國
立
交
通
大
學
電子工程學系 電子研究所
博 士 論 文
低溫強磁場橢偏儀之設計和量測及
低能隙量子井厚度相關之電子傳輸有效質量和遷移率
Low Temperature and High Magnetic Field Ellipsometer Design and
Measurement & Well-thickness Dependent Electron Transport Effective
Mass and Mobility in Low Bandgap Quantum Wells
研 究 生：蘇聖凱
指導教授：李建平 教授
低溫強磁場橢偏儀之設計和量測及
低能隙量子井厚度相關之電子傳輸有效質量和遷移率
Low Temperature and High Magnetic Field Ellipsometer Design and
Measurement & Well-thickness Dependent Electron Transport Effective
Mass and Mobility in Low Bandgap Quantum Wells
研 究 生：蘇聖凱 Student：Sheng-Kai Su
指導教授：李建平 Advisor：Chien-Ping Lee
國 立 交 通 大 學
電子工程學系 電子研究所
博 士 論 文
A DissertationSubmitted to Department of Electronics Engineering and Institute of Electronics
College of Electrical and Computer Engineering National Chiao Tung University
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy in
Electronics Engineering
June 2014
Hsinchu, Taiwan, Republic of China
中華民國一○三年六月
i
低溫強磁場橢偏儀之設計和量測及低能隙量子井厚度相關之電子傳輸有效
質量和遷移率
學生：蘇聖凱
指導教授
：李建平 博士
國立交通大學
電子工程學系 電子研究所 博士班
摘
要
論文的第一部分涵蓋了光譜橢偏儀在近紅外波段（700-1000nm）可使樣品置於低溫 （～4.2 K）和強磁場（磁場最高達 14T）下量測的設計和實現。論文中將詳細探討在低 溫環境下系統的光學和機械的各個組成部分。我們把主要的光學原件都結合在一個可插 入到常規的長頸液態氦杜爾瓶的探測器上，這樣的設計使得系統擁有很長的空宇光路徑 （～ 1.8 m × 2）。在偏振光的解析方面，我們使用偏振片-樣品-（四分之一波片）-旋轉偏振片的橢偏儀配置。在光路方面，我們用兩個介電反射鏡，一個在樣品前，另一 在樣品之後；而在樣品座下的兩軸壓電驅動傾角器則可用來調控反射光的方向，使光能 順利反射回旋轉偏振片而被量測。系統的功能性量測結果和其隨機誤差的分析都將在文 中展示。我們用此自行設計的橢偏儀系統探索砷化鎵極化子在磁場下傳播的特性。藉 此，我們可同時量測到砷化鎵激子橢偏光譜的振幅和相位頻譜以及其相位譜在能量接近 砷化鎵激子躍遷時，光左、右旋的轉變。更重要的，我們藉此量測方法觀察到有趣且未 曾被發表過的砷化鎵激子光譜的精細結構，且這些精細結構的磁光行為無法由已知特性 的激子態做解釋。鑑於此，我們把樣品的表面和磊晶界面都當作砷化鎵極化子的邊界， 如此可解釋這些精細結構的由來並歸咎其為多重極化子模態間的干涉結果。而對於此干
ii 涉精細結構的磁響應，我們提出了一個同時考慮極化子的空間色散和激子的中心運動與 相對運動耦合所導致的有效質量提升的模型對其做定性的解釋。 在論文的第二部分，我們提出了一個設計特殊半導體量子井的想法，這種量子井相對於 一般的量子井而言；即使厚度減小，仍能提供較小的電子傳輸有效質量和較高的遷移 率。在理論計算中，我們同時考慮了能帶的非拋效應和量子井能障所帶來的影響。在低 溫情況下，主要的散射機制包括界面粗糙度，合金無序和雜質的散射均被用來評估電子 在量子井中的遷移率。四種不同組合的低能隙量子井的結果和比較都將在文中展現。通 過適當選取合適的能帶組合的量子井和其能障的材料，此新穎的傳輸特性可被實現。
iii
Low Temperature and High Magnetic Field Ellipsometer Design and
Measurement & Well-thickness Dependent Electron Transport Effective
Mass and Mobility in Low Bandgap Quantum Wells
Student：Sheng-Kai Su
Advisor：Dr. Chien-Ping Lee
Department of Electronics Engineering & Institute of Electronics
National Chiao Tung University
ABSTRACT
The first part of this thesis covers the design and implementation of a spectral ellipsometer at near-infrared wavelength (700-1000nm) for samples placed in high magnetic fields (up to 14 Tesla) at low temperatures (~4.2 K). It details both the optical and mechanical aspects of the system in the low temperature environment. The main optical components are integrated in a probe, which can be inserted into a conventional long-neck He dewar and has a very long free-space optical path (~1.8 m x 2). A polarizer-sample-(quarter-wave plate)-rotating analyzer configuration was employed. Two dielectric mirrors, one before and one after the sample in the optical path, helped to reflect the light back to the analyzer and a two-axis piezo-driven goniometer under the sample holder was used to control the direction of the reflected light. Functional test results and analysis on the random error of the system are shown. The properties of GaAs polariton propagating in magnetic field have been explored using this self-designed ellipsometry system. We obtained both the amplitude and phase ellipsometric spectra simultaneously and observed helicity transformation at energies near the GaAs exciton transitions in the phase spectra. Interesting fine structures, which have not been reported before, have been observed and their magneto-optical behavior cannot be accounted by the known properties of excitonic states. Treating the surface and the growth interface as boundaries, we attribute the fine structures to the interference among various polariton modes.
iv
A model considering both the polariton spatial dispersion and the exciton effective mass enhancement induced by the coupling of the exciton center of mass and relative motions is proposed to explain the magnetic response of the interference ellipsometry spectra.
In the second part of this thesis, we propose an ideal to design a special kind of semiconductor quantum wells, which, in contrary to conventional quantum wells, are able to provide smaller electron transport effective mass and higher mobility when the quantum well thickness is decreased. The theory used accounts for both the nonparabolicity effect and the influence of the barrier. Major scattering mechanisms at low temperatures, including the scatterings by the interface roughness, the alloy disorder, and impurities have been considered in mobility calculations. The results of four different combinations of quantum wells are shown and compared. By properly choosing the well/barrier materials with proper band lineups, the novel transport property is achievable.
誌
謝
我要感謝在我完成研究計畫的過程中許多給予我幫助和支持的人。我首先要感謝我 的指導教授，李建平老師，給予我很大的自由，很充沛的資源去完成我本來從未以為我 能完成的事。與老師談話時常能激發我的一些想法，事實上，論文中第二部分的想法就 是源自於一次午餐閒聊時老師的提問。感謝多年來老師不論在研究還是生活上的提點和 信心的給予讓我可以安心的走在想要走的道路上。 我還要感謝中興大學物理系的孫允武老師。沒有他的幫助，我將無法順利完成低溫 強磁場橢偏儀的建製。感謝孫老師對於專業知識以及期刊論文寫作的幫助和指導。從他 身上我學到做科學研究以及論文寫作的嚴謹態度。儀器的完成還仰賴許多人的幫助。感 謝李良箴博士在儀器機械和低溫系統部分給予的幫助，巫朝陽同學在儀器控制和自動化 方面的付出，郭鴻榮學弟幫忙畫的 3D 設計圖以及宋育泰工程師對於實驗上的幫助。
I would like to thank Professor Oleksandr Voskoboynikov, whose humor and meaningful discussions, helping me overcome many difficulties in theoretical calculations.
還要感謝林聖迪老師午後的 coffee time“閒聊＂和給予的許多研究上和未來規劃上 的建議。感謝實驗室的學長: 羅明城、林大鈞、凌鴻緒、陳建旭的帶領和指導以及同學、 學弟妹們: 潘建宏、李宗霖、林岳明、林建宏、許宏任、李依珊 et al.的幫助。 最後我要感謝我的父母，他們的付出和支持，讓我即使已近而立之年仍無憂於“養 家活口＂的重責而能埋首於書本和研究之中。歲月無聲，縱使帶走厚厚的青春，卻也留 下許多回憶。那些辛苦、挫折與小小的成就，現今「也無風雨也無晴」了。
中華明國一○三年五月 蘇聖凱 謹誌於 國立交通大學電子所 v
Contents
Abstract (Chinese) i
Abstract iii
Acknowledgements v
List of Figures ix
List of Tables xii
Abbreviations xiii
Physical Constants xiv
Symbols xv
I
Low Temperature and High Magnetic Field
Ellipsome-ter Design and Measurement
1
1 Introduction 2
1.1 Ellipsometry . . . 2
1.2 Exciton-Polariton Light Semiconductor Coupling . . . 4
1.3 Scope of Part I . . . 5
2 Ellipsometry System Design & Measurements 7 2.1 Description of The Complete System . . . 7
2.2 Operation Principle . . . 9
2.3 Mechanical Design . . . 11
2.4 Experimental Procedure . . . 13
2.5 Data Acquisition and Reduction . . . 17
2.6 Error Analysis . . . 20
Contents vii
3 Results 24
3.1 GaAs Substrate . . . 25
3.2 GaAs 1-μm Epitaxial Layer . . . 25
4 Theory & Models 30 4.1 Direct-Excitons in Semiconductor . . . 30
4.1.1 Multi-Band Theory . . . 30
4.1.2 One-Band Model . . . 32
4.2 Excitons in Magnetic Fields . . . 34
4.2.1 One-Band Excitons in Magnetic Fields . . . 34
4.2.2 Multi-Band Excitons in Magnetic Fields . . . 35
4.3 Exciton-Polariton with Spatial Dispersion . . . 38
4.3.1 Isotropic . . . 38
4.3.2 Anisotropic Exciton Effective Mass . . . 40
4.4 Oblique Incidence of s & p Polarized Light . . . 41
4.4.1 s-polarized. . . 42
4.4.2 p-polarized. . . 43
4.5 Transfer Matrix for Non-Spatial Dispersive Material . . . 44
4.6 Transfer Matrix for Spatial Dispersive Material . . . 45
4.6.1 s-polarized. . . 45
4.6.2 p-polarized. . . 47
4.7 Calculation of Reflection Coefficients . . . 49
5 Discussion 51 5.1 GaAs Substrate . . . 51
5.2 GaAs Epitaxial Layer . . . 52
5.2.1 Dips A and B . . . 52
5.2.2 Dips C and D . . . 53
6 Conclusion 60 6.1 Low Temperature and High Magnetic Field Ellipsometry System . . 60
6.2 Recommendations for Future Research . . . 61
II
Well-thickness Dependent Electron Transport
Effec-tive Mass and Mobility in Low Bandgap Quantum Wells
64
7 Introduction 65 8 Theory & Calculation Methods 67 8.1 The Basic Idea . . . 678.2 Electron Transport Effective Mass . . . 68
8.3 Material Parameters . . . 70
8.3.1 Strain . . . 72
Contents viii 8.3.2.1 InGaAs system . . . 74 8.3.2.2 GaAsSb system . . . 75 8.3.2.3 InAsSb system . . . 76 8.3.2.4 GaInPSb system . . . 77 8.4 Scattering Mechanisms . . . 79
8.4.1 Interface Roughness Scattering (IRS) . . . 79
8.4.2 Alloy Disorder Scattering (ADS) . . . 80
8.4.3 Ionized Impurity Scattering . . . 81
9 Results & Discussion 82 9.1 Electron Transport Effective Mass . . . 82
9.2 Electron Transport Mobility . . . 86
10 Conclusion 91 10.1 Well-thickness Dependent Electron Transport Effective Mass and Mobility . . . 91
10.2 Recommendations for Future Research . . . 91
A Kerr Measurement & Generalized Ellipsometry 93
B Error Analysis of Ellipsometry Parameters 96
C Extreme Value and Its Corresponding Analyzer angle of a RAE
Intensity Distribution 98
D Finite-Well with Effective Mass Mismatch at Heterojuctions 100
List of Figures
2.1 Diagram of the long-neck liquid-Helium dewar with a liquid-Nitrogen
jacket. . . 7
2.2 Schematic diagram of the ellipsometery system . . . 8
2.3 Overall view of the ellipsometer insert. . . 12
2.4 Design of the ellipsometer insert. . . 13
2.5 The first step of the experimental procedure: sample preparation and optical alignment. . . 14
2.6 The second step of the experimental procedure: pumping. . . 15
2.7 The third step of the experimental procedure: cooling. . . 16
2.8 The final step of the experimental procedure: system ready. . . 17
2.9 Flowchart of the data acquisition . . . 18
2.10 Γ (a) and Λ (b) (in Eqs. (2.12) and (2.14)) as a function of Ψσ pp and ΨM pp . . . 20
2.11 Histograms of Ψeff pp and Δeffpp shown as bar charts. . . 23
3.1 Sample structures. . . 24
3.2 Spectra of Ψσ pp and Δσpp of intrinsic GaAs substrate for different magnetic fields at 4.2 K. . . 26
3.3 Comparison of Δσ pp spectra of GaAs substrate and GaAs epitaxial layer grown by MBE at zero magnetic field. . . 27
3.4 (a) Spectra of Δ and Ψ of GaAs epitaxial layer for magnetic fields of 0 T and 1 T at 4.2 K. (b) The raw data of the rotating-analyser ellipsometry measurements in polar coordinates at 1 T, 4.2 K for the energies 1.5157 eV and 1.5160 eV.. . . 28
3.5 Spectra of Δ and Ψ of GaAs epitaxial layer at 14 T, 4.2 K. . . 29
3.6 Spectra of Δ shifted to exclude the differences of the diamagnetic shift for magnetic fields varied from 2 to 14 T at 4.2 K. . . 29
4.1 Diagram of a exciton propagating in semiconductor. . . 32
4.2 Diagram of the directions of the applied magnetic field B and the crystal. . . 37
4.3 Dispersion relation of exciton-polariton. . . 40
4.4 Diagram of light incident to a surface separating vacuum and spatial dispersive medium . . . 41
4.5 Diagram of a multi-layer system . . . 45
5.1 Measured peak energy versus magnetic field (GaAs substrate). . . . 52 ix
List of Figures x
5.2 The measured energy shift of the dips A and B as functions of magnetic fields. . . 53
5.3 Diagram of the purposed model and measured SIMS data. . . 55
5.4 Calculated Δ spectra for various d1, d2, l, and different types of
ABCs (d). . . 56
5.5 Calculated Δ, Ψ, and normal-incidence reflectance specta. . . 57
5.6 Calculated energy difference (ΔE) of the dips C and D as a function of exciton effective mass Mex . . . 58
5.7 Calculated Δ spectra for various exciton effective mass Mex and
2Mex with different damping ν and 0.5ν. . . 59
6.1 Diagram of the ideal to measure exciton translational effective mass in various electric and magnetic fields. . . 63
8.1 Conduction band diagram to illustrate the basic idea. . . 68
8.2 Band lineup diagram of GaAs/GaIn0.25As. . . 74
8.3 Conduction band edge Ec and electron effective mass for different
concentration of GaAsSb. . . 75
8.4 Band lineup diagram of GaAsSb0.49/GaAsSb0.22. . . 76
8.5 Conduction band edge Ec and electron effective mass for different
concentration of InAsSb. . . 77
8.6 Band lineup diagram of InAsSb0.25/InAs. . . 78
8.7 Band lineup diagram of Ga0.3InSb0.9P/Ga0.2InSb0.8P. . . 79
9.1 Calculation results of the electron transport effective mass mt, the
confinement energy E0, and the transport mobility μ as functions
of the well width l for the strain on the well and on the barrier of the case GaAs/GaIn0.25As. . . 83
9.2 Calculation results of the electron transport effective mass mt, the
confinement energy E0, and the transport mobility μ as functions
of the well width l for the strain on the well and on the barrier of the case GaAsSb0.49/GaAsSb0.22. . . 84
9.3 Calculation results of the electron transport effective mass mt, the
confinement energy E0, and the transport mobility μ as functions
of the well width l for the strain on the well and on the barrier of the case InAsSb0.25/InAs.. . . 85
9.4 Calculation results of the electron transport effective mass mt, the
confinement energy E0, and the transport mobility μ as functions
of the well width l for the strain on the well and on the barrier of the case Ga0.3InSb0.9P/Ga0.2InSb0.8P. . . 86
9.5 Absolute value of the differential of the energy to the QW thickness 88
9.6 Integration of the wavefunction absolute value square R |ψ|2dz and to the fourth R |ψ|4dz . . . 89
List of Figures xi
9.7 (a) Illustration diagram of the electron wavefunction. (b) Mobil-ity determined by the remote impurMobil-ity scattering as functions of the well thickness for different distance of the delta doping to the bottom of the QW zi . . . 90
A.1 Diagram of the Kerr measurement . . . 93
A.2 Comparison of Ψσ
ppand Δσppspectra of GaAs epitaxial layer at 12 T
with and without the contribution of Rsp and Rps . . . 95
C.1 Comparison of the elliptical intensity distributions with and without a quarter wave plate. . . 99
List of Tables
4.1 Special cases of ABC for s polarized light. . . 47
4.2 Special cases of ABC for p polarized light. . . 49
8.1 k∙p band parameters for III-V semiconductors. . . . 71
8.2 Bowing parameters for III-V ternary compounds. . . 71
8.3 Parameters of GaAs/GaIn0.25As band alignment.. . . 74
8.4 Parameters of GaAsSb0.49/ GaAsSb0.22 band alignment. . . 76
8.5 Parameters of InAsSb0.25/ InAs band alignment. . . 78
8.6 Parameters of Ga0.3InSb0.9P (Barrier)/ Ga0.2InSb0.8P (Well) band alignment. . . 78
C.1 Maximum and minimum values of the elliptical intensity distribu-tion for Ac in different domains. . . 99
Abbreviations
ABC Additional Boundary Condition
ADS Alloy Disorder Scattering
Amp Amplifier
BIS Background Impurity Scattering
CM Center of Mass
DVM Digital Voltage Meter
FWHM Full Width at Half Maximum IRS Interface Roughness Scattering
MBE Molecular Beam Epitaxy
PEM Photo-Elastic Modulator
QW Quantum Well
RAE Rotating Analyzer Ellipsometry RIS Remote Impurity Scattering
SI Semi Insulating
SIMS Secondary Ion Mass Spectrometer
Physical Constants
Speed of Light in Vacuum c = 2.99792× 108 ms−1
Mass of Electron m0 = 9.10938× 10−31 kg
Planck’s Constant ~ = 1.05457 × 10−34 J s
Charge of Electron e− = 1.60218× 10−19 C Permeability of Free Space μ0 = 4π× 10−7 NA−1 Permittivity of Free Space ε0 = 8.8542× 10−12 Fm−1
Bohr Radius a0 = 0.5292× 10−10 m
Bohr magneton μB = 927.4× 10−26 JT−1
Symbols
A analyzer angle degree (◦)
A sample area m2
alc lattice constant ˚A
B magnetic field (induction) of magnets T
D displacement field FV m−2 E electric field V m−1 E electron energy eV Eg bandgap energy eV H magnetic field A m−1 I light intensity W
k electron in-plane wavevector m−1
l QW thickness nm
Mex exciton center of motion effective mass m0
me,h electron/hole effective mass m0
P polarization FV m−2
p momentum J s m−1
p polarization parallel to incident planes
q wavevector m−1
q change of the electron in-plane wavevector m−1
r space coordinate m
r reflection coefficient
s polarization perpendicular to incident planes
T temperature K
Symbols xvi
t time s
V potential energy eV
Y admittance S
Z impedance Ω
Π polarizer angle degree (◦)
Ψ real part of ellipsometry parameter degree (◦)
θ incidence angle degree (◦)
θ scattering angle degree (◦)
ψ wavefunction
Δ imaginary part of ellipsometry parameter degree (◦)
ω angular frequency s−1
ε dielectric function Fm−1
strain tensor χ susceptibility
τ scattering relaxation time s
μ reduce mass m0
Dedicated to my parents
Part I
Low Temperature and High
Magnetic Field Ellipsometer
Design and Measurement
Chapter 1
Introduction
Taking precise and careful measurements of subtle phenomena in semiconductors is very important in understanding the physical properties of the material. In the first part of the thesis, we describe the design and construction of an ellipsometer that can be used in high magnetic field and low temperatures. This is an unique scientific instrument, which is able to perform measurements that could not be done before. A lot of innovative ideas have been used and many difficult obstacles have been overcome in this challenging project. This is the first and the only ellipometer in the world that can be operated in such harsh environment. The system has been fully tested and characterized. We have used it in the study of exciton-polariton properties of GaAs. It has provided information that could not be achieved otherwise.
1.1
Ellipsometry
Ellipsometry is a technique to measure the change of a polarization state of light after being interacted with sample [1–3]. From analysing the polarization change, we can explore the light-matter interaction and deduce parameters such as over-layer/film thickness and complex dielectric functions [1–3]. More specifically, the spectroscopic ellipsometry is able to measure wavelength dependent complex di-electric functions providing additional information on the electronic states and the band structure of crystalline materials. This is especially important for semicon-ductor analysis and is useful for optoelectronic applications. Unlike reflectance and photoemission spectra, spectroscopic ellipsometry simultaneously provides both
Chapter 1. Introduction 3 real and imaginary parts of ellipsometric spectra connected to the complex dielec-tric function, which fully characterizes the inherent wavelength dependence linear response (both the absorption and dispersion) of materials.
Its high sensitivity integrated with a high magnetic field (B ) and low tempera-ture (T ) environment for the sample has attracted researchers to utilize such a technique to explore interesting quantum electron systems of which the optical response can show very intriguing effect induced by B. For many of these studies, especially on new phenomena of materials and new structures, it is desirable that these measurements are performed at low temperatures (T ) and high magnetic fields (B). For example, nonmagnetic semiconductor nano-object based artificial materials have been proposed to exhibit magnetic-material-like collective optical responses which can be revealed by low-T and high-B ellipsometric measurements [4–8]. Another area of interest is the spatial dispersive exciton-polariton related problems [9–13], for which the obliquely incident geometry nature and high-B fields provide extra information of the momentum space parallel to the interface and the effect of spin configurations. We believe such a comprehensive study will help to develop the microscopic theory of exciton-polariton behaviors in magnetic fields. Therefore a versatile but miniature ellipsometer that can be fitted into and operated in a low temperature and high magnetic field environment is of great importance for the studies mentioned above.
A generalized magneto-optical ellipsometry system was proposed by Berger et al. [14] to obtain complex refraction index and magneto-optical coupling constant simultaneously. Later, Neuber et al. [15] showed a temperature-varying general-ized spectral magneto-optical ellipsometer design with a He-flow cryostat and a small electromagnet of a few tens of mT. Schubert et al. [16] extended this tech-nique to far-infrared and a higher magnetic field of a few Tesla to characterize the carrier properties of n-GaAs; Hofmann et al. [17] further pushed the technique to terahertz frequency range. Mok et al. [18] presented a variable-angle vector-magneto-optical generalized ellipsometer with field magnitude up to 400 mT at room temperature. However, all the previous systems had relatively short optical path designs and were equipped with small coil magnets or split-coil superconduct-ing magnets, which limited the strength of the applied magnetic fields. To extend the operation range of the spectroscopic ellipsometry systems to higher magnetic fields, a new approach to design a ellipsometer is presented in this thesis.
Chapter 1. Introduction 4
1.2
Exciton-Polariton Light Semiconductor
Cou-pling
As mentioned above, the self-designed ellipsometer performs measurements differ-ent from common reflectance and transmission experimdiffer-ents for the sample in the low temperature and high magnetic field environment. We use it to investigate the properties of GaAs exciton-polariton propagation in high magnetic fields. The term “exciton-polariton”, was first introduced by J. J. Hopfield [19] to refer to the coupling of the electromagnetic field and the excitonic polarization field in semi-conductors. This concept has exerted considerable influences on researches in the linear optical regime [20–23] and on future applications such as plasma-gain lasing [24, 25], low energy switches [26–28], single photon sources [29–31], etc. When spatial dispersion of polaritons [32, 33] in the presence of interfaces is considered, the polariton problem becomes more complicated. Lack of knowledge on the ex-citon polarization near interfaces leads to various types of “additional boundary condition”(ABC) [32, 34–36] be proposed to match the multi-polariton modes in the material and the single electromagnetic wave at an interface.
Over the past 20 years there have been a lot of researches on polariton effects in GaAs QWs for fundamental and practical reasons. These effects have been ex-plained by the interference among multi-polariton modes and the exciton center-of-mass (CM) motion quantization. And the mechanisms exchange dominating for different QW thickness comparing to the photon wavelength and the exciton Bohr radius in the material [37, 38]. For a thick quantum well (QW), some features have been observed and have been explained as due to the interference among multi-polariton modes [39]. When the layer thickness is reduced, the exciton center-of-mass (CM) motion quantization has been expected to play a more im-portant role [38]. However, to recognize all the features induced by these coupled mechanisms is complicated. Recently, more elaborate microscopic models [10–13] have been developed to explain the interplay of the interference of polariton and the CM motion quantization for intermediate sample length and have an excellent agreement with recent experimental results [9]. However, no such treatments with magnetic fields have been published up to now.
Applying magnetic fields to semiconductors has attracted many interests in study-ing the behavior of magneto-exciton and bestudy-ing available to obtain the important
Chapter 1. Introduction 5 parameters (diamagnetic shift coefficient, g-factor, band-parameters, etc.) of ma-terial [40]. The last decade has seen growing importance placed on research in the magnetic field induced phenomena due to the coupling between the quantized ex-citon CM and relative motions [41–44]. Magnetic fields are not only a perturbation but also lead coupling of exciton CM motion and internal motion [41,42,45, 46], and possibly induces mixing of the exciton s-state and other states [43,44]. This is especially important for the sample that the exciton can move freely in mag-netic fields. However, all the experiments published previously were done with a normal-incidence configuration that can not provide a wavevector parallel to the sample surface and obliquely moving excitons in magnetic fields.
1.3
Scope of Part I
Having an ellipsometer that can operate at low temperatures and high magnetic fields makes it possible to perform experiments mentioned above in a different way. The first part of the research (Chapter 2) details the design, construct and characterization of the spectroscopic ellipsometry system that can be inserted into an Oxford long-neck low temperature dewar, which is equipped with a supercon-ducting magnet with field up to 14 T. This ellipsometer employs free space optics to bring the polarized light in and out of the sample stage, which is placed ∼1.6 m deep into the cryostat. Specific mechanical design to control the light beam and stabilize the overall system is presented. The system is fully tested and functional, and has provided magneto-optical spectroscopic information of our samples with high resolution and clarity.
The second part of this work involves measuring III-V semiconductor samples using the self-designed ellipsometer. More specifically, we study the behavior of GaAs polariton in a substrate and a 1-μm layer grown by molecular beam epitaxy (MBE) in high magnetic fields at 4.2 K. Unlike the previous measurements done with a normal incident angle, our measurements were performed with an oblique (60◦) incident angle under variable magnetic fields up to 14 Tesla (T) and were able to show simultaneously both the phase and the amplitude ratios of the two principal axis reflectivities.
From the measurement done with taking the ratio of the two principal-axis re-flectivities and a large wavevector component parallel to the interfaces of a GaAs
Chapter 1. Introduction 6 epilayer, fine structures in the ellipsometry spectra were observed. The peculiar structures, which cannot be seen in our GaAs substrate sample and have not been reported in the previous measurements of GaAs epilayer [47–52], were present even at 0 T. The fine structures evolved with the magnetic field in a fashion that cannot be explained by the traditional models of the magnetic field induced quantization. Treating our sample as a slab of GaAs with two boundaries , which are the surface and the epilayer-substrate interface, and using the theory of the dispersive polari-ton with Pekar’s ABC [32,36], we were able to explain qualitatively the observed fine structures in the ellipsometry spectra as a result of the interference among various polariton modes in the slab. The behavior of the spectra under magnetic fields was explained as a possible result of the enhancement of exciton effective mass [45, 46], which is due to the coupling of the exciton CM and relative mo-tions in the presence of magnetic field. We hope the results and their explanamo-tions provided in this work can stimulate an appropriate theoretical description.
Chapter 2
Ellipsometry System Design &
Measurements
2.1
Description of The Complete System
Figure 2.1: Diagram of the long-neck Helium dewar with a liquid-Nitrogen jacket.
Chapter 2. Ellipsometry System Design & Measurements 8
Figure 2.2: Schematic diagram of the ellipsometery system, which can be fit into a long-neck cryogenic dewar with a small-bore high-field superconducting
magnet.
To overcome the constraints of the small bore size (50 mm) of the high-field (14 T) superconducting magnet and a long optical path set by the long-neck liquid-Helium dewar with a liquid-Nitrogen jacket (Fig. 2.1), we designed a multi-reflection sample stage mounted at the end of an insert. The laser beam is brought in through free-space along the insert. In this way, we are able to maintain a large incident angle and high polarization stability of the laser beam into the sample as required by an ellipsometry system. To make sure the laser beam traveling from the top window of the dewar can go back to the same window after being reflected by the sample at the center of the magnet that is placed near the bottom of the dewar, we use two dielectric mirrors besides the sample under test to form a triple-reflection configuration.
Figure 2.2shows the complete schematic diagram of the ellipsometry system. The whole system was constructed as an insert for the low temperature dewar. It consists of a polarization generation part, a polarization detection part, a signal processing part, and a sample stage that holds the sample and the two dielectric mirrors in a low-temperature environment. A Ti-sapphire laser provides the coher-ent light in the wavelength range of 700 nm to 1000 nm and its FWHM linewidth
Chapter 2. Ellipsometry System Design & Measurements 9 is about 3 ˚A. The laser beam is firstly split into two by a beam-sampler. One beam is coupled into a monochromator for wavelength measurement, and the other is coupled into a 10-m long single-mode fiber ended with a collimator. The light out of the collimator passes through a Glan-Laser calcite polarizer with an extinction ratio larger than 105 and then becomes a linearly polarized light. One of the laser beams being branched out by the beam-splitter is detected by a silicon photodiode (D1) to monitor the fluctuation of polarization and intensity, while the other beam is guided into the cryogenic dewar through a window at room temperature. The incident beam is reflected by a dielectric mirror before reaching the sample. The outgoing beam is reflected by another dielectric mirror and then goes to the polar-ization detection part placed outside the dewar. The polarpolar-ization detection part includes a quarter-wave plate, a rotating analyzer, and another silicon photodiode (D2). The photocurrents from D1 and D2 are measured by lock-in amplifiers and current preamplifiers respectively in the signal processing part. All reflections, two from the mirrors and one from the sample, have a 60◦ incident angle and share the same incident plane, making the incoming beam and outgoing beam parallel to each other. The holder containing the mirrors and sample is attached to a two-axis piezoelectric goniometer which can tune the sample orientation in-situ.
2.2
Operation Principle
The polarization state of a light beam can be expressed in the form of Jones vector [1–3]. For a triple-reflection process, the output and input Jones vectors are related by pout sout ! = r C pp rCsp rC ps rssC ! rB pp rBsp rB ps rBss ! rA pp rspA rA ps rAss ! pin sin ! ≡ r eff pp reffsp reff ps reffss ! pin sin ! = reff pin sin ! , (2.1)
where p and s are the electric field components parallel and perpendicular to the incident plane, and r is the complex reflection coefficient with subscripts and su-perscripts specifying the polarization states and reflecting materials, respectively. For example, rB
ps represents the ratio of the s-component of the light reflected by the sample in the middle and the p-component of the incident light. The effective reflection matrix, reff, represents the overall result of all three reflections. The
Chapter 2. Ellipsometry System Design & Measurements 10 ratio of the output and input ratios of the p- and s- polarization components is defined as [53]
ρ = pout/sout pin/sin
= (r eff
pp/reffss) + (rspeff/rsseff)(pin/sin)−1 1 + (reff
pp/rsseff)(rpseff/rppeff)(pin/sin)
= R eff pp+ Reffsp(pin/sin)−1 1 + Reff ppRpseff(pin/sin) , (2.2) where [16, 53] Reffpp = r eff pp reff ss
= tan ΨeffppeiΔeffpp; (2.3a)
Reffsp = r eff sp reff ss
= tan ΨeffspeiΔeffsp; (2.3b)
Reff ps = reff ps reff pp
= tan ΨeffpseiΔ eff
ps, (2.3c)
tan Ψ and Δ stand for the amplitude ratio and the phase difference of the reflec-tivities.
To extract the reflection information from the triple-reflection measurement, we must obtain the effect of the dielectric mirrors first. This can be done through a calibration procedure by using three identical mirrors, i.e., using the same di-electric mirror (M) to replace the sample. The overall result of the three identical reflections then has the form
(rM pp)3+ 2fppM+ fssM rspMFM rM psFM (rssM)3+ 2fssM+ fppM ! , (2.4) where fM pp = rMpprpsMrspM, fssM = rssMrMpsrMsp, and FM = (rMpp)2 + rpsMrspM+ rMpprssM+ (rssM)2. Here we assume the reflection coefficients to be the same due to the same material and identical incident angle. The ellipsometry parameters for each mirror can be obtained with neglecting the product of rM
psrspM terms that are at least three orders smaller than the product of diagonal terms in general.
RMpp = (Rcalpp)1/3; (2.5a)
RspM= Rspcal[(Rcalpp)2+ Rcalpp + 1]−1; (2.5b) RMps = Rcalps[(Rcalpp)−2+ (Rcalpp)−1+ 1]−1. (2.5c)
Chapter 2. Ellipsometry System Design & Measurements 11 The superscript “cal” represents the three identical mirror calibration measure-ment.
For a measurement where the second mirror is replaced by a sample (σ), the effective reflectivity matrix becomes
(rM pp)2rppσ rMpprssMrspσ + gsp gps+ rM pprssMrpsσ (rMss)2rσss ! , (2.6)
where gsp/ps = rMsp/ps(rMssrσss+rMpprσpp). The ellipsometry parameters of the sample (σ) are then obtained by Eg. (2.3) as
Rσpp = R eff pp (RM pp)2 ; (2.7a) Rσsp = [Reffsp − RM sp(1 + Reff pp RM pp )](RMpp)−1; (2.7b) Rσps = [Reffps − RpsM(1 + R M pp Reff pp )]RMpp. (2.7c)
The mirror contributions RM
pp, RMsp, and RMps can be deduced from the three-mirror calibration procedure in Eq. (2.5).
2.3
Mechanical Design
The main part of the ellipsomety system is designed as a long insert (Fig. 2.3) to fit into an Oxford He long-neck dewar (Fig. 2.1). Because of the long distance (∼1.8 m) between the center of magnetic field and the outlet window and the extreme conditions in the dewar, the mechanical design of the ellipsometer has to be done very carefully to ensure stable operation and ruggedness under harsh conditions. It also needs to be flexible enough to allow fine adjustment of the optical path to make sure that the light beam travels through free space to the sample and back to the polarization detection part accurately. The main mechanical support of the insert is provided by four parallel 6 mm-outside-diameter stainless-steel tubes, which are kept in place and separated by several aluminum spacer plates. In each of the spacer plate, there are two cross-shaped holes to allow the laser beams to travel through. Such a frame structure avoids the possibility of bending the insert frame and also suppresses mechanical vibrations. The top of the frame structure,
Chapter 2. Ellipsometry System Design & Measurements 12
Figure 2.3: Overall view of the ellipsometer insert.
as shown in Fig. 2.4(a), is connected to a manifold that supports the optical head and provides necessary ports for pumping and electrical feedthroughs. The optical head mainly consists of two 25 cm ×10 cm ×1 cm parallel optical breadboards that hold the optical components and detectors. The bottom of the insert, as shown in Fig. 2.4(b), includes a sample stage, a two-axis piezoelectric goniometer and a protective housing, whose material are all titanium. Between components of different materials, beryllium-copper washers are used to reduce the deformation caused by different contractions at low temperatures. To ensure the reflected beam reaching the polarization detection part, the sample stage is mounted on a two-axis piezoelectric goniometer with its rotation center coinciding with the sample position to provide a very precise and instantaneous angle-control without sample displacement.
Chapter 2. Ellipsometry System Design & Measurements 13
Figure 2.4: Design of the ellipsometer insert. (a) The optical head consists of all polarization optical elements in the room temperature environment. (b) The bottom of the insert, positioned in 4.2 K environment, includes a sample
stage, a two-axis piezoelectric goniometer and a protective housing.
A thin-wall stainless steel jacket (see Fig. 2.3) with a 50 mm inner diameter is used to protect the insert and also allows the sliding seal to be used during cooling. The outer jacket must be fit to the insert by carefully tuning the 8 screws in the connection flange on the manifold to avoid any contact between them. Four springs are placed below the horizontal plate of the optical head to counterbalance the weight of the whole insert and also buffer the vibration from the dewar.
2.4
Experimental Procedure
In real measurement, two semi-insulating GaP substrates are used as the dielectric mirrors because GaP possesses a larger band gap than most III-V materials and a
Chapter 2. Ellipsometry System Design & Measurements 14
Figure 2.5: The first step of the experimental procedure: sample preparation and optical alignment. The alignment was done by fine tuning the 3-axis col-limator mount in the optical head and the 2-axis goniometer in the bottom of the sample stage with the help of two cross-shape holes shown in the picture.
Chapter 2. Ellipsometry System Design & Measurements 15
Figure 2.6: The second step of the experimental procedure: pumping. The outer jacket, which is a thin wall (∼ 1 mm) stainless steel tube, was connection to the flange on the manifold carefully by tuning the 8 screws shown in the picture. The pressure inside the insert was pumped down to ∼ 10−5 mbar, and
Chapter 2. Ellipsometry System Design & Measurements 16
Figure 2.7: The third step of the experimental procedure: cooling. The insert was lowered down into the low temperature dewar slowly via a sliding seal shown
in the picture.
reasonably large refractive index. In addition, no significant photo and magneto-optical response is observed for GaP under magnetic fields in the measurement band. These two GaP side mirrors and sample was mounted on the sample stage carefully by grease. To compensate the thickness difference of GaP and GaAs substrates, a glass slide was put under the GaAs substrate.
The optical alignment procedure was carried out by fine tuning the three-axis collimator mount on the top of the optical head (see Fig. 2.4 (a)) for the input light beam and the two-axis piezoelectric goniometer at the bottom of the sam-ple stage (see Fig. 2.4 (b) and Fig. 2.5) for the reflected light beam. After the alignment, the insert was carefully sealed by the outer jacket with an O-ring and bolts and then pumped down to ∼ 10−5 mbar (Fig. 2.6). During the time of pumping, the intensity of the reflected light measured by the detector was main-tained stable before and after sealed to insure no contact between the insert and the jacket1. Before the probe being inserted into the dewar, ∼ 25 mbar Helium
gas was introduced into the insert for heat exchange.
Chapter 2. Ellipsometry System Design & Measurements 17
Figure 2.8: The final step of the experimental procedure: system ready.
In the cooling procedure, the insert was lowered down into the low temperature dewar slowly via a sliding seal (Fig. 2.7) and at the same time the goniometer was adjusted instantaneously to keep the reflected light signal stable. The cooling procedure need to be manipulate slowly enough to avoid rapid changes in temper-ature. Once the whole probe was totally inserted into the dewar and the optical head was poised on the elevated table like in Fig. 2.8, the system was idle till thermal equilibrium and was ready to perform a measurement.
2.5
Data Acquisition and Reduction
Figure2.9shows the data acquisition procedure. The parameters including the po-larizer angle Π, the ranges and the steps of the laser wavelength, and the magnetic field are set at the beginning. The superconducting magnet can be switched to the persistent mode to save Helium consumption once the field strength reaches the setting value. The wavelength of the Ti-sapphire laser is auto-controlled by a step
Chapter 2. Ellipsometry System Design & Measurements 18
Figure 2.9: Flowchart of the data acquisition.
motor and is characterized by a monochromator together with the ellipsometry parameter measurement.
A polarizer-sample-rotating analyzer ellipsometry (RAE) configuration is used to measure the ellipsometry parameters. The measured light intensity can be ex-pressed as [1–3, 53] (see Appendix A)
I(Π, A) = |Ein|2|rss|2cos2Π|(Rpp+ R
sptan Π) cos A
+(RpsRpp+ tan Π) sin A 2, (2.8)
where Ein is the complex amplitude of the electric field before the polarizer. A and Π are the angles between the incident plane and the analyzer and polarizer transmission axes, respectively. This equation can be further simplified to
Chapter 2. Ellipsometry System Design & Measurements 19
I(Π, A) = ID[1 + α cos(2A) + β sin(2A)], (2.9a) where [53] α = |Rpp+ Rsptan Π| 2 − |RpsRpp+ tan Π|2 |Rpp+ Rsptan Π|2+|RpsRpp+ tan Π|2 , (2.9b) and
β = 2Re{(Rpp+ Rsptan Π)(RspRpp+ tan Π)} |Rpp+ Rsptan Π|2+|RpsRpp+ tan Π|2
. (2.9c)
The ellipsometry parameters depend only on α and β, and are independent of the average intensity ID. We can obtain α and β from the Fourier expansion of I(Π, A) [54]: ID = 1 N N X i=1 Ii; (2.10a) α = 2 IDN N X i=1 Iicos 2Ai; (2.10b) β = 2 IDN N X i=1 Iisin 2Ai. (2.10c)
where Ii is the intensity at Ai, and N is the total number of analyzer angles measured. We can use the Kerr measurement (see Appendix A) or follow the maturely developed method [16, 53] by choosing several azimuthal settings Π and using Eq. (2.9) to determine all the ellipsometry parameters Rpp, Rps, and Rsp. In some of our studies, the Rppresponse function near the exciton transition energy at low temperatures and high magnetic fields is important. The contribution from measured Rps and Rsp to Rpp spectra is small (see Appendix A) and does not influence the data fitting and interpretation shown in Chapter 5. For simplicity and clarity, we will focus on the response of Rpp and neglect Rps and Rsp in the following data analysis. The Eqs. (2.9b) and (2.9c) become the same as in the standard ellipsometry situation [1–3].
α = tan
2Ψpp− tan2Π
tan2Ψpp+ tan2Π, (2.11a)
and
β = 2 tan Π tan Ψppcos Δpp
Chapter 2. Ellipsometry System Design & Measurements 20
2.6
Error Analysis
For precise detection of the changes of Rσ
pp of the measured spectra under various external magnetic fields, to minimize the random error of the RAE technique with a multi-reflection configuration is important. It can be shown that making the light before the rotating analyzer circularly polarized gives better precision for RAE [55] (see Appendix B). Therefore, we choose a polarizer angle Π close to the measured ellipsometry angle Ψeff
pp to balance the intensities of p- and s-polarized components after three times of reflection.
Figure 2.10: Γ (a) and Λ (b) (in Eqs. (2.12) and (2.14)) as a function of Ψσpp and ΨM
Chapter 2. Ellipsometry System Design & Measurements 21 To analyze the precision of the measured result, we first find the relationship between the error of Ψeff
pp and the error of Ψσpp by taking the derivative of the real part of Eq. (2.7a).
dΨeffpp = sec 2Ψσ pptan2ΨMpp 1 + tan2Ψσ pptan2ΨMpp dΨσpp≡ ΓdΨσpp, (2.12) where Γ represents the coefficient that relates the two error quantities. Figure 2.10
(a) shows Γ as a function of Ψσ
pp and ΨMpp. For a given error in the triple reflection measurement, the error in the sample’s parameter can be minimized by choosing proper Ψσ
pp and ΨMpp to maximize Γ. Thus, we can choose the mirrors once we know the dielectric constant of the sample. For example, if Ψσ
pp equals 22◦ (for GaAs at a wavelength near 800 nm and a 60◦ incident angle), we can choose ΨM pp close to 45◦ (see Fig. 2.10(a)) to minimize the error (dΨσ
pp) of the deduced sample parameter.
However, the error in the triple reflection measurement, dΨeff
pp, depends on the polarizer angle Π and is related to the error in the Fourier component α by (see Appendix B)
dΨeffpp = [cos 2Ψeff
pp(tan2Π + tan2Ψeffpp)2 4 tan Ψeff
pptan2Π
]dα. (2.13)
By using this relationship and choosing Π = Ψeff
pp for best precision, Eq. (2.12) can be rewritten as dΨσpp= tan 2ΨM pptan Ψσpp Γ[1 + (tan2ΨM pptan Ψσpp)2] dα ≡ Λdα. (2.14) The error dΨσ
pp is now related to the error in the Fourier coefficient dα (see Ap-pendix B) by a factor Λ, which depends on the parameter Γ and the Ψpp values of the mirrors and the sample. Figure 2.10 (b) shows the factor Λ as a function of Ψσ
pp and ΨMpp. For a typical sample (ex. ΨGaAspp ∼ 22◦), Λ is a slowly varying function of ΨM
pp. The precision is not sensitive to the choice of a mirror when the polarizer angle Π is chosen close to Ψeff
pp.
The precision of the phase difference Δpp can be shown as (see Eq. (B.1b) in Appendix B)
dΔeffpp = 1 sin Δeff
pp(1− α2)3/2
Chapter 2. Ellipsometry System Design & Measurements 22 It can be improved by adding a quarter-wave plate in front of the analyzer espe-cially for measuring dielectric samples whose Δpp is close to 0◦ or 180◦. Then, Eq. (2.11b) becomes
sin Δeffpp =−sgn(P )√ β
1− α2 (2.16)
for the fast axis of a quarter-wave plate parallel to the incident plane. From Eq. (2.16), the sign of Δpp can be distinguished by adding a quarter-wave plate in front of a rotating analyzer. This is important especially for the measurements with photon energy close to the exciton or polariton states (see Chapter 3). To test the performance of this system at low temperature, we have taken data for more than 200 periods of rotations of the analyzer. The resulted standard deviations (shown in Fig. 2.11) of Ψeff
pp and Δeffpp are 0.04◦ and 0.3◦(while Ψσpp and Δσ
pp are ∼ 0.25◦ and 0.3◦) for Ψeffpp = 3.05◦ , Δeffpp = 100.2◦, and Π = 4◦. We can then use the standard deviations (dΨeff
pp and dΔeffpp ) and the known parameters (α, β and dA) to obtain the intensity fluctuation δ defined as dI/ID in Appendix B) using the coupled coupled Eqs. (B.3a) and (B.3b). We found that the contribution of δ is larger than that of the error of the analyzer angle dA and the residue from the asymmetry of Ai in real measurements. It can be attributed to the mechanical vibration through such a long framework of the insert. Although the random errors are not as small as that of conventional ellipsometers, the precision of this system is good enough for many of the proposed studies [4–8], and more importantly, it can operate at much higher magnetic fields than the ones reported previously.
Chapter 2. Ellipsometry System Design & Measurements 23
Figure 2.11: Histograms of Ψeff
pp and Δeffpp shown as bar charts. (a) The stan-dard deviation of Ψeff
pp is 0.04◦at average of 3.05◦ (b) The standard deviation of Δeff
Chapter 3
Results
We have used this self-designed ellipsometer to measure the samples including a semi-insulating GaAs (001) wafer and a 1-μm un-doped MBE grown layer on the substrate (Fig. 3.1). The results of them will shown in this chapter.
Figure 3.1: Sample structures. (a) Semi-insulating (S.I.) GaAs wafer (∼ 350 μm); and (b) MBE grown 1μm un-doped GaAs layer on S.I. GaAs wafer (∼ 350
μm).
Chapter 3. Results 25
3.1
GaAs Substrate
The Ψσ
pp and Δσpp spectra of GaAs substrate from 1.514 to 1.532 eV at 4.2 K are shown in Figs. 3.2(a) and 3.2(b) respectively for different magnetic fields up to 14 Tesla. In Fig. 3.2(b), the asymmetric curves in Δσ
pp spectra across the 180◦ line indicate that the phase differences change sign from right-handed (negative helicity) to left-handed (positive helicity) polarizations [56] near the transitions. The raw RAE data with a quarter-wave plate in front of the rotating analyzer at wavelength 812.12 nm and 812.4 nm (marked by the dashed cycles in Fig. 3.2(b)) are plotted in a polar coordinate in Fig 3.2(c). The sign change of Δσ
pp can be determined from the orientations of the long-axes (the dash lines) of the polar curves located in different quadrants (see Appendix C).
Semi-insulating GaAs substrates, because of the high defect density, usually have broadened spectra due to a large damping. Thus it is difficult to observe the fine structure of level splitting at high fields and even the signal from the zero-field exciton becomes very weak. In comparison, we show in Fig. 3.3 the ellipsomet-ric measurement of a high-quality GaAs layer grown by molecular beam epitaxy (MBE). The layer, grown on a (100) semi-insulating GaAs substrate, was undoped and had a thickness of 1-μm. Unlike what was seen from the substrate alone, the result from the epilayer shows discernible features in the ellipsometry spectra even at zero magnetic field as shown in Fig. 3.3, and some of the features exhibit intrigu-ing behaviors when high magnetic fields are applied. These features, which have not been reported before, can now be studied and observed using our ellipsometry system. The data will be showed in the next section.
3.2
GaAs 1-μm Epitaxial Layer
Figure 3.4 (a) shows the measured phase difference Δ and the amplitude ratio Ψ (inset) of the sample (to distinguish from the results of the GaAs substrate and for convenience, we drop the superscript “σ ”and subscript “pp ” in the following) at 0 T and 1 T. Fine structures with multiple peaks and dips are observed in the spectra and become apparent when the magnetic field is applied. In Fig 3.4 (a), similar to the results of the GaAs substrate, the measured curves in the Δ spectra across 180◦ (dash-line) indicates the sign change of Δ. The raw RAE data measured at 1 T
Chapter 3. Results 26
Figure 3.2: Spectra of (a) Ψσpp and (b) Δσpp of intrinsic GaAs substrate for different magnetic fields at 4.2 K. (c) Intensity versus analyzer angle at 812.12 nm (solid points) and 812.40 nm (open circles) in polar coordinates at 14 T.
Chapter 3. Results 27
Figure 3.3: Comparison of Δσ
pp spectra of GaAs substrate (dash-line) and GaAs epitaxial layer (solid-line) grown by MBE at zero magnetic field.
with the fast axis of the quarter-wave plate parallel to the incident plane at energies 1.5157 eV and 1.516 eV are showed in polar plots in Fig 3.4 (b). The orientation of the long-axes (dot-lines) of the two elliptical intensity distributions located in different quadrants represents that the handedness of the two polarisation states are different (Appendix C). The polarisation states turn from elliptical clockwise to counterclockwise for the energy approaching from 1.5157 eV to 1.516 eV in the Δ spectrum of 1 T.
When magnetic field is increased (≥ 6 T), more dips and peaks appear in the spectra. We show, for instance, the spectra of Ψ and Δ at 14 T in Fig. 3.5. It shows the corresponding property of the real (Ψ) and imaginary (Δ) spectra, which ensures the reliability of the measurement. We denote appropriately the dips as A, B, C, and D in the spectrum Δ the order from lower energy to higher energy.
To observe how the fine structures evolve with varied magnetic fields, we show the spectra of Δ with the magnetic field varied from 2 to 14 T in Fig. 3.6. The spectra are shifted in energy to exclude the diamagnetic shift differences. While the dips B, C, and D are observed starting from 0 T, the dip A appears only above 6 T. The separation of the dips A and B becomes larger when the magnetic field increases, but the dips C and D become closer first and then nearly maintain a certain distance when the magnetic field increases.
Chapter 3. Results 28
Figure 3.4: (a) Spectra of Δ and Ψ (the inset) of GaAs epitaxial layer for magnetic fields of 0 T and 1 T at 4.2 K. (b) The raw data of the rotating-analyser ellipsometry measurements in polar coordinates at 1 T, 4.2 K for the
Chapter 3. Results 29
Figure 3.5: Spectra of Ψ and Δ at 14 T, 4.2 K. Four dips are denoted as A, B, C, and D in the spectrum of Δ. The four dot-lines show the corresponding
property of the spectra Ψ and Δ.
Figure 3.6: Spectra of Δ for magnetic fields varied from 2 to 14 T at 4.2 K. The spectra are shifted to exclude the differences of the diamagnetic shift. The arrows represent the increasing separation between the dips A and B, and the
Chapter 4
Theory & Models
4.1
Direct-Excitons in Semiconductor
4.1.1
Multi-Band Theory
In this section, we review the direct-exciton model constructed by Baldereschi and Lipari [57] as the band mixing effect are accounted. The Hamiltonian (neglecting the CM motion and the electron spin) can be written as:
Hex(p) = H6LK×6(p) + [E0 + p2 2me − e2 4πεbr]I, (4.1) where HLK
6×6 describe the hole kinetic energy near k =0 with the relative electron-hole momentum p, and r is the distance of a electron and a electron-hole. The second part of Eq. (4.1) is a 6×6 diagonal matrix including the band-edge energy E0, the electron kinetic energy and the Coulomb interaction. For the diamond structure, Eq. (4.1) can be represented in a matrix [57]
P + Q L† M 0 (i/√2)L −(i/√2)M L† P − Q 0 M −(i/√2)Q i(p3/2)L M† 0 P − Q −L −i(p3/2)L† −(i/√2)Q 0 M† −L† P + Q −(i/√2)M† −(i/√2)L†
−(i/√2)L† (i/√2)Q i(p3/2)L (i/√2)M P + Δso 0
(i/√2)M† −i(p3/2)L† (i/√2)Q (i/√2)L 0 P + Δso
, (4.2) 30
Chapter 4. Theory & Models 31 and can be separated as
Hex(p) = Hs+ Hd, (4.3) where Hs = P 0 0 0 0 0 0 P 0 0 0 0 0 0 P 0 0 0 0 0 0 P 0 0 0 0 0 0 P + Δso 0 0 0 0 0 0 P + Δso , (4.4a) and Hd= Q L† M 0 (i/√2)L −(i/√2)M L† −Q 0 M −(i/√2)Q i(p3/2)L M† 0 −Q −L −i(p3/2)L† −(i/√2)Q 0 M† −L† Q −(i/√2)M† −(i/√2)L†
−(i/√2)L† (i/√2)Q i(p3/2)L (i/√2)M 0 0
(i/√2)M† −i(p3/2)L† (i/√2)Q (i/√2)L 0 0
, (4.4b) with Δso being the spin-orbit splitting, and the coefficients
P = p 2 2μ − e2 4πεbr (s-like), (4.5a) Q = p 2 x+ p2y − 2p2z 2μ1 (d-like), (4.5b) L =−i(px− ipy)pz 2μ2 (d-like), (4.5c) M =√3p 2 x− p2y 2μ1 − i pxpy 2μ2 (d-like). (4.5d)
The masses μ, μ1, and μ2 are related to the Luttinger parameters (γ1, γ2, γ3) as follows 1 μ = 1 me + γ1 m0, (4.6a) 1 μ1 = γ2 m0, (4.6b) 1 μ2 = 2 √ 3γ3 m0, (4.6c)
Chapter 4. Theory & Models 32 where m0 is the free-electron mass. Equations (4.4a) and (4.4b) represent the exciton Coulomb interaction formed by the isotropic hole and the electron-anisotropic hole respectively. The separation of the full Hamiltonian to the s-like and d-s-like parts renders the perturbation of it being possible. Because μ is much smaller than μ1 and μ2 (me m0), equation (4.5a) is more important than Eqs. (4.5b), (4.5c), and (4.5d). Therefore, we can treat the anisotropic part Hd as a perturbation of the hydrogen-atom-like part Hs. This represents that the exciton is just like a hydrogen atom with a small distortion. For Zinc-Blende structures, the Hamiltonian (4.3) should be added another contribution from inversion-asymmetry. However, this effect is small and negligible.
4.1.2
One-Band Model
Figure 4.1: Diagram of a exciton propagating in semiconductor. It moves from r to r0 during the time t to t0.
From 4.1.1, we know that the exciton is like a hydrogen atom with a small dis-tortion (Fig. 4.1 ). Hence, in this subsection, we use one-band effective mass Schr¨odinger equation [E0 + p 2 e 2me + p2 h 2mh − e2
4πεb|re− rh|]ψ(re, rh) = Eψ(re, rh), (4.7) as a good approximation to describe the exciton. Where re, rh, pe, ph, mh, E0, E and ψ(re, rh) are the electron coordinate, hole coordinate, electron momentum, hole momentum, hole effective mass, band edge energy, exciton energy, and exciton (envelope) wave function, respectively. Equation (4.7) can be rewritten as
[ P 2 2Mex + p2 2μ − e2 4πεbr]ψ(r, R) = (E− E0)ψ(r, R), (4.8)
Chapter 4. Theory & Models 33 using the relative coordinate
r = re− rh, (4.9a)
and the center-of-mass coordinate
R =mere+ mhrh
Mex , (4.9b)
where
P =− i~∇R, (4.9c)
p =− i~∇r, (4.9d)
Mex = (me+ mh), (4.9e)
μ = memh
Mex , (4.9f)
The exciton wave function can be found from Eq. (4.8), which is independent of R, and written in the form
ψ(r, R) = exp(iqex∙ R)φ(r), (4.10)
where qex = P/~. Substituting Eq. (4.10) into Eq. (4.8) yields a hydrogen-atom-like equation 1 [p 2 2μ − e2 4πεbr]φ(r) = (E− E0− ~2q2 ex 2Mex)φ(r), (4.11)
whose energy levels are
En=−R ex
n2 , (4.12a)
where the exciton Rydberg energy
Rex= μe
4
2(4πεb)2~2, (4.12b)
and the exciton Bohr radius
aex = 4πεb~ 2
μe2 . (4.12c)
To define the relative strength of the Coulombic and magnetic terms, we define
γ = ~ωc
2Rex (4.12d)
= ~eB
2Rexμ.
1Actually, just like the diagonal terms of H
Chapter 4. Theory & Models 34 Thus, the exciton energy
E = E0 + En+ ~ 2q2 ex 2Mex, or (4.12e) ~ω0 = ~ωT + ~ 2q2 ex 2Mex , (4.12f)
where ~ωT is the energy to create a motionless exciton.
4.2
Excitons in Magnetic Fields
4.2.1
One-Band Excitons in Magnetic Fields
We first consider the simple one-band excitons in a magnetic field [58], equation (4.7) becomes [E0+(pe− eA) 2 2me + (ph− eA)2 2mh − e2 4πεb|re− rh| ]ψ(re, rh) = Eψ(re, rh), (4.13) where A is the vector potential. Using the relative and CM coordinates, Eq. (4.13) can be rewritten as [p 2 2μ + e( 1 mh − 1 me)A∙ p + e2 2μA∙ A − e2 4πεbr (4.14) − 2e~ Mexqex∙ A + P2 2Mex]φ(r) = (E− E0)φ(r). When the Lorentz gauge
A = 1
2B × r (4.15)
is applied, Eq. (4.14) becomes [p 2 2μ + e 2( 1 mh − 1 me)B ∙ L + e2 8μ |B × r| 2 − e 2 4πεbr (4.16) − Mexe~ (qex× B ∙ r) + P 2 2Mex]φ(r) = (E− E0)φ(r),
where L is the angular momentum, the second term is the Zeeman term2, the
third term is the diamagnetic operator, the fifth and sixth terms depend upon the
2Electron and hole spins are not considered here, their contribution to the Zeeman term is
Chapter 4. Theory & Models 35 exciton motion and are usually neglected due to their small values. If only the s-states are considered, the Eq. (4.16) becomes
[p 2 2μ − e2 4πεbr + e2B2 8μ (x 2+ y2)]φ(r) = (E − E0)φ(r), (4.17) for B applied parallel to the z direction.
4.2.2
Multi-Band Excitons in Magnetic Fields
For considering the band mixing of excitons in magnetic fields, we discuss the theory developed by K. Cho et al. [59] for zinc-blend crystals in this subsection. The effective Hamiltonian in a magnetic field was obtained by considering the symmetry of invariant terms [59] and was compared to the result obtained by perturbation method done by Altarelli and Lipari [60]. The Hamiltonian is given by [59]
H = Eb + eΔ1J∙ ~σ + eΔ2(σxJx3 + σyJy3+ σzJz3) +egcμB~σ∙B (4.18) −2μB[eκJ ∙ B+eq(BxJx3+ ByJy3+ BzJz3)]
+(ea ex 2c ) 2 1 μ0[c1B 2+ c2(J
∙ B)2 + c3(BxBy{JxJy} + ByBz{JyJz} + BzBx{JzJx})], where ~σ and J are the effective spin operator for the electron and hole respectively, {JxJy} = (JxJy + JyJx)/2 etc., the axes x, y, z refer to three h001i axes of the crystal, and the nine parameters:Eb, eΔ1, eΔ2,egc,eκ, eq, c1,c2, c3 , which determine the exciton energies, can be expressed in terms of more fundamental material parameters by comparing to the perturbative expression [60]. The second and third terms in Eq. (4.18) stand for the exchange interaction, the fourth and fifth terms are the Zeeman terms, and the latest term accounts for the diamagnetic shift. The coefficients eΔ2, q, and c3e are the sources of the anisotropy.
For analysing the splitting patterns, we can express Eq. (4.18) as a matrix and select the quantization axis of the basis along the direction of the magnetic field B in the ζ axis. We choice the basis Jt, Jt
ζ
that can diagonalize the second term in Eq. (4.18), where
Chapter 4. Theory & Models 36 The relation between Jt, Jt
ζ and 1 2,± 1 2 × |J, Jζi is [59] |2, 2i = 32,32 12,12 , (4.20a) |2, 1i = √ 3 2 32,12 12,12 +1 2 32,32 12,−12 , (4.20b) |2, 0i = 1 2 32,−12 12,12 +√1 2 32,12 12,−12 , (4.20c) |2, −1i = 1 2 32,−32 12,12 + √ 3 2 32,−12 12,−12 , (4.20d) |2, −2i = 32,−32 12,−12 , (4.20e) |1, 1i = −1 2 32,12 12,12 + √ 3 2 32,32 12,−12 , (4.20f) |1, 0i = √1 2 32,−12 12,12 − √1 2 32,12 12,−12 , (4.20g) |1, −1i = √ 3 2 32,−32 12,12 − 1 2 32,−12 12,−12 . (4.20h)
The states |1, 1i, |1, 0i, and |1, −1i are dipole active for σ+, π, and σ− polariza-tions3 respectively. If the magnetic field (ζ axis) is applied along h001i directions
of the crystal (see Fig. 4.2), the matrix can be expressed as
|2, 2i |2, −2i |2, 1i |1, 1i |2, −1i |1, −1i |2, 0i |1, 0i
|2, 2i I11 I12 0 0 0 0 0 0
|2, −2i I21 I22 0 0 0 0 0 0
|2, 1i 0 0 II11 II12 0 0 0 0
|1, 1i 0 0 II21 II22 III11 III12 0 0
|2, −1i 0 0 0 0 III21 III22 0 0
|1, −1i 0 0 0 0 0 0 0 0 |2, 0i 0 0 0 0 0 0 IV11 IV12 |1, 0i 0 0 0 0 0 0 IV21 IV22 (4.21) 3σ
+, and σ− are the right-handed and left-handed circular polarizations in the Faraday
Chapter 4. Theory & Models 37
Figure 4.2: Diagramof the directions of the applied magnetic field B and the crystal. The axes x, y, and z are the crystal lattice coordinate; and the axis ζ
is the coordinate of the applied magnetic field. where I = (Eb+ c) 1 0 0 1 ! + (3 4Δ1e + c2) 1 0 0 1 ! + (1 2gc− 3κ) 1 0 0 −1 ! (4.22a) + 1 16 27( eΔ2− 4q) 12 eΔ2 12 eΔ2 27( eΔ2+ 4q) ! , II = (Eb+ c− 2κ) 1 0 0 1 ! +1 4Δ1e 3 0 0 −5 ! (4.22b) +1 4(gc+ 2κ− 2c2) 1 −√3 −√3 −1 ! + 1 16 15( eΔ2− 2q) −26√3q −26√3q −41( eΔ2+ 2q) ! ,
Chapter 4. Theory & Models 38 III = (Eb+ c− 2κ) 1 0 0 1 ! +1 4Δ1e 3 0 0 −5 ! (4.22c) −1 4(gc+ 2κ + 2c2) 1 −√3 −√3 −1 ! + 1 16 15( eΔ2+ 2q) 26√3q 26√3q −41( eΔ2− 2q) ! , IV = (Eb+ c) 1 0 0 1 ! + 3 4Δ1e − c2 κ + 1 2gc κ +1 2gc −54 Δ1e − c2 ! (4.22d) + 1 16 39 eΔ2 4q 4q −41 eΔ2 ! ,
where the magnetic field related parameters are
gc =egcμBB, (4.22e) q =qμBB,e (4.22f) κ =eκμBB, (4.22g) c = 1 2γ 2Rex(c 1 + 5 4c2), (4.22h) c2 = 1 2γ 2Rexc 2. (4.22i)
The existence of the exchange interaction ( eΔ1, eΔ2) splits the degenerate states, and the applied magnetic field leads coupling between diploe active and inactive states as shown in Eqs. (4.22b), (4.22c), and (4.22d). Thus, the dark (diploe inactive) states are observable under high magnetic fields.
4.3
Exciton-Polariton with Spatial Dispersion
4.3.1
Isotropic
For phenomenologically understanding the spectra of ellipsometry measurements near an excitonic transition, we consider the exciton-polariton model with spatial dispersion and in general, both p and s polarization in arbitrary incidence. When
