Investigation of Group IV Low-Dimensional Nanostructureszyang/Resume/Thesis/ZhengYangMasterT… ·...

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


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


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

iii

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


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

2


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


EED /1)(1 ∝ρ


)()(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

3


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


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

5


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.

7


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


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].

9


References

[1] S. Iijima, Helical microtubules of graphitic carbon, Nature 354, 56-58 (1991).

[2] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, and R. E. Smalley, C60:

Buckminsterfullerene, Nature 318, 162-163 (1985).

[3] S. Iijima and T. Ichihashi, Single-shell carbon nanotubes of 1-nm diameter,

Nature 363, 603-605 (1993).

[4] D. S. Bethune, C. H. Kiang, M. S. de Vries, G. Gorman, R. Savoy, J. Vazquez,

and R. Beyers, Cobalt-catalysed growth of carbon nanotubes with

single-atomic-layer walls, Nature 363, 605-607 (1993).

[5] i) C. T. White and T. N. Todorov, Nanotubes go ballistic, Nature 411, 649-651

(2001); ii) S. Frank, P. Poncharal, Z. L. Wang, and W. A. de Heer, Carbon

nanotube quantum resistors, Science 280, 1744-1746 (1998).

[6] P. Kim, L. Shi, A. Majumdar, and P. L. McEuen, Thermal transport

measurements of individual multiwalled nanotubes, Phys. Rev. Lett. 87,

215502 (2001).

[7] i) M. M. J. Treacy, T. W. Ebbesen and J. M. Gibson, Exceptionally high Young’s

modulus observed for individual carbon nanotubes, Nature 381, 678-680

(1996); ii) E. W. Wong, P. E. Sheehan, and C. M. Lieber, Nanobeam mechanics:

elasticity, strength, and toughness of nanorods and nanotubes, Science 277,

1971-1975 (1997); iii) M. F. Yu, O. Lourie, M. J. Dyer, K. Moloni, T. F. Kelly,

and R. S. Ruoff, Strength and breaking mechanism of multiwalled carbon

nanotubes under tensile load, Science 287, 637-640 (2000).

[8] i) S. Fan, M. G. Chapline, N. R. Franklin, T. W. Tombler, A. M. Cassell, and H.

Dai, Self-oriented regular arrays of carbon nanotubes and their field emission

properties, Science 283, 512-514 (1999); ii) J. A. Misewich, R. Martel, Ph.

Avouris, J. C, Tsang, S. Heinze, and J. Tersoff, Electrically induced optical

emission from a carbon nanotube FET, Science 300, 783-786 (2003).

[9] i) A. Y. Kasumov, R. Deblock, M. Kociak, B. Reulet, H. Bouchiat, I. I. Khodos,

10


Y. B. Gorbatov, V. T. Volkov, C. Journet, and M. Burghard, Supercurrents

through single-walled carbon nanotubes, Science 284, 1508-1511 (1999); ii) Z.

K. Tang, L. Zhang, N. Wang, X. X. Zhang, G. H. Wen, G. D. Li, J. N. Wang, C.

T. Chan, and P. Sheng, Superconductivity in 4 Angstrom Single-Walled Carbon

Nanotubes, Science 292, 2462-2465 (2001).

[10] i) A. C. Dillon, K. M. Jones, T. A. Bekkedahl, C. H. Kiang, D. S. Bethune, and

M. J. Heben, Storage of hydrogen in single-walled carbon nanotubes, Nature

386, 377-379 (1997); ii) C. Liu, Y. Y. Fan, M. Liu, H. T. Cong, H. M. Cheng,

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


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


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
c
C80

Figur
e 2.2 Schematics of fulle
1

r

4

(n, m)=(10, 5)enes and carbon nanotubes. [1]


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


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


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


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


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


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

http://cnst.rice.edu/reshome.html

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


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


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


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


−=

+=

−=

+=

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


∑+

+⋅−=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


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


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


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


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


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


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


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


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


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


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


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


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


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.

40


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

41


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


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


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


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


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


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


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


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

50


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%±

51


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


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.

53


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


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

,

56


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.

57


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.

58


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

59


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)


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.

60


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

61


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

62


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

63


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


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

65


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)


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.

66


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.,

67


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


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


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


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


References

[1] C.V. Raman and K.S. Krishnan, The optical analog of the Compton effect,

Nature 121, 711 (1928).

[2] i) J. Zi, H. Büscher, C. Falter, W. Ludwig, K. Zhang, and X. Xie, Raman shifts

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

self-organized Ge dot superlattice’” [Appl. Phys. Lett. 75, 3572 (1999)], Appl.

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.

72


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.

B 68, 125302 (2003).

[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

73


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

spatial correlations in quantum dot multilayers: Raman scattering interferences,

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.

B 35, 2243-2251 (1987).

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


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


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


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


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


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


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


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

82


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

83


References

] G. D. Gilliand, Photoluminescence spectroscopy of crystalline semiconductors,

Mater. Sci. Eng. R 18

[1

, 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


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


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


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


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


(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

108

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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