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Nanjing University
Investigation of Group IV Low-Dimensional Nanostructures
By
YANG ZHENG
Department of Physics, Nanjing University
Supervisor: Professor SHI YI
Feb. 2004 Nanjing, China
研 究 生 毕 业 论 文
(申请硕士学位)
论 文 题 目 IV 族 低 维 纳 米 结 构 的 研 究
作 者 姓 名 杨 铮
学 科 名 称 微 电 子 学 与 固 体 电 子 学
研 究 方 向 半 导 体 低 维 纳 米 结 构
指 导 教 师 施 毅 教 授
2004 年 2 月
南京大学研究生毕业论文英文摘要首页用纸
THESIS:Investigation of Group IV Low-Dimensional Nanostructures
SPECIALIZATION: Microelectronics and Solid-State Electronics
POSTGRADUATE: YANG Zheng
MENTOR: Prof. SHI Yi
With the development of science and technology, the research of low-dimensional nanostructures has been one of the core parts of nanoscience. Two cases of Group IV low-dimensional nanostructures were investigated in the thesis—carbon nanotubes (CNTs, quasi-1D) and Ge/Si quantum dot (quasi-0D) superlattices (QDSLs). The thesis is composed of two parts accordingly. In the first part, the geometric and band structure of CNTs have been investigated, and the progress of CNT-based electronics has been reviewed. In the second part, the optical properties of Ge/Si QDSLs have been studied through Raman scattering and photoluminescence (PL) spectra measurements. The results and conclusions of the thesis were listed below.
The symmetry of CNTs was analyzed, based on the geometric structure of CNTs. It was found that rotation axes of arbitrary fold could be obtained in some armchair or zigzag CNTs, which turn the existence of a set of point groups once only in the mathematic theory to real existences in natural molecules. The index n of zigzag (n, 0) and armchair (n, n) CNTs stands for their highest fold of the rotation axes, which together with their other symmetry elements make up the point group Dnh. The band structure of CNTs was calculated and discussed on the basis of TBA and band structure of graphite. The formulas and diagrams of the energy dispersion of the CNTs were presented. Furthermore, the progress of CNT-based electronics was reviewed. The base of CNT-based electronics’ application in the future—obtaining massive and high-density transistor arrays and separating semiconducting from metallic CNTs more effectively––was emphasized.
The Raman scattering measurements were performed on the Ge/Si QDSLs. The Ge-Ge, Ge-Si, and Si-Si peaks were observed in the spectra, which were arisen from the optical modes of the Ge in the QDs, the Ge-Si alloys, and Si substrate. The effects of the phonon confinement and strain in the Ge QDs can induce the red- and blue-shift of the Ge optical mode. The composition and strain in the QDs can be evaluated from the frequency-shift. Low-frequency Raman scattering peaks were first time observed in the non-resonant Raman scattering mode, which were arisen from the folded acoustic phonons in the Ge QDSLs. And it was found in the experiments that the intensity of the low-frequency Raman scattering peaks was closely related to the Ge and Si layer thickness and the number of the periods of the Ge QDSLs, the smaller periods, the lower intensity of the Raman peaks. The PL measurements were performed on the Ge/Si QDSLs. The Si-TO and PL peaks from the Ge QDs and the wetting layers were observed in the spectra. The theory of temperature-dependence of PL intensity in nanocrystalline semiconductors was first time verified by experiments in Ge/Si QDs. The temperature-dependence of the PL intensity has been fitted, from which a new approach could be obtained to estimate the heights and the electron effective masses of the QDs.
南京大学研究生毕业论文中文摘要首页用纸
毕业论文题目: IV 族低维纳米结构的研究
微电子学与固体电子学 专业 2001 级硕士生姓名: 杨 铮
指导教师(姓名、职称): 施 毅 教授
随着科学的发展,微观领域和宏观领域科研工作都不断向纵深发展,对纳
米结构,特别是低维纳米结构的研究已经成为纳米科学的核心组成部分。本文
工作研究了两类 IV 族低维纳米结构:准一维的碳纳米管和准零维的 Ge/Si量子
点超晶格。论文分为两个部分:第一部分研究了碳纳米管的几何结构与能带结
构,综述和展望了碳纳米管电子学;第二部分通过 Raman 光谱和荧光光谱测量
研究了自组装 Ge/Si 量子点超晶格的光学特性。本文的主要内容和结论如下:
在讨论碳纳米管的几何性质的基础上,对碳纳米管的对称性进行了详细研
究。首次发现任意度转动对称轴都可以在相应的齿型或椅型碳纳米管中找到对
应物,从而使一系列原本仅在数学理论上存在的点群在自然界中找到了对应物。
齿型碳纳米管(n, 0)和椅型碳纳米管(n, n)的指数 n 代表了碳纳米管的最高对称
转动轴的度数,它们的对称元素构成 Dnh 点群。在紧束缚近似和石墨能带计算
的基础上,对碳纳米管的能带结构进行了计算和研究,给出了碳纳米管的能量
色散关系公式和曲线。进而,综述了碳纳米管电子学的最新进展,分析了碳纳
米管电子学走向实用化的前提,特别强调碳纳米管晶体管阵列的大规模可控生
长组装和更有效的金属型与半导体型碳纳米管的可控分离。
论文对 Ge/Si 量子点超晶格进行 Raman 光谱的测量研究,成功观测到分别
来自于量子点中 Ge 的光学模,Ge-Si 合金和 Si 衬底 Ge-Ge,Ge-Si 和 Si-Si 峰。
通过对光学模的分析研究得到样品中的组份和应变等重要性质,指出样品中的
声子限制效应和应变将会使 Ge 光学模发生红移和篮移,通过对平移量大小的
分析,可以对样品中的组份和应变进行评估。首次在非共振 Raman 模式下观测
到低频声学模,阐明了 Raman 谱中的低频声学模来源于超晶格中的声学声子折
叠,并从实验中观测到其强度随周期数增大而增强。对 Ge/Si 量子点超晶格进
行荧光光谱的测量研究,观测到 Si 的 TO 发光峰,来自于 Ge 量子点的发光峰,
以及来自于 Ge 浸润层的发光峰。首次在 Ge/Si 量子点样品中验证了荧光光谱对
峰强的依赖关系,并通过对 Ge/Si 量子点超晶格的变温荧光光谱的拟合及分析,
提出了对 Ge/Si 量子点尺寸和其电子有效质量新的测评方法。
“One may say the eternal mystery of the world is its comprehensibility.”
--Albert Einstein (1879-1955)
To my parents
Investigation of Group IV Low-Dimensional Nanostructures
Part I
Geometric and Band Structure of Carbon Nanotubes
Part II
Optical Spectra of Self-Assembled Ge/Si Quantum Dot Superlattices
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Contents
Contents
Prologue 1 Nanostructure 1 Dimensionality 2 Group IV Elements 3 Carbon Nanotubes & Ge/Si Quantum Dot Superlattices 4 Structure of the Thesis 5
Part I Geometric and Band Structure of
Carbon Nanotubes (CNTs)
Chapter 1 Introduction to Carbon Nanotubes 7 1.1 Basic concepts of Carbon Nanotubes 7 1.2 Properties of Carbon Nanotubes 8 1.3 Processing of Carbon Nanotubes 9 References 10
Chapter 2 Geometric Structure of Carbon Nanotubes 13 2.1 Geometric Structure of Carbon Nanotubes 13 2.2 A Series of Novel Point Groups: the Symmetry in CNTs 16 2.3 Conclusions 19 References 20
Chapter 3 Band Structure of Carbon Nanotubes 21 3.1 Tight Binding Approximation 21 3.2 Band Structure of Graphite 24 3.3 Band Structure of Carbon Nanotubes 27 3.4 Results and Discussion 27 3.5 Conclusions 31 References 32
Chapter 4 Carbon Nanotubes for Electronics 33 4.1 Carbon Nanotube-Based Junctions 33 4.2 Carbon Nanotube-Based Transistors 35 4.3 Carbon Nanotube-Based Circuits 36 4.4 Prospects 39 References 40
i
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Contents
Part II Optical Spectra of Self-Assembled Ge/Si
Quantum Dot Superlattices (QDSLs)
Chapter 5 Self-Assembled Ge/Si QDSLs 43 5.1 Growth of Ge Quantum Dots 43 5.1.1 Molecular Beam Epitaxy 43 5.1.2 Stranski-Krastanow Growth Mode 46 5.1.3 Self-Assembled Growth of Ge Quantum Dots on Si 47 5.2 Applications of Ge Quantum Dots 48 5.3 Self-Assembled Ge/Si QDSLs 50 References 54
Chapter 6 Raman Scattering in Ge/Si QDSLs 55 6.0 Basic Concepts of Raman Spectroscopy 55 6.1 Raman Spectra of Ge/Si QDSLs 58 6.2 Optical Phonons in Ge/Si QDSLs 59 6.2.1 Experimental Results 59 6.2.2 Discussion 62 6.2.3 Conclusions 64 6.3 Acoustic Phonons in Ge/Si QDSLs 64 6.3.1 Experimental Results and Discussion 65 6.3.2 Conclusions 71 References 72
Chapter 7 Photoluminescence in Ge/Si QDSLs 75 7.0 Basic Concepts of PL 75 7.1 PL Spectra of Ge/Si QDSLs 77 7.2 Temperature-Dependent PL Spectra of Ge/Si QDSLs 78 7.2.1 Experimental Results and Discussion 78 7.2.2 Conclusions 81 7.3 PL Spectra of More Ge/Si QDSLs Samples 81 7.3.1 Experimental Results 81 7.3.2 Discussion and Conclusions 83 References 84
Epilogue 85 Main Conclusions 85 Future Work and Prospects 87
ii
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Contents
Appendix Appendix A Point Groups 89 A.1 Thirty-Two Crystal Point Groups 89 A.2 Frequently Used Non-crystal Point Groups 90Appendix B Calculations in Chapter Three 91Appendix C Programs in Chapter Three 93 C.1 Energy Dispersion of Graphite 93 C.2 Energy Dispersion of Armchair Carbon Nanotubes 93 C.3 Energy Dispersion of Zigzag Carbon Nanotubes 94Appendix D Calculations in Section 6.2 95Appendix E Details for Equation 7.1 96Appendix F Details for Figure 7.5 98Appendix G Index of Tables 99Appendix H Index of Figures 100Appendix I List of Publications (First Author) 103 I.1 Regular Papers/Articles 103 I.2 Conference Abstracts/Papers 104 I.3 Book Chapters (Translated) 105Appendix J Publications ↔ Graduation Thesis 106
Résumé 107
Acknowledgements 108
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Prologue
Prologue
This thesis is made up of two parts. The first part will be focused on carbon
nanotubes (CNTs), a kind of quasi one-dimensional nanostructure. While in the
second part, another novel low-dimensional semiconductor nanostructure—Ge/Si
quantum dot superlattices (QDSLs)—will be investigated, which is not only a kind
of quasi zero-dimensional (quantum dot), but also a kind of quasi two-dimensional
(superlattice) nanostructure. In the prologue, it will begin with three
concepts—“nanostructure”, “dimensionality”, and “group IV elements”, which are
closely related to the thesis. Then, the two cases of low-dimensional group IV
nanostructures—carbon nanotubes and Ge/Si quantum dot superlattices—will be
introduced briefly, which are the main contents of the thesis. Finally, the structure of
the thesis will be presented.
Nanostructure
Nanostructure is “a material structure assembled from a layer or cluster of atoms
with size of the order of nanometers. A number of methods exist for the synthesis of
nanostructured materials. They include synthesis from atomic or molecular
precursors (chemical or physical vapor deposition, gas condensation, chemical
precipitation, aerosol reactions, biological templating), from processing of bulk
precursors (mechanical attrition, crystallization from the amorphous state, phase
separation), and from nature (biological systems).” (From McGraw-Hill's
AccessScience at www.accessscience.com)
Nanostructure science and technology is a broad and interdisciplinary area of
research and development activity that has been growing explosively worldwide in
the past few years. It has the potential for revolutionizing the ways in which
materials and products are created and the range and nature of functionalities that
1
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Prologue
can be accessed. Worldwide study of research and development status and trends in
nanoparticles, nanostructured materials, and nanodevices (or more concisely,
nanostructure science and technology) was carried out in the past few years. The
goals are to get nanostructure materials for novel performance. It represents the
beginning of a revolutionary new age in our ability to manipulate materials for the
good of humanity. The synthesis and control of materials in nanometer dimensions
can access new material properties and device characteristics in unprecedented ways,
and work is rapidly expanding worldwide in exploiting the opportunities offered
through nanostructuring.
Nowadays, nanostructure science and technology mainly focuses on such topics as
quantum dots, quantum wires, quantum wells, superlattices, clusters and so on.
Nanostructures in different dimensionality have remarkably different properties.
Dimensionality
Bulk Quantum Walls Quantum Wires Quantum Dots
a. b. c. d.
ρ 2D(E
)
ρ 1D(E
)
ρ 0D(E
)
ρ 3D(E
)
Energy Energy Energy Energy
Figure 0.1 Density of states with different dimensionalilies.
The motion of electrons in nanostructures with reduced dimensionality is confined.
Electrons in two-dimensional (2D) systems such as quantum wells (Figure0.1b), are
confined in one direction; electrons in one-dimensional (1D) systems such as
quantum wires (Figure0.1c), are confined in two directions; electrons in
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Prologue
zero-dimensional (0D) systems such as quantum dots (Figure0.1d), are confined in
three directions.
Dimensionality affects the energy levels and the density of states (DOS) in
nanostructures. From dimensional considerations, the different energy dependent
behaviors of the DOS for nanostructure with different dimensionality can be
obtained. In the bulk materials (3D), the DOS
EED ∝)(3ρ
in 2D systems, the DOS
constED =)(2ρ
in 1D systems, the DOS
EED /1)(1 ∝ρ
in 0D systems, the DOS
)()(0 EED δρ ∝
where )(Eδ is a delta function of energy.
Group IV Elements
The IVA column in the periodic table are made up of five elements--Carbon (C),
Silicon (Si), Germanium (Ge), Tin (Sn) and Lead (Pb), among which three elements
are related to my thesis. Carbon nanotubes are composed of carbon, which will be
discussed in the first part of the thesis, while Ge/Si quantum dot superlattices in the
second part are composed Germanium and Silicon.
Carbon is one of the most abundant elements on earth. It can be found in many
forms ranging from coal, petroleum, to limestone and dolomite. Carbon is the
principle element in all-living things. Carbon is used in everyday life. When talking
about carbon, most people think of diamond and graphite—the two conventional
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Prologue
matters composed of carbon atoms. In diamonds, each carbon atom is a tetrahedral
sp3 bonded to four other carbon atoms. Graphite is a sp2 shaped hybridized molecule.
The layers of graphite are held together by van der Waals force that is one of the
weakest bonding forces in bonding. A new era in carbon materials began when in the
mid-1980s the family of buckminsterfullerenes were discovered followed by the
discovery of carbon nanotubes in 1991. The discorery of these structures set in
motion a new world-wide research boom that seems still to be growing.
Silicon makes up 28% of the earth's crust by weight, and is the second most
abundant element, exceeded only by Oxygen. Silicon-based microelectronics is the
base of modern science and technology. Germanium is a rare element and has the
similar electronic properties to silicon. But Germanium has a narrower band gap
than silicon. Both Silicon and Germanium are indirect bandgap semiconductors.
They have low radiative efficiency and are not appropriate for optoelectronic
devices in their bulk form. But when they became small into quantum dot or alloy
with each other, the circumstance will change, which is called Energy Band
Engineering.
Carbon Nanotubes & Ge/Si Quantum Dot Superlattices
Carbon nanotubes can be thought of as cylinders constructed from rolled up
graphitic sheets. Carbon nanotubes have an impressive list of attributes. They can
behave like metals or semiconductors, depending not only on the diameter but also
on the helicity. They can conduct electricity better than copper, can transmit heat
better than diamond. Small-diameter single-walled carbon nanotubes are quite stiff
and exceptionally strong, meaning that they have a high Young’s modulus and high
tensile strength. In the long term, perhaps the most valuable applications of carbon
nanotubes will take further advantage of their unique electronic properties. Carbon
nanotubes can in principle play the same role as silicon does in electronic circuits,
4
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Prologue
but at a molecular scale where silicon and other standard semiconductors cease to
work. The electronic
Ge/Si quantum dot superlattices are a kind novel nanostructure grown by molecular
beam epitaxy in Stranski-Krastanow self-assembling growth mode. In Ge/Si
quantum dot superlattices, they consist of some periods of bilayers, in which the
quantum dot layers are separated by Si spacer layers. Thus the superlattice structure
forms. Ge/Si quantum dot superlattices have lots of fascinating applications, such as
mid-infrared photodetectors, lasers, resonant tunneling diodes, thermoelectric cooler,
cellular automata, and quantum computer, etc.
Structure of the Thesis
The thesis is made up of two parts. The first part of this thesis was made up of four
chapters (from Chapter 1 to 4). In Chapter 1, the discovery, basic properties,
applications, synthesis and processing of CNTs were briefly introduced. In Chapter 2,
the symmetry of CNTs was analyzed in detail based on the geometric structure of
CNTs. It was found that rotation axes of arbitrary fold could be obtained in CNTs,
which turn the existence of a series of point groups only in the mathematic theory to
real existence in nature. In Chapter 3, the band structure of CNTs was calculated and
discussed. In Chapter 4, the basic concepts and recent progress in CNT-based
electronics were reported.
The second part of this thesis was made up of three chapters (form Chapter 5 to 7).
In Chapter 5, the basic knowledge of growth and applications of self-assembled
Ge/Si QDSLs was briefly introduced. The parameters of our Ge/Si QDSLs samples
were presented. In Chapter 6, the Raman scattering measurements were performed
on the Ge/Si QDSLs. The Raman spectra could be divided into two regions—the
optical and acoustic mode. The composition and strain in the samples can be
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Prologue
investigated from the optical mode. We first time observed the low-frequency
acoustic modes in Ge/Si QDSLs by non-resonant Raman scattering mode. In
Chapter 7, the photoluminescence measurements were performed on the Ge/Si
QDSLs. The relation between the dimension and effective electron mass of Ge
quantum dots can be obtained from the temperature-dependent photoluminescence
measurements.
Besides two parts (seven chapters), the thesis also included prologue, epilogue, and
some appendixes. In the prologue, three concepts—nanostructure, dimensionality,
and Group IV elements—were briefly discussed, and the structure of thesis was
presented. In the epilogue, the main conclusions of the thesis were summarized
again and the future experiments and applications of the CNTs and Ge/Si QDSLs
were prospected. Some explanation and calculations in detail for the contents of the
thesis were presented in the appendixes.
6
Part I
Geometric and Band Structure of Carbon Nanotubes (CNTs)
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter One
Chapter One
Introduction to Carbon Nanotubes
Sumio Iijima first noticed odd nanoscopic threads lying in a smear of soot through a
transmission electron microscope (TEM) at the NEC Fundamental Research
Laboratory in Tsukuba, Japan, in 1991. [1] Made of pure carbon, as regular and
symmetric as crystals, these exquisitely thin, impressively long macromolecules
soon became known as carbon nanotubes, and they have been the object of intense
scientific study ever since. Carbon nanotube is another fascinating discovery after
the discovery of C60 fullerene in 1985 [2], which made three scientists get 1996 Nobel
Prize of Chemistry.
1.1 Basic Concepts of Carbon Nanotubes
Carbon nanotubes can be thought of as cylinders constructed from rolled up
graphitic sheets. A single-walled carbon nanotube is rolled up by only one sheet of
graphite and multi-walled carbon nanotube consists of several concentric tubes
rolled up by sheets of graphite. The carbon nanotubes observed by Iijima in 1991
were multi-walled carbon nanotubes. Two years later, in 1993, single-walled carbon
nanotubes were first observed by Iijima’s group at NEC [3] and Donald Bethune’s
group at IBM’s Almaden Research Center in California [4] independently.
High-resolution transmission electron microscope (HRTEM) images of multi-walled
carbon nanotubes observed by Iijima in 1991 were shown in Figure 1.1. A
cross-section of each carbon nanotube is illustrated below the HRTEM images. The
carbon nanotubes shown in Figure 1.1(a), (b), and (c) consist of five, two, and seven
graphitic sheets, respectively. The diameters of tube (a), (b), and (c) are 6.7, 5.5 and
6.5 nm, respectively. The separation between an outer and inner tube is 0.34 nm,
which matches that in bulk graphite.
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter One
Figure 1.1 High-resolution transmission electron microscope (HRTEM) images of multi-walled carbon nanotubes. A cross-section of each nanotube is illustrated. (a) Nanotube consisting of five graphitic sheets, diameter 6.7 nm. (b) Two-sheet nanotube, diameter 5.5 nm. (c) Seven-sheet nanotube, diameter 6.5 nm. [1]
1.2 Properties of Carbon Nanotubes
Carbon nanotubes have an impressive list of attributes. They can behave like metals
or semiconductors, depending not only on the diameter but also on the helicity,
which will be discussed in detail in Section 3.4. They can conduct electricity better
than copper, can transmit heat better than diamond. Because of the nearly
one-dimensional electronic structure, electronic transport in metallic carbon
nanotubes occurs ballistically (i.e., without scattering) over long nanotube lengths,
enabling them to carry high currents with essentially no heating. [5] Phonons also
propagate easily along the nanotube. [6] And they rank among the strongest materials
known. [7] Small-diameter single-walled carbon nanotubes are quite stiff and
exceptionally strong, meaning that they have a high Young’s modulus and high
tensile strength.
Furthermore, carbon nanotubes have lots of other fascinating properties and
8
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter One
applications, such as excellent field emission [8], superconductivity [9], hydrogen
storage [10], sensors [11] and probes [12] etc.
In the long term, perhaps the most valuable applications will take further advantage
of carbon nanotubes’ unique electronic properties. Carbon nanotubes can in principle
play the same role as silicon does in electronic circuits, but at a molecular scale
where silicon and other standard semiconductors cease to work. The electronic
application of carbon nanotube will be discussed in detail separately in Chapter 4
(Carbon Nanotubes for Electronics).
1.3 Processing of Carbon Nanotubes
Carbon nanotubes are usually made by arc-discharge [3,4,13], pulsed laser
vaporization (PLV, also called laser ablation) [14], or chemical vapor deposition
(CVD) [15]. Arc-discharge and laser ablation methods for the growth of carbon
nanotubes have been actively pursued in the past ten years. Both methods involve
the condensation of carbon atoms generated from evaporation of solid carbon
sources. The temperatures involved in these methods are close to the melting
temperature of graphite, 3000-4000 ºC. The growth process of CVD involves
heating a catalyst material to high temperature (500-900 ºC or so) in a tube furnace
and flowing a hydrocarbon gas through the tube reactor for a period of time.
The properties of metallic carbon nanotubes are much different from those of
semiconducting ones. Processing the various mixtures of carbon nanotubes—
separating the metallic from the semiconducting carbon nanotubes—is very
important. In recent years, some separation methods have been found, such as
selectively destroying metallic carbon nanotubes by electrical heating [16], separating
metallic from semiconducting carbon nanotubes through their different relative
dielectric constants [17], and chemical method [18].
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter One
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and M. S. Dresselhaus, Hydrogen storage in single-walled carbon nanotubes at
room temperature, Science 286, 1127-1129 (1999).
[11] J. Kong, N. R. Franklin, C. Zhou, M. G. Chapline, S. Peng, K. Cho, and H. Dai,
Nanotube molecular wires as chemical sensors, Science 287, 622-625 (2000).
[12] H. Dai, J. H. Hafner, A. G. Rinzler, D. T. Colbert, and R. E. Smalley, Nanotube
as nanoprobes in scanning probe microscopy, Nature 384, 147-150 (1996).
[13] i) T. W. Ebbesen and P. M. Ajayan, Large-scale synthesis of carbon nanotubes,
Nature 358, 220-222 (1992); ii) C. Journet, W. K. Maser, P. Bernier, A. Loiseau,
M. Chapelle, S. Lefrant, P. Deniard, R. Lee, and J. E. Fischer, Large-scale
production of single-walled carbon nanotubes by the electric-arc technique,
Nature 388, 756-758 (1997).
[14] i) A. Thess, R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, C. Xu, Y. H. Lee, S.
G. Kim, A. G. Rinzler, D. T. Colbert, G. E. Scuseria, D. Tománek, J. E. Fischer,
and R. E. Smalley, Crystalline ropes of metallic carbon nanotubes, Science 273,
483-487 (1996); ii) P. C. Eklund, B. K. Pradhan, U. J. Kim, Q. Xiong, J. E.
Fischer, A. D. Friedman, B. C. Holloway, K. Jordan, and M. W. Smith,
Large-scale production of single-walled carbon nanotubes using ultrafast pulses
from a free electron laser, Nano Letters 2, 561-566 (2002).
[15] Z. F. Ren, Z. P. Huang, J. W. Xu, J. H. Wang, P. Bush, M. P. Siegal, and P. N.
Provencio, Synthesis of Large Arrays of Well-Aligned Carbon Nanotubes on
Glass, Science 282, 1105-1107 (1998).
[16] P. G. Collins, M. S. Arnold, and Ph. Avouris, Engineering carbon nanotubes
11
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter One
and nanotube circuits using electrical breakdown, Science 292, 706-709 (2001).
[17] R. Krupke, F. Hennrich, H. v. Löhneysen, and M. M. Kappes, Separation of
metallic from semiconducting single-walled carbon nanotubes, Science 301,
344-347 (2003).
[18] Z. Chen, X. Du, M. H. Du, C. D. Ranchen, H. P. Cheng, and A. G. Rinzler,
Bulk separative enrichment in metallic or semiconducting single-walled carbon
nanotubes, Nano Letters 3, 1245-1249 (2003).
And several review articles on carbon nanotubes,
[19] T. W. Ebbesen, Carbon nanotubes, Physics Today, 26-32 (June, 1996).
[20] C. Dekker, Carbon nanotubes as molecular quantum wires, Physics Today,
22-28 (May, 1999).
[21] Special Issue (on Carbon Nanotubes), Physics World, 22-53 (June, 2000).
[22] P. G. Collins and Ph. Avouris, Nanotubes for electronics, Scientific American,
62-69 (December, 2000).
[23] R. H. Baughman, A. A. Zakhidov, and Walt A. de Heer, Carbon
nanotubes—the route toward applications, Science 297, 787-792 (2002).
[24] A. Hirsch, Functionalization of single-walled carbon nanotubes, Angew.
Chem. Int. Ed. 41, 1853-1859 (2002)
12
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Two
Chapter Two
Geometric Structure of Carbon Nanotubes
It has been more than ten years passed since the discovery of carbon nanotubes, lots
of properties of them have been investigated thoroughly by the scientists. And
carbon nanotubes’ remarkable applications in the future attracted many people. But
one of the most basic properties of carbon nanotubes—the geometric symmetry is
not discussed completely. Though the symmetry elements in them are not as
abundant as those in some molecules, such as C60, carbon nanotubes provide us a
series of novel symmetry elements and a series of novel point groups, which include
arbitrary-fold rotation axes and make the counterpart of point group in mathematic
theory found in real molecules. In this chapter, the geometric structure of carbon
nanotubes will be discussed firstly. Then a series of point groups based on the novel
symmetry elements of carbon nanotubes will be reported, which turn the existence
of the point groups only in the mathematic theory to real existence in molecules. By
cutting out the sheet along the lines perpendicular to Ch and rolling up the sheet in
the direction of the vector, a carbon nanotube can be obtained.
2.1 Geometric Structure of Carbon Nanotubes
Carbon nanotubes (CNTs) can be thought of as cylinders constructed from rolled up
graphite sheets. Single-walled carbon nanotube (SWNT) is rolled up by only one
sheet of graphite and multi-walled carbon nanotube (MWNT) consists of several
concentric tubes rolled up by sheets of graphite. In this and the next chapter, the
discussion is focused on SWNT.
Figure 2.1 shows the graphitic lattice, in which x and y denote the coordinates. And
a1 and a2 are unit vectors. The vector Ch that is called chiral vector, is defined
pointing from one carbon site to another equivalent site in the hexagonal lattice.
13
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Two
Figure 2.1 Graphite lattice, x and y consist the coordinates, and a1 and a2 the unit vectors. The geometric structure of carbon nanotubes can be defined by the chiral vector Ch or by a pair of integral indexes (n, m).
d
a(n, m)=(5, 5) C60 e
b(n, m)=(9, 0)C70
f
cC80
Figur
e 2.2 Schematics of fulle1
r
4
(n, m)=(10, 5)enes and carbon nanotubes. [1]
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Two
By cutting out the sheet along the lines perpendicular to Ch and rolling up the sheet
in the direction of the vector, a carbon nanotube can be obtained. Thus the chiral
vector Ch becomes the circumference of one of the cross-section circles of the
carbon nanotubes. The angle φ between the chiral vector and x axis is called chiral
angle. The vector T, which is orthogonal to Ch, stands for the axis direction of
carbon nanotube. The vector Ch can be related to the unit vectors a1 and a2 as Ch =
na1 + ma2, so the pair of integral indexes (n, m) defines the carbon nanotube. As
vector Ch pointing outside the area between the dashed line (n, 0) and the dotted line
(n, n) have an equivalent vector inside these two lines, all possible carbon nanotubes
are uniquely defined with the restriction m ≤ n.
A (n, m) carbon nanotube corresponds to a diameter
ππcch amnmn
d −++==
)(3 22C (2.1)
and a chiral angle
+−
=)(3
arctgmn
mnφ (2.2)
where ac-c = 1.42 Å is the length of carbon-carbon bond in graphite.
A sheet rolled up along (n, 0) and (n, n) directions both result in a non-chiral carbon
nanotube. The kind of carbon nanotubes rolled up along the dotted line (n, n)
direction i.e. φ=0˚ are called armchair carbon nanotubes while carbon nanotubes
rolled up along the dashed line (n, 0) direction i.e. φ=30˚ are called zigzag carbon
nanotubes. The carbon nanotubes rolled up along the direction between (n, 0) and (n,
n) are called chiral carbon nanotubes. The schematics of (5, 5) armchair, (9, 0)
zigzag, and (10, 5) chiral carbon nanotubes are shown in Figure 2.2 (d), (e), and (f),
respectively.
15
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Two
2.2 A Series of Novel Point Groups: the Symmetry in CNTs
Since the periodic arrangement of atoms in crystal lattice, the symmetry of crystal is
circumscribed. There are only 2-, 3-, 4- and 6-fold rotation axes in crystal, from
which together with reflection and inversion thirty-two kinds of crystal point group
are obtained (Please see Appendix A.1). Since the discovery of C60, people found
5-fold rotation axes in them orthogonal to each of their regular pentagon planes as
shown in Figure 2.2 (a). And C60 has very high symmetry that belongs to point
group Ih. The point group Ih has 120 group elements (Please see Appendix A.2),
which is the point group that has the highest symmetry. And ih CII ⊗= , where I is
the point group which the regular icosahedron belongs to. It is the discovery of C60
that turn Ih’s only existence in the mathematic theory to real existence in natural
molecules. And the 5-fold rotation axes can also be found in other fullerenes, such as
C70 and C80 which have 5-fold rotation axes perpendicular to their regular pentagon
planes on each bottom as shown in Figure 2.2 (b) and (c). Other symmetry elements
such as 7-fold axes have been found step by step in different kinds of molecules. The
symmetry elements in carbon nanotubes are very singular and interesting, all kinds
of n-fold axes can be found in some carbon nanotubes, where n is an integer larger
than two. The symmetry elements in carbon nanotubes will be discussed below
based on the geometric structure.
As shown in Figure 2.3 (b), the carbon nanotube (9, 0) is rolled up along the chiral
vector Ch. Its cross section is a regular enneagon (Figure 2.3 (d)), from which its
symmetry can be studied. The middle cross-section is a hσ reflection plane. There
are nine vσ reflection planes perpendicular to the cross-section, such as the plane
across point A and the middle point between E and F. There are eight proper rotation
axes along the direction of carbon nanotube (i.e., perpendicular to the cross-section),
which are two axes, two axes, two C axes, and two axes. There are 9C 29C 1
349C
16
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Two
also nine proper rotation axes 2C ′ in the middle cross-section plane. And there are
eight improper rotation axes along the direction of carbon nanotube, which are two
axes, two axes, two axes and two axes, where 9S 29S 1
3S 49S nhn CS σ= .
Together with identity element E, the point group D9h are made up.
Figure 2.3 The schematics of rolled up graphite sheet and cross sections of (5, 5) armchair and (9, 0) zigzag carbon nanotubes.
When we observe the cross-section of a zigzag carbon nanotube (n, 0), we find that
there is a regular polygon consisting of n equivalent carbon atoms. Thus there is an
n-fold axis orthogonal to the cross-section. So we obtain the conclusion that we can
get arbitrary-fold rotation axes from different zigzag carbon nanotubes, for example
there is a 9-fold axis along a (9, 0) zigzag carbon nanotube. By studying the
symmetry of regular polygons, we obtain that zigzag carbon nanotubes belong to
Dnh point group, where n is an integer larger than two. In the classic theory, only D2h,
D4h and D6h are in the thirty-two crystal point groups, while the others only exist in
17
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Two
the mathematic theory. Now all Dnh point groups are found in the symmetry of
natural molecules.
a
b
Figure 2.4 The cross-sections of two series of armchair and zigzag carbon nanotubes, from which we can understand the geometric symmetry elements belonging to group Dnh.
For the armchair carbon nanotubes, the discussion is a little more complex. As
shown in Figure 2.3 (a) and (c), the armchair carbon nanotube (5, 5) is rolled up
along chiral vector Ch. In its cross-section, there are two different kinds of carbon
atoms, each kind of atoms make up a regular pentagon. The two pentagons are
congruent and they can coincide after rotation. Thus we can just discuss the five
carbon atoms consisting of pentagon instead of ten atoms. Just like the discussion in
carbon nanotube (9, 0), the symmetry of armchair carbon nanotube (5, 5) belongs
to point group D5h. We get the conclusion that armchair carbon nanotube (n, n) also
belongs to the Dnh point group, there are 2n carbon atoms in the cross-section of an
armchair carbon nanotube (n, n), which form two congruent regular polygons of n
sides. Then the following discussion is the same as zigzag carbon nanotube. Figure
2.4(a) and (b) give a series of cross section of zigzag and armchair carbon nanotubes,
from which we can understand the symmetry in carbon nanotubes more thoroughly.
Let’s focus on the point group Dnh. This kind of group has 4n group elements,
consisting of 2n proper rotations, one horizontal reflection hσ , n vertical reflection
18
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Two
vσ , (n−1) improper rotation and inversion. When n is an odd integer, group
and it has (n+3) classes, which are hn CD 1⊗=
(,, nn vh
nhD
,E 2,)1(,)1 CnSnC ′− ϕϕ−σσ , where there are both (n−1)/2 classes in
and . When n is an even integer, group ϕC ϕS innh CDD ⊗= and it has (n+6) classes,
which are 22 22,)2 CnCnS ′′′,ϕ(,)1 nC −ϕ
ϕS
(,,2
,2
, ninndvh −σσ,E σ
ϕC
, where there are n/2
classes in and (n/2−1) classes in . The character table of D∞h group is given
in Table 2.1.
Table 2.1 Character table of D∞h group.
D∞h E 2Cφ C′2 i 2iCφ iC′2 A1g 1 1 1 1 1 1 A1u 1 1 1 −1 −1 −1 A2g 1 1 −1 1 1 −1 A2u 1 1 −1 −1 −1 1 E1g 2 2cosφ 0 −2 2cosφ 0 E1u 2 2cosφ 0 −2 −2cosφ 0 E2g 2 2cos2φ 0 −2 2cos2φ 0 E2u 2 2cos2φ 0 −2 −2cos2φ 0
…… …… …… …… …… …… ……
2.3 Conclusions
The index n of zigzag (n, 0) and armchair (n, n) carbon nanotubes stands for the
highest fold of the rotation axes among the symmetry elements in carbon nanotubes.
This n-fold axis together with other symmetry elements in carbon nanotubes make
up the point group Dnh. And all kinds of n-fold axes can be found in some zigzag or
armchair carbon nanotubes, where n is an integer larger than two.
Chapter Two was submitted to Physics Letters A and Acta Physica Sinica.
19
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Two
References
[1] http://cnst.rice.edu/reshome.html.
[2] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon
Nanotubes (Lodon: Imperial College Press), 48-53, (1998).
[3] M. S. Dresselhaus, G. Dresselhaus, and R. Saito, Carbon fibers based on C60
and their symmetry, Phys. Rev. B 45, 6234-6242 (1992).
[4] P. Delaney, H. J. Choi, J. Ihm, S. G. Louie, and M. L. Cohen, Broken symmetry
and pseudogaps in ropes of carbon nanotubes, Nature 391, 466-468 (1998).
[5] M. Damnjanovic, I. Milosevic, T. Vukovic, and R. Sredanovic, Full symmetry,
optical activity, and potentials of single-wall and multiwall nanotubes, Phys.
Rev. B 60, 2728-2739 (1999).
20
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
Chapter Three
Band Structure of Carbon Nanotubes
Since the discovery of carbon nanotubes (CNTs), their extraordinary properties,
especially their electronic properties and remarkable perspective applications have
attracted many scientists. Lots of experiments have been done to demonstrate that
CNT could play a pivotal role in the upcoming revolution of silicon-based
microelectronics. Some CNT-based electronic devices and circuits have been
discovered and developed, such as CNT-based field effect transistors (FETs),
CNT-based single electron transistors (SETs) and CNT-based logic circuits. The
basis of developing CNT-based electronic devices and circuits is the electronic
structure of CNTs, especially the band structure of CNTs. The related study has been
done, using tight binding approximation [2] and effective mass approximation [3] soon
after the discovery of CNTs.
During the last decade, there have been lots of changes in describing CNTs.
Nowadays, there have been uniform indexes and symbols in describing CNTs’
geometric structure and standards for classifying CNTs. Although the old theories
are surely still correct, they cannot be yet used directly for their different indexes,
and they are not thorough. In this chapter, the band structure of CNTs in terms of the
uniform indexes is discussed, in which the conclusions can be used directly by the
experimenters and electronic engineers. Firstly, we shall begin with tight binding
approximation, and secondly, the band structure of graphite. Finally the band
structure will be calculated and discussed.
3.1 Tight Binding Approximation
Tight Binding Approximation (TBA) is a frequently used theory in calculating band
structure. The basic functions of the TBA are the one-particle eigen-functions of the
21
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
valence electrons of the free atoms, more strictly, of the atoms composing the crystal
under consideration. These eigen-functions are called atomic orbitals, they are
denoted by
)()( αα ϕϕ tRrrR −−≡ j
j (3.1)
where j is a general quantum number index, coordinate origin is located at R and
atomic core is located at . The orbitals considered above are orthonormalized,
i.e. one has
αtR +
( ) RRRR ′′′′′
′ = δδδϕϕ αααα
jjjj | (3.2)
In order to represent the eigenstates of the valence electrons of a crystal, one needs,
rigorously speaking, all orbitals of the cores of its atoms since only the totality of all
orbitals forms a complete basis set in Hilbert space. However, not all of these
contribute in an essential manner. The largest contributions are to be expected from
orbitals forming the valence shells of the free atoms. Within the TBA one takes
only these orbitals into account. This corresponds to a perturbation-theoretic
treatment of the Hamiltonian matrix with respect to the atomic orbital basis; only
matrix elements between valence orbitals are considered while those involving other
orbitals are neglected.
Although the two orbitals are localized in different spatial regions, and the integral
over the product of the two, the so-called overlap integral, turns out to be relatively
small, it may not be neglected because its influence on the energy eigenvalues is of
the same order of magnitude as the matrix elements of the Hamiltonian between
orbitals at different centers. The latter elements are essential because they are
responsible for the bonding between atoms in a crystal and for the splitting of the
atomic energy levels into bands. The non-orthogonality overlap integrals must
therefore also be taken into account. This may be done directly, by writing down and
solving the eigenvalue problem for the crystal Hamiltonian in the non-orthogonal
basis set of the atomic orbitals. This procedure is, however, quite inconvenient
22
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
because the matrix of overlap integrals has to be calculated explicitly and
diagonalized together with the Hamiltoninan matrix. It is more useful to employ a
set of orthogonalized orbitals by forming linear combinations of the . )(rRαϕ j
To represent the Bloch type eigenfunctions by means of atomic orbitals , it is
convenient to transform the latter into Bloch type orbitals . This is done by
means of the k-dependent unitary transformation
)(rRαϕ j
)(rkαϕ j
∑ +⋅=R
RtRk
k rr )(1)( )( αα ϕϕ α jij eN
(3.3)
The are called Bloch sums of atomic orbitals. And the orthogonality of the
orbitals results in the orthogonality of their Bloch sums , such that
)(rkαϕ j
Rαϕ j )(r )(rk
αϕ j
( ) kkkk ′′′′′
′ = δδδϕϕ αααα
jjjj | (3.4)
Consider the Schrödinger equation of the crystal
)()()()(2
)(ˆ 22
rkrrr kkkn
nnn EV
mH ψψψ =
+∇−=
h (3.5)
In the TBA, the eigen-functions of the Schrödinger equation are written as
linear combinations of Bloch sums
)(rknψ
(kαϕ j )r
( )∑∑ ==α
α
α
αα ϕψαϕψj
jn
j
jjn jc )(|)()( rkrr kkkkk (3.6)
where the linear combination coefficient ( )nj j kk k ψαα |=c . Employing Equation (3.6)
in the Schrödinger Equation (3.5), we obtain
( ) ( )nn
j
n jEjjHj kk kkkkk ψαψαααα
|)(||ˆ| =′′′′∑′′
(3.7)
where the matrix elements of the Hamiltonian are given by the expressions
∑′
−+′⋅ ′′′=′′ ′
R
ttRk Rkk αααα αα jHjejHj i |ˆ|0|ˆ| )( (3.8)
with
∫∫ ′′′′
′ −′−−≡=′′′ )(ˆ)()(ˆ)(|ˆ|0 **0 αα
αα ϕϕϕϕαα tRrtrrrrrR R jjjj HdHdjHj (3.9)
23
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
3.2 Band Structure of Graphite
Now the band structure of graphite is to be calculated using TBA. Since the spacing
of the lattice planes of graphite (3.37Å) is much larger than the hexagonal spacing
(1.42Å) in the layer, a first approximation in the treatment of graphite may be
obtained by neglecting the interactions between planes.
Graphite possesses four valence electrons, three of which form σ covalent bonds
with neighboring atoms in the plane through sp2 hybrids. The fourth electron that is
considered to be in the 2pz state forms delocalized π bond with all 2pz electrons of
other carbon atoms in the plane. The three electrons forming covalent bonds will not
play a part in the conductivity of graphite, the band structure of graphite is therefore
only determined by the 2pz state electrons.
a b
Figure 3.1 The vector space and reciprocal vector space of graphite, two lattices of regular hexagons that can be congruent after a rotation of 30°.
The vectors , and reciprocal vectors , of graphite are shown in Figure
3.1, as
1a 2a 1b 2b
24
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
−=
+=
−=
+=
jib
jib
jia
jia
aa
aaaa
aa
ππ
ππ
23
2
23
2
223
223
2
1
2
1 (3.10)
where lattice constant Å46.23 == −ccaa .
There are two kinds of atoms A and B which are not equivalent in the graphite,
shown in Figure 3.1 (a). Here, the kind of A atoms are located at R while the kind
of B atoms are located at tR + , 3ait = . Using Equation (3.6), assuming
and , we obtain λ== BA cc kk ,1 t=tt = BA ,0
)()()( rrr kkkBAn λϕϕψ += (3.11)
where )(rkϕ stands for the wave function of 2pz electrons atomic orbital.
Employing Equation (3.11) into Schrödinger Equation (3.5), just like Equation (3.7),
we can get
=+=+
)()(
2221
1211
kk
n
n
EHHEHHλλ
λ (3.12)
where
)(ˆ)(
)(ˆ)()(ˆ)(*2112
2211
rr
rrrr
kk
kkkk
BA
BBAA
HHH
HHHH
ϕϕ
ϕϕϕϕ
≡=
≡=≡ (3.13)
Eliminating λ we obtain the secular equation
0)(
)(
2221
1211 =−
−k
k
n
n
EHHHEH
(3.14)
from which we get
1211)( HHEn ±=k (3.15)
Using Equation (3.8) and (3.9), we obtain
∑ ⋅−′−=R
RkieEH 0011 γ (3.16)
where ∫∫ −−=′= rRrrrr dHdHE )(ˆ)(,)(ˆ *0
20 ϕϕγϕ , and
25
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
∑+
+⋅−=tR
tRk )(012
ieH γ (3.17)
where . Employing the nearest neighboring vector ∫ −−−= rtRrr dH )(ˆ)(*0 ϕϕγ
),0(),2,23(),2,23( aaaaanb ±−±±=R
in Equation (3.16), we obtain
+′−= )cos()
2cos()
23cos(22 0011 ak
akakEH yyxγ (3.18)
and employing the nearest neighboring vector
)2,32(),2,32(),0,3()( aaaaanb −−−=+ tR
in Equation (3.17), we obtain
−+−= )
32exp()
2cos(2)
3exp(012
akiakakiH xyxγ (3.19)
thus
++= )
23cos()
2cos(4)
2(cos41 22
02
12akakak
H xyyγ (3.20)
Since 0γ is multiple of 0γ ′ , the second part of Equation (3.18) is often neglected in
general discussions. Now we get the energy dispersion of graphite, that is
)2
3cos()2
cos(4)2
(cos41)( 200
akakakEE xyyn
graphite ++±= γk (3.21)
Figure 3.2 Energy dispersion of graphite
26
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
3.3 Band Structure of Carbon Nanotubes
If the CNT is sufficiently long, the effect of the ends can be neglected. Starting from
the band structure of graphite we can calculate the band structure of CNTs by
applying periodic boundary conditions. From the discussions of the geometric
structure of CNTs, it is well-known that a SWNT is rolled up from a sheet of
graphite to form a cylinder, so in the circumferential direction waves have to obey
the periodic boundary conditions
)Z(2 ∈=⋅ qqh πCk (3.22)
where is the wave vector. Employing k
jiaaC amnamnmnh )(21)(
23
21 −++=+=
in Equation (3.8), it is obtained
)Z(4)()(3 ∈=−++ qqakmnakmn yx π (3.23)
Then, the energy dispersion of CNTs can be formulize as
+
−−
+
+±=
)(2)(4
cos2
cos42
cos41)( 20 mn
akmnqakakkE yyy
yqtube
πγ (3.24a)
or
−+−
+
−+−
+±=2
3cos)(2
3)(4cos4)(2
3)(4cos41)( 20
akmn
akmnqmn
akmnqkE xxxx
qtube
ππγ
(3.24b)
3.4 Results and Discussion
According to the energy dispersion of CNTs, we can get that the energy gap of CNTs
are zero only when the indexes satisfy
)(3 Zppmn ∈=− (3.25)
27
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
So, it is easy to identify that CNTs whose indexes satisfy Equation (3.25) are
metallic and CNTs whose indexes do not satisfy Equation (3.25) are
semiconducting.
Figure 3.3 Distribution of metallic and semiconducting carbon nanotubes
Thus, one third are metallic and two thirds are semiconducting among all zigzag
CNTs. This is different from armchair CNTs which are all metallic.
From the geometric structure of CNTs, it has been noted that there are important
kinds of CNTs, which are armchair and zigzag CNTs, having particular electronic
properties among all CNTs.
The indexes of armchair CNTs are (n, n), meeting with Equation (3.25) whatever n
is. So, all the armchair CNTs are metallic. Employing m = n in Equation (3.24a), we
can get the energy dispersion of armchair CNTs:
)2,,1(cos2
cos42
cos41)( 20 nq
nqakak
kE yyy
qarmchair
L=
+
+±=
πγ (3.26)
28
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
0.0 0.2 0.4 0.6 0.8 1.0
-3
-2
-1
0
1
2
3
E(γ 0)
ky(π / a)0.0 0.2 0.4 0.6 0.8 1.0
-3
-2
-1
0
1
2
3E(γ 0)
ky (π / a)
(6, 6) (5, 5)
Figure 3.4 Energy dispersion of armchair CNTs (5, 5) and (6, 6), which are both metallic.
The energy dispersions of armchair CNTs (5, 5) and (6, 6) are shown in Figure 3.4,
It can be seen that there are cross-parts between the bands.
The indexes of zigzag CNTs are (n, 0), meeting with Equation (3.25) when n=3p,
where p is an integer. Employing the boundary conditions of zigzag CNTs
qanky π2= in Equation (3.21), we can get the energy dispersion of zigzag CNTs.
)2,,1(2
3coscos4cos41)( 20 nqak
nq
nqkE x
xqzigzag
L=
+
+±=
ππγ (3.27)
The energy dispersion of zigzag CNTs (6, 0), (7, 0), (8, 0), and (9, 0) are shown in
Figure 3.5. It can be found that there are cross parts between the bands in (6, 0) and
(9, 0) zigzag CNTs while none in (7, 0) and (8, 0) zigzag CNTs. So (6, 0) and (9, 0)
zigzag CNTs are metallic while the other two are semiconducting.
29
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
0.0 0.2 0.4 0.6 0.8 1.0
-3
-2
-1
0
1
2
3
E(γ 0)
kx)3/( aπ
0.0 0.2 0.4 0.6 0.8 1.0
-3
-2
-1
0
1
2
3
E(γ 0)
kx )3/( aπ
(7, 0) (6, 0)
0.0 0.2 0.4 0.6 0.8 1.0
-3
-2
-1
0
1
2
3
E(γ 0)
kx)3/( aπ
(8, 0)
0.0 0.2 0.4 0.6 0.8 1.0
-3
-2
-1
0
1
2
3
E(γ 0)
kx)3/( aπ
(9, 0)
Figure 3.5 Energy dispersion of zigzag CNTs (6, 0), (7, 0), (8, 0) and (9, 0). The CNTs (6, 0) and (9, 0) are metallic while CNTs (7, 0) and (8, 0) are semiconducting.
30
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
3.5 Conclusions
The band structures of carbon nanotubes were calculated on the basis of band
structure of graphite by tight binding approximation. The energy dispersion of
carbon nanotubes was formulized in Equation (3.24). One third of the carbon
nanotubes whose indexes meet with Equation (3.25) are metallic, while the other
two thirds are semiconducting. The energy dispersion of armchair carbon nanotubes
was formulized in Equation (3.26). All the armchair carbon nanotubes are metallic.
The energy dispersion of zigzag carbon nanotubes was formulized in Equation
(3.27). Also, only one third of zigzag nanotubes are semiconducting.
Part of Chapter Three was published as an abstract on the Proceedings of 2002 Chinese Materials Research Symposium.
31
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Three
References
[1] P. R. Wallace, The Band Theory of Graphite, Phys. Rev. 71, 622~634 (1947).
[2] M. S. Dresselhaus, G. Dresselhaus, and R. Saito, Electronic structure of
graphene tubules based on C60, Phys. Rev. B 46, 1804-1811 (1992).
[3] H. Ajiki and Tsuneya Ando, Electronic states of carbon nanotubes, J. Phys. Soc.
Jpn., 62, 1255-1266 (1993).
[4] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon
Nanotubes (Lodon: Imperial College Press), 61-64, (1998).
32
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Four
Chapter Four
Carbon Nanotubes for Electronics
In the 20th century, silicon based microelectronics brought a series of industrial
revolutions and transformed our lives a lot. Today, although the silicon
microelectronics industry is already pushing the critical dimensions of transistors in
commercial chips below 200 nanometers—about 400 atoms wide—engineers face
lager obstacles in continuing this miniaturization. Within this decade, the materials
and processes on which the computer revolution has been built will begin to hit
fundamental physical limits. Scientists all over the world work hard from day to day,
in order to find the solutions to the problem and the way out of the conventional
silicon microelectronics. Carbon nanotube is one of the answers! Experiments over
the past decade have given researchers hope that wires and functional devices tens of
nanometers or smaller in size could be made from carbon nanotubes and
incorporated into electronic circuits that work far faster and on much less power than
those existing today’s silicon microelectronics. Some CNT-based electronic devices
and circuits have been discovered and developed, such as CNT-based FETs,
CNT-based SETs, and CNT-based logic circuits, which will be discussed in this
chapter. Though the prospects of CNT-based electronics are fascinating, there is still
some problems that need to solve before the CNT-based circuits come into real
application, which will be discussed in the last section of this chapter.
4.1 Carbon Nanotube-Based Junctions
Through some special mechanisms, carbon nanotubes can form on-tube and
cross-tube junctions, which might be used in nanoelectronics in the future. The
on-tube and cross-tube CNT-based junctions will be introduced briefly below. Then
some other CNT-based junctions will also be mentioned.
33
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Four
On-Tube (Intramolecular) Junctions
It has been shown theoretically [1-3, 7] and in experiments [4-6] that the introduction of
pentagon-heptagon pair defects into the hexagonal network of carbon nanotubes can
join two half-tubes of different helicity seamlessly with each other and construct
on-tube junctions. Thus the helicity of the bulk carbon nanotube has been changed
and its electronic structure has fundamentally been altered. These junctions have
great potential for applications since they are of nanoscale dimensions and made
entirely of a single element. Since carbon nanotubes are metals or semiconductors
depending sensitively on their structures, these on-tube junctions could behave as
the desired nanoscale metal-semiconductor Schottky barriers, semiconductor
heterojunctions, or metal-metal junctions with novel properties, and that they could
be the building blocks of nanoscale electronic devices.
Crossed-Tube Junctions
While it remains a challenge to controllably grow nanotube intramolecular junctions
for possible electronic applications, a potentially viable alternative would be to
construct junctions between two different nanotubes, i.e. having two nanotubes
crossing each other in contact. The crossed-tube junction consisting of two naturally
occurring crossed nanotubes with electrical contacts at each end of each nanotube
has been measured. [8] Both metal-metal and semiconductor-semiconductor junctions
exhibit high tunneling conductances on the order of 0.1 e2/h, where e and h are mass
of free electron and Planck’s constant, respectively. Theoretical study indicates that
the contact force between the nanotubes is responsible for the high transmission
probability of the junctions. Metal-semiconductor junctions show asymmetry in the
I-V curves and these results appear to be understood well from the formation of a
Schottky barrier at the junction. Crossed-nanotube junctions can also be created by
an electron beam [9] and mechanical manipulation using the tip of an atomic force
microscope [10].
34
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Four
Besides on-tube and crossed-tube junctions, some other junctions can also be formed
by carbon nanotubes, such as pn junctions through modulated chemical doping of an
individual [11] and Schottky junctions through contacting a semiconducting nanotube
with a metal [12]. They also have the potential applications in nanoelectronics in the
future.
4.2 Carbon Nanotube-Based Transistors
Field effect transistor (FET) is the most basic element in the electronic circuits.
Carbon nanotube-based FET (CNTFET) was fabricated in 1998 [14, 15], which made
turning carbon nanotube-based circuits into reality possible. In this kind of FET, a
semiconducting carbon nanotube lies across two metallic contacts fabricated on the
top of the silicon substrate, and a voltage is applied to the gate to move carriers onto
the tube. It is found that the nanotube can be “turned on” by applying a negative bias
to the gate, which induces holes on the initially non-conducting tube. This device is
thus analogous to a p-type MOSFET, with the nanotube replacing silicon inversion
layer as the material that hosts charge carriers. The resistance of the device can be
changed by many orders of magnitude, and it operates at room temperature—a
property that has eluded most other nanoscale devices. The schematic of carbon
nanotube-based FET is shown in Figure 4.1.
It has been found that the gate electrode can change the conductivity of the carbon
nanotube channel in an CNTFET by a factor of one million of more, comparable to
Si MOSFETs. Because of its tiny size, the CNTFET could switch reliably using
much less power than a Si-based device.
In recent years, different kinds of novel CNTFETs have been fabricated, such as
electrolyte gated CNTFET [16, 17], high-κ dielectrics CNTFET [18] and ballistic
CNTFET [19], which all have their own good qualities and performances.
35
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Four
Carbon Nanotube
Figure 4.1 Schematic of CNT-based FET.
And room-temperature single-electron transistor (SET) was realized within
individual metallic single-wall carbon molecules in 2001. [20] SETs have been
proposed as a future alternative to conventional silicon electronic components.
However, most SETs operate at cryogenic temperatures, which strongly limits their
practical application. The carbon nanotube-based SET working at room-temperature
is also a breakthrough in single electron device area.
4.3 Carbon Nanotube-Based Circuits
Figure 4.2 Demonstration of voltage transport characteristics and schematics (insets) of CNT-based inverter, NOR gate, SRAM, and ring oscillator in RTL style. [24]
36
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Four
It was predicted that carbon nanotubes could play a pivotal role in the upcoming
revolution of conventional silicon microelectronics. [21, 22] Controllable manufacture
carbon nanotube-based circuits is one of the most key step. Fortunately, it is coming
into reality step by step. In 2000, the carbon nanotube RLC circuits were reported. [23]
But it was not controllable but random. The breakthrough came in 2001. Several
kinds of carbon nanotube-based circuits in resistor-transistor logic (RTL) style [24]
and carbon nanotube based complementary inverter [25] were fabricated one after the
other. In 2002, kinds of carbon nanotube based circuits in complementary logic style
were fabricated. [26]
Voltage transport characteristics and schematics of carbon nanotube-based inverter,
NOR gate, SRAM, and ring oscillator in RTL style were shown in Figure 4.2. These
RTL carbon nanotube-based circuits show favorable characteristics. [24] Though just
in RTL style, the realization of digital logic with CNTFET circuits represents an
important step toward carbon nanotube-based nanoelectronics.
As we know, RTL is not a good logic style for integrated circuits. We need
complementary logic. But intrinsic CNTFETs are all p-type. The complementary
circuits hadn’t come into being until n-type CNTFETs were fabricated. After lots of
experiments, it was found that n-type CNTFETs could be made by vacuum
annealing or doping [25].
Schematics of carbon nanotube-based inter- and intramolecular complementary
inverters were shown in Figure 4.3 (a) and (c), respectively. And voltage transport
characteristics of carbon nanotube-based inter- and intramolecular complementary
inverters were shown in Figure 4.3 (b) and (d), respectively. They have good output
characteristics, and a gain larger than one is possible. [25]
37
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Four
Figucomp
FiguAND
a
c
re 4.3 Schematics and voltage transport characlementary inverters. [25]
re 4.4 Schematics and voltage transport chara gates in complementary logic style. [26]
38
b
teris
cter
d
tics of CNT-based inter- and intra-molecular
istics of CNT-based NOR, OR, NAND and
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Four
Schematics and voltage transport characteristics of carbon nanotube-based NOR,
OR, NAND and AND gates in complementary logic style were shown in Figure 4.4,
respectively. The CNTFET arrays in these circuits are assembled via a patterned
chemical synthesis approach, and from the synthesis, simple computing operations
are made possible by using the high percentage of semiconducting SWNT-FETs. [26]
Memories are also the key elements in electronics. Different kinds of carbon
nanotube-based memories have been fabricated in recent years. [27-29]
4.4 Prospects
Lots of progress has been made in CNT-based electronics till now, especially the
complementary logic circuits based on CNTs were fabricated. But many tasks and
challenges sill lie ahead of real application of CNT-based circuits, including
enabling device functions in non-vacuum conditions, obtaining massive and
high-density transistor arrays, and interconnecting carbon nanotubes on-chip. And
another key problem is how to separate semiconducting CNTs from metallic CNTs
more effectively. After all these problems are solved, a magnificent system of
electronics based on CNTs will come into being.
In Chapter Four, part of Section 4.1 & 4.2 was published as a review article on Res. & Prog. of Solid State Electronics in Chinese, and part of Section 4.3 & 4.4 were published as a review article on Physics in Chinese.
39
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Four
References
[1] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Tunneling conductance of
connected carbon nanotubes, Phys. Rev. B 53, 2044-2050 (1996).
[2] L. Chico, V. H. Crespi, L. X. Benedict, S. G. Louie, and M. L. Cohen, Pure
carbon nanoscale devices: nanotube heterojunctions, Phys. Rev. Lett. 76,
971-974 (1996)
[3] A. Rochefort and Ph. Avouris, Quantum size effects in carbon nanotube
intramolecular junctions, Nano Letters 2, 253-256 (2002).
[4] Z. Yao, H. W. Ch. Postma, L. Balents, and C. Dekker, Carbon nanotube
intramolecular junctions, Nature 402, 273-276 (1999).
[5] M. Ouyang, J. L. Huang, C. L. Cheung, and C. M. Lieber, Atomically resolved
single-walled carbon nanotube intramolecular junctions, Science 291, 97-100
(2001).
[6] i) J. Li, C. Papadopoulos, and J. Xu, Growing Y-junction carbon nanotubes,
Nature 402, 253-254 (1999); ii) C. Papadopoulos, A. Rakitin, J. Li, A. S.
Vedeneev, and J. M. Xu, Electronic transport in Y-junction carbon nanotubes,
Phys. Rev. Lett. 85, 3476-3479 (2000).
[7] A. N. Andriotis, M. Menon, D. Srivastava, and L. Chernozatonskii,
Rectification properties of carbon nanotube “Y-junctions”, Phys. Rev. Lett. 87,
066802 (2001).
[8] M. S. Fuhrer, J. Nygård, L. Shih, M. Forero, Y. G. Yoon, M. S. C. Mazzoni, H. J.
Choi, J. Ihm, S. G. Louie, A. Zettl, P. L. McEuen, Crossed Nanotube Junctions,
Science 288, 494-496 (2000).
[9] F. Banhart, The formation of a connection between carbon nanotubes in an
electron beam, Nano Letters 1, 329-332 (2001).
[10] i) H. W. Ch. Postma, M. de Jonge, Z. Yao, and C. Dekker, Electrical transport
through carbon nanotube junctions created by mechanical manipulation, Phys.
Rev. B 62, R10653-R10656 (2000); ii) A. Nojeh, G. W. Lakatos, S. Peng, K.
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Four
Cho, and R. F. W. Pease, A carbon nanotube cross structure as a nanoscale
quantum device, Nano Letters 3, 1187-1190 (2003).
[11] C. Zhou, J. Kong, E. Yenilmez, and H. Dai, Modulated chemical doping of
individual carbon nanotubes, Science 290, 1552-1555 (2000).
[12] F. Léonard and J. Tersoff, Role of Fermi-level pinning in nanotube Schottky
diodes, Phys. Rev. Lett. 84, 4693-4696 (2000).
[13] P. L. McEuen, Carbon-based electronics, Nature 393, 15-17 (1998).
[14] S. J. Tans, A. R. M. Verschueren, and C. Dekker, Room-temperature transistor
based on a single carbon nanotube, Nature 393, 49-51 (1998).
[15] R. Martel, T. Schmidt, H. R. Shea, T. Hertel, and Ph. Avouris, Single- and
multi-wall carbon nanotube field-effect transistors, Appl. Phys. Lett. 73,
2447-2449 (1998).
[16] M. Krüger, M. R. Buitelaar, T. Nussbaumer, C. Schönenberger, and L. Forró,
Electrochemicall carbon nanotube field-effect transistor, Appl. Phys. Lett. 78,
1291-1293 (2001).
[17] S. Rosenblatt, Y. Yaish, J. Park, J. Gore, V. Sazonova, and P. L. McEuen, High
performance electrolyte gated carbon nanotube transistors, Nano Letters 2,
869-872 (2002).
[18] A. Javey, H. Kim, M. Brink, Q. Wang, A. Ural, J. Guo, P. Mcintyre, P. McEuen,
M. Lundstrom, and H. Dai, High-κ dielectrics for advanced carbon-nanotube
transistors and logic gates, Nature Materials 1, 241-246 (2002).
[19] A. Javey, J. Guo, Q. Wang, M. Lundstrom, and H. Dai, Ballistic carbon
nanotube field-effect transistors, Nature 424, 654-657 (2003).
[20] H. W. Ch. Postma, T. Teepen, Z. Yao, M. Grifoni, and C. Dekker, Carbon
nanotube single-electron transistor at room temperature, Science 278, 77-78
(2001).
[21] S. Satio, Carbon nanotube for next-generation electronics devices, Science
293, 76-79 (1997).
[22] P. L. McEuen, Carbon-based electronics, Nature 393, 15-17 (1998).
[23] N. A. Prokudina, E. R. Shishchenko, O. Joo, D. Y. Kim, and S. H. Han, Carbon
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Four
nanotube RLC circuits, Adv. Mater. 12, 1444-1447 (2000).
[24] A. Bachtold, P. Hadley, T. Nakanishi, and C. Dekker, Logic circuits with
carbon nanotube transistors, Science 294, 1317-1320 (2001).
[25] V. Derycke, R. Martel, J. Appenzeller, and Ph. Avouris, Carbon nanotube
inter- and intramolecular logic gates, Nano Letters 1, 453-456 (2001).
[26] A. Javey, Q. Wang, A. Ural, Y. Li, and H. Dai, Carbon nanotube transistor
arrays for multistage complementary logic and ring oscillators, Nano Letters 2,
929-932 (2002).
[27] T. Rueches, K. Kim, E. Joselevich, G. Y. Tseng, C. L. Cheung, and C. M.
Lieber, Carbon nanotube-based nonvolatile random access memory for
molecular computing, Science 289, 94-97 (2000).
[28] M. S. Fuhrer, B. M. Kim, T. Durkop, and T. Brintlinger, High-mobility
nanotube transistor memory, Nano Letters 2, 755-759 (2002).
[29] M. Radosavljević, M. Freitag, K. V. Thadani, and A. T. Johnson, Nonvolatile
molecular memory elements based on ambipolar nanotube field effect
transistors, Nano Letters 2, 761-764 (2002).
And some review articles on “carbon nanotube based electronics”,
[30] Z. Yang, Y. Shi, S. L. Gu, B. Shen, R. Zhang, and Y. D. Zheng, Carbon
nanotube-based electronic devices, Res. & Prog. of Solid State Electronics, 22,
131-136 (2002). (In Chinese)
[31] Z. Yang, Y. Shi, S. L. Gu, B. Shen, R. Zhang, and Y. D. Zheng, Carbon
nanotube for electronics, Physics, 31, 624-628 (2002). (In Chinese)
[32] T. W. Ebbesen, Carbon nanotubes, Physics Today, 26-32 (June, 1996).
[33] C. Dekker, Carbon nanotubes as molecular quantum wires, Physics Today,
22-28 (May, 1999).
[34] Special Issue (on Carbon Nanotubes), Physics World, 22-53 (June, 2000).
[35] P. G. Collins and Ph. Avouris, Nanotubes for electronics, Scientific American,
62-69 (December, 2000).
[36] R. H. Baughman, A. A. Zakhidov, and Walt A. de Heer, Carbon
nanotubes—the route toward applications, Science 297, 787-792 (2002).
42
Part II
Optical Spectra of Self-Assembled Ge/Si Quantum Dot Superlattices
(Ge/Si QDSLs)
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
Chapter Five
Self-Assembled Ge/Si QDSLs
Self-assembled Ge quantum dots (QDs) grown by molecular beam epitaxy (MBE)
and chemical vapor deposition (CVD) have attracted a great deal of interest for
several years. Ge quantum dots may be used for lots of applications, such as
mid-infrared photodetectors, lasers, resonant tunneling diodes, thermoelectric cooler,
cellular automata, and quantum computer, etc. In the following chapters, the
self-assembled Ge quantum dots on Si grown by MBE in Stranski-Krastanow mode
–self-assembled Ge/Si quantum dot superlattices (QDSLs)—will be focused on. In
this chapter, the growth and applications of Ge quantum dots will be introduced and
discussed firstly, then the detail information about the samples used in the following
experiments will be presented. Finally, the experiments carried on the samples in the
following chapters will be mentioned briefly.
5.1 Growth of Ge Quantum Dots
5.1.1 Molecular Beam Epitaxy
Molecular beam epitaxy (MBE) was developed in the early 1970s as a means of
growing high-purity epitaxial layers of compound semiconductors [1, 2]. Since that
time it has evolved into a popular technique for growing III-V compound
semiconductors as well as several other materials. MBE can produce high-quality
layers with very abrupt interfaces and good control of thickness, doping, and
composition. Because of the high degree of control possible with MBE, it is a
valuable tool in the development of sophisticated electronic and optoelectronic
materials and devices.
43
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
In MBE, the constituent elements of a semiconductor in the form of ‘molecular
beams’ are deposited onto a heated crystalline substrate to form thin epitaxial layers.
The ‘molecular beams’ are typically from thermally evaporated elemental sources,
but other sources include metal-organic group III precursors (MOMBE), gaseous
group V hydride or organic precursors (gas-source MBE), or some combination
(chemical beam epitaxy or CBE). To obtain high-purity layers, it is critical that the
material sources are extremely pure and that the entire process is done in an
ultra-high vacuum environment. Another important feature is that growth rates are
typically on the order of a few angstroms per second and the beams can be shuttered
in a fraction of a second, allowing for nearly atomically abrupt transitions from one
material to another.
The nuclear part of an MBE system is the growth chamber. The growth chamber of
a generic MBE system and several of its subsystems are illustrated in Figure 5.1.
Figure 5.1 Diagram of a typical MBE system growth chamber.
44
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
Samples are loaded onto the growth chamber sample holder/heater via a
magnetically coupled transfer rod. The sample holder rotates on two axes, which are
drawn in Figure 5.1. Before growth, the sample holder is flipped around from the
loading position so that the sample faces the material sources. For improved layer
uniformity, the sample holder is designed for continual azimuthal rotation (CAR) of
the sample. The CAR also has an ion gauge mounted on the side opposite the sample
which can read the chamber pressure, or be placed facing the sources to measure
beam equivalent pressure (BEP) of the material sources. A liquid nitrogen cooled
cryoshroud is located between the chamber walls and the CAR and acts as an
effective pump for many of the residual gasses in the chamber. The substrate holder
and all other parts that are heated are made of materials such as Ta, Mo, and
pyrolytic boron nitride (PBN) which do not decompose or outgas impurities even
when heated to 1400ºC. To monitor the residual gasses, analyze the source beams,
and check for leaks, a quadrupole mass spectrometer (QMS) is mounted in the
vicinity of the CAR. The material sources, or effusions cells, are independently
heated until the desired material flux is achieved. Changes in the temperature of a
cell as small as 0.5ºC can lead to flux changes on the order of one percent. Computer
controlled shutters are positioned in front of each of the effusion cells to be able to
shutter the flux reaching the sample within a fraction of a second.
One of the most useful tools for in-situ monitoring of the growth is reflection
high-energy electron diffraction (RHEED). It can be used to calibrate growth rates,
observe removal of oxides from the surface, calibrate the substrate temperature,
monitor the arrangement of the surface atoms, determine the proper overpressure,
give feedback on surface morphology, and provide information about growth
kinetics. The RHEED gun emits ~10KeV electrons which strike the surface at a
shallow angle (~0.5-2 degrees), making it a sensitive probe of the semiconductor
surface. Electrons reflect from the surface and strike a phosphor screen forming a
pattern consisting of a specular reflection and a diffraction pattern which is
indicative of the surface crystallography. A camera monitors the screen and can
45
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
record instantaneous pictures or measure the intensity of a given pixel as a function
of time.
5.1.2 Stranski-Krnstanow Growth Mode
There are three known modes of heteroepitaxial growth: Frank-van der Merwe
(F-vdM)[3], Volmer-Weber (V-W)[4], and Stranski-Krastanow (S-K) [5]; these may be
loosely described as layer-by-layer (2D), island growth (3D), and layer-by-layer plus
islands (2D followed by 3D). The schematic diagrams of the three epitaxial growth
modes are shown in Figure 5.2.
F-vdM V-W S-K
Figure 5.2 Schematics of the three epitaxial growth modes.
F-vdM growth mode, which is simply the successive addition of 2D layers to the
substrate crystal, occurs when the epilayer and substrate have matched lattice
constants. It is the most widely used epitaxial growth process in semiconductor
device production. For highly mismatched lattice constants between epitaxial
material and substrate, V-W growth mode occurs. In V-W, the epitaxial material
minimizes its free energy by trading increased surface area for decreased interface
area, forming an island structure like water droplets on glass directly on an unwetted
substrate surface.
46
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
S-K growth mode may occur when the epitaxial material is slightly mismatched to
the substrate. In slightly mismatched systems, the epitaxial film will be strained, so
that its in-plane lattice constant fits the lattice constant of the substrate. Growth will
continue pseudomorphically until the accumulated elastic strain energy is high
enough to form dislocations. The thickness at which dislocations are formed is
essentially determined by the extent of lattice mismatch. The partial strain relaxation
for systems with epitaxial lattice misfit can also take place through reorganization of
the epilayer material. In S-K, the growth of a pseudomorphic 2D layer is followed
by reorganization of the surface material in which 3D island are formed. Most of the
material will accumulate in the islands, and only a thin wetting layer will remain of
2D growth.
It was found that Ge dots could be self-assembly grown on Si substrate[6], which will
be discussed in the next subsection.
5.1.3 Self-Assembled Growth of Ge Quantum Dots on Si
Self-assembled Ge quantum dots can be grown by MBE in the S-K growth mode.
The lattice constants of Ge and Si are 5.66 and 5.43 Å, respectively. This is the case
of slightly mismatch, which is about 4%.
Figure 5.3 Schematic of the lattice mismatch between Ge and Si and the strain layer.
47
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
Because of the ~ 4% lattice mismatch between Ge and Si, Ge grown on Si substrate
grows in a layer-by-layer mode for only several layers (generally 3~6 monolayers),
which form the wetting layers, then 3D islands form.
For practical applications it is very important that the islands formed have the same
size. Luckily enough, The S-K growth mode seems to have some self-limiting
process, which results in a rather uniform size and shape distribution of the islands.
On planar Si, the uniformity of the dots has also been found to depend critically on
the growth parameters, such as growth temperature, growth rates, Ge deposited
coverage and holding time at the growth temperature after Ge deposition.
5.2 Applications of Ge Quantum Dots [7, 8]
The practical value of Ge quantum dots is in their utility for various applications.
There are mainly four aspects to the interest in self-assembled Ge/Si quantum dots.
First, like all other quantum dots, the electron confinement within quantum dot
yields interesting electronic properties. The allowed electron energy levels are
different than in bulk Ge or Si. Potential applications include diodes, lasers, and
photodetectors with novel properties such as higher efficiency, lower threshold, or
useful frequencies of operation.
Second, self-assembly is a good alternative to conventional methods of producing
micro- and nanoelectronic structures. Typically, electronic devices are created by
photolithography, a difficult and expensive process. Self-assembly eliminates the
need for photolithography in making quantum structures. As reducing the feature
sizes of microelectronics is a continual goal, self-assembled structures may also
benefit by being smaller than the limiting size of photolithograph.
48
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
Third, SiGe materials are easily integrated with the predominantly Si-based
microelectronics industry—unlike the growth of other semiconductor materials such
as the Group III-V compound GaAs—Ge has the advantage of being a Group IV
material, allowing it to directly substitute for Si in crystal structures.
A fourth aspect of interest in SiGe quantum dots also involves the compatibility of
Ge with Si. Device makers would like to design structures with flat, relatively thick
Ge-rich layers.
Electronic Applications
The indirect band gap and the heavy masses of electrons do not make Si an ideal
candidate for fabrication of resonant tunneling diodes. Ge quantum dots could be
used to fabricate improved tunneling diodes with a reduced valley current density
because of the delta density of states. And Ge quantum dot can be used to realize
novel cellular automata, a class of device/circuit which may minimize the problems
of interconnect in today’s CMOS circuits.
Optoelectronic Applications
Multi-layered Ge quantum dots can be used to fabricate novel quantm dot
photodectors operating in the mid-infrared range. Efficient photo-emitters and
perhaps Ge lasers might be even possible using Ge quantum dots. Multi-layered Ge
dot superlattices may be used as a gain media in which interband transitions in
indirect semiconductors like Si and Ge are assisted by phonons to make up the
momentum difference between the initial and final states.
Thermoelectric Applications
Thermoelectric materials with high figure of merit can enable novel thermoelectric
devices with efficient solid-state refrigeration and power conversion. Ge quantum
dot structure may have great potential in thermoelectric applications. It was found
49
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
that quantum dots might effectively confine phonons or could even strongly scatter
phonons, which result further in their thermal conductivity reduction. Meanwhile,
Ge quantum dots have delta density of states and quantum confinement of electrons
(holes) could also add up to the additional improvements of the thermoelectric
power factor the figure of merit. Thus, Ge quantum dot may represent a fine
example of the “phonon-blocking & electron-transmitting” structures.
Quantum Information Applications
Ge quantum dots also have potential applications in quantum computer, just like
some other quantum dots. The basic engineering prerequisites for a successful
implementation of quantum computation device are: creation of the quantum bit
(qubit) with decoherence time significantly longer than computation cycle, unitary
rotations of the qubits and ability to control interaction between qubits. These may
be all achieved in the future on Ge quantum dots.
5.3 Self-Assembled Ge/Si QDSLs
Eleven samples, labeled A to J, were grown by solid-source molecular beam epitaxy
on Si (100) substrate with Stranski-Kranstanov growth mode. All of these samples
consisted of 100nm Si buffer layers, followed by several period bilayers, in which
Ge dot layers are separated by 20-nm-thick Si spacer layers. And The Ge quantum
dots in the samples are vertically correlated and no cap Si layer is used for all these
samples. The differences among these samples are the Ge layer thickness, the
number of the periods, and the growth temperature. Samples 136, 138, and 137 were
grown at 600 °C with 22 periods of Ge and Si bilayers, and the Ge coverages of
these samples were 6, 12, and 15 Å, respectively. Samples 210, 226, and 187 were
grown at 540 °C with 10 periods, and the Ge coverages of these samples were 12, 15,
and 18 Å, respectively. Samples 269, 264, 265, 266, and 183 were grown at 540 °C
with the same Ge layer thickness of 15 Å, and consisted of 2, 5, 20, 35, and 50
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
periods, respectively. The growth parameters and structural data of these samples are
summarized in Table 5.1.
Table 5.1 Growth parameters and structural data of the samples.
Sample Ge layer thickness (nm)
Si layer thickness (nm)
Growth Temperature (°C) Periods
136 0.6 20 600 22
138 1.2 20 600 22
137 1.5 20 600 22
210 1.2 20 540 10
226 1.5 20 540 10
187 1.8 20 540 10
269 1.5 20 540 2
264 1.5 20 540 5
265 1.5 20 540 20
266 1.5 20 540 35
183 1.5 20 540 50
Figure 5.4 (a) shows a two-dimensional (2D) atomic force microscope (AFM) image
of a typical highly uniform self-assembled Ge quantum dots sample (not listed in
table 5.1) grown at 600 °C. Figure 5.4 (b) is the three-dimensional (3D) AFM image
of the same sample. The Ge layer nominal thickness is 15 Å. The dots are all
dome-shaped with the base size and the height of about 70 nm and 15 nm,
respectively. The areal density of the dots is about 3 × 108 cm-2 and the height
deviation of the dots is about . 3%±
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
a
0.60.40.20 µm
b
Figure 5.4 The (a) 2D and (b) 3D AFM images of a typical uniform self-assembled Ge quantum dots sample at the growth temperature of 600 °C. The Ge thickness is about 1.5nm. The base size and the height of the dots are about 70 and 15 nm, respectively.
Figure 5.5 A typical cross-sectional TEM image of a 10-period self-assembled Ge quantum dot superlattices sample grown at 540 °C. The thickness of Ge and Si spacer layer are 1.2nm and 20 nm, respectively. Vertical correlation is clearly seen.
52
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
Figure 5.5 shows a cross-sectional transmission electron microscope (TEM) image
the following two chapters, it will be focused on the optical properties of the
of a 10-period self-assembled Ge QD SL sample (sample 210 in Table 5.1) grown at
540 °C. The Ge layer nominal thickness is 1.2 nm and the Si spacer layer thickness
is 20 nm. Vertically correlated islands are evident. The origin of vertical correlation
is attributed to preferential nucleation due to the inhomogeneous strain field induced
by buried dots.
In
samples presented above. The experimental methods are Raman scattering and
photoluminescence measurements. From spectra of the experiments, lots of
information and conclusions about the samples can be obtained and drawn, which is
the main contents of the Chapter 6 and 7.
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Five
References
[1] A. Cho, Film Deposition by Molecular Beam Techniques, J. Vac. Sci. Tech. 8,
S31-S38 (1971).
[2] A. Cho and J. Arthur, Molecular Beam Epitaxy, Porg. Solid-State Chem. 10,
157-192 (1975).
[3] F. C. Frank and J. H. van der Merwe, Proc. Roy. Soc. London A 198, 205
(1949).
[4] M. Volmer and A. Weber, Z. Phys. Chem. 119, 277 (1926).
[5] I. N. Stranski and Von L. Krastanow, Akad. Wiss. Lit. Mainz Math. –Natur.
K1. IIb 146, 797 (1939).
[6] i) D. J. Eaglesham and M. Cerullo, Dislocation-Free Stranski-Krastanow
Growth of Ge on Si (100), Phys. Rev. Lett. 64, 1943-1946 (1990); ii) Y. W. Mo,
D. E. Savage, B. S. Swartzentruber, M. G. Lagally, Kinetic Pathway in
Stranski-Krastanow Growth of Ge on Si (001), Phys. Rev. Lett. 65, 1020-1023
(1990).
[7] P. Schittenhelm, C. Engel, Findeis, G. Abstreiter, A. A Darhuber, G. Bauer, A. O.
Kosogov, and P. Werner, Self-assembled Ge dots: Growth, characterization,
ordering and applications, J. Vac. Sci. Tech. B 16, 1575-1580 (1998).
[8] K. L. Wang, J. L. Liu, and G. Jin, Self-assembled Ge quantum dots on Si and
their applications, J. Crystal Growth 237-239, 1892-1897 (2002).
54
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
Chapter Six
Raman Scattering in Ge/Si QDSLs
To study the phonon mechanisms in Ge quantum dots is indispensable since
phonons are very important for heat transport and luminescence in indirect band gap
semiconductors. And it is useful for fabricating novel devices in the future. In this
chapter, we report the study of optical phonon modes and their relations to phonon
confinement and strain in Ge quantum dot superlattices, and the properties of
acoustic phonon mode by Raman scattering measurements. The effects of the
phonon confinement and the strain within the Ge dots can induce the red- and
blue-shift of the optical phonon modes. From analyses of the frequency shift, it is
found that strain relaxations in Ge quantum dot superlattices are not only from Ge/Si
interdiffusion but also from other reasons such as dot morphology transition. The
low-frequency Raman peaks from the folded acoustic phonons was first time found
in non-resonant mode. The relation between the intensity of the Raman peaks and
periods of the samples is discussed.
6.0 Basic Concepts of Raman Spectroscopy
Raman Scattering Effect
A light scattering process is an interaction of a primary light quantum with atoms,
molecules, or some other elementary excitations in crystals, solids or other matters,
by which a secondary light quantum is produced, with a different phase and
polarization and maybe another energy when compared to that of the primary light
quantum. The scattering process occurs with an extremely short time delay.
An elastic scattering process produces radiation with the same energy as that of the
primary light, such as Rayleigh, Mie or Tyndall scattering. In the elastic scattering,
55
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
the incident and scattered light have the same frequency. An inelastic scattering
process produces secondary light quanta with different energy. During the
interaction of the primary light quantum with the elementary excitations in the
matter, the energy of the elementary excitations may be exchanged and a secondary
light quantum of lower or higher energy is emitted. The energy difference is equal to
the energy of the elementary excitations. These elementary excitations can be
phonons, magnons, plasmons, etc. Raman scattering is the inelastic light scattering.
When light is scattered by any form of matter, the energies of the majority of the
photons are unchanged by the process, which is elastic (i.e. Rayleigh scattering peak
in the scattered light spectrum). However, about one in one million photons or less,
loose or gain energy that corresponds to eigen-energy of the elementary excitation in
the scattering matters. This can be observed as additional peaks in the scattered light
spectrum. The spectral peaks with lower and higher energy than the incident light
are known as Stokes and anti-Stokes peaks respectively. Most routine Raman
experiments use the red-shifted Stokes peaks only, because they are more intense at
room temperatures.
Explanations of Raman Scattering
Raman effect can be explained by classical and quantum theory, both of which will
be discussed in this subsection. In classical theory, the atoms in the crystal are
polarized by the electromagnetic field of the incident light. The polarization vector
Pv
is proportional to the electric field Ev
, and the coefficient α called
polarizability,
EPvv
⋅=α
where α is modulated by the vibration of the crystal lattice,
)cos(000 rq vv ⋅−∆+=∆+= tqωααααα
Assuming the frequency and wave vector of the incident light are ω and kv
,
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
respectively,
)cos( rkEE vvvv⋅−= to ω .
Thus,
( ) ( )( ) ({ }rqkrqkE
rkErkErqPvvvvvvv )
vvvvvvvvv
⋅−−−+⋅+−+∆+
⋅−=⋅−⋅⋅−∆+=
)()(cos)()(cos21
)cos()cos()cos(
0
000
tt
ttt
qqo
ooq
ωωωωα
ωαωωαα.
The first term in the equation above describes Rayleigh scattering (elastic); the
second term Stokes Raman scattering, the frequency of the scattering light decreases
to )( qωω − ; the third term anti-Stokes Raman scattering, the frequency of the
scattering light increases to )( qωω + .
Figure 6.1 shows the energy level diagrams for Rayleigh, Stokes, and Anti-Stokes
scattering, which are the basic explanation of Raman scattering.
Lωh Sωh Lωh Sωh
qωh
Lωh Sωh
qωhLS ωω = qLS ωωω −= qLS ωωω +=
Anti-Stokes Stokes Rayleigh
Figure 6.1 Energy level diagrams for Rayleigh, Stokes, and Anti-Stokes scattering.
The frequency of the incident light is denoted as Lω while that of the scattering
light is Sω After the incident photons (light quanta) are absorbed by the crystal, the
electrons in the crystal are excited from an initial state to a virtual state. Then these
electrons in the virtual state transit to the final state, radiating scattering photons.
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
6.1 Raman Spectra of Ge/Si QDSLs
Raman spectrum of a typical self-assembled Ge QDSLs sample is shown in Figure
6.2. The optical and acoustic phonon modes can be investigated through the high-
and low-frequency regime of the Raman spectrum, respectively. The Si-Si, Ge-Ge
and Si-Ge optical phonon modes are found in the high-frequency region of Raman
spectra, which are arisen from the Si substrate, the optical phonon mode of Ge, and
the SiGe alloy in the wetting layers, respectively. The effects of the phonon
confinement and strain within the Ge dots can induce the red- and blue-shift of the
optical phonon modes. Strain relaxations in Ge QDSLs are not only from Ge/Si
interdiffusion but also from other reasons such as dot morphology transition. The
low-frequency Raman peaks are arisen from the folded acoustic phonons. The two
regions in the Raman spectra, i.e. the optical and acoustic phonons will be discussed
in the following two sections.
0 10 20 30 40 250 300 350 400 450 500 550
Si-Si
Si-Ge
Optical Mode
Ram
an In
tens
ity (a
.u.)
Raman Shift (cm-1)
Acoustic Mode
Ge-Ge
Figure 6.2 Raman spectrum of a typical self-assembled Ge quantum dot superlattices sample.
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
6.2 Optical Phonons in Ge/Si QDSLs
Self-assembled Ge quantum dots (QDs) have attracted growing interest because they
show optical and electronic properties much different from bulk solid-state materials.
Both fundamental physical properties and potential applications in novel devices of
Ge QDs have been widely investigated. To study the phonon mechanism in Ge QDs
is indispensable since phonons are very important for luminescence in indirect band
gap semiconductors. In this section, we report on the investigation of optical phonon
mode and its relation to the phonon confinement and strain in Ge quantum dot
superlattices by Raman scattering measurements. Optical phonon confinement
effects have been extensively investigated in two-dimensional (2D) semiconductor
heterostructures and one-dimensional (1D) systems. Owing to the difficulties of
experimental observation caused by fluctuations of size, shape, and orientation in
zero-dimensional (0D) systems, however, the confinement effects are not so much to
be investigated. So far, only a few relative works have been reported on Si [2] and
Ge[3]. In the present experiments, we find that the Ge-Ge peak in the Raman
spectra are shifted slightly to their bulk value 300cm-1, which is deduced that it is
mainly attributed to the lateral compressive strain and phonon confinement in the Ge
QDs. The main purpose of this letter is to understand the characteristics of the strain
and the optical phonon confinement in self-assembled Ge QDSLs. The
experimentally observed optical phonon frequency shift values are analyzed with
quantitative calculations.
6.2.1 Experimental Results
Five samples used in this work, labeled A, B, C, D, and E, were grown by
solid-source molecular beam epitaxy (MBE) system with S-K growth mode.
Samples A and B were both grown at 600 °C with 22 periods of Ge and Si bilayers,
and the Ge coverages of the two samples were 15 and 6 Å, respectively. For sample
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
B, it is only with Ge strain layer. Samples C, D, and E were all grown at 540 °C with
10 periods of Ge and Si bilayers, and the Ge coverages of these samples were 12, 15,
and 18 Å, respectively. The Si layer thickness of 20 nm was used for all samples.
The structural data of these samples are summarized in Table 6.1.
Table 6.1 Structural data of samples used in optical phonon Raman measurements.
Sample No. Ge layer thickness (nm)
Si layer thickness (nm)
Growth temperature (°C) Periods
A 137 1.5 20 600 22 B 136 0.6 20 600 22 C 210 1.2 20 540 10 D 226 1.5 20 540 10 E 187 1.8 20 540 10
(e)
0.25µm
(d)
(c) (a)
0.5µm 0.5µm 0.5µm
(b)
Figure 6.3 The 2D AFM images of (a) samples A, (b) sample C, (c) sample E, and (d) sample D; and (e) the 3D AFM image on the same spot of sample D.
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
Figure 6.3 (a), (b), (c), and (d) show the 2D AFM images of samples A, C, E, and D;
and (e) the 3D AFM image on the same spot of sample D. The average dot height
and base of sample D are determined to be 10 and 90 nm, respectively.
Raman scattering measurements were performed using a JY T64000 Raman system
at room temperatures. All spectra were excited by the 488 nm line of an Ar ion laser
in the backscattering, the spectral resolution is about 0.5 cm−1.
250 300 350 400 450 500 550 600
Sample B
Si-SiLOC
Si-Si
Si-Ge
Ram
an In
tens
ity (a
.u.)
Raman Shift (cm-1)
Ge-Ge
Sample A
Figure 6.4 Raman spectra of the samples A and B. Sample A includes 22 periods of Ge dots, while sample B just with Ge wetting layers.
Figure 6.4 shows the Raman spectra of samples A and B. In the curve of sample A,
besides the strong Si substrate signal at 520cm−1, Ge-Ge, Si-Ge, and local Si-Si
(Si-SiLOC) vibrational peaks can be seen at 299, 417, and 436cm−1, respectively. The
appearance of the Si-Ge and Si-SiLOC vibrational peaks implies the formation of
SiGe alloy in the wetting layers and the existence of strain in Si underneath the dots.
The Ge-Ge peak arises from the Ge QDs, which stands for the optical mode of the
dots. For sample B, the Ge-Ge mode is much weaker. The difference of the Ge-Ge
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
modes between the two samples suggests that the Ge-Ge mode is mainly from the
Ge QDs rather than the wetting layers.
Figure 6.5 shows the Raman spectra of samples C, D, and E in the spectral region of
Si-Ge and Ge-Ge optical phonons. Similar Si-Ge and Ge-Ge lines are observed for
these samples. The vertical dotted line is fixed at 0ω =300cm−1, which stands for the
optical phonon position for bulk crystalline Ge. The frequency positions of the
Ge-Ge optical phonons in dot samples are at about 300cm−1, the exactly values for
samples C, D, and E are 299, 298, and 297 cm−1, respectively.
250 300 350 400 450
Sample E
Sample C
Ram
an In
tens
ity (a
.u.)
Raman Shift (cm-1)
297299
298
Sample D
Figure 6.5 Raman spectra of the samples C, D, and E. The frequency positions of the Ge-Ge peak in the samples are shifted slightly to their bulk value (300cm-1, the vertical dotted line).
6.2.2 Discussion
There are mainly two possible physical origins, in principle, which can cause a
Raman shift of optical phonons. The first one is resulted from the strain: the lattice
mismatch of Si and Ge leads to a compressive strain on the dots in the lateral
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
directions, which induces a Ge-Ge mode shift to the higher frequency side (blue-
shift). The second reason is the phonon confinement from the spatial limitations of
superlattices, causing a shift of optical phonons towards lower frequency (red-shift).
The blue-shift of Ge optical mode due to the strain within the Ge QDSLs can be
expressed as
xxstrain pCC
q εω
ω
−=∆
11
12
0
1 (6.1)
where 0ω is the frequency of the Ge zone-center LO phonon; p and q the Ge
deformation potentials; C11, and C12 elastic coefficients; xxε the biaxial strain.
Here, 0ω =0.565×1014s−1, p=−4.7×1027s−2, q=−6.167×1027s−2, C11=1288 kbar,
C12=482.5 kbar[4]. For fully strained pure Ge on Si, 042.0) −=( −= SiGexx aaSiaε
with and the lattice constants of Si and Ge, it obtains .
But as a matter of fact the dots in the samples are not fully strained due to the strain
relaxation from the atomic intermixing at the Si/Ge interface. So
Sia Gea 1cm4.17 −=∆ strainω
xxε should be
written as xxGeSiGe aaa
−−
1)
SiGe ax)
xxGeSi −1
1(
( , where is the lattice parameter of the
SiGe alloy with the Ge concentration x and can be determined by Vegard’s law
xxGeSia−1
GeSiax1
xax
−+=−
. And the Ge concentration x can be calculated from the
relative intensity of Ge-Ge and Si-Ge optical mode peaks in Raman spectra. For
samples C, D, and E, x equal 0.5 and . (Please see Appendix D) 1cm−∆ω 8=strain
In the present experiments, the Ge optical modes were found at about 300cm−1 in the
Raman spectra. It is the phonon confinement that causes a red-shift of Ge optical
mode. From the simple linear chain model, it can be easily understood why the
red-shift come into being: the spatial limitations of superlattices (the height of the
QDs) cause a shift of confined optical phonons towards lower frequency. From the
Richter, Wang, and Ley model (RWLM)[2, 5], which was once successfully applied to
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
study the phonon confinement in Si, the frequency red-shift due to phonon
confinement can be estimated approximately. Using RWLM, the frequency red-shift
can be expressed as γ
ω
−=∆ − d
aA Ge
conph (6.2)
where d is the height of the Ge QDs, Ge lattice constant, and A and γ the
constant parameters with A=52.3 cm
Gea
−1 and γ=1.586, respectively. Employing the
average dot height 10 nm into Equation (2), we obtain that red-shift of the samples
equal about 1 cm−1. Compared with the blue-shift caused by strain, this value is a
minor one, i.e. the calculated value of Ge optical phonon frequency is still larger
than the experimental one. So it suggests that there are some additional strain
relaxation mechanisms besides Ge/Si interdiffusion, such as strain relaxation from
dot morphology transition. For example, the more Ge QDs transform from the
pyramid shape to the dome shape, the more strain relaxation relaxes. [4]
6.2.3 Conclusions
In summary, we have reported on the investigation of Raman scattering in the
self-assembled Ge quantum dot superlattices, the Si-Si, Ge-Ge, Si-Ge and Si-SiLOC
peaks were found in the Raman spectra, which were arisen from the Si substrate, the
optical phonon modes of Ge, the SiGe alloy in the wetting layers and the existence
of strain in Si underneath the QDs, respectively. The effects of the phonon
confinement and strain within the Ge QDs can induce the red-shift and blued-shift of
the optical phonon modes. Strain relaxation in Ge QDs superlattices is not only from
Ge/Si interdiffusion but also from other reasons such as dot morphology transition.
6.3 Acoustic Phonons in Ge/Si QDSLs
In recent years, a great deal of attention has been paid to the phonons and electrons
64
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
in self-assembled Ge quantum dots (QDs) using optical spectroscopy techniques,
such as Raman scattering and photoluminescence. To study the phonon and electron
transport in such low-dimensional structures is of great interest for both fundamental
physics and potential applications. The Raman scattering and resonant Raman
scattering in self-assembled Ge QD SLs were first reported by Liu et al. [3] and
Kowk et al. [6], respectively. Then most of the published Raman studies on
self-assembled Ge QDs were limited to the optical phonon frequency range [4, 7, 10-13].
It was demonstrated that valuable information about strain and Si/Ge interdiffusion
in the Ge QDs could be derived from the Ge-Ge optical phonon mode (around 300
cm-1) in the Raman spectra [4, 10-13]. Few studies on Raman scattering by acoustic
phonons in self-assembled Ge QDs have been reported until Liu et al. [7] observed a
series of peaks in the range from 60 to 150 cm-1 in the Raman spectra. Although
several explanations were considered in Yu’s comment [8] on this work and in the
response of Liu et al. [9], no definitive identification could be provided yet. And
Milekhin et al.6 observed a series of doublet peaks below 100 cm-1 in the Raman
spectra of Ge QD SLs, which were attributed to the folded longitudinal acoustic (LA)
phonons in the QD SLs and explained by the Rytov’s model [17]. However, Raman
spectra of self-assembled Ge QD SLs below 60 cm-1 have only been observed in
resonant Raman scattering by Cazayous et al. [14] and Milekhin et al. [15], and no
publishment has reported the spectra in non-resonant Raman scattering since the
signals were rather weak. In this section, we report the study of the non-resonant
Raman spectra of self-assembled Ge QDs below 60 cm-1.
6.3.1 Experimental Results and Discussion
Ten samples used in the experiments, labeled A to J, were grown by solid-source
molecular beam epitaxy on Si (100) substrate with S-K growth mode. All of these
samples consisted of 100nm Si buffer layers, followed by some period bilayers, in
which Ge layers were separated by 20-nm-thick Si spacer layers. The Ge QDs in the
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
samples were vertically correlated and no cap Si layer was used for all these samples.
The differences among these samples are the Ge layer thickness, the number of the
periods, and the growth temperature. Samples A, B, and C were grown at 600 °C
with 22 periods of Ge and Si bilayers, and the Ge coverages of these samples were 6,
12, and 15 Å, respectively. Samples D and E were grown at 540 °C with 10 periods,
and the Ge coverages of the two samples were 12 and 15 Å, respectively. Samples F,
G, H, I and J were grown at 540 °C with the same Ge layer thickness of 15 Å, and
consisted 2, 5, 20, 35, and 50 periods, respectively. The growth parameters and the
structural data of these ten samples are summarized in Table 6.2.
Table 6.2 Structural data of samples used in low-frequency Raman measurements.
Sample No. Ge layer thickness (nm)
Si layer thickness (nm)
Growth temperature (°C) Periods
A 136 0.6 20 600 22 B 138 1.2 20 600 22 C 137 1.5 20 600 22 D 210 1.2 20 540 10 E 226 1.5 20 540 10 F 269 1.5 20 540 2 G 264 1.5 20 540 5 H 265 1.5 20 540 20 I 266 1.5 20 540 35 J 183 1.5 20 540 50
Figure 6.6 shows a cross-sectional TEM image of a 20-period self-assembled Ge QD
SLs sample grown at 540 °C. The Ge layer nominal thickness is 1.5 nm and the Si
spacer layer thickness is 20 nm. Raman scattering measurements were performed
with a JY T64000 Raman system in backscattering geometry at room temperature.
All the spectra were excited by the 514 nm line of an Ar ion laser and recorded with
a liquid-nitrogen-cooled charge-coupled device camera. The spectra were obtained
using the same excitation power and data accumulation time. The spectra resolution
is about 1 cm-1.
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
Figure 6.6 A typical cross-sectional TEM image of a 20-period self-assembled Ge quantum dot superlattices sample at the growth temperature of 540 °C (sample H in table 6.2). The thickness of Ge and Si spacer layer are 1.5nm and 20 nm, respectively. Vertical correlation is clearly seen.
5 10 15 20 25 30 35 40 45
Ram
an In
tens
ity (a
.u.)
Raman Shift (cm-1)
Sample C
Sample A
Si sub
Sample B
Figure 6.7 Low-frequency Raman scattering spectra of the samples A, B, C, and an identical Si substrate. Samples B and C both include 22 periods of Ge QDs, while A just with Ge wetting layers.
Figure 6.7 shows the low-frequency Raman spectra of the samples A, B, C, and an
identical Si substrate. As stated previously, samples B and C are 22-period dot
samples, while the Ge layer thickness in sample A is too thin to form any QDs, i.e.,
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
it is a sample only with Ge wetting layers. As seen from Figure 6.7, the Raman
scattering peaks can be clearly found at 15 and 16 cm-1 for samples B and C,
respectively, and no obvious Raman scattering peaks in sample A and Si substrate.
We assume that the low-frequency Raman scattering peaks also originate from the
folded acoustic phonons in the samples related to the periodicity SLs, just like the
Raman peaks in Milekhin et al.’s experiments. [10] But in our experiments, we found
the peaks in lower frequency range using non-resonant Raman scattering. Rytov’s
elastic continuum model [17] has been applied for folded acoustic in Ge QD SLs. [10]
In the model, the acoustic phonon dispersion can be approximately written as
)sin()sin(2
1)cos()cos()cos(2
2
1
12
2
2
1
1
vd
vd
kk
vd
vdqd ωωωω +
−= (6.3)
where and 21 ddd +=22
11
ρρ
vvk = ; and , and , 1v 2v 1d 2d 1ρ and 2ρ are
sound velocity, thickness and density in Ge and Si layers, respectively. These
physical parameters can be obtained from Ref. 14. Thus in Figure 6.7, no
low-frequency Raman scattering peaks were found in Si substrate, and the peaks
were too weak to see in sample A, since its largest difference between and .
The intensity of the low-frequency Raman scattering peaks of sample C were much
higher than sample B. There is a ratio about 0.3 (B versus C, and both were
subtracted the signal from the Si substrate). The Ge QDs in sample C are larger
(mainly having larger heights) since its larger Ge layer nominal thickness and
accordingly smaller difference between and than those in sample B. We
assume that the difference of the Raman peak intensity between the two samples
(having same periods) mainly attribute to the difference between and . This
rule can also be observed from Figure 6.8. Three obvious Raman peaks were found
both in sample D and E, locating at 10, 18, 24 cm
1d 2d
1d 2d
1d 2d
-1 and 10, 18, 25 cm-1 for sample D
and E, respectively. The three Raman peak intensity ratios of sample D to E are 0.5,
0.6, and 0.8 from left (low-frequency) to right (high-frequency) respectively. And
some interesting theoretical works on how the Raman scattering spectra of the Ge
68
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
QD SLs depend on the QDs’ size and the number of QDs layers have already been
reported by Cazayous et al. [16], but there has been very few detail experimental
reports yet besides ours. However, due to the relatively large dot height of our
samples and the non-resonant Raman scattering mode, the strict periodicity cannot
be seen in the Raman spectra and the number of Raman peaks was small.
10 15 20 25 30
Ram
an In
tens
ity (a
.u.)
Raman Shift (cm-1)
Sample D
Si Sub
Sample E
Figure 6.8 Low-frequency Raman scattering spectra of the samples D and E. Samples D and E both include 10 periods of Ge QDs.
Figure 6.9 (a) shows the low-frequency Raman scattering spectra of a series of
samples F to J, which have almost the same Ge QDs size but different number of Ge
QDs layers. As stated previously, samples F, G, H, I, and J include 2, 5, 20, 35, and
50 periods of Ge QDs, respectively. From Figure 6.9(a), it was found that the Raman
peak intensity increased with the increase of the number of the Ge QD layers, i.e. the
periods of the SLs of different samples. It is assumed that the Raman scattering
amplitudes associate with each QDs layer when the Ge QDs in different layers are
vertical correlated [14, 16]. The Ge QDs’ self-assemble during growth of lattice
mismatched Si/Ge layers, providing effective strain relief. When QD layers are
stacked, the buried dots influence the nucleation in the subsequent layers. This
69
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
interaction occurs via elastic strain fields and induces vertical QD alignment. So the
Ge QDs in different layers are vertical correlated.
Figure 6.9 (a) Lnormalized ratioscattering peaks and J include 2, 5
5 10 15 20 25 30 35 40 45
Sample F Sample H Sample I Sample J
Ram
an In
tens
ity (a
.u.)
Raman Shift (cm-1)
13
4
5
1
345
(a)
Sample G
2
2
o o,
3rd peak
20 35 500.0
0.2
0.4
0.6
0.8
1.0
Inte
nsity
Rat
io (N
orm
aliz
ed)
Periods
1st and 2nd peaks
(b)
w-frequency Raman scattering spectra of the samples F, G, H, I, and J and (b) the of the first (from low frequency to high frequency), second, and third Raman f the samples H and I to the corresponding peaks of the sample J. Samples F, G, H, I, 20, 35, and 50 periods of Ge QDs, respectively.
70
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
Sample F has just 2 Ge QD layers, so the Raman peaks are too weak to be observed.
Sample G has 5 Ge QD layers and only one low-frequency Raman scattering peak
was found. There were all three low-frequency Raman scattering peaks of sample H,
I, and J, and the intensity of the Raman peaks increase with the increase of the
number of the Ge QDs layers. Figure 6.9(b) shows the graph of the normalized ratio
of the first (from low-frequency to high-frequency), second, and third Raman peaks
of the samples H and I to the corresponding peaks of the sample J versus the number
of Ge QDs layers.
6.3.2 Conclusions
In conclusion, we have reported the low-frequency Raman scattering spectra of
self-assembled Ge QD SLs. Low-frequency Raman scattering peaks were observed
and we assumed that they were arisen from the folded acoustic phonons in the Ge
QD SLs. And it was found in our experiments that the intensity of the low-frequency
Raman scattering peaks was closely related to the Ge and Si layer thickness and the
number of the periods of the Ge QD SLs, the smaller periods, the lower intensity of
the Raman peaks.
In Chapter Six, Section 6.1 was published as an abstract on the Proceedings of 2003 Chinese Semiconductor Symposium; Section 6.2 was published on Chinese Physics Letters; and Section 6.3 was submitted to Applied Physics Letters.
71
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Six
References
[1] C.V. Raman and K.S. Krishnan, The optical analog of the Compton effect,
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in Si nanocrystals, Appl. Phys. Lett. 69, 200-202 (1996); ii) J. Zi, K. Zhang,
and X. Xie, Comparison of models for spectra of Si nanocrystals, Phys. Rev. B
55, 9263-9266 (1997); iii) V. Paillard, P. Puech, M. A. Laguna, R. Cartes, B.
Kohn, and F. Huisken, Improved one-phonon confinement model for an
accurate size determination of silicon nanocrystals, J. Appl. Phys. 86,
1921-1924 (1999).
[3] i) J. L. Liu, Y. S. Tang, K. L. Wang, T. Radetic, and R. Gronsky, Raman
scattering from a self-organized Ge dot superlattice, Appl. Phys. Lett. 74,
1863-1865 (1999); ii) A. V. Kolobov and K. Tanaka, Comment on “ Raman
scattering from a self-organized Ge dot superlattice” [Appl. Phys. Lett. 74,
1863 (1999)], Appl. Phys. Lett. 75, 3572-3573 (1999); iii) J. L. Liu, Y. S. Tang,
and K. L. Wang, Response to “Comment on ‘ Raman scattering from a
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Phys. Lett. 75, 3574-3575 (1999).
[4] J. L. Liu, J. Wan, Z. M. Jiang, A. Khitun, K. L. Wang, and D. P. Yu, Optical
phonons in self-assembled Ge quantum dot Superlattices: Strain relaxation
effects, J. Appl. Phys. 92 6804-6808 (2002).
[5] H. Richter, Z. P. Wang, and L. Ley, The one phonon Raman spectrum in
microcrystalline silicon, Solid State Commun. 39, 625-629 (1981).
[6] S. H. Kwok, P. Y. Yu, C. H. Tung, Y. H. Zhang, M. F. Li, C. S. Peng, and J. M.
Zhou, Confinement and electron-phonon interactions of the E1 exciton in
self-organized Ge quantum dots, Phys. Rev. B 59, 4980-4984 (1999).
[7] J. L. Liu, G. Jin, Y. S. Tang, Y. H. Luo, K. L. Wang, and D. P. Yu, Optical and
acoustic phonon modes in self-organized Ge quantum dot superlattices, Appl.
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Phys. Lett. 76, 586-588 (2000).
[8] P. Y. Yu, Comment on “Optical and acoustic phonon modes in self-organized Ge
quantum dot superlattices” [Appl. Phys. Lett. 76, 586 (2000)], Appl. Phys. Lett.
78, 1160-1161 (2001).
[9] J. L. Liu, G. Jin, Y. S. Tang, Y. H. Luo, K. L. Wang, and D. P. Yu, Response to
“Comment on ‘Optical and acoustic phonon modes in self-organized Ge
quantum dot superlattices’” [Appl. Phys. Lett. 78, 1160 (2001)], Appl. Phys.
Lett. 78, 1162-1163 (2001).
[10] i) A. G. Milekhin, N. P. Stepina, A. I. Yakinmov, A. I. Nikiforov, S. Schulze,
and D. R. T. Zahn, Raman scattering of Ge dot superlattices, Euro. Phys. J. B
16, 355-359 (2000); ii) A. G. Milekhin, N. P. Stepina, A. I. Yakinmov, A. I.
Nikiforov, S. Schulze, and D. R. T .Zahn, Raman scattering study of Ge dot
superlattices, Appl. Surf. Science 175-176, 629-635 (2001); iii) A. G. Milekhin,
A. I. Nikiforov, O. P. Pchelyakov, S. Schulze, and D. R. T. Zahn, Phonons in
Ge/Si superlattices with Ge quantum dots, JEPT Lett. 73, 461-464 (2001); iv)
A. G. Milekhin, A. I. Nikiforov, O. P. Pchelyakov, S. Schulze, and D. R. T. Zahn,
Phonons in self-assembled Ge/Si structures, Physica E 13, 982-985 (2002).
[11] M. Cazayous, J. Groenen, F. Demangeot, R. Sirvin, and M. Caumont, T.
Remmele, M. Albrecht, S. Christiansen, M. Becher, H. P. Strunk, and H. Wawra,
Strain and composition in self-assembled SiGe islands by Raman spectroscopy,
J. Appl. Phys. 91, 6772 (2002).
[12] P. H. Tan, K. Brunner, D. Bougeard, and G. Abstreiter, Raman characterization
of strain and composition in small-sized self-assembled Si/Ge dots, Phys. Rev.
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[13] Z. Yang, Y. Shi, J. L. Liu, B. Yan, Z. X. Huang, L. Pu, Y. D. Zheng, and K. L.
Wang, Strain and phonon confinement in self-assembled Ge quantum dot
superlattices, Chin. Phys. Lett. 20, 2001-2003 (2003).
[14] i) M. Cazayous, J. R. Huntzinger, J. Groenen, A. Mlayah, S. Christiansen, H. P.
Strunk, O. G. Schmidt, and K. Eberl, Resonant Raman scattering by acoustical
phonons in Ge/Si self-assembled quantum dots: Interferences and ordering
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effects, Phys. Rev. B 62, 7243-7248 (2000); ii) M. Cazayous, J. Groenen, J. R.
Huntzinger, A. Mlayah, and O. G. Schmidt, Spatial correlations and Raman
scattering interferences in self-assembled quantum dot multilayers, Phys. Rev.
B 64, 033306 (2001); iii) M. Cazayous, J. Groenen, J. R. Huntzinger, A. Mlayah,
U. Denker, and O. G. Schmidt, A new tool for measuring island dimensions and
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Mater. Sci. Eng. B 88, 173 (2002).
[15] A. G. Milekhin, A. I. Nikiforov, O. P. Pchelyakov, S. Schulze, and D. R. T.
Zahn, Size-selective Raman scattering in self-assembled Ge/Si quantum dot
superlattices, Nanotechnology 13, 55-58 (2002).
[16] M. Cazayous, J. Groenen, A. Zwick, A. Mlayah, R. Carles, J. L. Bischoff, and
D. Dentel, Resonant Raman scattering by acoustic phonons in self-assembled
quantum-dot multilayers: From a few layers to superlattices, Phys. Rev. B 66,
195320 (2002).
[17] S. M. Rytov, Akoust. Zh. 2, 71 (1956).
[18] D. J. Lockwood, M. W. C. Dharma-Wardana, J. M. Baribear, D. C. Houghton,
Folded acoustic phonons in Si/GexSi1-x strained-layer superlattices, Phys. Rev.
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74
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Seven
Chapter Seven
Photoluminescence in Ge/Si QDSLs
Many studies have been performed on the optical properties of Ge/Si quantum dots.
The study of photoluminescence (PL) in self-Assembled Ge/Si quantum dots is one
the most direct approach to investigate their optical properties. And it is the basis of
the optical applications of self-Assembled Ge/Si quantum dots, such as
photodetectors, light emitting diode, photodiode, etc. In this chapter, the
photoluminescence in Ge/Si QDSLs will be reported. Temperature-dependent
photoluminescence measurements on self-assembled Ge/Si QDSLs were carried out
and investigated. In the photoluminescence spectra, the photoluminescence peaks
from Si TO-phonon assisted recombination and recombination in Ge QDs were
discussed. The temperature-dependence of the photoluminescence intensity has been
fitted and analyzed, from which some characteristic parameters of the Ge QDs, such
as the dimension and effective mass of the electrons, have been estimated. It is the
first time to discuss the relation between the dimensions of the QDs and the effective
mass of the electrons in the QDs.
7.0 Basic Concepts of Photoluminescence
Photoluminescence spectroscopy is a contactless, non-destructive method of probing
the electronic structure of materials. Specifically, light is directed onto a sample,
where it is absorbed and imparts excess energy into the materials in a process called
“photo-excitation”. One way this excess energy can be dissipated by the sample is
through the emission of light, or luminescence. In the case of photo-excitation, this
luminescence is called photoluminescence. Then intensity and spectral content of
this photoluminescence is a direct measure of various important material properties.
Specifically, photo-excitation causes electrons within the materials to move into
permissible excited states. When these electrons return to their equilibrium states,
75
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Seven
the excess energy is released and may include the emission of light (a radiative
process) or may not (a non-radiative process). The energy of the emitted light—or
photoluminescence—is related to the difference in energy levels between the two
electron states involved in the transition—that is, of the emitted light is related to the
relative contribution of the radiative process. Figure 7.1 shows the schematic of the
process of photoluminescence process.
Continuum states Laser energy
PL Intensity
k
E
hωemissionhωexcitation
Electronic bound state
VB
CB
Figure 7.1 Schematic of photoluminescence process.
Photoluminescence spectroscopy has been widely recognized for a long time as a
useful tool for characterizing the quality of semiconductor materials as well as for
elucidating the physics that may accompany radiative recombination.
Photoluminescence spectroscopy is useful in quantifying: optical emission
efficiencies, composition of the material such as alloy, impurity content, and layer
thickness. Photoluminescence spectroscopy have been widely and frequently used in
the research work of the physical properties as bandgap determination, impurity
levels and defect detection, recombination mechanisms, and material quality. In the
following sections, the photoluminescence in self-assembled Ge/Si QDSLs will be
reported and discussed.
76
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Seven
7.1 PL Spectra of Ge/Si QDSLs
Photoluminescence measurements were carried out in a 20-period self-assembled
Ge/Si QDSLs sample. The growth temperature was 540 °C. The thickness of Ge and
Si spacer layer were 1.5nm and 20 nm, respectively. PL measurements were
performed with an FTIR-T60 system. The spectrum was excited by the 514 nm line
laser and recorded with liquid-nitrogen-cooled Ge based CCD camera. The
measurements were performed at 10 K. Figure 7.2 shows the PL spectrum.
0.6 0.7 0.8 0.9 1.0 1.1 1.2
PL In
tens
ity (a
rb. u
nit)
Photon Energy (eV)
Si-TO
Ge QDs
TO+NP
T=10K
Figure 7.2 A typical PL spectrum of a 20-period self-assembled Ge QDSLs sample. Grown at 540 °C. The thicknesses of Ge and Si spacer layer were 1.5nm and 20 nm, respectively.
The strong peak centered around 1.1 eV is the Si transverse optical (TO) phonon
assisted recombination. The peak located at higher energy than Si TO peak arises
from other recombination in Si, such as Si transverse acoustic (TA) peak and Si
no-phonon (NP) peak. The broad peaks at lower energies (from 0.7 to 1.0 eV)
labeled “Ge dots” (inside the dashed-line rectangle) are attributed to electron-hole
77
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Seven
recombination within the dots or at the interface between the dots and surrounding
Si [2, 5]. These peaks can be decomposed into two peaks that correspond to NP- and
TO-phonon assisted recombination [3, 4]. The peaks between the Si TO-peak and the
broad Ge dot peak arise from the Ge wetting layers. PL study has provided some
insight about the band structure of the sample [3, 4]. The frequency shift of the Ge dot
peak can be used to estimate the degree of Si/Ge intermixing.
The temperature-dependent PL measurements will be discussed in the next section,
from which more information about the Ge/Si QDSLs can be obtained.
7.2 Temperature-Dependent PL Spectra of Ge/Si QDSLs
In this section, the temperature-dependence of the PL intensity has been reported
and fitted by the Arrhenius and Berthelot type function. Some valuable parameters
were obtained through the fitted curves, one of which was closely related to the
dimensionality and effective mass of electron of the QDs.
7.2.1 Experimental Results and Discussion
Photoluminescence measurements were carried out in a 20-period self-assembled
Ge/Si QDSLs sample. The growth temperature was 540 °C. The thickness of Ge and
Si spacer layer were 1.5nm and 20 nm, respectively. Figure 7.3 shows the 2D and
3D AFM image of this Ge/Si QDSLs sample.
PL measurements were performed with an FTIR-T60 system. The spectra were all
excited by the 514 nm line laser and recorded with liquid-nitrogen-cooled Ge based
CCD camera. Figure 7.4 shows the PL spectra. The temperature-dependent PL
spectra are presented in Figure 7.4. The sample is the same as the one in Figure 7.2.
The PL curves of the sample were recorded at 10, 30, 50, 80, 120, 160, and 200 K.
78
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Seven
Figure 7.3 Typical 2D and 3D AFM images of a uniform self-assembled Ge/Si QDSLs sample at the growth temperature of 540 °C. The thickness of Ge and Si spacer layer were 1.5nm and 20 nm, respectively.
0.6 0.7 0.8 0.9 1.0 1.1 1.2
PL In
tens
ity (a
rb. u
nit)
Photon Energy (eV)
10K
30K
50K
80K
160K
200K
120K
Figure 7.4 Temperature-dependent PL spectra of the same sample in Figure 7.2. The PL curves were recorded at 10, 30, 50, 80, 120, 160, and 200K.
It has been reported that the temperature-dependence of the PL intensity in
nanocrystalline semiconductors is of the combination of Arrhenius and Berthelot
79
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Seven
type temperature dependence both in theoretical works [6] and silicon clusters [7]. We
find that the rules also exist in the Ge/Si quantum dots. The temperature-dependence
of the PL intensity is given by [6]
(TI
)exp(1
1)
0
TT
TTI r
B
+⋅+=
ν (7.1)
where ν is characteristic reduced frequency and TB and Tr are characteristic
ur experimental data were fitted using this model and the results were represented
temperatures. In our experiments, I(T) and I0 are the integrated PL intensity at
temperature T and 10K, respectively.
O
in Figure 7.5, which was the temperature-dependence of the integrated PL intensity.
The dots are the experimental data and the solid line is the fitted curves.
0 20 40 60 80 100 120 140 160 180 200 220
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
I T / I 10
K
Temperature (K)
Experimental Data Fitted Curves
Figure 7.5 The temperature-dependence of the PL intensity. The dots are the experimental data. The solid line is the fitted curves.
our fitted curve, the characteristic parameters TB, Tr, and
In ν equal to 31.6K, 0K,
and 0.025, respectively. So it is no contribution of the radiative term in the fitted
curve. The characteristic temperature TB is given by the expression [6]
80
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Seven
BeB kma
T *222π=
2h (7.2)
where a is the confinement size of the QDs and is the effective mass of the
ele
.2.2 Conclusions
was the first time that the theory of temperature-dependence of the PL intensity in
.3 PL Spectra of More Ge/Si QDSLs Samples
.3.1 Experimental Results
our samples used in the photoluminescence experiments, labeled A to D, were
*em
electron in the QDs. The average height of our QDs sample is about 10nm, thus the
*em equals about 0.014me, where me is the rest electron mass. The value of this
ctron effective mass cannot be compared with that in bulk Ge, due to Ge/Si
interdiffusion. The degree of the interdiffusion can be estimated by the Raman
scattering measurements. (Please see Section 6.2 for detail.)
7
It
nanocrystalline semiconductors was verified by the experiments in Ge/Si quantum
dots samples. The temperature-dependence of the PL intensity has been reported and
fitted, from which the characteristic temperature TB could be obtained. And the
height and the electron effective mass of the QDs could be estimated through TB.
7
7
F
grown by solid-source MBE on Si (100) substrate with S-K growth mode. All of
these samples consisted of 100nm Si buffer layers, followed by some period bilayers,
in which Ge layers were separated by 20-nm-thick Si spacer layers. The differences
among these samples are the Ge layer thickness, the number of the periods, and the
growth temperature. The growth parameters and the structural data of these samples
are summarized in Table 7.1.
81
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Seven
Photoluminescence measurements on samples A, B, C, and D were performed with
Table 7.1 Structural data of samples used in PL measurements.
Sample No. thickness (nm) thickness (nm) temp ) Periods
an FTIR-T60 system. The spectra were all excited by the 514 nm line laser and
recorded with liquid-nitrogen-cooled Ge based CCD camera. Figure 7.6 shows the
photoluminescence spectra.
Ge layer Si layer Growth erature (°C
1.5 20 600
B 226 1.5 20 540 10
C 265 1.5 20 540 20
A 137 22
D 266 1.5 20 540 35
Figure 7.6 The PL spectra of samples A, B, C, and D at 10 K. The structural data of these samples are summarized in Table 7.1.
0.6 0.7 0.8 0.9 1.0 1.1 1.2
PL In
tens
ity (a
rb. u
nit)
Photon Energy (eV)
C
A
BD
Si-sub
T=10K
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Seven
7.3.2 Discussion and Conclusion
From the experiments performed above, it was found that the photoluminescence
intensity and peak-position of the Ge/Si quantum dots are closely related to the
growth parameters and structural data of the Ge/Si QDSLs samples. It is assumed
that this mainly affected by the Ge composition in the quantum dots and the
dimensions the quantum dots. These conclusions still need more experiments to
verify. After photoluminescence experiments of more samples (all the samples
presented in Table 5.1) are done, more precise and quantitative conclusions will be
drawn and presented.
In Chapter Seven, Section 7.2 was submitted to Chinese Journal of Semiconductors
nd part of Section 7.1 & 7.2 was submitted to Materials Letters. a
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Chapter Seven
References
] G. D. Gilliand, Photoluminescence spectroscopy of crystalline semiconductors,
Mater. Sci. Eng. R 18
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, 99-399 (1997).
] G. Abstreiter, P. Schittenhelm, C. Engel, E. Silveria, A. Zrenner, D. Meertens,
and W. Jäger, Growth and characterization of self-assembled Ge-rich islands
on Si, Semicond. Sci. Technol. 11
[2
, 1521-1528 (1996).
] L. P. Rokhinson, D. C. Tsui, J. L. Benton, and Y. H. Xie, Infrared and
photoluminescence spectroscopy of p-doped self-assembled Ge dots on Si, Appl.
Phys. Lett. 75
[3
, 2413-2415 (1999).
] i) J. Wan, G. L. Jin, Z. M. Jiang, Y. H. Luo, J. L. Liu, and K. L. Wang, Band
alignments and photon-induced carrier transfer form wetting layers to Ge
[4
islands grown on Si(001), Appl. Phys. Lett. 78, 1763-1765 (2001); ii) J. Wan, Y.
H. Luo, Z. M. Jiang, G. Jin, J. L. Liu, K. L. Wang, X. Z. Liao, and J. Zou,
Effects of interdiffusion on the band alignment of GeSi dots, Appl. Phys. Lett.
79, 1980-1982 (2001).
[5] M. W. Dashiell, U. Denker, C. Müller, G. Costantini, C. Manzano, K. Kern, and
O. G. Schmidt, Photoluminescence of ultrasmall Ge quantum dots grown by
molecular-beam epitaxy at low temperatures, Appl. Phys. Lett. 80, 1279-1281
(2002).
[6] M. Kapoor, V. A. Singh, and G. K. Johri, Origin of the anomalous temperature
dependence of luminescence in semiconductor nanocrystallites, Phys. Rev. B
61, 1941-1945 (2000).
[7] H. Rinnert and M. Vergnat, Influence of the temperature on the
photoluminescence of silicon clusters embedded in a silicon oxide matrix,
Physica E 16, 382-387 (2003).
84
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Epilogue
Epilogue
Main Conclusions
In the thesis, the geometric and band structure of carbon nanotubes were
investigated; the carbon nanotube-based electronics is reviewed and its prospects
were pointed out; the Raman scattering and photoluminescence measurements were
performed on the Ge/Si quantum dot superlattices, valuable information was
obtained. The main conclusions of this thesis were summarized again below.
(1) The index n of zigzag (n, 0) and armchair (n, n) carbon nanotubes stands for the
highest fold of the rotation axes among the symmetry elements in carbon
nanotubes. This n-fold axis together with other symmetry elements in carbon
nanotubes make up the point group Dnh. And all kinds of n-fold axes can be
found in some zigzag or armchair carbon nanotubes, where n is an integer larger
than two. (Chapter 2)
(2) The band structures of carbon nanotubes were calculated on the basis of band
structure of graphite by tight binding approximation. The energy dispersion of
carbon nanotubes was formulized in Equation (3.24). One third of the carbon
nanotubes whose indexes meet with Equation (3.25) are metallic, while the other
two thirds are semiconducting. The energy dispersion of armchair carbon
nanotubes was formulized in Equation (3.26). All the armchair carbon nanotubes
are metallic. The energy dispersion of zigzag carbon nanotubes was formulized
in Equation (3.27). Also, only one third of zigzag nanotubes are semiconducting.
(Chapter 3)
(3) Lots of progress has been made in CNT-based electronics till now, especially the
complementary logic circuits based on CNTs were fabricated. But many tasks
85
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Epilogue
and challenges sill lie ahead of real application of CNT-based circuits, including
enabling device functions in non-vacuum conditions, obtaining massive and
high-density transistor arrays, and interconnecting carbon nanotubes on-chip.
And another key problem is how to separate semiconducting CNTs from metallic
CNTs more effectively. After all these problems are solved, a magnificent system
of electronics base CNTs will come into being. (Chapter 4)
(4) We have reported on the investigation of Raman scattering in the self-assembled
Ge quantum dot superlattices, the Si-Si, Ge-Ge, Si-Ge and Si-SiLOC peaks were
found in the Raman spectra, which were arisen from the Si substrate, the optical
phonon modes of Ge, the SiGe alloy in the wetting layers and the existence of
strain in Si underneath the QDs, respectively. The effects of the phonon
confinement and strain within the Ge QDs can induce the red-shift and
blued-shift of the optical phonon modes. Strain relaxation in Ge QDs
superlattices is not only from Ge/Si interdiffusion but also from other reasons
such as dot morphology transition. (Section 6.2, Chapter 6)
(5) We have reported the low-frequency Raman scattering spectra of self-assembled
Ge QD SLs. Low-frequency Raman scattering peaks were observed and we
assumed that they were arisen from the folded acoustic phonons in the Ge QD
SLs. And it was found in our experiments that the intensity of the low-frequency
Raman scattering peaks was closely related to the Ge and Si layer thickness and
the number of the periods of the Ge QD SLs, the smaller periods, the lower
intensity of the Raman peaks. (Section 6.3, Chapter 6)
(6) In the PL spectra of self-assembled Ge/Si QDSLs, the strong peak centered
around 1.1 eV is the Si-TO phonon assisted recombination. The peak located at
higher energy than Si-TO peak arises from other recombination in Si, such as
Si-TA peak and Si-NP peak. The broad peaks at lower energies are attributed to
electron-hole recombination within the dots or at the interface between the dots
86
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Epilogue
and surrounding Si. The peaks between the Si TO-peak and the broad Ge dot
peak are arisenfrom the Ge wetting layers. (Section 7.1, Chapter 7)
(7) Theory of temperature-dependence of the PL intensity in nanocrystalline
semiconductors was verified by the experiments in Ge/Si quantum dots samples.
The temperature-dependence of the PL intensity has been reported and fitted,
from which the characteristic temperature TB could be obtained. And the height
and the electron effective mass of the QDs could be estimated through TB.
(Section 7.2, Chapter 7)
(8) It was found that the photoluminescence intensity and peak-position of the Ge/Si
quantum dots are closely related to the growth parameters and structural data of
the Ge/Si QDSLs samples. It is assumed that this mainly affected by the Ge
composition in the quantum dots and the dimensions the quantum dots. These
conclusions still need more experiments to verify. After photoluminescence
experiments of more samples are done, more precise and quantitative
conclusions will be drawn and presented. (Section 7.3, Chapter 7)
Future Work and Prospects
In the Part II of this thesis, the optical spectra of Ge/Si quantum dot (QD)
superlattices (SLs) were discussed, from which some properties of the Ge/Si QDSLs
could be investigated, such as phonons and optical properties. The mainly
experimental techniques are Raman scattering and photoluminescence (PL)
spectroscopy. Besides these two methods, some more characterization will be
carried out in the future to get more information about the properties of the Ge/Si
QDSLs, especially the electrical experiments. The experiments on electronic Raman
scattering, Hall measurements, and C-V (capacitance-voltage) measurements will
be done subsequently. Thus some electronic properties of the Ge/Si QDSLs can be
87
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Epilogue
obtained, such as the electronic states, mobility and carrier concentration.
And there are still some work to do just in Raman and PL characterization of Ge/Si
QDSLs, such as the correlation of Raman and PL spectra, the work continued to
Section 7.3 based on more samples’ PL spectra, and the fitting work of more
samples’ temperature-dependent PL spectra.
Regarding carbon nanotubes (CNTs), there are still a lot of theoretical and
experimental works to do, among which the most fascinating thing is the emergence
of CNT-based products, such as CNT-based integrated circuits, CNT-based
superconducting devices, CNT-based hydrogen storage container and CNT-based
TFT (thin film transistor) display. But I think it will still take a few years to realize
it.
The uniformity is the real problem before both the CNTs and Ge/Si QDSLs can have
any practical applications. In the future application, we need CNTs of uniform
properties and QDs in uniform size and shape. Nowadays, lots of techniques have
been developed to obtain CNTs and Ge/Si QDSLs with good uniformity. It will be
more spectacular tomorrow!
88
Appendix
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix A
APPENDIX A
Point Groups
A.1 Thirty-Two Crystal Point Groups
International Symbol Lattice
Schöenflies
Symbol Full Short Symmetry Elements (Operations)
Order
C1 1 1 E 1Triclinic S2 (Ci) 1 1 iE, 2
C2 2 2 2,CE 2C1h (Cs) m m hE σ, 2
Monoclinic C2h 2/m 2/m hiCE σ,,, 2 4
D2 (V) 222 222 22 2,, CCE ′ 4C2v mm2 mm2 vCE σ2,, 2 8
Orthorhom
bic D2h (Vh) (2/m)(2/m)(2/m) mmm vhiCCE σσ 2,,,2,, 22 ′ 8C3 3 3 32, CE 3
S6 (C3i) 3 3 63 2,,2, SiCE 6D3 32 32 23 3,2, CCE ′ 6C3v 3m 3m vCE σ3,2, 3 6
Trigonal
D3d )/2(3 m m3 vSiCCE σ3,2,,3,2, 623 ′ 12C4 4 4 24 ,2, CCE 4S4 4 4 42 2,, SCE 4
C4h 4/m 4/m hSiCCE σ,2,,,2, 424 8D4 422 422 2224 2,2,,2, CCCCE ′′′ 8C4v 4mm 4mm dvCCE σσ 2,2,,2, 24 8
D2d (Vd) 24m 24m dSCCE σ2,2,,2, 424 8
Tetragonal
D4h (4/m)(2/m)(2/m) 4/mmm dvhSiCCCCE σσσ 2,2,,2,,2,2,,2, 42224 ′′′ 16C6 6 6 236 ,2,2, CCCE 6
C3h (S3) 6 6 hSSCE σ,2,2,2, 633 6C6h 6/m 6/m hSSiCCCE σ,2,2,,,2,2, 63236 12D6 622 622 22236 3,3,,2,2, CCCCCE ′′′ 12C6v 6mm 6mm dvCCCE σσ 3,3,,2,2, 236 12D3h 26m 26m vhSCCE σσ 3,,2,3,2, 323 ′ 12
Hexagnoal
D6h (6/m)(2/m)(2/m) 6/mmm dvhSSiCCCCCE σσσ 3,3,,2,2,,3,3,,2,2, 6322236 ′′′ 24T 23 23 23 3,8, CCE 12Th 3)/2( m m3 hSiCCE σ3,8,,3,8, 623 24O 432 432 4223 6,6,3,8, CCCCE ′ 24Td m34 m34 423 6,6,3,8, SCCE dσ 24
Cubic
Oh )/2(3)/4( mm m3m 464223 6,6,3,8,,6,6,3,8, SSiCCCCE dh σσ′ 48
89
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix A
A.2 Frequently Used Non-crystal Point Groups
Schöenflies Symbol
International Symbol
Symmetry Elements (Operations) Order
C5 5 45
35
255 ,,,, CCCCE 5
S8 78
34
582
3848 ,,,,,,, SCSCSCSE 8
D5 2255 5,2,2, CCCE 10
C5v vCCE σ5,2,2, 255 10
C5h 45
35
255
45
35
255 ,,,,,,,,, SSSSCCCCE hσ 10
D4d dCCSCSE σ4,4,,2,2,2, 223848 ′ 16
D5d dSSiCCCE σ5,2,2,,5,2,2, 310102
255 20
D5h dh SSCCCE σσ 5,2,2,,5,2,2, 2552
255 20
D6d dCCSCSCSE σ6,6,,2,2,2,2,2, 2251234612 ′ 24
I 23255 15,20,12,12, CCCCE 60
Ih σ15,20,12,12,,15,20,12,12, 63101023
255 SSSiCCCCE 120
C∞ ∞ φ∞CE 2, ∞
C∞h ∞/m φφ∞∞ SiCE 2,,2, ∞
C∞v ∞m vCE σφ ∞∞ ,2, ∞ D∞h ∞/mm 2,2,,,2, CSiCE v ∞∞ ∞∞
φφ σ ∞
90
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix B
APPENDIX B
Calculations in Chapter Three
Equation (7)
Employing ( )∑=α
αϕψαψj
jnn j )(|)( rkr kkk into )()()(ˆ rkr kkn
nn EH ψψ =
( ) ( )∑∑ =α
α
α
α ϕψαϕψαj
jnn
j
jn jEHj )(|)()(ˆ| rkkrk kkkk
left multiplied then integrating (i.e. left multiplied )(*
rkαϕ ′′j |kα′′j )
( ) ( )
( ) ( )nnjj
j
nn
j
nn
j
n
jEjE
jjjEjjHj
kk
kk
kkkk
kkkkkkk
ψαδδψα
ααψαψααα
ααα
αα
|)(|)(
||)(||ˆ|
′′==
′′=′′
′′∑
∑∑
Equation (8)
From Bloch Theorem (i.e. ) )()( rRr kRk
kαα ϕϕ jij e ⋅−=− )()( Rrr k
Rkk +′=′ ⋅− αα ϕϕ jij e
and ∑∑ −−== +⋅+⋅
R
tRk
RR
tRkk tRrrr )(1)(1)( )()(
ααα ϕϕϕ αα
jijij e
Ne
N
then ∑∑ ′=−′=+′ +⋅+⋅
R
tRk
R
tRkk rtrRr )(1)(1)( 0
)()( αα
α ϕϕϕ αα jij
ij eN
eN
so ∑ ′=+′=′ ⋅⋅−
R
tkk
Rkk rRrr )(1)()( 0
ααα ϕϕϕ α jijij eN
e
i.e., ∑∑ ⋅+⋅ ==≡R
tk
RR
tRkk rrrk )(1)(1)( 0
)( ααα ϕϕϕα αα jijij eN
eN
j
and ∑′
′′′
′+′⋅′′ =≡′′R
RtRk
k rrk )(1)( )( αα ϕϕα α jij eN
j
finally, 11|ˆ|0
)(|ˆ|)(1|ˆ|
)(
0)(
=′′′=
=′′
∑∑
∑∑
′
−′+′⋅
′
′′′
−′+′⋅
RR
ttRk
R RR
ttRk
R
rrkk
NjHje
HeN
jHj
i
jji
αα
ϕϕαα
αα
αα αα
91
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix B
Equation (11)
From Equation (6), assuming and λ== BA cc kk ,1 ttt == BA ,0 ,then
)()()()()( rrrrr kkkkkkkBABBAAn cc λϕϕϕϕψ +=+=
Equation (16)
∫∫
∑∑
∑∑
−−=−−≡′
=≡
′−≡−+=
−==≡
≠
⋅
≠
⋅
⋅⋅
rRrrRrr
rrrr
Rrrrr
Rrrrrrr
R
Rk
R
Rk
R
Rk
RR
Rkkk
dHH
dHHEwhere
eEHeH
HeHeHH
ii
iAAiAA
)(ˆ)()(ˆ)(
)(ˆ)(ˆ)(
)(ˆ)()(ˆ)(
)(ˆ)()(ˆ)()(ˆ)(
*0
20
000
0
011
ϕϕϕϕγ
ϕϕϕ
γϕϕϕϕ
ϕϕϕϕϕϕ
Equation (17)
∫
∑∑
∑
−−−=−−−≡
−≡−−=
=≡
+
+⋅
+
+⋅
+
+⋅
rtRrrtRrr
tRrr
rrrr
tR
tRk
tR
tRk
tRR
tRkkk
dHHwhere
eHe
HeHH
ii
BAiBA
)(ˆ)()(ˆ)(
)(ˆ)(
)(ˆ)()(ˆ)(
*0
)(0
)(
0)(
12
ϕϕϕϕγ
γϕϕ
ϕϕϕϕ
Equation (18)
( ) ( )
+′−=
+−++′−=
−++
−−+
−+
+−+
+
′−=
′−= ∑ ⋅−
)cos()2
cos()2
3cos(22
)cos(2)22
3cos(2)22
3cos(2
expexp)22
3(exp
)22
3(exp)22
3(exp)22
3(exp
00
00
00
0011
akakakE
akakakakakE
aikaikakaki
akakiakaki
akaki
E
eEH
yyx
yyxyx
yyyx
yxyxyx
i
s
nb
γ
γ
γ
γR
Rk
Equation (20)
++=
+−
−+=
)2
3cos()
2cos(4)
2(cos41
)32
exp()2
cos(2)3
exp()32
exp()2
cos(2)3
exp(
220
20
212
akakak
aki
akaki
aki
akakiH
xyy
xyxxyx
γ
γ
92
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix C
APPENDIX C
Programs in Chapter Three
C.1 Energy Dispersion of Graphite %----Constants and Variables----% gammar=1; number=2; kxa=(-pi:pi/50:pi).*(1/1.732); kya=(-pi:pi/50:pi); %----Calculation----% Nx=length(kxa); Ny=length(kya); for i=1:Nx for j=1:Ny T1=cos(kya(j)/2*number); T2=cos(1.732*kxa(i)/2*number); E1(i,j)=gammar*sqrt(1+4*(T1*T1)+4*(T1*T2)); end end E2=-E1; %----Drawing----% surf(kya,kxa,E1); hold; surfc(kya,kxa,E2); axis([-pi pi -pi pi -3 3]); set(gca,'xtick',[-pi:pi/3:pi]); set(gca,'ytick',[-pi:pi/3:pi]); set(gca,'ztick',[-3:1:3]); colormap cool; hold off;
C.2 Energy Dispersion of Armchair Carbon Nanotubes %----Constants and Variables----% n=5;
93
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix C
a=pi; gamma0=1; %----Calculation----% k=0:1/1000:1; for q=1:6 for j=1:1001
E1(q,j)=gamma0*sqrt(1+4*cos(k(j)*a/2)*cos(k(j)*a/2)+4*cos(k(j)*a/2)* cos((q-1)*pi/n));
end end E2=-E1; E(1:6,:)=E1;E(7:12,:)=E2; %----Drawing----% plot(k,E); axis([0 1 -3.5 3.5]); xlabel('ky');ylabel('E(gamma0)');
C.3 Energy Dispersion of Zigzag Carbon Nanotubes %----Constants and Variables----% n=6; a=pi/sqrt(3); gamma0=1; %----Calculation----% k=0:1/1000:1; for q=1:7 for j=1:1001
E1(q,j)=gamma0*sqrt(1+4*cos(pi*(q-1)/n)*cos(pi*(q-1)/n)+4*cos(pi*(q-1)/n)*cos(sqrt(3)*k(j)*a/2));
end end E2=-E1; E(1:7,:)=E1;E(8:14,:)=E2; %----Drawing----% plot(k,E); axis([0 1 -3.5 3.5]); xlabel('kx');ylabel('E(gamma0)');
94
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix D
APPENDIX D
Calculations in Section 6.2
( ))(2
1
0yyxxzzstrain qp εεε
ωω ++=∆ PRB, 5, 580 (1972)
xxstrain pCCq ε
ωω
−=∆
11
12
0
1
)cm300(~s1065.5 11130
−−×=ω
xxzzxxyy CC εεεε
11
122and −==
)cm(5.414 1−×−≅∆ xxstrain εω
Elastic Coefficients
kbar5.482kbar1288
12
11
==
CC
Ge deformation potentials.
227
227
s10167.6s107.4
−
−
×−=
×−=
qp
Frequency of the bulk Ge zone center LO phonon.
Biaxial strain model.
Si
GeSi
aaa
xx−
=ε
nm566.0nm543.0
Ge
Si
==
aa
042.0−=xxε )cm(4.17 1−=∆ strainω
Fully Strained
Partially Strained (Due to the strain relaxation from Ge/Si interdiffusion)
SiGeGeSi )1(-1
axxaaxx
−+=
Vegard’s Law
xxxaaa
aaxx
xx
xx
04.096.011
)1(11
Ge
Si
-1
-1
GeSi
GeGeSi
+−≅
−+−=
−=ε
)96.0(Ge
Si ≅aa
)1(2SiGe
GeGe
−≅
xxB
II
6.11x
+≅
?=ε ?=∆ω strainxxSiSiGeGe
1II2.3=B
95
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix E
APPENDIX E
Details for Equation 7.1*
In early studies on the temperature dependence of the luminescence intensity in
nano-crystalline and amorphous semiconductors, it was found that the radiative
process had a temperature dependence of the Arrhenius type
)exp(TTR r
rr −=ν (E.1)
where represents the radiative recombination rate, T is a characteristic
activation temperature and
rR r
rν a characteristic frequency.
Recent studies show that the temperature dependence of the luminescence intensity
in nano-crystalline and amorphous semiconductors are also related to nonradiative
(or hopping) process, which has an anomalous Berthelot type of temperature
dependence,
)exp(B
Bhop TTR −=ν (E.2)
where represents the hopping escape rate, is the characteristic Berthelot
temperature associated with escape process and
hopR BT
Bν a characteristic frequency.
The luminescence decay time τ —a useful quantity for study—can be expressed in
terms of the competition between the radiative decay and hopping decay dynamics.
hopr RR +=τ1 (E.3)
The intensity of the photoluminescence (PL) line is expressed as
rRtNtI )()( = (E.4)
96
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix E
where is the population of the excited carriers at time , )(tN t
])(exp[)( 0ptNtN
τ−= (E.5)
where . Thus 10 << p
])(exp[),( 0p
rtRNtIτ
τ −= (E.6)
The time-integrated PL intensity is given by
)/)/1((
)1(
])[(])[(])(exp[
])(exp[
),()(
000
0
0
110
00
0
ppNIRIpp
RN
tdttpRN
dttRN
dttII
r
r
ppppr
pr
Γ≡=
Γ=
⋅⋅−=
−=
=
∫
∫
∫
∞ −
∞
∞
τ
ττττ
ττ
ττ
(E.7)†
Thus
)exp(1
1)(
0
00
TT
TT
IR
RI
RRRITI
r
B
r
hophopr
r
+⋅+=
+=
+=
ν
(E.8)
where rB ννν /= .
* M. Kapoor, V. A. Singh, and G. K. Johri, Origin of the anomalous temperature
dependence of luminescence in semiconductor nanocrystallites, Phys. Rev. B 61,
1941-1945 (2000).
† Yang Zheng, Frequently used progression and integrals in statistical physics (in
Chinese), College Physics, 22, 25~28, (2003).
97
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix F
APPENDIX F
Details for Figure 7.5
Integrated PL intensity in Figure 7.4
Temperature (K) Integrated Intensity (Area)
10 20.35766
30 18.56416
50 17.90006
80 15.69283
120 9.59143
160 4.05017
200 1.29694
Normalized
Temperature (K) Integrated Intensity (Area)
10 1
30 0.9119
50 0.8793
80 0.7709
120 0.4712
160 0.1990
200 0.0637
98
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix G
APPENDIX G
Index of Tables
Table 2.1 Character table of D∞h group. Page 19
Table 5.1 Growth parameters and structural data of the samples. Page 51
Table 6.1 Structural data of samples used in optical phonon Raman measurements.
Page 60 Table 6.2 Structural data of samples used in low-frequency Raman measurements.
Page 66 Table 7.1 Structural data of samples used in Photoluminescence measurements.
Page 82
99
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix H
APPENDIX H
Index of Figures
Figure 0.1 Density of states in different dimensionalilies. Page 2
Figure 1.1 High-resolution transmission electron microscope (HRTEM) images of
multi-walled carbon nanotubes. A cross-section of each nanotube is
illustrated. (a) Nanotube consisting of five graphitic sheets, diameter 6.7
nm. (b) Two-sheet nanotube, diameter 5.5 nm. (c) Seven-sheet nanotube,
diameter 6.5 nm. [1] Page 8
Figure 2.1 Graphite lattice, x and y denotes the coordinates and a1 and a2 the unit
vectors. The geometric structure of carbon nanotubes can be defined by
chiral vector Ch or by a pair of integral indexes (n, m). Page 14
Figure 2.2 Schematics of fullerenes and carbon nanotubes. [1] Page 14
Figure 2.3 The schematics of rolled up graphite sheet and cross-sections of (5, 5)
armchair and (9, 0) zigzag carbon nanotubes. Page 17
Figure 2.4 The cross-sections of two series of armchair and zigzag carbon
nanotubes, from which the geometric symmetry elements belonging to
group Dnh can be analyzed and understood. Page 18
Figure 3.1 The vector space and reciprocal vector space of graphite, two lattices of
regular hexagons that can be congruent after a rotation of 30°. Page 24
Figure 3.2 Energy dispersion of graphite. Page 26
Figure 3.3 Distribution of metallic and semiconducting carbon nanotubes. Page 28
Figure 3.4 The energy dispersions of armchair carbon nanotubes (5, 5) and (6, 6),
which are both metallic. Page 29
Figure 3.5 The energy dispersion of zigzag CNTs (6, 0), (7, 0), (8, 0) and (9, 0). The
CNTs (6, 0) and (9, 0) are metallic while CNTs (7, 0) and (8, 0) are
semiconducting. Page 30
Figure 4.1 The schematic of CNT-based FET. Page 36
100
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix H
Figure 4.2 Demonstration of voltage transport characteristics and schematics (insets)
of CNT-based inverter, NOR gate, SRAM, and ring oscillator in RTL
style. [24] Page 36
Figure 4.3 Schematics and voltage transport characteristics of CNT-based inter- and
intra-molecular complementary inverters. [25] Page 38
Figure 4.4 Schematics and voltage transport characteristics of CNT-based NOR, OR,
NAND and AND gates in complementary logic style. [26] Page 38
Figure 5.1 Diagram of a typical MBE system growth chamber. Page 44
Figure 5.2 Schematics of the three epitaxial growth modes. Page 46
Figure 5.3 Schematic of the lattice mismatch between Ge and Si and the strain layer.
Page 47
Figure 5.4 The (a) 2D and (b) 3D AFM images of a typical uniform self-assembled
Ge quantum dots sample at the growth temperature of 600 °C. The Ge
thickness is about 1.5nm. The base size and the height of the dots are
about 70 and 15 nm, respectively. Page 52
Figure 5.5 A typical cross-sectional TEM image of a 10-period self-assembled Ge
quantum dot superlattices sample grown at 540 °C. The thickness of Ge
and Si spacer layer are 1.2nm and 20 nm, respectively. Vertical
correlation is clearly seen. Page 52
Figure 6.1 Energy level diagrams for Rayleigh, Stokes, and Anti-Stokes scattering.
Page 57
Figure 6.2 Raman spectrum of a typical self-assembled Ge quantum dot
superlattices sample. Page 58
Figure 6.3 The 2D AFM images of (a) samples A, (b) sample C, (c) sample E, and
(d) sample D; and (e) the 3D AFM image on the same spot of sample D.
Page 60
Figure 6.4 Raman spectra of the samples A and B. Sample A includes 22 periods of
Ge dots, while sample B just with Ge wetting layers. Page 61
Figure 6.5 Raman spectra of the samples C, D, and E. The frequency positions of
the Ge-Ge peak in the samples are shifted slightly to their bulk value
101
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Appendix H
(300cm-1, the vertical dotted line). Page 62
Figure 6.6 A typical cross-sectional TEM image of a 20-period self-assembled Ge
quantum dot superlattices sample (sample H in table 6.2). at the growth
temperature of 540 °C. The thickness of Ge and Si spacer layer are
1.5nm and 20 nm, respectively. Vertical correlation is clearly seen.
Page 67
Figure 6.7 Low-frequency Raman scattering spectra of the samples A, B, C, and an
identical Si substrate. Samples B and C both include 22 periods of Ge
QDs, while A just with Ge wetting layers. Page 67
Figure 6.8 Low-frequency Raman scattering spectra of the samples D and E.
Samples D and E both include 10 periods of Ge QDs. Page 69
Figure 6.9 (a) Low-frequency Raman scattering spectra of the samples F, G, H, I,
and J and (b) the normalized ratio of the first (from low frequency to
high frequency), second, and third Raman scattering peaks of the
samples H and I to the corresponding peaks of the sample J. Samples F,
G, H, I, and J include 2, 5, 20, 35, and 50 periods of Ge QDs,
respectively. Page 70
Figure 7.1 Schematic of photoluminescence process. Page 76
Figure 7.2 A typical PL spectrum of a 20-period self-assembled Ge QDSLs sample
grown at 540 °C. The thicknesses of Ge and Si spacer layer were 1.5nm
and 20 nm, respectively. Page 77
Figure 7.3 Typical 2D and 3D AFM images of a uniform self-assembled Ge/Si
QDSLs sample at the growth temperature of 540 °C. The thickness of
Ge and Si spacer layer were 1.5nm and 20 nm, respectively. Page 79
Figure 7.4 Temperature-dependent PL spectra of the same sample in Fig.7.2. The PL
curves were recorded at 10, 30, 50, 80, 120, 160, and 200K. Page 79
Figure 7.5 The temperature-dependence of the PL intensity. The dots are the
experimental data. The solid line is the fitted curves. Page 80
Figure 7.6 The PL spectra of samples A, B, C, and D at 10 K. The structural data of
these samples are summarized in table 7.1. Page 82
102
Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Acknowledgements
Acknowledgements
本文是在施毅教授的悉心指导和亲切关怀下完成的。施老师以渊博的知识、
深厚的学术造诣、宽阔的科学视野和富有启发性的指导引领作者进入了半导体
低维纳米结构的科研领域,并使得本论文得以顺利完成。
三年多来,施毅老师对作者在学业上的严格要求、思想上的亲切教诲、生
活上的备至关怀使作者在科研探索的道路上不断前进。施老师不倦的科学追求,
严谨的治学态度,高尚无私的品德和非凡的人格魅力让作者受益无穷,终生难
忘。在作者刚刚进入实验室时,作为一个刚刚学完“四大力学”和仅看过一些
科研文献的本科生,对于实验知识特别是实验技能的掌握和了解是相当匮乏的。
正是在施老师循循善诱的不断指导下作者才得以真正走入科学研究的神圣殿
堂。记得有一次施老师刚刚从 California 大学 Berkeley 分校归来,需要处理的
事情纷繁复杂,他还是从百忙之中抽出时间来教我们这些本科生一些基本实验
技能,我还清楚地记得连切硅片我都是在那时学会的。
三年多来,作者还得到了本实验室郑有炓教授,张荣教授,沈波教授,顾
书林教授,江若琏教授,韩平教授,陈敦军副教授,谢自力副教授,修向前副
教授和朱顺民工程师等老师的指导和帮助,在此作者表示深深的感谢!特别是
郑有炓老师和张荣老师,郑老师是实验室的学术带头人,而作者是在张老师的
推荐下才进入本实验室的。另外,实验室的这些老师们除了在平时工作中给予
过作者帮助外,大多数还教过作者一些课程使作者从中受益,真正是作者的“老
师”。沈波老师教的《电磁学》是最基本和应用最广的物理课程之一;张荣老师
教的《半导体物理》带领作者第一次真正进入了半导体领域;顾书林老师教的
《半导体器件》使作者了解到了理论和实践的结合;郑有炓老师教的《半导体
低维结构》促进了作者对半导体领域近几十年来最新的科研进展情况的了解和
掌握,并且是对作者平时科研工作指导意义最大的课程。
三年多来,作者同本实验室濮林博后,辛煜博后,杨红官博士,鄢波博士,
刘宏博士,田俊博士,闾锦博士,陈杰智博士,肖洁硕士,赵立青硕士,黄凯
硕士,黄壮雄硕士,王军转硕士和周慧梅硕士等同学结下了深情厚谊,在此对
他们陪伴我渡过的美好时光以及对我的帮助深表感谢!濮林博后指导作者进行
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Yang Zheng--Investigation of Group IV Low-Dimensional Nanostructures Acknowledgements
过一些文章修改的工作;杨红官博士给予作者不少帮助,特别是在作者作为本
科生刚刚进入实验室不久;鄢波博士经常和作者一起做实验,使作者从中学到
了很多知识;闾锦博士是作者平时在实验室接触最多的同学,从他身上也学到
了很多东西;肖洁硕士在计算软件运用方面给予作者很多帮助;黄壮雄硕士是
平时实验室中和作者讨论最多的同学,作者从中收益不少;陈杰智博士和赵立
青硕士是作者最经常的足球伙伴。另外,感谢实验室的周建军硕士帮助作者进
行样品的快速退火,感谢叶建东博士在 AFM 图像处理上提供的帮助。
作者特别感谢 California 大学电气工程系的 J. L. Liu 教授和 K. L. Wang 教
授提供了锗硅量子点超晶格的样品。特别感谢 J. L. Liu 教授对我的几篇文章的
指点。
感谢南京大学化学系的胡征教授和王喜章博士提供和帮助进行碳纳米管
的 PECVD 生长。感谢中国科学院物理所王恩哥研究员和白雪冬研究员提供和
帮助进行碳纳米管的热丝法生长。
感谢中国科学院南京地质古生物所的茅永强高级工程师协助 SEM 表征。
感谢南京大学物理系李雪飞老师协助 AFM 表征。感谢南京大学物理系沈剑沧
老师和分析中心陈强老师协助 Raman 测量。
感谢中国科学院半导体所杨富华研究员和陈涌海研究员提供变温PL测量,
感谢边历峰博士和屈玉华硕士她俩亲自帮助作者进行低温和变温 PL 测量。
感谢中国科学院物理所闻海虎研究员提供变温 Hall 测量,特别感谢高红博
士帮助作者进行 Hall 测量,记得实验当天由于次日仪器安排紧张,她一直帮助
作者将实验进行到凌晨一点。
特别感谢南京大学分析中心的程光煦教授在 Raman 理论和实验方面的指
导和帮助。
感谢南京大学物理系李正中教授对作者的一篇关于碳纳米管群的文章提的
一些指导性意见,感谢刘法教授抽空阅读了这篇文章,并从“群”的角度同作
者进行了有益讨论。
最后,我要感谢我的父母,感谢他们的养育之恩,感谢他们这么多年来对
我的学业的支持!
杨 铮
2004 年 2 月于南京大学
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