Wu

指導教授: 盧廷昌 教授 Advisor: Prof. Tien-chang Lu

郭浩中教授 Prof. Hao-chung Kuo

國 立 交 通 大 學 電機資訊學院

顯示科技研究所

國立交通大學

顯示科技研究所

指導教授: 盧廷昌教授 Advisor: Prof. Tien-chang Lu

國立交通大學電機資訊學院

碩士論文

中華民國九十九年七月

₍

₎