Nanocomputing computational physics for nanoscience and nanotechnology

RPS Nano Computing “preface” 2008/10/5 viiPREFACE The contents of this book are based on the material of Nano Com-puting course I taught at National Tsing Hua University since 2004.. RPS

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James J Y Hsu

NANOCOMPUTI NG

Computational Physics for

Nanoscience and Nanotechnology

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NANOCOMPUTING

Computational Physics for Nanoscience and Nanotechnology

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RPS Nano Computing “preface” 2008/10/5 vii

PREFACE

The contents of this book are based on the material of Nano

Com-puting course I taught at National Tsing Hua University since 2004

Nanotechnology is catching attention and gaining importance in both

academia and industry alike, and students are very much interested in

this emerging topic There is the need to have a coherent presentation

on the related disciplines, namely, theoretical physics, computer

sci-ence, applied mathematics, and engineering study In considering the

importance of the four technologies for the future, Nano Technology

(NT), Biomedical Technology (BT), Information Technology (IT),

and Ecology Technology (ET), the course is designed to give breadth

on related subjects, but keep depth on computation and physics

On the theoretical side, we cover the Mesoscopic Physics and

Nonlinear Many Body Physics On the computer science, Object

Ori-ented Programming and Parallel Computing are incorporated On the

applied mathematics, Asymptology and Algorithm are reviewed For

the engineering training, some applications and MATLAB are

pre-sented Students are introduced to the multiscales and multisciences

from this book, and are requested to solve all the problems by either

MATLAB or C++

The target audience for the book is students at the senior and

graduate level The emphasis of this book is to teach students to solve

problems from the features and characteristics of the problem itself,

and not from a presumed methodology or a predefined tool It tries

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RPS Nano Computing “preface” 2008/10/5 viii

Preface

to avoid the students from falling into the mind frame of what the

old saying, “If you are a hammer, everything else is a nail.” The

rightful problem solving mentality is let the problem reveal where the

solution might be, and study the clues to find the answers Therefore,

start from the asymptotic analysis once the problem is translated

into a mathematical equation, and get all the hints possible even if a

numerical solution is inevitable

This book is organized as follows: It introduces the issues in

nanoscience, reviews the mathematical tools both numerical and

ana-lytical, and then applies the tools to more advanced problems through

a repetition of the ideas and an increase in the level of sophistication

so as to allow a deeper understanding of the physics and the

prob-lem solving techniques Finally, it applies the scientific knowledge

for practical applications The ultimate goal of this book is to

pre-pare students with enough background to start working on a research

dissertation in theoretical nanoscience

James J Y HsuMarch 2008

viii James J Y Hsu

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RPS Nano Computing “preface” 2008/10/5 v

ACKNOWLEDGEMENTS

I would like to thank Professor T L Lin for suggesting the course

title, and ESS faculty and students for giving me the opportunity to

teach this course The interaction with Professor C H Tsai’s

Car-bon Nanotube group was most beneficial Many insightful help from

colleagues, post-doctors and students at both NCKU and NTHU

are gratefully acknowledged Some derivations and programs were

aided by Yee Mou Kao, Young-Chung Hsue, Chun Hung Lin, Eugene

Pogorelov, Chieh-Wen Lo, Ying-Chi Chung, Chi-Yeh Chen, Robert

Weng, Wellin Yang, Lichung Ko, and Cheng Hao Wu This book

was proofread by Dr Fay Sheu I also thank my wife, Dr Yen-Hwa

Hsu, and my daughters, Ingrid and Jessica, for their support to let me

concentrate on research in Taiwan for the past few years

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CONTENTS

1.1 Tools for Measurement — To See is to Believe 4

1.2 Carbon Tells Us First 7

1.3 Mother Nature Knows Best 10

1.4 Challenges in the New Millennium 12

Chapter Two Tools for Analysis 19 2.1 MATLAB 20

2.2 Program Control 29

2.3 Asymptology 33

Chapter Three Mesoscopic Systems 59 3.1 Review on Quantum Physics 59

3.2 Quantum Chemistry 78

3.3 Molecular Biology 88

3.4 Condensed Matter Physics 91

Chapter Four Analytical Chapter 115 4.1 Multiple Time Scales 116

4.2 Multiple Space Scales 124

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Contents

5.1 Recursion and Divide-and-Conquer 136

5.2 Probabilistic Algorithm 139

5.3 Evaluation and Search 150

5.4 Molecular Dynamics 159

5.5 Finite Element Method 164

Chapter Six Nonlinear Many Body Physics and Transport 187 6.1 Density Functional Theory 189

6.2 Correlation and Coherence 199

6.3 Green’s Function Method 204

6.4 Transport 218

Chapter Seven OOP, MPI and Parallel Computing 227 7.1 C++ and Object Oriented Programming 228

7.2 Message Passing Interface 233

7.3 OpenMP 242

Chapter Eight Low Dimensionality and Nanostructures 245 8.1 Quantum Dot and Quantum Wire 245

8.2 Nanostructure Electronic Properties 252

Chapter Nine Special Topics 261 9.1 Plasmon 261

9.2 Quantum Hall Effect 277

9.3 Chaos and Stochasticity 284

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RPS Nano Computing “Ch01” 2008/9/25 1

Chapter One

LITTLE BIG SCIENCE

“Look deep into nature, and then you will understand

everything better.”

Albert Einstein (1879–1955)

In a talk given in 1959, Richard Feynman asked, “Why cannot

we write the entire 24 volumes of the Encyclopedia Brittanica

on the head of a pin?” He went ahead to suggest that devices

and materials could someday be manipulated to atomic

speci-fications, and “The principles of physics, as far as I can see, do not

speak against the possibility of maneuvering things atom by atom.”

Nevertheless, there was not much progress in this direction after

perhaps that owing to the lack of instruments needed to perform what

was desired More importantly, scientists could not really “see” what

nature was doing This, however, changed in the 1980s when progress

was made on detection devices capable of looking deep into nature

The electron optics and the scanning tunneling microscope (STM)

won Ernst Ruska, Gerd Binnig and Heinrich Rohrer the Nobel Prize

in 1986 The other developments such as the atomic force microscopy

(AFM) helped open up the nano domain

Nanoscience provides new approaches to material science and

material engineering, and offers a great opportunity to upgrade

exist-ing industries from bottom up When manipulated from the atomic

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Little Big Science

level, ample examples of products can be magically improved Of the

four major leading industries in this century — electronics and IT

(Information Technology), biomedical, energy, and transportation —

nano science will impact greatly on them In particular, there will

be optimized materials, biomaterials, and smart materials Not only

will it be a new technology, but also a new man-nature relationship

Scientists and engineers have to rethink the environmental, health,

and ethical issues The ever-improved man-made materials will

opti-mize to meet the conflicting demands and to provide solutions for the

resource-hungry human society to reduce, reuse and recycle

A matured product is, as a rule, optimized and versatile A case

in point is the cellular phone It is made with ever-greater

function-alities such as digital video camera and Internet connectivity, not

to mention the clock, the alarm, and the address book This in fact

makes the traditional wrist watch obsolete, and also opens up many

Internet services and applications Nature has no shortage of

biologi-cal examples of optimized and versatile construct, perfected through

tens of thousands of years of evolution Some species of squid is

capable of rocketing at 20 miles per hour by ejecting a jet of water

Such force is derived through its muscular contraction coupled with

a smooth outer lining of least hydro-resistance Its muscles

maxi-mize the elasticity and minimaxi-mize the viscosity Another example is

the spider’s web which provides a net that resists the wear and tear

under the sun, the rain and the wind It is not perfect but economically

optimized

Biomaterials in the greater sense include medicine The

ulti-mate success in the stem cell technologies will create bioulti-materi-

biomateri-als with a patient’s signature to allow for cell therapy or organ

transplant Another exciting breakthrough in 2007 is the discovery

that human skin cells can be re-programmed to become “induced

pluripotent stem cells” After the human genome project was

com-pleted in 2002, the book of life was opened for further reading as

a recipe to prolong life The race for genomic medicine has just

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begun Drug discovery will benefit tremendously from nano

com-puting For example, once the 3D structure of a protein is known,

potential inhibitors capable of blocking specific active sites can be

screened through computer simulation This will greatly shorten the

time to market, and reduce the cost of animal tests and clinical trials

And the dream is to have personalized medicine, prescribed

accord-ing to an individual’s genetic makeup, perhaps even accommodataccord-ing

in human genomes the 0.1% differences due to single nucleotide

polymorphism (SNP).

A purpose of nanobio studies is to be able to confirm many

find-ings inferred from the inductive biological methods, with results

developed from reductive physics methods from the first principles

On the other hand, the Princeton group at the Biologically Inspired

Materials Institute proclaims the possibility of creating the

self-healing skin to alleviate the problems such as the failed protective

tiles on the space shuttle Columbia in 2003 This is a biomaterial

similar to living beings like blood clotting to protect a cut in the

skin In the ultimate sense, it will be able to program the material

to confer with the intelligence of an agent, a robot or a catalyst

to make things happen as needed Biodegradable plastics could be

the least of these examples There is no shortage of examples of

intelligence in living organisms, just think of how a fertilized egg

is programmed to hatch into a chick, or a dandelion flower spreads

its seeds

Will humans be one day smart enough to design drugs that defeat

drug resistance by microorganisms and viruses? So far the strategy of

drug design is to find a way of inhibiting or killing the virus

Unfor-tunately the survived virus such as HIV may find proper mechanisms

to defeat the purpose If the drug takes advantage of similar

mecha-nisms from the virus, viz., mutating according to what the enemy is

doing and developing a new medicine strategy accordingly, we would

have a smart medicine This could be the ultimate smart material man

could make

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1.1 Tools for Measurement — to See is to Believe

Observations enabled the ancient Chinese to record events of comets,

solar eclipses, and supernovae, and to practice acupuncture and herbal

medicine Scientists, from the onset of modern scientific thought in

Greece, struggled for centuries to learn about the sizes of atoms and

molecules Recent developments in microscopy allow scientists to

see and manipulate particles of nanometer dimensions, thus signifies

the beginning of nanotechnology Microscopy, in its many forms, is

one of the most important techniques used to study the size, shape

and characteristics of small objects These include Scanning

Tunnel-ing Microscopy, Electron Microscopy, Atomic Force Microscopy,

Soft X-Ray Microscopy and Optical Microscopy.

The scanning tunneling microscope (STM) provides a

three-dimensional profile of a surface at the atomic scale It is one of the

most powerful and widely employed tools for surface analysis, very

useful in characterizing roughness and defects and determining the

size and conformation of molecules and aggregates on the surface

The STM utilizes quantum tunneling to draw up an electron current A

stylus, or atomically sharp tip, scans the surface of a sample at certain

distance The study of surfaces has important physical, chemical and

biological implications, ranging from the studies of semiconductors,

microelectronics, high precision optical components, metals, surface

chemistry, to those of enzymatic effect and viral infection The STM

works best with conducting materials, but it is also possible to affix

organic molecules on a surface and study the deposited structures

The atomic force microscope (AFM) measures topography with

a force probe Invented by Binnig, Quate and Gerber in 1986, AFM

can be used for surveying the material surface or measuring electric,

magnetic, and other physical or chemical properties in the

nanometer-scale The AFM operates by measuring attractive or repulsive forces

between a tip and the sample to achieve atomic-scale resolution It

is equipped with sensitive and sharp tips, flexible cantilevers, optical

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Tools for Measurement — to See is to Believe

lever, and force feedback circuit The optical lever operates by

reflect-ing a laser beam off the cantilever, thus greatly magnifies motions of

the tip since the cantilever-to-detector distance generally measures

thousands of times the length of the cantilever The force is not

mea-sured directly, but calculated by measuring the deflection of the lever

with the knowledge of the stiffness of the cantilever

The AFM combines the optical probe with the atomic interactions

to achieve greater resolution than the traditional optical microscopy

Both AFM and optical microscopy are powerful tools for gaining

information about structure and function of biomolecules and living

cells Owing to the fact that light cannot pass through an aperture

smaller than its wavelength, the so-called diffraction limit, optical

microscopy has its limitations Optical microscopy however attracted

great attention in the nano regime in 1998 when Ebbesen et al.

demonstrated that a nano layer of silver aggregates enhances the

near-field strength and consequently its resolution Optical microscopy, in

general, is relatively inexpensive and reliable It requires little sample

preparation and works at room temperature and atmospheric pressure

These factors make near-field optics a favored tool as a biosensor in

studying DNA, RNA, or protein.

Light scattering over molecules has been well understood to result

from the elastic scattering of Rayleigh and the inelastic scattering of

Raman The incident photon energy can excite vibrational modes of

the molecules A spectral analysis of the scattered light could reveal

molecular structure Raman scattering has applications in remote

monitoring for pollutants, and widely used to examine for example,

the diameter of carbon nano tube, single or multi-walled

The neutron’s magnetic moment is an ideal probe to study

mag-netic structures in condensed matter physics It can be used to

inves-tigate solid state magnetism, magnetic nanostructures, structural and

magnetic disorder, spin fluctuations and excitations in complex or

nano-structured magnetic systems and highly correlated electron

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systems Small angle x-ray scattering (SAXS) can be an

analyti-cal tool to examine the structural characterization of solid and fluid

materials in the nanometer range SAXS is applied to investigate

structural details in the 0.5 to 50 nm size range in materials such as:

nanopowders, proteins, viruses, DNA complexes, polymer films and

fibers, catalyst surface, and liquid crystals

The microarray is one promising device that will help unravel

the secret of life By far, it is one of the better tools to observe

how a biological system is doing Presently, extracting microarray

data to get meaningful information, however, remains inadequate or

even elusive This may change as the methods of its analysis

con-tinue to improve The impact will be strongly felt once this research

tool becomes practical and effective for clinical applications It has

promising applications in determining a patient’s gene profile as well

as in monitoring the progress during drug treatment The era of

per-sonalized medical care will finally arrive when a patient’s genome

can be deciphered within reasonable time duration A few other tools

such as mass spectrometer, AFM, and Near-Field Optics may be

good candidates to help accomplish that It is widely believed that

thousands of genes and their products (i.e., RNA and proteins) in

a given living organism function in a complicated and orchestrated

way However, traditional methods in molecular biology generally

work on a “one gene, one experiment” basis, which means that the

throughput is very limited and the “whole picture” of gene function

is hard to obtain The DNA microarray that is attracting tremendous

interests has the promising capability of monitoring the entire genome

on a single chip Researchers can have a better picture of the

inter-actions among thousands of genes from the microarray experiment

Base-pairing (i.e., A-T and G-C for DNA; A-U and G-C for RNA) or

hybridization is the underlining principle of DNA microarray

Microarray is a powerful research tool and would be a very

impor-tant clinical diagnostics method Microarray designs might be

cate-gorized into genotype arrays, expression arrays, and protein arrays

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Carbon Tells Us First

In each category, many more varieties are readily available Protein

microarrays have for example, antibody array, antigen array, lysate

array, surface antigen array, human cytosine detection array, allergy

antigen array, protein domain array, small molecular array,

enzyme-protein array, etc The sensitivity, specificity (correlation with the

observable), and reliability define the quality of an array design It

has been used to do early detection of molecular signatures of cancer

disease, to understand the metabolism and protein regulatory

func-tions, to study drug resistant mechanisms during cancer treatment,

and to profile patient prognosis signatures The multi-gene based

approach found the prognosis signature, which consists of genes that

function in regulation of cell cycle, invasion, metastasis and

angio-genesis Patients having tumors with the poor prognosis signature

tend to develop distant metastases shortly afterwards From the

pro-filing, proper course of treatment can be prescribed Some patients

may require surgery only; whereas patients with very poor prognosis

signature may need radiation therapy and chemotherapy following

surgery The method defining the reporter genes that predict distant

metastases appears to have wider validity to other cancer cases, and

obviously is a research topic of great importance

Nanoscience and nanotechnology have evolved to encompass

multi-disciplinary inputs from physics, biology, chemistry and

engi-neering The field is richly benefited from information

technol-ogy, electronics and mechanics; they provide the ultimate tools

for measurement They also develop microsystems with

multi-functionalities, optimized as examplied in microarrays In fact, in

microarray, what is achieved might be thought of as the first in its

kind, a biochip that is equivalent to “the lab on a chip”

1.2 Carbon Tells Us First

Although water is the universal medium for life on Earth, most of the

chemicals that make up living organisms are based on the element

carbon Of all chemical elements, carbon is unparallel in its ability to

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form molecules that are large, complex, and diverse, and this

molec-ular diversity has made possible the diversity of living beings that

have evolved on Earth Carbon atoms are the most versatile building

blocks of molecules The organic chemistry is the study of carbon

compounds The famous carbon family includes carbon nano tube

(CNT), graphite, and diamond In 1985 Robert F Curl, Harold

W Kroto and Richard E Smalley discovered the bucky ball, or

fullerene, a striking compound of carbon atoms arranged in a closed

shell or cage It resembles the geodesic dome designed by the

Amer-ican architect R Buckminster Fuller for the 1967 Montreal World

Exhibition The researchers named the newly-discovered structure

buckminsterfullerene after Fuller The carbon bucky ball C60 (see

Function C60 in 2.1.4) serves as a good example of the greatness and

beauty of nanostructures The fullerene may be considered as a zero

dimensional entity, the CNT a one dimensional entity, the graphite

two dimensional, and diamond three dimensional Fullerenes are

formed when vaporized carbon atoms condense in the inert gas A

cluster of 60 carbon atoms (cf P 30), C60, is the most abundant, and

a molecular structure of great symmetry Its cage structure may be

ideal for drug delivery and its size may be just right as an inhibitor

to attach to the active site of an enzyme It may be made into

per-fect reproducible quantum dot for mass production The discovery

of fullerene and carbon nanotube (CNT) aroused renewed interest

in nanotechnologies making Feynman’s prediction come closer to

reality

After S Iijima published his Nature paper on carbon nanotubes

in 1991, researchers have been fascinated by these nanostructures and

their extraordinary electrical and mechanical properties The many

potential usages of CNTs envisioned include: as field emitters for

flat-panel display, as field effect transistor (FET), or as nano

sen-sors affixed with reaction-specific molecules, and as tips for scanning

probe microscopy There are also potential applications in hydrogen

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Carbon Tells Us First

storage, drug delivery, molecular wires and high-strength

compos-ites The high aspect ratio of CNTs with an intrinsic diameter as

small as 0.7 nm and with a length extending to several microns makes

them ideal as tips for scanning probe Since the diameter of the tip

determines the imaging resolution, CNT tips provide high

resolu-tion The length of CNT tips permits the tracing of rough surfaces

with steep and deep features Furthermore, the extraordinary strength

and the ability to retain structural integrity after deformation

(elas-tically buckling) make CNT scanning probes very robust They can

withstand hard wearing better than the conventional silicon scanning

probes Attaching various tips to CNT may produce scanning probes

that are magnetic, electric, or mechanical The composite materials,

containing carbon nano tubes, may offer many unprecedented

qual-ities The CNT sheet has been shown to hold up weight thousands

times its own Building sensors on the tube may yield wide

applica-tions in biomedicine The flat-panel display with CNT as the electron

emitter offers great pixel resolution Similarly CNT may be used

for the ink jet printing The nano technology doctrine would seek

per mass performance in strength, capability, intelligence as well as

multi-functionalities CNT might just be the material to deliver this

The most expensive form of carbon is diamond, which is

crys-talline carbon found in nature Man-made diamond sheet or diamond

rock, produced by plasma-enhanced, chemical-vapor deposition, is

getting better in the size, cost and speed Artificial diamonds are

widely used in industry As with natural diamond, they possess many

fascinating properties Diamond is extremely low in chemical

reactiv-ity, in thermal expansion and in electrical conductance It is extremely

high in thermal conductance, tensile strength, and mechanical

hard-ness Nano diamond has been applied to enhance resistance to

mate-rial wear, corrosion, and abrasion, and to reduce the porosity

For years, silicon has been the keystone for manufacturing

pro-cessors, memory chips and other chips It is expected to last for at

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least another decade, thanks to breakthroughs in chip making

tech-niques and materials If 19th century is the century of iron, then 20th

century is the century of silicon It can be anticipated that 21st century

would be the century of carbon On the other hand, if 19th century

is the century of macrotechnology, then 20th century is the century

of microtechnology It can be anticipated that 21st century would be

the century of nanotechnology

1.3 Mother Nature Knows Best

The lotus effect is a good example of nature’s ingenuity Although

lotus grows in muddy rivers and lakes, the leaves remain clean

The surface of lotus leaves is superhydrophobic and water droplets

falling onto the leaves bead up As the droplets roll off, dust

par-ticles absorbed by the droplets are carried off with them This is

how the lotus leaves appear to be self-cleaning As it turns out, the

leaves are covered with micrometer-size surface structures called

papillae that are themselves coated with hydrophobic wax crystals in

nanometer diameter By contrast, the non-stick Teflon surface would

have become increasingly sticky upon prolonged use, not to mention

the possibility of its cancer-causing effect That shows how Mother

Nature knows better

The marine cone snail has shells ranging in sizes from less than

an inch to 9 inches long The snail unwinds itself in the shell to come

out of the opening A sharp stinger stabs the prey much like a harpoon,

but loaded with paralyzing, fast-acting venom The composition of

their venom differs greatly among species and also within the same

species The venom contains hundreds of different toxins that are

small peptides of typically 12–30 amino acid residues The

three-dimensional structures of these peptides are highly constrained due

to their high density of disulfide bonds The snails are a treasure trove

of novel chemical compounds Several toxins with clinical values

are being studied, synthesized and tested The toxins appear to be

highly evolved and precision targeted at their prey Already, scientists

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Mother Nature Knows Best

have found toxins that can serve as analgesic, anti-convulsive, and

as inhibitors of neurotransmitters Someone’s poison may be another

person’s elixir

A good percentage of animals find their way by tracing the

chem-ical trail Animals are masters at sensing chemchem-ical messages, whereas

we are rather ignorant The ants spread a chemical to designate the

path for other ants to follow Dogs can be trained to sniff baggage

at airport to find illicit drugs or even money trail They are reported

to be capable of detecting bladder cancer from urine samples, or to

find survivors in the wreckage of a devastating earthquake Scientists

are still fascinated by how birds and butterflies travel southbound in

winter and northbound in summer Salmons never go to school to

learn navigation, yet they could travel thousands of miles across the

Pacific Ocean without parental guidance We certainly can learn a lot

from nature and other residents on earth

Nature has good hints about clean energy Plants have long

adopted the strategy of utilizing the solar energy to sustain life

Sci-entists are able to produce hydrogen by using bacteria to breakdown

organic waste The fuel cells and the solar cells are strong candidates

as renewable energy sources Fuel cells would meet the size

speci-fication as a nano energy supply unit The energy consumption may

be reduced in certain areas with the nano-dimensioned products, but

the energy demand will inevitably continue to grow There is the

con-cern of whether these renewable energy sources could be sufficient

In this regard, nuclear energy delivers at an energy density million

times the typical chemical energy source In fact, it generates the least

amount of environmental waste By comparison, coal-mining takes

a severe toll on human lives, adds a high cost on landscape

reclama-tion Burning coals also produces the green house effect because it

yields carbon dioxide and sulfur dioxide As an energy source, coal

is a thermal pollutant To learn from nature, we might add that after

all, the very source of solar energy is precisely the nuclear energy

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The choice is in the hands of our younger generation and the Earth

will feel it

Elephants and sharks can grow a few sets of teeth during their

lifetime Growing another set of teeth in humans might be the last

item on the wish list for scientists to make good use of stem cells The

discovery of stem cells, the raw material for cell construct, opens the

door to cell therapy and organ transplant that will not only prolong

life but also make life healthy at the later time It is truly the fountain

of youth waiting to be fully appreciated

1.4 Challenges in the New Millennium

The human dominance on earth may not be a blessing when human

beings pay no respect to nature Dinosaurs dominated the earth and

became extinct Humans may have already overpowered nature and

may have caused enough harm to lead to eventual self-destruction

if the trend continues We have stockpiled so many atomic weapons

that would destroy the entire earth many times over The damage

to the ozone layer could increase the chance of skin cancer by the

increased exposure to UV radiation The excessive use of

chemi-cals, especially insecticides, pesticides and disinfectants, continues

to pollute the environment It is important that we don’t overlook the

possible correlation of this with the increased asthma or leukemia in

the younger children We made mistakes regarding the safe usage of

medicine by over-prescribing antibiotics That speeds up the

resis-tance by the pathogens We continue to produce and even use

chemi-cal weapons despite the chemi-call to ban them We promote meat growth by

using hormones, antibiotics, and with feedstock derived from animals

of questionable origin In the early 1990s, the outbreak of the

“mad-cow disease” in UK and the resulting epidemic in parts of Europe can

be traced to feed supplements contaminated with the scapie agent, a

prion, not to mention the industrial pollution, which is well

docu-mented as a major risk to human health There is a close link between

cancer and industrial pollution, as any epidemiologist will attest The

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Challenges in the New Millennium

pollutants also endanger the wildlife If the heat waste from industries

and automobiles reaches a critical level, the weather may no longer

be solely determined by the natural four seasons For some people,

whether the more frequent, sporadic temperature extremes, dry and

wet rainfalls, the gigantic landslides, the humongous hurricanes, and

the record loss of ice in the Arctic correlate with the increased energy

consumption, may still need more conclusive evidence For many

others the global warming cannot be dealt with efforts that are too

little and too late, as advised by the UN It will challenge to this

generation of scientists to find out the solutions as the fate of future

human generations is at stake

With down sizing, we may consume much less resources and

energy, and produce much less waste As with many technologies

that do have undesirable effects, the nano particles do have the bio

incompatibility that may be a cause for concern as being

carcino-genic since they can slip into the human body without being detected

and their active chemical surface may hinder the metabolic pathway

The proper packaging from the nano scale to the micro scale, and

then onto the system application will have to be safeguarded against

undesirable consequences Nano technologies will evidently provide

better solutions than most of the existing ones If the 19th century

is the century of chemistry, and the 20th century is the century of

physics, the 21st century will undoubtedly be the century of biology

and nanotechnologies Humans have arrived at this level of accuracy

and dimension

The advancement in instrumentation however, does not

automat-ically transform the process into the desired precision, whether in

manufacturing or in surgical procedures Achieving that goal must

rely on analysis, computation, and understanding Numerical

simu-lation techniques have become increasingly powerful tools in science

and engineering disciplines In biology terminologies, the in-vivo

(inside the biological system) and the in-vitro (inside the lab) are

often expensive and time consuming, and there are things inaccessible

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to experiments, but may be examined in-silico (inside the silicon

chip) Computational molecular biology, computational chemistry

and computational physics are well connected with the nano

comput-ing science The paradigm of the engineercomput-ing focus will shift from

the traditional discipline to one with scientific background in

biol-ogy, chemistry, physics plus computer science Many issues remain

as immediate challenges for computational scientists in the nano

regime Little is currently known about the structural growth of small

clusters of any element The electronic properties of excitation

spec-tra, of clusters of ionic or covalent materials remain a challenge

The ab-initio calculations are limited at the moment The computer

simulation, for example, for the growth of carbon nanotube (CNT)

remains elusive in helping the experimental realization of single

wall CNT, or controlling the tube chirality Diamond growth under

plasma vapor chemical deposition (PVCD) would benefit from the

same

Software will be important to nanotechnology and to the enabling

technologies along the way The built-in intelligence and the

intel-ligence to build will be the only way to make nano molecules to

micro then to macro systems to function according to performance

specifications Algorithms such as simulated annealing by

reward-ing performance and penalizreward-ing the disservice might work when

the understanding of the fundamental mechanism is lacking On

the other hand, the very existence of molecular brains in nature is

persuasive for us to explore the nature’s design of molecular

com-puter Along the way, we may find intelligence, wisdom, or even our

soul

Further Reading

It is certain that biology will have a great impact on the nanoscience

for many years to come and vice versa Students are encouraged to

read the book Biology by Neil Campbell, Jane Reece and Lawrence

Mitchell (1999)

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Exercise

1 The work function is typically half of the ionization energy of

the last electron in the single atom The latter is 11.2 eV for the

carbon atom It is therefore reasonable to assume that a CNT has

a work function of around 5.6 V Apply a voltage of 6 V to an open

end CNT of zigzag type with 6 atoms in each circumference,

what is the expected current? Note that an electron at 1 eV of

energy would travel at a velocity of 4.19× 107cm/sec, and one

electron traveling at the speed of light would give a current of

4.8× 10−9 ampere

2 Mohs’ scale of mineral hardness characterizes the scratch

resis-tance of various minerals through the ability of a harder

mate-rial to scratch a softer It was created, in 1812, by the German

mineralogist Friedrich Mohs and is one of several definitions of

hardness in materials science Diamond has a Mohs’ scale of 15,

boron carbide 14, and silicon carbide 13 Give an argument to

estimate the hardness of fullerene in comparison with these three

materials

3 The AFM measures the forces between a sharp probing tip and a

sample surface Images are taken by scanning the sample relative

to the probing tip while measuring the deflection of the cantilever

as a function of lateral position Typical spring constants of the

cantilever are between 0.001 N/m to 100 N/m and typical forces

between tip and sample range from 10−11N to 10−6N The force

can be thought to arise from changes in the electromagnetic (EM)

wave energy, which are caused by bringing the tip close to the

surface When no observed surface is present, these waves are

singly scattered from the tip and escape to infinity When the

space is restricted by the scanned surface, using the uncertainty

principle xp = , we may find the energy change equal

toε = cp = c/x At a separation of 1 Å, what is the force

generated?

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4 A light beam undergoes diffraction and spreads in diameter This

effect limits the minimum size d of light spot formed at the focus

of a lens, known as the diffraction limit, given by d = 2.44λf /a,

whereλ is the wavelength of the light, f is the focal length of the

lens, and a is the diameter of the light beam, which will be taken

as the atomic size, namely∼1 Å Given the light wavelength λ ≈

5000 Å, what is the focal length to observe the lattice spacing on a

film? Given a typical focal length, say on the order of centimeters,

what is the wavelength that will allow the lattice spacing to be

observed?

5 Find the top 10 major atomic elements in the human body

6 List all the atomic elements in the nucleotides

7 List all the atomic elements in the amino acids

8 The cone snail venom is composed of 20 amino acids What is

the venom’s average weight?

9 Estimate the solar energy that is received on earth per square

meter per day The distance between the sun and the earth is

1.5× 1011meters The diameter of the earth is 1.3× 107meters,

and that of the sun is 1.4× 109meters Assume that the sun has

the blackbody radiation emitted at the surface temperature of

6000◦C.

10 There is 1370 watts per square meter, arriving at the atmosphere

of the earth as the average heat flux from the sun Assuming all

these energies are reaching the surface of the earth and are being

absorbed without reflection by a cross section ofπR2, where R,

the radius of the earth, is 6366 km, estimate the maximum solar

power deposited on the earth

11 The worldwide energy consumption in year 2006 is roughly 500

quadrillion BTU One quad (short for quadrillion BTU) is defined

as 1015 BTU, which is about one exajoule (1.055 × 1018 J)

One quadrillion BTU is 1015BTU So 500 quadrillion BTU is

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5× 1020Joule Estimate the power consumption on the earth in

watt

12 Taking into account the conversion efficiency of power usage,

compare the power consumption rate on the earth by humans

with the solar power Remember the heat has been accumulating

through the years since the beginning of the Industrial

Revolu-tion, and that the energy difference between summer and winter

is the important factor to compare as far as the natural four

sea-sons is concerned Make a conclusion on whether we are entering

an era when human power consumption is capable of changing

the four seasons? This may serve as the first principle proof of

whether the global warming is real

13 Estimate the amount of coal and uranium 235 needed to

gen-erate the same amount of energy in problem 11 Estimate the

amount of carbon dioxide released Estimate the distance a car

could travel with this energy assuming the weight of a car is

3500 pounds or 1600 kilograms A typical automobile tire has

an average coefficient of rolling friction of 0.015

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RPS Nano Computing “ch02” 2008/9/25 19

Chapter Two

TOOLS FOR ANALYSIS

“It is the nature of all greatness

not to be exact.”

Edmund Burke (1729–1797)

A s the scientific problem becomes more complex, it is

less likely to find an exact closed-form solution in terms

of known functions Resorting to numerical analysis isoften inevitable Nevertheless, there are advantages of

an analytical solution however primitive The physical law would

have the clarity and the insightfulness from the compact

analyti-cal solution Asymptology thus plays an important role in

extend-ing to regimes where exact closed-form is lackextend-ing Furthermore, the

numeric method provides the ultimate alternative solution, which

may otherwise be difficult to envision To be sure that a theory is

cor-rect and transparent to professionals, students should be aware that

analysis by analytical or numerical means alone is often insufficient

Numerical computation, Experimental observation, Analytical

cal-culation, plus Physical picture persuasion, properly abbreviated as

NEAP, depicted in the drawing at the four corners of a tetrahedron,

represents elements needed to elucidate the physics principles This

will not only help ensure a correct conclusion, but also in itself a

prac-tical process to scientific discovery Scientific computing, especially

modeling and simulation, can be an important part of knowledge

cre-ation You may want to start with brainstorming within yourself or

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Tools for Analysis

colleagues, explaining the ideas with hand waving arguments,

fol-lowed by the back of the envelop calculations and in the end solving

the problem with a formal theory, and practicing NEAP to publish a

successful paper

In selecting the programming languages Fortran (Formula

Trans-lator) and C++ will do the job Java, PERL, PhP are internet-ready

tools for expanding to the server technology The issues for

devel-oping computer programs are usability, scalability, portability and

versatility Other concerns are the robustness, development cost (time

to market), maintenance and user interface The built-in functions

in MATLAB with its software suites, greatly ease the programming

effort This is rather useful especially when a scientific or engineering

problem is not well defined, and quick answers on a few conjectures

are most desirable MATLAB reduces the overhead in programming,

the time to get the right answer, and the effort to visualize the result

Unless a problem has its mathematical formulation well defined, it is

impractical to start programming a large code This is where

MAT-LAB would make a great difference To develop a large application

program, C++ is still the recommended choice The C or C++

lan-guage lets the user inherit Graphical User Interface (GUI) objects It

could be very beneficial especially when codes are being designed

for commercial purpose

2.1 MATLAB

The name MATLAB is derived from MATrix LABoratory It is a

matrix-based language and a convenient tool in manipulating vectors

and matrixes The use of the vector operation in MATLAB greatly

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MATLAB

speeds up the performance, compared to a scalar operation with a

do/while/for loop The use of the scripted M-file, with the extension

‘.m’, is an excellent way to run a program But for beginners, it is

easier to execute commands in the interactive window By default,

the output of a command is automatically printed to the window,

and the results are immediately displayed on screen MATLAB is a

convenient interpreter rather than a compiler; it is therefore easy to

debug It is a high-level language with many mathematical

program-ming facilities Its graphic tools help you present a solution in striking

displays It is simple enough for a non-programmer to go beyond the

spreadsheet, yet sophisticated enough for a serious programmer to

implement–even a three-dimensional code in finite element method

(FEM) The 64-bit architecture greatly enhances its scientific

com-puting capability The much-awaited parallel MATLAB, to run on

distributed memory, is also available

MATLAB’s strengths include algorithms, matrix manipulation

and graphical tools Although it is not designed for symbolic

com-putation, it makes up for this weakness by allowing the direct link to

Maple, a software with integrated numerics and symbolics MATLAB

has many mathematical functions These include integral (INT),

dif-ferentiation (DIFF), limit (LIMIT), Taylor expansion (TAYLOR),

summation (SYMSUM), factorization (FACTOR), polynomial roots

(ROOTS), etc Throughout this book, an uppercase name will be

used to denote a function name Students should know that MATLAB

recognizes a function name in lower case only

The output of each command line is automatic unless a

semi-colon (;) ends the line This is by far the smartest design in all

available languages By contrast, C coding would require, for

exam-ple, a ‘PRINTF’ function to output a text line, say ‘printf(“Hello,

World!”\n);’whereas C++ may use ‘COUT’, as in ‘cout“Hello,

World!”endl;’ JAVA coding requires importing the applet and

graphics to draw a string at the position following the string:

‘g.drawString (“Hello, World!”, 50,25)’ FORTRAN needs the

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Tools for Analysis

“WRITE” function, as in ‘write(“Hello, World!”)’ In MATLAB, a

line can be commented out by adding % at the front There is no

nota-tion for commenting out a secnota-tion, as in C++, which is performed by

using the ‘/*’ and ‘*/’ pair This is one of MATLAB’s shortcomings

and does cause some inconvenience

MATLAB’s ability to create a vector or a matrix and to evaluate

numerically with the built-in functions is very satisfying Problems

can be solved with the least number of lines of coding The matrix

functions provided include inversion (INV), Fast Fourier Transform

(FFT), eigenvalue and eigenfunction solver (EIG), singular value

decomposition (SVD), and cubic spline data interpolation (SPLINE),

etc The ease in getting the impressive graphical output is rather

rewarding

2.1.1 Symbolic Analysis

To get a sense of how the programming in MATLAB works, we

shall start with a few examples and perform symbolic analysis At

the MATLAB prompt, we may define the variables first by ‘SYMS’.

Sometimes we need to define the types of the variable, either ‘REAL’,

‘INTEGER’, ‘COMPLEX’, or ‘POSITIVE’ These can be placed

following the sequence of the variables Let us examine the following:

syms x;

limit(sin(x)/x,x,0)

ans = 1

% Define a symbolic variable x;

% Evaluate sin(π) The answer is returned in c and is given by

less than "eps" (epsilon), the current limit of precision, which

is considered to be zero No need to declare c as it is defined as the output.

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roots([3, -4, 1])

ans = 1.0000

0.3333

% Find roots of a polynomial 3x2− 4x + 1.

% The answer is x = 1 and x = 1/3.

% SOLVE can handle more than polynomials.

% Define a 2x2 unit matrix

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% Indeed it is true, but you need to simplify the algebra.

Note that if the variable is not defined, then ‘ANS’ is the default

variable, which could be used for the next evaluation It is a temporary

holder of the output and will be replaced in the subsequent operations

There are other functions that would come in handy for the symbolic

operations The ‘SIMPLE’ and ‘SIMPLIFY’ would reduce the

alge-braic expression to a more manageable form, and “PRETTY’ would

format the equation to a typesetting style

2.1.2 Vector Calculation

The matrix operation in MATLAB is both convenient and powerful

for scientific and engineering analysis Many useful matrix

construc-tions are provided: ‘EYE’ for identity matrix, ‘RAND’ for random

number matrix, ‘ZEROS’ and ‘ONES’ for matrixes with elements

of zeros and ones MATLAB has the operator overloaded with

con-venient notation A ‘*’ represents the normal matrix multiplication,

whereas ‘.*’represents the element-element multiplication The same

principle applies to division, ‘/’vs ‘./’, and power, ‘ˆ’vs ‘.ˆ’ The

expo-nential, logarithmic, transcendental and other more elaborate

math-ematic functions are by default operating on an element-by-element

% More on the colon Note that you can use it to get slices of

a vector or get the whole thing Equivalent to a wild card in choosing elements.

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this case, switches between row and column vectors.

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% The two expressions give the same result.

To get help, at the command prompt type ‘help FUNCTION’

Here FUNCTION is the name of the particular function you seek

help For example, type ‘help rand’ at the command prompt lets you

know how to use the random-number generation command Thus

for “rand(‘state’, sum(100*clock));”, the random number generation

may be reset to a different state every time This is particularly useful

when you are doing Monte Carlo or Simulated Annealing

calcu-lations There are vector operators such as the inner product ‘DOT’,

the cross product ‘CROSS’, the determinant ‘DET’, the inversion

‘INV’ and the eigenvalues and eigenvectors of a matrix ‘EIG’ It is

most convenient to find the solution to a matrix equation AX= B

by the simple operation ‘X=A\B’, although this operation could be

slow for large matrix

2.1.3 Graphical Presentation

A picture is worth a thousand words MATLAB contains a wide

vari-ety of techniques to display data graphically Sometimes to see is to

understand since having the output of your programming effort in the

visual form may reveal the inadequacy in programming or conceptual

blind spots For example, if you are looking for the minimum value

of F(x,y), it is possible that what you in fact get is a local minimum

rather than the true value, owing to your choice of the algorithm A

3D plot of z=F(x,y) would easily reveal the shortcomings, if present

To experiment with MATLAB’s graphics capabilities, the simplest

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MATLAB

one is the plot command, ‘PLOT’, as in ‘x=0.01:0.1:10;

plot(x,exp(-x));’ This will produce an x-y plot of the function y=exp(-x) from

x=0 to x=10 Here are a few examples:

% Declare wave functions

the atomic number Z.

% Define 1S and 2S orbitals.

% Set the grids on.

% Contour plot of anharmonic oscillator with the energy

% define max velocity

% get velocity and space vectors

% load x, y into X, Y matrixes

% Plotting (X,V) demonstrates well the function overload in general The argument can be vector or matrix.

% Electrostatic potential plot

% Create the phi values in (x,y) plane

% Replicate the x,y coordinates into the square matrixes

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% Evaluate the phi value due

to two negative charges placed along the x-axis, and two positive charges on the y-axis

% Request a contour plot

% Draw a helix structure with

X, Y, Z defined by the parametric dependency on t.

% Helical coil rotating along the axis

% Save as an avi file Execute the file outside MATLAB to play the movie.

% Creat avi object by AVIFILE is

% Electrostatic potential plot

% Create the phi values in (x,y) plane

% Replicate the x,y coordinates into the square matrixes

% Evaluate the phi value due

to two negative charges placed along the x-axis, and two positive charges on the y-axis

% Request a contour plot

% ask the user to input the atomic number

Z=

2

28 James J Y Hsu

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