The novelty and surface to volume ratio dependent electron band structure in semiconductor nanowire

THE NOVELTY AND
SURFACE-TO-VOLUME-RATIO DEPENDENT
ELECTRON BAND STRUCTURE IN
SEMICONDUCTOR NANOWIRE

YAO DONGLAI
(Master of Science)


A THESIS SUBMITTED
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


DEPARTMENT OF PHYSICS


NATIONAL UNIVERSITY OF SINGAPORE

2011



THE NOVELTY AND SURFACE-TO-VOLUME-RATIO DEPENDENT
ELECTRON BAND STRUCTURE IN SEMICONDUCTOR NANOWIRE


YAO DONGLAI
2011
Acknowledgement i

Acknowledgement

This thesis summarizes my research work that has been done since I came to
Professor Li Baowen’s group in 2006. During my PhD study, I have worked
with quite a lot of people whose contribution in assorted ways to the research
and the making of this thesis deserved special mention. It is my pleasure to
show my gratitude to them all in my humble acknowledgment.

In the first place I would like to record my gratitude to Li Baowen for his
supervision, advice, and guidance from the very early stage of this research as
well as giving me extraordinary experiences through out the work. Above all
and the most needed, he provided me unflinching encouragement and support
in various ways. Anytime I was in confusion or lost direction in my study, he
will rectify my mistake and guide me to the right way. His truly scientist
intuition has made him as a constant oasis of ideas and passions in science,
which exceptionally inspire and enrich my growth as a student and a
researcher. I also thank him for giving me continuous support and help on
applying NUS research scholarship and President Graduates Fellowship,
China Overseas Excellent Graduates Awards, Research Assistant position in
National University of Singapore and IME Astar Singapore, which support me
Acknowledgement ii


from the very base during my whole candidature of my PhD. Without him, I
can never reach here. I thank him from my deep heart.

I gratefully acknowledge Professor Zhang Gang, my co-supervisor, for his
advice, supervision, and crucial contribution, which made him a backbone of
this research and so to this thesis. His involvement with his originality has

triggered and nourished my intellectual maturity that I will benefit from, for a
long time to come. Professor Zhang, I am grateful in every possible way and
hope to keep up our collaboration in the future. Furthermore, I thank him for
using his precious times to read this thesis and gave his critical comments
about it. I am indebted to him more than he knows.

Many thanks go in particular to Professor Wang Jian-sheng, Professor Gong
Jiangbin. I am much appreciated for their valuable advice in science
discussion, supervision in courses of computational physics, advanced
quantum dynamics. I have also benefited by advice and guidance from
Porfessor Wang, who also kindly grants me his time even for answering some
of my unintelligent questions.

I would like to thank Department of Physics, Centre for Computation Physics
Acknowledgement iii


in National University of Singapore (NUS). It is them who provide me such a
good research environment and financial support. A lot of thanks go to Dr.
Zhang Xinhuai and Shin Gen, who gives me a lot of help on the high
performance computing in SVU and CCSE in NUS.

I also benefited a lot from Professor Li Zhenya, Professor Gao Lei, Professor
Shen Mingrong, Professor Jiang Qin, Professor Wu Yinzhong, Professor Zhu
Shiqun, Professor Gu Jihua, Professor Gan Zhaoqiang, Professor Nin
Zhaoyuan, Professor Fang Liang, Professor Mu Xiaoyong and President Zhu
Xiulin during my bachelor and master study in Suzhou University. I specially
thank Professor Li Zhenya for his mentorship during my master degree, and
Professor Gao Lei for his recommendation to NUS. Many thanks go to
Professor Guo Guangyu in National Taiwan University for his patient

explanation and detailed instructions on my very first step in the field of
ab-init computational physics.

To all the group member: Wang Lei, Wu Gang, Lan Jinhua, Li Nianbei, Yang
Nuo, Wu Xiang, Chen Jie, Zhang Lifa, Ren Jie, Shi Lihong, Ni Xiaoxi, Zhang
Kaiwen, Xie Rongguo, Xu Xiangfang, Zhu Guimei, Zhang Xun, Ma Jing,
Feng Lin, I thank you so much for your useful discussion, sincere comments,
Acknowledgement iv


and instructive suggestions not only in the weekly group meeting but also in
our personal conversation. I am proud to record that I had several years to
work with you all.

Where would I be without my family? My parents deserve special mention for
their inseparable support. My mother, Yu Huiyu, in the first place is the person
who put the fundament my learning character, showing me the joy of
intellectual pursuit ever since I was a child. My father, Yao Huaxiang, is the
one who sincerely raised me with his caring and gently love.

Words fail me to express my appreciation to my wife, Hu Wei, whose
dedication, love and persistent confidence in me, has taken the load off my
shoulder. I owe her for being unselfishly let her intelligence, passions, and
ambitions collide with mine. Therefore, I would also thank my parents in-law
for letting me take her hand in marriage, and accepting me as a member of the
family, warmly.

Finally, I would like to thank everybody who was important to the successful
realization of thesis, as well as expressing my apology that I could not mention
personally one by one.

Table of Content


Acknowledgement ···································································· i

Abstract (Summery) ································································· v

Publications ·········································································· vii

List of Tables ······································································· viii

List of Figures ······································································· ix

Chapter 1 Introduction ······················································· 1
1.1 General background from nanotechnology to silicon nanowires.
·············································································· 1
1.2 Literature review ···················································· 3
1.3 Introduction to our work ··········································· 6
References ······························································· 10

Chapter 2 Modelling and Methodology ·································· 13
2.1 Density Functional Theory········································ 13
2.2 Tight Binding Method ············································· 19
2.3 Density Functional Tight Binding ······························· 21
2.4 DFT applied to Silicon nanowires ······························· 22
2.5 Discussion ··························································· 24
References ······························································· 26

Chapter 3 A Universal Expression of Band Gap for Silicon Nanowires
of Different Cross-Section Geometries ····································· 31

3.1 Introduction ························································· 32
3.2 SVR(Surface-to-Volume Ratio) ·································· 32
3.3 Density Functional Tight Binding (Methodology) ············ 33
3.4. Results and discussion ············································ 36
3.5 Conclusions ························································· 42
References ······························································· 44

Chapter 4 Impacts of size and cross-sectional shape on surface lattice
constant and electron effective mass of silicon nanowires ··············· 53
4.1 Introduction ························································· 54
4.2 Methodology ························································ 56
4.3 Results and discussion ············································· 59
4.4 Conclusions ························································· 64
References ······························································· 66

Chapter 5 Direct to Indirect Band Gap Transition in [110] Silicon
Nanowires ······································································· 73
5.1 Introduction ························································· 74
5.2 Density Functional Theory and DMol3 ························· 76
5.3 Results and discussion ············································· 77
5.4 Conclusions ························································· 81
References ······························································· 83

Chapter 6 Conclusion and Future Research ····························· 89
6.1 Conclusion ·························································· 89
6.2 Future Research ···················································· 92
References ······························································· 96

Abstract v

ABSTRACT

THE NOVELTY AND SURFACE-TO-VOLUME-RATIO DEPENDENT
ELECTRON BAND STRUCTURE IN SEMICONDUCTOR NANOWIRE

By Donglai Yao

In the field of nanotechnology, we focus this thesis on the novelty and
surface-to-volume ratio dependent electronic band structure in semiconductor
nanowires by means of first principle calculation. Silicon nanowires (SiNWs)
in [110] growth direction is main research object, whose cross-sectional
geometrics and surface-to-volume ratio dependence on the electronic band gap,
effective mass are covered in this thesis. We have found that there is a
universal band gap expression which is only related to surface-to-volume ratio
for nanowires with dimension up to 7 nm. Most interestingly, this expression
is a linear dependence of band gap on surface-to-volume ratio, which is
independent of the specific cross sectional shape. We also explore the electron
effective mass of [110] silicon nanowires with different cross sectional shapes.
We found that the electron effective mass decreases with the SiNW transverse
dimension (cross sectional area) increases. With the same cross sectional area,
Abstract vi

the triangular cross section SiNW has larger electron effective mass than that
of rectangular cross section SiNW. We also trying to find the direct to indirect
band gap transition in [110] SiNWs. We successfully estimated the critical
dimension where this direct-indirect band gap transition takes place by using
the gauge of SVR and the DFT calculation results. It is found that tri-SiNW
has the largest transition dimension up to 14 nm in diameter.
Publication List vii

Publication List:


1. A Universal Band Gap Expression for Silicon Nanowires of
Different Cross-Sectional Geometries, Donglai Yao, Gang Zhang,
and Baowen Li, Nano Letters, 2008, 8 (12), 4557-4561

2. Impacts of Cross-Sectional Shape and Size on Electron Effective
Mass of Silicon Nanowires, Donglai Yao, Gang Zhang, Guo-Qiang
Lo and Baowen Li, Applied Physics Letters, 94, 113113, 2009

3. Size dependent thermoelectric properties of silicon nanowires,
Lihong Shi, Donglai Yao, Gang Zhang, Baowen Li, Applied
Physics Letters, 95, 063102, 2009

4. Large Thermoelectric Figure of Merit In Si1-xGex Nanowires
Lihong Shi, Donglai Yao, Gang Zhang, Baowen Li, Accepted for
publication on Applied Physics Letters, 2010

5. Direct to Indirect Band Gap Transition in [110] Silicon Nanowires,
Donglai Yao, Gang Zhang, and Baowen Li, submitted

List of Table viii

LIST OF TABLES

Table:


Table 3.1: Transverse dimension D (in nm), cross section area A (in nm

2
) and the number
of atoms N in the supercell in our calculations. The dimension D is defined as the largest
distance between the terminating hydrogen atoms in the cross section plane. ············ 48

Table 5.1: ΔE versus SVR relationship, critical SVR and diameters for tri-, rect-, and
hex-SiNWs studied in this work. ································································ 88

List of Figures ix


LIST OF FIGURES

Figures:

Figure 3.1: (Color online) Schematic diagrams of the SiNWs used in our
calculations. From left to right, they are the tri-, rect- and hex-SiNWs. In
tri-SiNW, the angle α is 70.6º and β is 54.7º, where this structure is in
accordance with the nanowires studied in the experimental work in Ref. 23.
The blue dotted lines represent the virtual cages used to construct the SiNWs.
Si and H atoms are represented in yellow and white, respectively. ············ 47

Figure 3.2: Energy band structure for tri-SiNWs with transverse dimension of
(a) D=1.79 nm and (b) D=4.09 nm. The valence band maximum has been
shifted to zero. The blue dotted lines are drawn to guide the eyes. ············ 49

Figure 3.3: Energy band structure for rect-SiNWs with transverse dimension
of (a) D=1.66 nm and (b) D=3.85 nm. The valence band maximum has been
shifted to zero. The blue dotted lines are drawn to guide the eyes. ············ 50

Figure 3.4: Energy band structure for hex-SiNWs with transverse dimension
List of Figures x


of (a) D=1.40 nm and (b) D=3.71 nm. The valence band maximum has been
shifted to zero. The blue dotted lines are drawn to guide the eyes. ············ 51

Figure 3.5: (a) Band gap versus the transverse dimension D. (b) Band gap
versus SVR. The red solid line is the best-fit one with slope 0.37±0.01 eV-nm.
Inset of (b) is the band gap versus SVR relation based on the results from Ref.
15 for hex-SiNWs. ···································································· 52

Figure 4.1. Schematic diagrams of the tri-SiNWs and rect-SiNWs used in our
calculations. The blue dotted lines represent the virtual cages used to construct
the SiNWs. Si and H atoms are represented in yellow and grey, respectively.69

Figure 4.2. The surface lattice constant versus the transverse dimension of
SiNWs. a
0
=5.46 Å is the calculated lattice constant of bulk silicon. ·········· 70

Figure 4.3. (a) Energy band structure for tri-SiNWs with cross sectional area
A=1.9 nm
2
. (b) Energy band structure for rect-SiNWs with A=2.3 nm
2
. In (a)
and (b), the valence band maximum has been shifted to zero. (c) The lowest
CBM for tri-SiNWs with different transverse dimensions. (d) The lowest CBM
for rect-SiNWs with different transverse dimensions. In (c) and (d), the CBM

List of Figures xi


has been shifted to zero. The symbols are DFTB simulation results, and the
solid lines are the best-fit ones with parabolic approximation. ················· 71

Figure 4.4. The electron effective mass versus the transverse cross sectional
area for tri- and rect-SiNWs. m
e
is the mass of free electron. ·················· 72

Figure 5.1: Cross-section view of the 3 types of SiNWs: Tri-SiNW (Triangular
cross-section SiNW), Rect-SiNW (Rectangular cross-section SiNW) and
Hex-SiNW (Hexagonal cross-section SiNW). a, b and c are the lateral facets,
which are (100), (110) and (111), respectively. In Tri-SiNW, the angle α is
70.6º and β is 54.7º. The blue dotted lines represent the virtual cages used to
construct the SiNWs. Si and H atoms are represented in yellow and white,
respectively. ············································································ 85

Figure 5.2: The dependence of conduction band edges on the SVR for
hex-NWs. Inset is the band structure of hex-NW with cross sections area of
7.29 nm
2
. ················································································ 86

Figure 5.3: (Color online) ΔE versus surface-to-volume ratio (SVR) for tri-,
rect-, and hex-SiNWs. ······························································· 87
[Chapter1.Introduction]1

Chapter 1

Introduction



1.1 General background from nanotechnology to silicon
nanowires.

Nanotechnology is defined as the field of science and technology devoted to
studies of the synthesis, properties and applications for structures and
materials with at least one critical dimension less than the scale of
approximately 100 nm [1] Nanotechnology is the ability to manipulate
individual atoms and molecules to produce nanostructured materials and
submicron objects that have applications in the real world. Nanotechnology
involves the production and application of physical, chemical and biological
systems at scales ranging from individual atoms or molecules to about 100
nanometers, as well as the integration of resulting nanostrucures into lager
systems. Nanotechology is likely to have a profound impact on our economy
and society in the early 21st century, perhaps comparable to that of
information technology or cellular and molecular biology. [2] Science and
[Chapter1.Introduction]2

technology research in nanotechnology promises breakthroughs in areas such
as materials and manufacturing, nanoelectronics, medicine and healthcare,
energy, biotechnology, information technology and national security. It is
widely felt that nanotechnology will be the next industrial revolution.

Nanowires are attracting much interest from those seeking to apply
nanotechnology and (especially) those investigating nanoscience. Nanowires,
unlike other low-dimensional systems, have two quantum-confined directions
but one unconfined direction available for electrical conduction. This allows

nanowires to be used in applications where electrical conduction, rather than
tunneling transport, is required. Because of their unique density of electronic
states, in the limit of small diameters nanowires are expected to exhibit
significantly different optical electrical and magnetic properties to their bulk
3-D crystalline counterparts. Increased surface area, very high density of
electronic stats and joint density of states near the energies of their van Hove
singularities, enhanced exiton binding energy, diameter-dependent band gap,
and increased surface scattering for electrons and phonons are just some of the
ways in which nanowires different from their corresponding bulk materials.
Yet the sizes of nanowires are typically large enough ( >1nm(~2 x Si-Si Bond)
in the quantum-confined direction) to result in local crystal structures that are
[Chapter1.Introduction]3

closely related to their parent materials, allowing theoretical predictions about
their properties to be made based on knowledge of their bulk prosperities. Not
only do nanowires exhibit many properties that are similar to, and others that
are distinctly different from, those of their bulk counterparts, nanowires also
have the advantage from an applications standpoint in that some of the
materials parameters critical for certain properties can be independently
controlled in nanowires but no in their bulk counterparts. Certain properties
can also be enhanced nonlinearly in small-diameter nanowires, by exploiting
the singular aspects of the 1-D electronic density of stats. Furthermore,
nanowires have been shown to provide a promising framework for applying
the “bottom-up” approach to design of nanostructures for nanoscience
investigations and for potential nanotechnology applications.

1.2 Literature reviews.

In “Nanowire Nanosensor for Highly Sensitive and Selective Detection of
biological and Chemical Species”, Cui et al.[3] reported their fundamental

work in the interdisciplinary field of nanowire biosensor. They use the
boron-doped silicon nanowires (SiNW) as the highly sensitive, real-time
electrically based sensors for biological and chemical species. They noted that
[Chapter1.Introduction]4

nanowire (NW) and nanotube (NT) can be regarded as possible two candidates
of the nano-scale biosensors. They prefer SiNW because silicon does have a
well established industrial base, and the dopant type and concentration of
SiNW can be well controlled. In addition, the massive existing knowledge of
chemical modification of oxide surfaces about SiNW can be exploited.

Bent et al. examined the reaction of the Si(100)-2x1 surface with
five-membered cyclic amines. They find out that the connection between
Si(100)-2x1 surface and organic molecules could be N-Si bond based on the
N-H dissociation. From their work, we can learn that not only the C-Si bond
could be formed at the Si surface, but also the N-Si bond. This give us a new
idea that we can attach the chemical functional groups, such as amino acid
( with N-H bond), as the linking molecules to the SiNW.[4]

In Ref. [5], N. Lorente, et al. studied the SiNW grown along the <100>
direction with a bulk Si Core using the density-functional calculations. They
found two kinds of surface reconstructions appear with energetical
equivalence. One of the reconstructions is found to be strongly metallic while
the other one is semimetallic. These results show us that doping is not required
in order to obtain good conducting Si nanowire. As we stated above in the
[Chapter1.Introduction]5

introduction part, it is one of our research interests: the effect of the surface
reconstruction.

Ponomareva et al. [6, 7] studied the structure stability, electronic properties
and quantum conductivity of small-diameter silicon nanowires. In their papers,
they studied quite a few different Silicon Nanowire structures with diameters
ranging from 1 to 6nm using the GTBMD scheme [8]. The different growth
directions ([111], [110], and [100]) of the nanowire are also investigated. They
found that the tetrahedral type nanowires oriented in the <111> direction are
the most stable. They also found that the cage-like nanowires have better
electrical conducting properties. Getting these results, we have the ideas that
the contributions of the surface energy play an important role in the stable
nanowire structure.

In Ref. [9], Rurali et al. gives us a detailed report on the size effects in
surface-reconstructed <100> and <110> silicon nanowires. They performed
ab-init calculation on the electronic structure to study the surface
reconstructions of <100> and <110> nanowires with different diameters. The
diameter of the small size nanowire is another important factor to influence the
band-structure and electronic structure of the nanowire.
[Chapter1.Introduction]6

1.3 Introduction to our work

Extensive investigations have been carried out on the synthesis, properties and
applications of SiNWs. Experimental technology has been developed to
control the growth of SiNWs not only in various growth orientations, but also
with various shapes of transverse cross section including rectangle (square),
hexagon (rough circle), and triangle.[10-14] A large number of theoretical and
experimental works have been done to explore the effect of chemical
passivation, surface reconstruction, and growth orientations on electronic
structures. [11, 15-20] However, compared with the study of these impacts on
electronic properties of SiNWs, much less has been done on the impacts of

cross sectional geometries. Focusing on the effect of the cross sectional
geometries, we have present our research result on this area in Chapter 3.

As the size of the materials is reduced to nanometer regime, the physical and
chemical properties of nano scale materials can be significantly changed.
One-dimensional systems, such as nanowires and nanotubes are of outstanding
current interest as one of the promising building blocks for future nanoscale
electronic, optoelectronic, and phononic devices. As the demands of more
compact devices emerge, silicon nanowires (SiNWs) have attracted extensive
[Chapter1.Introduction]7

attention due to their compatibility with Si-based electronic technology. The
fascinating potential applications [21] such as novel power device [22],
thermoelectric materials [23-26] and, biological and mechanical sensors [27,
28] have attracted wide research interests in recently years. Inspired by
experimental works, more and more theoretical efforts have been made to
understand the electronic properties of SiNWs. The impacts of the diameter,
surfaces reconstruction, and doping have been reported. [23-29] Further
developments of SiNW device design require theoretical tools that can provide
reasonable quantitative predictions for realistic structures both accurately and
time saving. This is hard to achieve with ab initio calculations which can only
be applied to systems of limited size. One possible way of improving this
situation is to use semi-empirical approach such as single band effective mass
approximation, which can handle much larger systems.

It is well known that the ultimate speed of integrated circuits depends on the
carrier mobility, which is inversely proportional to the effective mass. So the
concept of effective mass plays a key role in nanoscale electronics and
photonics device design. It has been found that the effective mass of SiNW
increases as the transverse dimension decreases, [30] and the effective mass

can also be modified by uniaxial strain. [31] Applying strain is a useful
[Chapter1.Introduction]8

method to modulate band structures and enhance the device performance. It is
also a very economical way and has the advantage of being compatible with
current CMOS process. The strain can be experimentally realized through
depositing a capping layer around the SiNW. At the interface between NW
core and the cover layer, the lattice constant mismatch couples to the
electronic as well as the optical properties of such system. So it is
indispensable to accurately evaluate surface lattice constant of SiNWs.
Although some research has been done, many important and fundamental
questions remain unsolved. For example, what is the quantum confinement
effect on surface lattice constant? And how does the surface lattice constant
and electron effective mass depend on nanowire cross-sectional shape? This
will be answered in Chapter 4.

Recently, SiNWs attract considerable attention for energy harvesting
applications, such as solar cell, due to their unique optical and electrical
characteristics. The good absorption for solar energy using SiNW arrays has
been demonstrated experimentally and theoretically. [32-34] It is well known
that bulk silicon has an indirect band gap, with the valence-band-maximum
(VBM) locates at the Γ point and the conduction-band-mimimum (CBM)
locates approximately 85% from Γ to X. The indirect band gap characteristic
[Chapter1.Introduction]9

limits the application of silicon in optoelectrics. However, it is demonstrated
both experimentally [35, 36] and theoretically [37, 38] that ultra-thin (1.3 nm
in diameter) hydrogen-terminated [110] and [111] SiNW are direct band gap
semiconductors. The fundamental band gap characteristic is key role in many
applications, such as light emission and absorption. It is indicated the

possibility of fabricating Si-based visible optical devices.
Obviously, there exists indirect to direct band gap transition. Using
first-principles calculations, it has been shown that the indirect-to-direct
energy gap transition for [111] SiNWs at a diameter of less than 2.2 nm. [37]
However, the situation with [110] SiNW is still unknown, which wires seem to
be the most promise candidate for the nanowires electronic device and
bio-sensor because the band gaps of [110] SiNWs is the smallest among those
of the [100], [112] and [111] wires of the same diameter [39]. Moreover, so far
there is no systematic report on the indirect-direct band gap transition, and its
dependence on the geometry of SiNWs. As the indirect band gap and
consequential weak light absorption remain the bottleneck for their application
in optoelectronics/solar PV, a detailed understanding of the indirect-to-direct
band transition via diameter is of primary importance to the development of
new applications for SiNWs. It could be extremely difficult to get this
information from experiments. In Chapter 5, we studied the direct to indirect
[Chapter1.Introduction] 10

band gap transition when the transverse cross section of SiNW is increased, by
using first-principle calculations. An extremely linear dependence on the
surface-to-volume ratio is found for both the lowest and second-lowest
conduction band edge.

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