LOCAL DENSITY OF STATES FOR ONE DIMENSIONAL APERIODIC

BINARY SEQUENCES USING LOCAL GREEN'S FUNCIlON METBOD

by

J. A Christopher Delaney

Department of Physics

Submitted in partial m e t of the requirements for the degree of

Master of Science in Physics

Fadty of Graduate Studies The University of Westem Ontario

Londo~, Ontario August 1996

O J. A Christopher Delaney 1996

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ABSTRACT

The goal of this thesis was to d e what the effect of aperiodic sequencing in

binary alloys would have on the electronic density of states. Our method of

evaluation is the local green's fimction (LGF) method which we just@ as being a

reasonable compromise between accuracy and computational practicalitycalitv This

method gives us the local density of states (LDOS) which we evaluate at the

surface. From the LDOS we can infer the structure ofthe density of states (DOS).

The well known cases of pure materials and periodic binary alloys are examined in

order to clearly itlustrate the advantages and limitations of our tool. We have

proposed a new tool to simplifL the evaluation of the LDOS for the Fibonacci

sequence that enables us to compute extremely large systems with a high degree of

speed and accuracy.

Our study of the LDOS of a Fihnacci sequence binary alloy has suggested to us

that the electronic density of states shows seSsimilarity. We showed this by

demonstratkg the reoccurrence of the original LDOS when we examined a portion

of the LDOS. Since we do not asstme se%simiIarity in any of our initial

assumptions, the existence of selfsimiIarity implies that we have generated a

fractal through sequence effects. It is our cfaim that this hctd is a Cantor set.

W e infer this on the basis of the structure of the LDOS,

I wish to express a profound feeling of gratitude to my supervisor Philip Tong for

his patient guidance and support throughout my studies at Western*

I would Like to thank NSERC br their financial support in the form of a Post-

Graduate Scholarship. I would like to thaak the University of Western Ontario for

their assistance in the form of graduate support. I would like to thank the

department of Mathematics at Lakehead University for their assistance in

employing me while I worked on this thesis.

I would like to thank the members of my advisory cormnittee for seeking to

understand my obscure thesis topic and provide guidance.

I would like to thank all those who inspired me to a career in science. This is the

final result oftheir efforts (and they may take that any way they like)! 0

Last, but not least, 1 would like to thank my family for standing by me both in

good times and bad. Thanks for all of the encouragement and support.

TABLE OF CONTENTS

CERTIFICATE OF EXAMINATION ABSTRACT ACKNO-S TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION

1.1 Purpose 1.2 Deasity of States Calculations 1.3 Previous Works 1 -4 Aiternative Models 1.5 Quasi-crystalline Structures 1 -6 Thin Film Structures 1 -7 Figures for Chapter 1

CHAPTER 2 MODEL AND METHOD

2.1 The Fibonacci Chain 2.2 The Cantor Set 2.3 The Tight Binding Model 2.4 SimpLifications Present in Model 2.5 The Characteristic Equation Method 2.6 The Greats Function Method 2.7 The Local Green's Function Method 2.8 Calculations for Large Systems 2.9 Why the LDOS is Lmponam 2.10 Figura for Chaper 2

HAPTEX 3 LOCAL DENSITY OF STATES FOR PURE PERIODIC STRUCTRES

3.1 Analytic Solutions for Pure Stnrctures 3.2 Bulk States for Pure Stnrctures 3.3 Bulk States for Periodic Structures 3 -4 Pure Materials as seen from the Surhce 3 -5 Periodic Alloy Materials as seen fiom the Surface 3 -6 General Notes 3.7 Figures for Chapter 3

Page ii iii iv v

1

i 4 7 11 12 13 15

16

16 18 21 24 25 26 28 32 34 38

ANL)

39

39 42 43 45 46 47 49

CHAPTER 4 THE LOCAL DENSITY OF STATES FOR FIBONACCI SEQUENCE STRUCTURES 72

4. I Band Edge Effects for a Fibollacci Sequence Structure 72 4.2 Resolution E E i s on a Fibonacci Chain Structure 74 4.3 Cantor Set Energy Spectrum 76 4-4 The Devil's Staircase 79 4.5 Figures for Chapter 4 82

CHAPTER 5 CONCLUSION 113

5.1 Conclu!3ions 5.2 Diiections for Further Study

APPENDIX I 117

REFERENCES 133

Chapter Introduction

This thesis project uses a ofresults tiom pure mathematics to shed light

on a physics problem. The existence of "exoticw sets has long been known in

mathematics, although the formal definition of such sets as htals has been quite

recent (Edgar 1990, Feder 1988). Along with the new field of non-linear dynamics

(chaos), these novel ways of representing sets have givm us powerful new ways to

model physical phenomena. This has led to a revolution in our understanding of

the universe that is very Herent than the classical Newtonian mechanistic view.

While this theory stiu includes Newtonian determinism, it does describe a class of

phenomenon that have to be treated differently fkom the classic types of problems.

Instead of ignoring certain complex patterns as "noisen or anomalies, we have

discovered that they do have an underlying order or structure. This structure may

be very complex and thus impossible to predict with accuracy (the problem with

studying chaos) but it does exist. Chaos differs from linear classical mechanics in

being very sensitive to initial conditions; very small differences may have an

enormous effect on the evolution of the system.

This new field of chaos and the closely related field of hctals are a

characteristic of noelinear systems in classical mechanics. A very interesting

question is whether or not we can find evidence of chaos in Quantum Mechanics.

After all, our theories would indicate that the macroscopic world is still based in

Quantum Mechaaics. However, the equations of Quantum Mechanics (Le. the

Schriidinger equation) are linear equations; a regime where chaotic and fractal

& i s typically do not occur. One possiie answer to this question lies in the

study of aperidcity in condensed matter systems- It is possible to argue that the

fiactai effcts might be introduced by the ~eqllencing of the eleznents in a binary

chain despite the behavior of the system being descflibed by linear equations. Or in

other words, the non-linearity and chaotic nature intrinsic to the physical system

may enter the ScbrWinger equation though the interaction potentid. This is one

of the key things we want to demonstrate in this thesis: that the arrangement of

atoms in a binary chain in an aperiodic Seqllence can actualy lead to fkactal effects.

From the idea that atomic sequencing can generate fractal effects, we

hypothesize that a certain specific arrangement of two types of atoms can cause

the energy eigen-values of the electrons bound to the atoms in this chain to

conform to a h c t a l structure (of which the Cantor set is going to be the specific

example once we get there). If true, this hypothesis would imply that we have a

new class of solutions for the energy spectra of a binary alloy. This fhctal energy

spectra would be very different @om what we have catdated and measured in

particles, atoms and solids with a periodic basis. The specific arrangement of

elements in a binary sequence to have order but not periodicity is called aperiodic

by some mathematicians and quasi-periodic by some physicists (although we have

people in both fields who use the term often 8ssoc1kted with the other field). The

two terms are used interchangeably in this thesis. We will formally define quasi-

periodicity in Chapter 2. For now, we will use a "working definition" as follows:

an aperiodic sequence possesses order described by a set of rules but it does not

repeat itself regardiess of how long it gets

The specific aperiodic arrangement studied in this thesis is called the

Fibonacci sequence. The Fibonacci sequence is related to the Golden Mean which

has been known since the days of ancient Greece, or possibly even earlier- It is a

well known and exteasivety studied example in mathematics of an ordered

sequence that does not manifest periodicity. A cIose examination of nature yieIds

many examples in wbich the Golden Mean appears as a basic unit of proportion

and the Fibonacci sequence is often surprisingly fundamental (Tong 2994).

The term hctal is a very recent one and dates to 1975 when it was coined

by Benoit Mandelbrot to descn'be certain types of "irregularn sets (Edgar 1990).

These "irregular" sets quickly became a p o w and fmhfid area of study. The

structure of the spectrum we are seeking to create here is one of the most basic

examples of a W; Mandelbrot called it the Cantor Dust (we call it the Cantor

set) as it seemed to lack any form of solidity. The Cantor set has an infinite

number of components and a d-e stnucture, but cannot be Localized to any

specific point along the structure. Every time one zooms in to Iook at a point, one

finds that it is composed of the & Cantor set, identical to the one which we

began with. This is one of the most important properties of hctals, the idea of

self-similarity. What this means is that the whole is contained within the parts and

so every part is "similar" to the whole!

It is the intersection of fractal theory with otherwise straightforward

calculations that makes this topic so interesting There are two main ideas behind

this thesis. One, is that we are interested in showing that by altering the linear

sequence of molecules (or atoms) in a binary system form a perfkdy periodic case

to that of an aperiodic case that we get a self-similar energy spectrum for the

electronic density of states (which consists ofthe binding energies of the electrons

in the aperiodic chain).

Two, is that we are interested in showing that we have, or might have, a

hctal present- The method we have chosen does not allow us to directly test for

the presence of a fractal in our results. We wiU. however, attempt to infix the

existence of a fractal &om the presence of the most firndamental of fractal

properties, that of seKWarityarity

1.2 Densitv of States Calculations

Let us begin by discussing the notion of density of states. It is well known

in Condensed Matter Physics that the equations descniig a sequence of atoms

(basically a one-dimensional chain) can be solved for an energy spectnrm of bound

electronic states. The placement of these eigen-values (the number per unit energy

for example) is known as the density of states 0 s ) ; or, more accurately, as the

electronic density of states as one can discuss the density of states of other things

besides electrons (phonons for example). This is one of the physical quantities that

is of great interest in Condensed Matter Physics (Ecoaomou 1983, KitteI 1986).

Calculations of the density of states are of great use in predicting a wide range of

macroscopic properties of a solid which range from the results of scattering

experiments &om the solid to the heat capacity of the solid. Of course, when

calculating reaI world examples, we typically have to go to two or tbree

dimensions (onedimensional atomic chains do not exist in nature as fiu as we

know, although we can adapt our results to model a series of thin tilms).

However, the important physics is demonstrated by the one-dimemiod example

and it is much easier to calculate than the multi-dimensional case. In this thesis,

we will confine ourselves to the discussion of the one-dimensiod case.

One staradard way to go about calculating the W S is to use the Green's

hction method. W e are not interested in the Green's bction method in the

standard sense. While Green's kct ion calcuktions are a possible method of

calculation for this problem, we reject their use because they are too long and

cumbersome. Instead we will use a closely related method of calculation, the local

Green's fimctiom The Green's hction has a singularity present at each and every

bound state. The presence of this singularity enables us to determine the energy

values at which bound states occur, the average number of bound states in a given

energy range and the overall structure of the bound states. When we adapt our

method of evaluating our model to the local Green's Wction method, we sti l l have

the chain of singularities but only as sampled at one point along the chain. The

main disadvantage of the local Green's fimction method of calculation is that it

reveals the energy states as "seen" f?om one lattice site. This meam that fhr away

states will be somewhat "masked" due to distance of the bound state fkom the

point at which we are calculating the energy specbum fiom Plus, since we must

sample at a specific atomic site, in a binary chain the energy spectrum of one of the

two types of elements will be less detailed as it will be less clearly "seen" fkom the

sample site (although the location of the eigen-values will still be correct). This

gives us the local density of states. In this thesis we are typically going to

examine the local density of states fiom one end of the chain (i.e. from the surfke

of the material).

We could deal with the problem of site sampling by bydating the Green's

bct ion at all sites and averaging the result to give the "trueH energy spectnm (the

energy spectnrm given by the DOS as opposed to the one given by the LDOS), but

we are forced to reject this approach due to the time we would spend in

calculating the Green's fimctions. Besides, as we will see, the effects of switching

to the local density of states h m the density of states does not mask the features

of interest in our systea We will also note that the fine details of the system are

not of great interest in the @odic case as the energy spectmm is wd behaved.

We wiU note that in a system o f b i i atoms arranged in a aitetnating sequence,

we get an energy gap with both a lower and an upper bound with all states

clustered above or below this gap. What taking the local density of states will do

is to "suppress" the features of one of the bands while laving the f a r e s of the

other band intact. This is an acceptable trade-off for the vastly simplified

calculations that are required for the local Green's fimction and so we will adopt

this method in this thesis,

In an infinite system we get singularities at the band edges of the DOS.

The singularities at the band edges are an inaccuracy due to the I-dimensional

approximation of a crystal (there are no actual I-dimensional crystals in nature).

They are caused by an infinite number of small wntriiutions fiom "fx away"

atoms. When there is any sort of disorder in the system, these singularities vanish

due to scattering effects resulting from the breakdown of periodicity. As we will

see, the shift &om an infiaite system to a finite system does not change the location

of the band edges and is not important in the case of an aperiodic chain The

height of the Green's fimction has some se~similar characteristics, but they are

f i d t to see and we have much better ways of proving saf-similarity. This is

especially true in our case as we are using the local Greeds function which firitber

obscures the self-similar height fatures of the Green's bction and makes it nearly

impossible to show this form of self-similarity with any degree of rigor.

In the case of an aperiodic system, the situation is a little more

complicated. We are studying a quasi-periodic system and as a result the energy

spectrum does not appear to be as clearly defined as in the periodic case (where we

have a clear structure). We no longer have the edge singdarities in the bands

(which are destroyed by scattering as soon as disorder is introduced). Nor do we

seem to have well d&ed bands like in the periodic case. One of the problems

fixing early researchers in this field was that every point on the spectrum is unique

(as opposed to an infinite periodic system in whicb every A and B site will be

identical for to a pure system where every point is identical). In our research we

will typically take the surfice as being the d&dt poiat we choose to consider as a

way of handling the diffidty of what site to sample at and thus avoid the question

of which site to choose as the point to generate our local Green's Gnction from

entirely.

1.3 Previous Works

In 1983 SteUan Ostlund, Ruhul Pandit, Da\rid Rand, Hans Joachim

Schetinhuber and Eric D. Siggia suggested that when the Schriidinger equation is

applied to an "almost-periodic" potential of infinite extent, a new type of state

occurred that was neither extended nor localized; they called these states "critical"

states (Ostlund, Paudit, Rand, Schellnhuber and Siggia 1983, O s t l d and Pan&

1984). This new form type of state was extremely interesting as it presented an

alternative to the two classic types of states (singular and continuous). Another

interesting fatwe about their adysis was the use of the renorxnahtion group

(RG) transformation to study for scaling properties that exist in such systems. The

RG method has one central advantage: it enables one to deal with large numbers

of sites. It is an extension of the transfff matrix method that greatly simplifies the

problem However, it is not our intention to deal with this method in this thesis,

beyond mentioning that it exists and has been used by previous authors as a way of

avoiding the problems of the charsrcteristic equation method (which we d o n in

Chapter 2).

In a paper exploring this topic (Tong 1994), Philip Tong began with a

discussion ofthe golden mean. It was first introduced in geometry by the relation

CB AC -=- where C is a point on the straight liw AB. The ratio AC AB

&-I Y=- 0.6 1 8. .. that satisfies the equation - CB = - is the value of the 2 A AB

golden mean.. The golden mean is intimately comected to the study of an

aperiodic sequence known as the Fihmcci sequence- If we use the Fibonacci

sequence to generate a binary chain (using A and B sites for example), the ratio

between the two different types of elements (ratio of A sites to B sites) that make

up the binary chain is y.

In part 7 of Tong's paper, in a section entitled "Almost-Periodicity (Quasi-

Periodicity)", is a discussion on an application of the Fibnacci sequence to

condensed matter physics. Up until the study of aperiodic systems, there were

only two types of wave functions known: singular (correspo~~ing to normalizable

wave-functions) and continuous (corresponding to unnormalizable wave-

functions). The singular wave-finction represents a system in which the spectra

consists of discrete points, which can be evaluated over their entire range. The

continuous wave-functions correspond to a spectnun in which the wavebctions

are nan-localized and cannot be completely evaluated. It was long accepted that

these two types of wav&naions represented all of the poss~-biiities (either a

wave-function was localized or it wasn't). The study of aperiodic sequencing

changed all ofthis.

The study of aperiodic systems (notably Fibonacci) showed the possi'bilty

of a third type of w8ve-fUnctions, the critical wave-fimction. This W o n is

neither singular nor watinuous but instead it is singular cootinuous (defiuing

exactly what this reaIly means is one ofthe central goals of this thesis). In order to

understand how it is possible to have this third type of wave-function, it is

necessary to consider the work of the mafhdc ian Georg Cantor. Cantor

posited a set in which there was an idmite number of members but when summed

up the total measure of the set was zero. The analogy that best descri'bes it is this:

take a line of arbitmy length aud remove the middle third; then remove the middle

third of the two liues that now stist; continue doing this for an intinite number of

repetitions. One ends up with an infinite number of Lines with a total length of zero

(see figure 1-1) . If we want to be more precise in our terminology, a length of

zero corresponds to the Lesbegue measure of the line being zero.

So we have posited a relation between the Golden Mean and the Fthnacci

sequence in our discussion above (aad we will go into more detail in chapter 2).

However, some authors have claimed that the Golden Mean has other relations to

our c o n d d matter system beyond defining the ratio of the two ~~ types

of atoms in our binary chain. Kohmoto noted that the ratio of atoms (type A to

type B) which are crossed when one draws a line across a 2-D quasi-crystal was

that of the Golden Mean. The result of this has the exact propties of a 1-D

Fibonacci chain. This would indicate that the 1-D atomic chain has the properties

of a line drawn through a 2-D quasi-cxystal. Chakrabarti proposed thtt y3 was the

relevant scale factor for his work where y is the Golden Mean. What he meant by

this was quite simple. When we "zoom in" on a hctal, we see the same hctal

repeated (this, in fact, is the definition of self-similarity). Chakrabarti's results

iadicated that this was the appropriate amount to zoom in by in order to see the

same image repeated exactly for an infinite system,

Another group that studied the problem of an aperiodic binary atomic chain

was that of Kohmoto and SutherIancf (Himnoto and Kohmoto 1992, Kohmoto,

Sutherland and Tang 19874 Kohoto, Sutherfand and Tang 1987B), who

suggested that a 1-D binery chain which was arranged in a Fibonacci sequence

would have aa eigen-value spectrum which corresponded to a Cantor set with a

Lebesque measure of zero. The immediate and interesting implication was that the

states, although infinite in number, are infinitely sparse. This would make standard

methods of solving the problem of an infiaite binary useless due to the unusual

properties that an aperiodic system possesses. The most common approach to

solving aa infinite (or semi-infinite) binary atomic chain is to guess a value for the

eigen-value E, substitute it into the Schrdinger equation and then see if the

integrated wave-function obeys the boundary conditions or not. If the integrated

wave-function does not obey the boundary conditions, then we try an improved

value of E. The problem here is that if the spectnun has a measure of zero (like

with the Cantor set which is our candidate fbr the spectrum), it megas that we can

never actuaIly find the correct value of E by guessing as no matter where we place

our trial eigen-value, it win not correspond to any of the eigen-values ofthe system

(i.e the binary atomic aperiodic chain). This is especially crucial in a system like

that has hctd propexties as they are notoriously sensitive to perturbations by the

introduction of even very small inaccuracies. This is different than when we

explore a periodic binary atomic chain where a s d error in the estimate will lead

to a small error in the final answer. In the case of the periodic chain. we might be

willing to settle for an answer that is within certain parameters (our error analysis).

However, whea we deal with an aperiodic system it is not clear that this approach

will not lead to m o t magnification.

Kohmoto and Sutheriand made the suggestion that the energy speanrm of

an aperiodic binary chain is a Cantor set. This @lies that the energy spectrum

wiU have fractal properties (as all Cantor set objects do). This is an important

point because it demonstrates that the bear Schriidinger equation can be used to

produce what is essentially a wn-linear phenomenon, solely on the basis of

sequence effects (the ordering of the atoms in the atomic chain).

1 -4 Alternative Models

Kohmoto, Sutherland and Tang studied a model in which the site energy

(the energy with which the electron is bound to an individual site) of the atoms

varied, the case in which there are two merent site atoms with &iffwent binding

energies. Another possible way to modei the variations of a Fibonacci seqyence

was studied by Chakrabarti, KarmaLar and Moitra In their modd they varied the

interaction energy instead of the site energy- This corresponds more to a modei in

which the atoms are all of the same type (thus they possess the same binding

energy for their electrons) but the inter atomic bonding varies depending on the

atom due to the lattice structure altering the distaace between various atoms. This

best corresponds to redly small chunks of matter forming crystalline structures

with five fold symmetry present Wore frustration prevents this formation fiom

growing further (see Quasi-crystalline Structures below). They were interested in

the local electronic phenomenon as represented by the local density of states as

determined by the l o d Green's hction-

Their work is very similar to ours but it wncesecafa on a diEerent type of

system. Their r d t s did show the presence of Cantor set behavior of the eigen-

value spectrum for a Fibonacci chain only with a completely different model. This

is of real interest, because it suggests that something more fhdamental is going on

here than just strange results Eiom a dque model This behavior seems to be

somehow inahrsic to F~haacci chain behavior- The literature supports the

supposition that aperiodic sequencing leads to the generation of eigen-value

spectra that have Cantor set properties as a general rule.

The contents ofthis thesis would have remained as pure speculation, had it

not been for a surprising discovery made in the last decade. The discovery that

made this field interesting to modern physicists was the discovery in 1986 of the

AlgLijCu icosahedral phase (Ashraff and Stinchcornbe 1988, Bak 1986, Ianot

1992, Kasner, Schwabe and Bottger 1995) which was a 2-D analog of our

problem. This provided a dear example of a system that should exhibit the

properties of our model. This discovery thus makes the theoretid work we have

done interesting as it now has a clear relation to real wodd phenomena. Since we

at last have an example of a crystai structure with five-fold symmetry (this means

that the lattice of the crystal would look the same five times as you rotated it

through a complete circle or once every 72O), we should have a real case where

this analysis may prove u W in predicting the properties ofthe crystal. This type

of symmetry (five-fold) was previously considered to be a forbidden symmetry.

This new type of crystal is based on a twenty sided Platonic Solid called an

icosahedron. This is critically important because, it demonstrates the macroscopic

existence of five-fold symmetry which, while the preferred ordering of a very small

system, was not expected to exist in a macroscopic system due to bond ftustratio~~

Bond hstration occurs when the structure of the crystal lattice of a solid is such

that it is impossible to place an even number of atoms around each atom in the

lattice. This produces strain that makes such a system inherently unstable at the

macroscopic level. While we could create artificial systems that corresponcted to

an arrangement of atoms in a F ~ b ~ c c i sequence (thin film semiconductor layered

compounds or dieiecfcic slab wave guides), here was the first case of this port of

atomic ordering sristing as a natural phenomenon.

The idea of how one would generate such a solid actually rests with the

mafhematician Roger Penrose (84 1986). He proposed overcoming the problem

of bond Erustration by dealing with a binary system in which two ditfkeat shapes

were used. In two dimensions this is known as Penrose tiling. It can certainly be

generalized to three dimensions. So the "trick" that nature uses to get five-fold

symmetry in a macroscopic solid is to use two difikzdy shaped constituents

which can be made to fit together without any bond frustration. This does mean,

however, that all studies of this kind must begin with a b i i system and not a

single atomic type crystal.

1.6 Thin Film Structures

While the discovery of quasi-crystals gave us a natural example of

aperidcity, we are able to manufacture aperiodic systems artificially in a

laboratory. We are able create a series of thin 61ms (a superlattice) of two

Merent types of compounds in the laboratory today. Typidy, one might

alternate between two dierent compounds and use layers of equal thickness. This

would result in a periodic system. However, we could also arrange these layers in

an aperiodic sequence instead. The result of this layering would be a m e d o g

ofthe one dimetlsiod model we have been studying. The principal difficuity is

that the effects we are studying tend to emerge only as the number of layers

becomes large. The sort of sizes we will ultimrtte1y consider are far too large to be

done in this manner.

h o t h a artificial structure could be created using while dielectric materials

in the manufact~ue of wave guides. Once again, we could alternate two different

types of dielectric material in an aperiodic sequence in order to produce a structure

with the properties of an aperiodic binary atomic chain. However, since the point

of a wave guide is to achieve tight control of the system in order to communicate

intbrmation accurately, the arrangement of a series of wave guides in a F~haacci

sequence would serve better as an example of "what not to do" thaa as a practical

item of interest. The reason behind building such devices seems to preclude

arranging the dielecaic slabs in an aperiodic sequence as an effkctive optioe

1-7 F i m e s for Chanter 1

Figure 1 - 1 : The triadic Cantor set

Chapter 2 : Model and Method

2.1 The Flbmcci Chain

Back in section 1.1 we introduced the concept of an aperiodic sequence.

There are many difFiient kinds of aperiodic sequences known to physicists and

mathematicians that c d d be used to model a binary atomic chain (the typical

range of choices is Fthnacci , Thue-Morse, periodic doubling or Rudia-Sbapiro) .

While an anaiysis of any ofthe various choices might provide valuable insight (and

have been discussed by other authors), the Fibnacci sequence has been chosen

because we hypothesize that it will give us an energy spectnrm that is a variant of

the Cantor set. The Fibnacci chain (a binary sequence of atomic sites generated

by the Fibonacci method) shows quasi-periodicity making it a good model for

quasi-crystalline structures (see section 1.7). This makes the Fibonacci sequence a

possible candidate to model naturally occurring (or potentially naturally occurring)

structures. In addition, layered compounds at any sequence have been fabricated

by physicists. Since the Frbnacci sequence is applicable to both quasi-crystals and

thin film structures, it is our best choice for examhation (as opposed to some other

types of aperiodic sequencing which are only applicable to thin film structures).

First we should begin by examining what a Fibomcci sequence is and the rules for

generating it.

The generation of a Fibonacci sequence of a binary system is achieved by a

simple rule. Let us begin with a single atom of type A Now, we generate the

Fibonacci chain by replacing A with B and B with B A So the chain, at each

successive replacement, looks as follows:

A

B

BA

BAB

BABBA

BABBABAB

BABBABABBABBA

BABBABABBABBABABBABAB

and so forth. This simple replacement scheme is all that we need to generate the

Fibonafci chaia. The length of the c&in at each step corresponds to a Fibonacci

number (1, 1,2,3,5,8, I3,2 1 ...) which are obtained by adding the two previous

numbers in the chain together- Another way to generate the chain can be found by

noticing that the chain at the nth step consists of the chain at n-1 followed by the

chain at a-2. So, to generate the chain at step n, one adds the chain at step n-1 to

the chain at step n-2 rather then going through a lengthy substitution process. If

we examine the sequence of atomic sites above, we note that the chain sequences

obey the basic recursion relation where Fn is the nth Fiboaacci chain:

which will also give us the Fibonacci numbers (we add the last number of sites and

the second last number of sites together) so long as F, is a Fibonacci number (i-e.

the number of sites is always a Fihaacci number under this scheme). The ratio of

the number of atoms of atomic type A to the number of atoms of atomic type B is

y (the Golden Ratio) in an infinite chain. This gives us an aperiodic chain that

never repeats regardless of the length (as will be seen in the following paragraph)

but has a definite order and is most certainly not random because the chah is

generated by a rule.

A typical example of a aperiodic ac t ion givm by mathematiciaas is the

sum of two harmonic fimctions that have an irrational ratio of their periods. So,

for example, the function @)=sin (2nx) + sin (27cax), where a is an irrational

number between 0 and 1, is an aperiodic functio~~ To show that a Fibonacci

sequence belongs to this class of aperiodic functions we first define what is called a

HulI h c t i o a The Hull firaction is defined as Vk= [(k+l)y]~ - @cy]~, where the

numbers in the square brackets are rounded down to the nearest integer, I means

that the numbers are integers and k is an integer. This will give us a function that

generates a sequence of 0's and 1's as k increases. If we replace the sequence of

0's and 1's with A's and B's w e get the Fibonacci sequence for a binary atomic

chain as shown above. Now, i f we replace k by a continuous real variable, say x,

then we get a periodic function with a period of (I-), and y is obviously irrational.

This means that we have two basic periods in Vk, on is 1 and the other is y. Since

y is irrational, no wmm~n integer multiple of y and i can possibly be found. This

means that we have what is lmown as two i n c o m m ~ t e periods (meaning no

rational common multiple can be found). The addition of two fimctiorrs with

incommensurate periods gives us an aperiodic fhctioa Thus, the Fibonacci

sequence can be used to construct an aperiodic function.

2.2 The Cantor Set

Back in Chapter 1 we presented the idea that a one dimensional chain of

atoms (or a cut across a two dimensional Penrose tiled surface) wuld possibly lead

to an energy spcmm that was a Camor set ofLekgue m- zero. W e this

is not proven, it has been i d i d h m numerical experiments (like the one we are

doing in thjs thesis). Now, befbre we discuss how we are going to use this

construction to generate the energy spectrum of an aperiodic binary atomic chain,

we should take a fw moments to introduce the Cantor set in a more formal way.

First of all, the Cantor set we desaibed back in Chapter 1 was the simpiest

type of Cantor set: the triadic Cantor s e t One a n easify determhe it's Lebesgue

measure as follows. Mer the nth stage of generating the triadic Cantor set we

have 211 disjoint intervals each of length 311. So the Lebesgue measure of the set C

(the Cantor set) as we let a go to M t y becomes

This means that we have a set with an infinite number of members (because 2n has

an infinite number of members as n*) with a total length of zero.

There are other forms of the Camor set and, in fjtct, we will encounter a

different type of Cantor set in this Thesis (our results will make it obvious that we

are not dealing with the traditional triadic form of the Cantor set). In our

numerical experiment to determine the eigen-value specmun of an aperiodic binary

atomic chain, we will d e the claim that our set of eigen-values is a slight

variation of the standard form of a Cantor set The eigen-value spectnnn that we

are trying to confirm the existence of is much closer to that formed by a line

divided in 5 sections in which the second and the fourth section have been

removed (see figure 2-1 for an illustration). This also gives us a Lebesgue measure

of zero as we have 3n disjoint intervals each of length 9. So this gives us L(C)=O

as n-w, as we get the of (3/S)n as n approaches infkity- Of course, this

assumes the line lengths are equal (they aren't in our experiment) aad that the ratio

of removed line to retajned h e is precisely 2 to 3 (we have no reason to assume

that this is the case in our experiment).

However, we can be hopefid when we note that it is also not necessary fbr

the hes to be of equal length so long as the relative ratio is preserved at each

stage for the Lebesgue measure to go to zero (Feder 1988). The key feature of a

Cantor set is that we remove a certain percentage of a line at iteration of our

generating procedure. So long as the amount removed is the same at each

iteration, we will have a system that hss an infinite number of collstitlleuts [m fm

unwuntably infinite) of measure zero. So, in e f f i it is this "self-similarityn

criterion that distinguishes the Cantor set. Like with all l-D self-similar fiurctiom,

any single line le~gth contains the whole set in the infinite case. As a result, in our

quest to demonstrate the presence of a Cantor set, it is the self-similarity feature is

the most important.

Ow of the authors in this field, M. Kohmoto, claims that the eigen-value

spectrum of a one dimensional binary atomic Fibonacci chain is a Cantor set. He

also claims that the ratio between the line lengths (the amount not removed at each

iteration when generating the Cantor set) and the total lea@& is y (the Golden

Mean). Now if this ratio is prewwed though an idkite number of repetitions

then we will still have L(C)tO just at a slightly Werent rate of convergence.

There are two important points to make at this stage. One is to emphasize

that we have not assumed a Cantor Set in any of our initial assumptions. Instead,

we took a Fibonacci Sequence of atoms and determined the LDOS. If the density

of states gives us a Cantor Set euergy specmun it is solely due to sequence eft is .

Two, it is important to recognize that we can wwr actually generate an W t e

sequence of sites and so we wiIl never actually generate a compIete replica of the

true Cantor set. Instead we must be satisfied with a system whose e E m s

approach that of a Cantor set as the number of sites increases towards intidy

(which means that macroscopic systems should show a very cbse approximation

to the true Cantor set). So what we are looking for is a system in which the

general fonn of the energy spectnun is seEsirmi for at least a fw mamrifidons

(i.e. we might not have the entire set cuntained in each segment, but we should

have some evidence that the fimction is actually seff-similar). This is the best proof

we can get of the presence of a Cantor Set.

2.3 The Tiaht Binding Model

In this problem of a one dimensional binary atomic Fihnacci chain, we are

using the tight binding model (TBM) (sometime referred to as the tight binding

approximation). The TBM assumes that the electrons in a solid are sdEciently

tightly bound that we need only consider nearest neighbors. This will be true in

many physical problems when the wave fimctioas at the individual atomic sites

decay to zero betbre they reach the second nearest neighbor- Remember, as well,

that in our one dimensional modd the spreading of the wave hction will be

blocked by the nearest neighbors and there are no other directions for interaction

to take place in. The Ti@ Binding Hamiltonian (iicluding only nearest neighbor

interaction) fot a binary alloy is:

wherc i is the site, +I, ~i is the site and is the & d m C ~ C W . &j

Fibonacci chain ( i i our modeI, in general, Ei represents the site energy of an atom

at the ith position m the chain).

The tight binding model is often used to d e s c n i the behavior of insul:aton

and semiconductors and would be inappropriate for a metaI (for which these

assumptions would be incorrect as the electrons in a metal are highly mobile). We

are mostly interested in pure binary crystals with the collstituents ananged in a

Fibonacci sequence (ahbough we will also ccmsider pure substances in this thesis).

So let us define what we mean by a binary doy.

In a binary alloy, we have a case in which two diffixeut types of atoms have

been mixed together (either randomly or in some deliberate scheme). The most

obvious example ofthis is the vcry simple periodic arrangement where we alternate

between two different types of atoms with diffixent binding energies. It is

necessary to make certain assumption here, and in our model we assume that the

two sites, A and B, have diffkrent bmding energies but that the interaction energy

is the same regardless of whether a site is type A or type B. This simplification has

been introduced to make calculation c u a s i d d y easier wbik costing us little in

accuracy as interaction effects will be p ~ c i p d y second order effects. Reamnk,

we are also principally interested in sequence effects and we do not need excessive

amounts of fine detail in the model so we have chosen to use the very simple TBM

in which the interaction energy is nearest neighbon only. Chakrabarti et a1 (1989)

used a TBM to study a similar system- However, instead of varying the site

energy, they used variable interadon energies to study their problem (loag and

short bond modef). So varying the site energy is not the only way to introduce

sequence effects, but it is the most m e .

We know &om Quantum Mechanics that the expectation value of the

Hamiltoniaa gives the energy eigeu-values as follows:

where H is the Hamiltonian given in (2.3) and Ei are the energy eigen-values that

occur when i=j (as the equation is d o n n l y equal to zero when i4). In our case,

by substituting for H above, we get:

where j++l and 63 is the Dirac delta fimction.

In the model proposed in this thesis, we are dealing with a 1-D binary chain

of atoms in which the tight binding approximation is asswned. The atoms

themselves are arranged in a Fibonacci chain of some arbitrary length generated in

the fahion that we saw back in section 2.1. W e are concerned with the eaergy

spectnun ofthis chain aud the density of states that redts from it. To detamine

the density of states (or more precisely the local density of states) we will use the

local Green's function,

2.4 Simplifications Reseat in Mode1

Because we are principally interested in the results of sequence & i s in

the eigen-value spectrum ofa one dimensional atomic chain, we have neglected a

number of complications that would be present in a real physical system.

Following Economou (1983) they can be summarized as follows:

1. We have assumed that there is only o m atomic orbital at each atomic site in our

binary atomic chain 6.e. only one valance dearon). This would rarely ($ever) be

true in a real, solid state system. The model of a system with more bound states at

a given atomic position would tend to make the system extremely complicated at

little gain in understanding as we are interested in sequence effects - not the

intricate details of the structure effects (which are harder to accurately represent

then they are worth).

2. The nearest neighbor approximation that we used is o h not enough in a real

system - second or third nearest neighbor effects might have relevance to the

problem in a real system However, the effects of second and third nearest

neighbors are much less of a problem in a one dimensional system where the wave

function of the electron is unlikely to extend to second or third nearest neighbors

because it is blocked by the nearest neighbors- Once again, while we s M d keep

the problem of interactions besides nearest neighbors in mind when gen-g to

a real system, it adds unnecessary complicatious to this thesis.

3. True atomic orbitals will tend to have nearly orthogonal wave functions instead

of the wave fimctions being actually orthogonal to each other. This introduces a

small inaccuracy that is not aivial to deal with (it makes the Green's Function

Method approach problematic ifone wishes to deal with this &kt).

It is quite hqortant to keep in mind that the TBM model bas these

drawbacks when genefajizing to an arbitrary real system Each of these are

complications that could in principle be introduced to a system in order to do a

more accurate calculation. Obvious1y, if the system bas very weak bonding, a

model along the lines of the Nearly Free Electron (NFE) model would be more

appropriate than anything considered here.

2.5 The Characteristic Eauation Method

Before we descrii the method used to evaluate the eigen-values of the

energy spectrum in this thesis, it would be instructive to consider how physicists

have traditiody gone about evaluating the DOS. Here we are solving the

Schrodinger equation fi = h!h or, we use Dirac notation,

H( Y n ) = El ~ n } for each of the eigen-values of the spectnun. The simplest

manner is to c o w equation (2.5) into matrix form and find the deteminate of

the matrk In the tight-binding approximation, we can represent the general wave

function as a summation of atomic orbitals. This gives us 'P = for our i

wave function for a T B U Now we form the matrix which gives us:

where the is the site energy, V-4 is the iatemction energy between the nearest

sites (which is assumed to be the same tbr dI m e s t neighbor interactions) and E

is the eigen-value we are solving for- For a binary alloy, Ei takes on only two

values (E A, E B ) depending on whether the atomic site in qudon is an A-site or

a B-site. Then we generate the characteristic ecptioa as a polynomial of order n

(where n is the order of the matrix) and solve for E. We then count the number of

states in each energy range and display the result as a histogram. In the abseace of

periodicityf this means that the matm in (2.6) is infinite in size. Even for a

truncation at 1,000 sites, we have a million element matrix. Obviously, such

immense matrices are not practical for calculation. This method is obviously too

cumbersome for large systems as s o h g the characteristic equation becomes more

diflicult and numerical stability declines. Besides, this technique tends to be

numerically unstable. Since the phenomenon we are ubmately looking for (the

presence of a Cantor set in the eigen-value spectnrm) is more pronounced as the

number of sites increases, we decide that this traditional technique is not what w e

want to use for our problem and move on to discussing the Green's hction

method.

2.6 The Green's Function Method

Now that we have generated the chain of atoms, d&ed the sequencing

and examined the Quantum Mechanics that govan the behavior of the chain, it is

necessary to develop the mechanism we will actually use to find the ektrouic

density of states for the atomic chain. The mechrrnism is based on the Green's

Fw1ction Method. This metbod was chosen because it has long been used to

determine the energy spectra for I-D TBM systems although with di.fRxent atomic

sequencing (Economou 1983, Pant and Tong 1980, Tong and Pant 1980).

Green's Functions are-best kuown as the solutions to the set of diffitial

equations defined by:

m which E is a complex variable. In (2.7), L<r) is a time indepeudent, linear,

Herrtlitian djffierential operator which obeys the eigen-value equation:

L ( d l + K ) = ~ n l + j ) (2-8)

where E, are the eigen-values. The set (&I can also be determined to be

orthonormal without loss ofgeneraiity and thus:

( o f l o j ) = sii (2-9)

The results of equation 2.8 and 2.9 are properties we would expect of our

soIutions to these differential equations. In Quantum Mechaaics, L is replaced by

H, the Hamiltouiaa So we are able to gemate a formal solution of the

Schredinger equation W=EY using G = - (so that each &gem-value witl be E - H

represented by a sungularity). In matrix form and for the TBM we have:

I 1 G=(YI-~)=-

E - H M

where M is a matrix of the form seen in equation 2.6.

The Green's Function Method is well studied (Economou 1983) when

applied to a periodic system. However, when dealing with an aperiodic system, we

are forced to consider a further simplification

2-7 The L m d Green's Function Method

In this thesis we are interested in the Iocal density of states for reasons of

computational efficiency. The density of states is defined as:

and the local density of states as:

where en is the nth eigen-due ofthe system (Economou 1983). The equations

are swnmed over all of the eigen-states present in the system. In a very small

system (i. e. very fm atoms in the chain), equations (2.1 1) and (2.1 2) are counting

bctions that lead to a series of discrete peaks. In an infinite system (i-e.

macroscopic), the energy eigen-values are so ctosely packed they tend to fonn a

smeared, continuous band

Now, if we were to use equation (2.12) to evaluate the local density of

states (LDOS) we would be back where we started, having to evaluate all of the

eigen-states in order to evaluate the LDOS of the Schrtjdioger equation.

However, it turns out that we can relate n(E,r_) to the imaginary pomon of the

Green's Function. W e can express the LM)S at a given atomic position as:

where G is the Green's Function and r is the site at which this function is evduated.

The important property of the Green's fimction is that it can be evaluated directly

without having to account for the individual eigen-states.

Now Iet us express the Green's function in the fonn that is most convenient

for our problem. We have:

which we can also be represented as:

where (2.15) is an alternate form of (2.14) that enables us to easily separate the

imaginary portion of the Green's Function.

The first atomic site can be represented by the Greents h c t i o n as the

continued hction:

E- E n - b 7 -

b E- E n - I - - * - -

where E* is the site energy at site n, b is the interaction energy between 2 sites and

E is the eigen-value (Heine 1980). Clearly, equation 2.16 wiU have many values at

which it will become a singular. So we will add a quantity ia to the site energy

where i is an imaginary number (fi) and o is a small number (if it is a large

number it will begin to outweigh the site energy and thus mask it). This

"broadeningn is introduced in order to "blur" the singularities of eqyation 2.16 into

a series ofpeaks Otherwise, the siogularities become so sharp that they cannot be

seen. We think of this as looking at the energy spectrum through a "fkq leas".

It is simple to justay this form of the local Green's W o a (LGF) fiom

simple matrix algebra for a tridiagonai matrk In the LGF problem we are

interested in the first diagonal element of the Green's Function Matrix (which

contains all of the information on the location ofsingdarities although they will be

distorted as they are the singtilatities as "seen tiom the first siten). If we take our

tri-diagonal matrix as:

in which D represents the diagonal elements and V represents the interaction

terms. Now the Green's Function at the first diagonal site is Gu = (M-L)ll ; the

(1,l) diagonal element of the inverse matrix of M.

If we have a 2x2 matrix such that M = then we can define the

inverse matrix as:

which we can apply to our nxn problem If we set a=D1, fk(V1.0,0,0,..-), x =(v~,o,o,o, ..JT and 6=M' (the M matrix reduced by one rank) then we get:

as the (1,I) element of the inverse matrix. Now we need to evaluate ( o ~ l ) ~

and we apply the same procedure which gives us:

which if we continue to evatuate will yidd the continued firacton we see in

equation (2.16). We are greatly assisfed by the zeros in the f3 and x vectors which

simplify the problem Now that we have the fom ofthe LGF, let us discuss how

we apply it.

This form ofG is easily seen to be a sofutioa for the LDOS by noting that

1 n(a, E) = -- I la)). &re we are expressing the diagonal 15 h((al +iru-H

elements of Gkr l ; E) as when r_=a. This form of G will give us a E t i w - H

series of infinitely sharp peaks when we attempt to evaluate the density of states

(which is a scaled form of the imaginary portion of G). Since we cannot see

infinitely sharp delta fbnctiom (and locating them is very cMECUIt) we introduce an

imaghuy term to broaden the peaks. So this is why we added the term ia, to the

E n terms where o is usually a very small quantity. Adding this co term gives us:

This is the final form of the equation that broadens the peaks as ifwe were using a

fUay lens to observe them and thus makes it possible to see them. In chapter 3 we

will examine the results that this scheme gives us for the

pure and periodic materials. But fmt, let us consider the

"well known" systems of

techniques used when the

atomic chain being evaluate by the LGF becomes very long.

2.8 Calculations for Large Svstems

While directly evaluating the continued tiaction is an & d v e method for

relatively small systems Cm this thesis it has been extended to atomic chains with as

many as 233 atomic sites), for extremely large systems it b m e s impossitbly

cumbetsome and numerical instabilities are a serious problem. However, in order

to model large atomic chains of over a thousand sites we need to consider an

atomic chain that is large enough to "approximate" some of the fkatwes of an

infinite system. In addition, we have reason to suspect that there might be sew-

similarity present in the energy spectrum and testing for &similarity requires that

we have to drastically reduce the energy scale more than one or two times. So we

wiU have to get clever to deal with the inverse of a matrix of a typical size of 6000

by 6000 elements.

The first thing that we need to note when considering a binary chain

arranged in the Fibonacci sequence is that each sequence is the sum of the previous

two sequences (we saw this in section 2.1). Now we are evaluating our function

at a d c e site (i.e. the beginning for a long chain of atomic sites). ' Ibis gives us

the LGF of equation 2.2 1. Directly evaluating equation 2.2 1 is impossible as

numerical instability would make the results meaningless. What we do instead is to

define a series of hctiods. First of all, we set the interaction energy ~=t?=l

(the nearest neighbor interaction) and scale all of the energy values in our system

to the interaction energy being equal to one (so we no longer have to account for it

as it always multiples one in our equations). For an atomic chain consisting of a

single B site we have G . =GI = 1 1

= - where we use b as a simplified E B - i ~ b

notion for the site energy minus the broadening For an A-site in the same atomic

chain we would use a =E A - io as our simplified notion. Now we can expand this

atomic chain with a single atom to a chain of BA (conesponding to the next stage

1 in the generation of a binary Fihnacci chain). This gives us Gz = - t

as our b - -

a

second order local Green's bction. For a chain of BAB, we get G3 = I

1 b - - 1

as our third order Local Green's Function W e can obviously conhue this process

for as many iterations as we need to. Now we dehe the ith local Green's function,

1 q, as Gi = Gi[z]z = 0 where G d z J = G ~ [ z ] = - - This means that

6-2

order to generate our LGF for a chain of arbitrary length, we use a recursion

1 relation with initid conditions. So we define G I = G b [ O ] = - and b

1 G 2 = Gr[G.[O]] = -

I as our initid conditions. Our recursion relation is b - -

a

G[z] = Gi - l[Gi - 2 [ ~ ] ] which we recognize as the recursion relation used to

generate a Fibonacci sequence we saw in equation 2.1. Then we can define

G 3 = Gb[Ga[Gb[O]]] = G 3[G 2 [ 0 ] ] as the first elemem in which we use the

recursion relation. Continuing the progression, we get G 4 = G 3[G 2[0]] as the

second element generated by the mansion relation+ We can contirme this

progression indefinitely.

Using this method we can generate G i[ z ] , where i is the Fibonacci

sequence number. So in order to evduate a biaary alloy arranged in a Fibonacci

sequence of 6765 sites we only need to evaluate 19 of these functiods (provided

we completely evaluate the polynomial at every stage). To evaluate more than a

million sites would only require 30 steps instead of the million recpired by the

approach discussed in section 2.7. This gives us vastly improved speed and

accuracy over co11ventioC18t techniques as we need to do fsr fwer evaluations over

which much less numericai error will be introduced.

2.9 Urhv the LDOS is Immrtant

In very d systems it is no great inc011venience to evaluate the total

density of states assuming a tight binding model me in all ddat ions in this

thesis). All that finding the DOS requires is to evaluate the LDOS (which is the

deusity of states taken a a specific atomic site) at every site and average the result

(to eliminate the position dependent ef€kcts from the system). For a chain with

only one type of atom that is infinite in extent this is easy as every site is identical.

Taking the LDOS automatically gives the DOS. For a periodic binary alloy which

is infinite we only have two types of sites and evaluating the LDOS at two adjacent

sites gives us the DOS. In both of these cases there is either little (periodic) or no

(pure) increase in computation time to generate the DOS and so there is no reason

why we should concern ourselves with the LDOS. These models are suitable for

the examination of bulk states (states fu fkom the surfkce of the material) and

W t e size is easily simulated by a @odic boundary condition

This m m e r of genexating the DOS begins to break down in two specific

circumstances; when we introduce disorder or when we examine the material war

the d a c e . Introduction of &&order begins to cause problems rapidly as sites

become unique since the chain seen on either side of a specific atom is unique for

each and eveq atom In this case it is necessary to evaluate the LDOS at every

site m order to generate the W S (which multiples the computation time by an

order oh) . Examining the energy spectmm near the d a c e causes the same sort

of problem (Le. every state is unique) as the distance of the atom hm the stdhce

will give it a unique position and thus a unique LDOS. Once again we are forced

to evaluate the LDOS at each and every site ifwe want the actual DOS. However,

the information on the positions ofthe eigen-values is contained in the LDOS but

it tends to distort tbe DOS based on position effects.

In the F ~ b n a c d sequence binary alloy we are dealing with the same sort of

problems we see in disordered systems. Since the nature of the Fihnacci

sequence is that it never repeats, it is obvious that every site must be unique.

Given this we typic* choose to generate the LDOS for the chain at the SUCface

(it is rtn arbitrary choice but it simplifies things immensely by giving us interactions

with other atoms only in one direction). The advantage of our method in section

2.8 is that we manage to use the nature of the Fibonacci sequence to vastly reduce

the number of computations needed to calculate the L W S for a long chain- The

question we need to ask is whether or not acamining ody the LDOS will be

enough to give us usable results. The alternative is impractical, as a calculation of

a system of 300,000 sites can take a half hour on a 486 computer. Multiplied

300,000 times results in a computation fkr too large to be practical.

The clue resides in what f ~ e s of the DOS that the LDOS tends to

distort. in a binary system we tend to bave two energy bands formed (one for each

type of atom). When we wduate the DOS of the system we will see each band

clearly and properly formed. When we evaluate the LDOS we are evaluating it at

one specific site. The band formed by contniutiom £kom the type of atom at the

site which we evaluate the LDOS will be it's proper shape, but the other band will

be suppressed or distorted What this feature of the LM)S meaus is that we

should get good results in one of the two bauds and we can always change the

location at which we are evaluating the LDOS to get a clear graph of the other

band. In fact, as we will see in section 4.3, the central featwe we are looking for

in our aperiodic binary atomic chains (the Cantor set) will be present in both bands

if we examine a system with enough sites.

This is a new approach which hes been made necesacy by the difliculties

posed by this type of problem Similar work has been attempted for disordered

systems (Heine 1980), but this is the first systematic attempt to deal with this exact

model of a I-D binary atomic Fibonacci chain using this technique. The use of the

LGF to get the LDOS is not the most desirable approach to the problem, but given

the constraints of computer time it is the most practical. In the next chapter, we

wiU examine the LDOS of the well understood cases ofthe periodic binary alloy

and pure systems. From the results for pure and periodic systems, we will be able

to predict the sorts of efftixs that this method will have when applied to a binary

alloy arranged in a Fibonacci sequence. As well, this study will enable us to gain a

clearer undemanding of the LGF method and compare our results to those

predicted adytically. As will be obvious, without our new technique for

calculating the LDOS for iarge systems we wodd be unable to get good results

fiom the LGF- We would be M e d to the number of sites we could use to around

the 1042th Fibonacci number. This would be far too fw sites for a good

representation of the LDOS for very large systems as we will see in chapter 3.

2.10 Figures for Chapter 2

Firmre 2- 1 : The five piece Cantor set.

Chapter 3 : Local Density of States for Pure and Periodic Structures

3.1 Analytic Solutions for Pure Structures

Both pure and periodic structwes cen be d v e d analyticallyY So, one of

the most important things we want to do is to ensure that the results that are

produced by our model are consistem with these expressions. Since the systems

we will study in chapter 4 do not have analytic sohtions (because of an absence of

periodicity), it is important to check our tool in an area where we do have

solutions that can be compared to the results of our calculations. This will give us

far greater confidence in the resuits we obtain when we use this tool in other

systems.

Fist, let us consider a semi-infinite chain (a chain which extends an infinite

distance away fiom a point on one side) composed ofa single type of atom (call it

A which has site energy a). W e can define the Local Green's Function (LGF) as:

( E i - a ) - b

b (&-a)-- * - *

a is the binding energy of the site, and b

atomic site in the chain Now, since this

is the

chain

where Ei ate the eigen-values,

interaction energy between each

extends i&iteIy away @om the

site, we can remove the first site &om the chain and we sti l l get G. This enables us

to express the LGF as G = 1

or we can express this as ( E - o ) - ~ ' G

-61 GL + ( E - a ) ~ = 1 which is clearly a form of the quadratic equation:

b2G2 - (E -a)G + 1 = 0 . W e can solve this using the quadratic formula and we get

( E -a) kJ(E-a;-4b2 G = as our LGF for a semi-infinhe chain of atoms.

2b2

Now, the DOS (we have the DOS here because the position doesn't matter in a

pure chain of atoms as aIi sites are the same) is defined as the -Im G. However, G

is only complex if (E-~P < 4b2 (which also gives us our maximum bandwidth).

the curve has it's maxirmun at the middle of the band when @-a)+ and goes to

zero at the band edges in the serni-idhite case. See figures 3-1 and 3-2 for the

shape of the a w e with b L 1 and a= 3and -2 respectively.

Now, if we consider an e t e chain instead, we get a different local

Green's fimctioa Our new function now must consider that the chain extends for

an infinite distance in both directions. The new equation is

Gulf= I

2 2 where we account for both a left and

...

a right hand infinite chain. Now,

chains in the expression fbr GX

*. -

once again, we can consider the two semi-infinite

to be G (the locat Green's firaction f i r the semi-

infinite case). This would give us G ha = I where G is the LGF for ( ~ - a ) - 2 ; ~

the semi-infinite case. In the semi-infinite case, we can represent the LGF as

G = ( E - ~ ) * ~ which we can substitute into our expression for 2d

is only going to be imagiaary if@-@ < 4b2 (which, once again defines the width

of the band). This meaas that our DOS for an infinite system will be

DOS= 2:

1 - The minimum of the equation will occur at E=a and the

DOS will go to infinity at the band edges (as we get the inverse of zero). This

form of the DOS is supported by the literature (Economou 1983, KitteI 1986).

See figures 3-3 and 3 4 for the shape of the curve for 61 and a= -3 and -2

respectively.

The expression here for the DOS are off by a scaiing fkctor (1) but give us 7r

two very important pieces of information. F I i of all, they define the behavior of

the DOS at the band edges relative to the bmd center- The band center is a

tnaxhum for the semi-infinite case and a minimum for the infinite case. Second,

they define what the band width is and w h it should be centered on. In both

cases the band will be centered on (La) and the width is defined by the condition

(E-~P <4b2 which is required for G to be h a g b y (and the DOS is the imaginary

portion of G). If was examine the four plots of the analytic hctions (figures 3-1

to 3-4), we note that the width is uniformly equal to 4 (which is what we would

predict given the condition (E-~P 4 and the values we have assigned to the

interaction energy-

3 -2 Bulk States for Pure Structures

Now that we have discussed the analytic solutions for pure materials, let us

begin our exploration ofthe resuhs of our numerical calculations by discussing the

LM)S of a pure structure. Because we are dealing with a pure material, the

LDOS at any site is the same throughout all of the sites in an infinite system (since

no matter which site is chosea the chain extends out an infinite distance on both

sides). This means that in this case the LDOS is equivalent to the DOS since there

will not be any position dependence when d sites are identical. We will notmalize

(i.e. scale everything in our problem to) the interaction energy, b in the analytic

solutions we discuss above, which is then one and set the site energy (a in our

discussion above) to be negative three (ii arbitrary energy units scaled so that the

interaction energy is one). Since we cannot actually solve n infinite chain, we take

a lattice with 89 sites and apply a periodic boundary condition at the ends (so that

It repeats itself indefinitely thus giving us an approximation of an infinite chain).

The energy band of -5.1 to 0.1 energy units (where positive one energy unit is

defined as the interaction energy). When we plot the LDOS (or DOS, they are

equivalent in this system) vs. energy, we get figure 3-5. We note that the DOS

becomes infinite at the band edges and forms a concave shape (just as we wouid

expect fiom our ansilytic solutions). The physical interpretation of this

phenomenon is due to the infinite number of infinitesimal wnrn%utions firom the

"far away" atoms in the infinite chain of atoms There is only one continuous

energy band. This concave shape for the DOS will remain reqprdless of variations

in the broadening tam cu (unless a becomes very large compared to the energy

terms in which case the shape of the curve is drowned out by the broadening). We

might note that we have singularities present at the band edges (making them very

well defined). Ifwe examine figure 3-6, we note that a tenfdd increase in a makes

the shape of the LDOS slightly less distinct

sharp). However, the shape shows a high

extremely closely to see even the slightest

(the band edges becomes a little less

degree of stability (one has to look

ditlkence). The singularitits at the

band edges (or the close approximations to singuiuities) are a cotuxpence of our

one dimensional d 1 .

If we vary the site energy from negative three to negative two arbitrary

energy w i t s and examine the same range3 we note that the center ofthe LDOS has

been moved fkom approximately negative three to appmrrimateiy negative two in

figure 3-7. This, once again, follows the analytic solutions which predict that the

center of the LWS wiU be at the same energy as the site energy of the atoms that

make up the atomic chain So we have well defined edges t&at are dependent on

the site energy which continues to follow the analytic predictions. One might also

note that the band width of approximately four is the same that we would predict

from the analytic solution in 3.1 which predict G is only imaginary when (E-~P <

4b2 (3, the band wid& must four times @ is fbw in this problem)-

3 -3 Bulk States for Periodic AUOY Structures

Now let us examine what will @pen a, instead of having a pure

substance, we have a system in which our one dimeasiod atomic chain is

composed of two different types of atoms which alternate with each other. Now

the LDOS should no longer be the same as the DOS because there are two

different types of sites and so not evey site is "identical" (so, immediately, we

realize that our analytic solutions are no longer a u* guide as they assumed that

all sites had identical site energies). The LDOS at an A site should be Merent

than the LDOS at a B-site. The U)OS ofa binary alloy splits into two bands, each

associated with one of the two different kinds of atoms. We wilt start our

examination of this type of system with a periodic chain beginning with site type B

and alternating with type A This will give us a chain of BABABABABA a d so

forth. In figure 3-8, the LWS is evaluated at a B site with ~ g - 7 units of energy

and we will see that alternating the system to a periodic b i atomic chain &om a

pure atomic chain has resulted in two distinct energy bands. Only o w of these

bands has a shape resembling that of figure 3-5. We have deliberately set the

energy ofthe atoms at the A and B sites far apart to avoid overlap in the bands.

The €3 site has an energy of nqative seven arbitrary energy units and the band

corresponding to it has the fkatures of the LDOS. The A site has an energy of

negative two arbitrary energy units and has had this shape greatly distorted. The

"deformation" of the higher (or upper) energy band is a consequence of the L W S

which is site dependent.

Now, let us examine what happens if we begin the chain with a type A site

instead ofa type B site. W e observe in figure 3-9 that we get the same locations

of energy bands, except now it is the A energy band that is broad and well d&ed

and the B energy band that no longer has the characteristic bowl shape. This

feature is a result of taking the Local Density of States; we are seeing the energy

spectrum as it is seen tiom a single atomic site. If we let the gap between the

bands decrease and then we get a much better defined upper band, as we see in

figure 3-10. Since we have narrowed the energy gap between the two sites, the

influence of the upper band has become much greater. We can continue to

decrease the energy separation between the two bands to the point that the bands

actually overlap in figure 3-1 1, and we have a continuous function in which the

two bands actually touch. Note that the height of the peak in the second band is

actually higher than one of the band edges seen in this figure. This is entirely due

to the extremely high level of interaction between the two diffkrent types of atomic

sites.

3.4 Pure Materials as seen tirom the Swfkce

Up until this point we have been discussing infinite chains of atoms, which

are an appropriate model for the bulk of a solid. For a finite solid, we have

surfaces which are often of interest. The singularities at the band edges are due to

the contniutions of "fx awayu atoms. At the d c e , our analytic solutions

predict that we no longer have these singulerities. [nstead, the LDOS forms a

"bell shape" like in figure 3-12. Here, even with over 6000 sites being used, the

band edges go to zero instead of M t y . The difference is that we are dealing

with a semi-infinite system and not a truly infinite system (the chain extends in only

one diedon and not two). The bell shape feature is extremely stabIe as the

broadening increases. The bell shape is exactly what one would predict corn our

examination of the analytic solution to this system back in section 3.1 (with a

maximum in the middle and zeros at the edges).

It is possible to destroy this stability as we see in figure 3-13, where we see

a set of peaks arising at very low levels of broadening What is happening here is

that we have increased our resolution to the point that we are beginning to isolate

individual groups of eigen-states. This f m e is an artifha of using only a finite

number of sites in our chain. If we decrease our broadening W e t , we see that

the LDOS becomes dominated by this peaky type of structure in figure 3-14. We

no longer see the smooth "bell shape" structure at all, but instead see groups of

eigen-values. By increasing the number of sites we could examine the LDOS at

this greater degree of resolution and stiu get the smooth curve, but we would still

reach a point in any M e chain where this sort of structure d become dominant.

Unfortunately, we cannot easily resort to an infinite chain as a remedy, as it is

impossible to create a truly infinite chain in a computer and we can no longer

impose a periodic boundary condidon here. In practice, when faced with the

appearance of individual peaks representiag eigen-values, we must either increase

the broadening (and thus decrease the resolution) or increase the number of sites

(and thus increase the computational cost to solve the problem).

3 -5 Periodic Allov Materials as seen from the Swfhce

When we look at Semi-infinite atomic chains for periodic structures (which

we will evaluate at the fkst site), we get the uasurprising result that the

singularities have been eliminated in the dominant region In figure 3-15, the

LDOS has a w e d the same bell shape we saw in pure materials. The general

structure of the upper band is unchanged (it still goes to zero on one side and to a

large value on the other).

Recall that in the previous section we noted that the LDOS began to break

up into discrete peaks when the resolution sufficiently sharp if the chain is not long

enough We repeat the analysis of this phenomenon we did in section 3.4 by

reducing the size of the chain in figure 3-16 to 178 sites from 13,530 sites. The

effect of reducing the number of atoms in our semi-infinite chain is dramatically

illustrated in figure 3-17 in which we examine what happens when we increase the

broademhg by a factor of 10. The series of sharp peaks collapse back into a

slightly jagged form of the LDOS for a periodic structure.

3 -6 Generai Notes

The caldatioas of the Local Density of States for a idkite periodic lattice

does give the exact Deosity of States for a pure system. In a periodic system, the

LDoS (evaluated at one end, the B-site in the binary alloy case) also gives the

correct locations of the bands but suppresses the features of one of the bands (the

one that one's calculations did not originate on).

If we remove the periodic boundary conditions for the chain of atoms and

substitute a finite chain for the idbite chain, we get the same positions of ow

bands but the band edge singularities are removed What has happened is that we

have removed the contriions of an iafinite number of " 6 ~ away" sites. While

the contniutions of these fa away sites is small, when an infinite number are

present it results in band edge singularities.

We can easily see by comparing figures 3-5 and 3-6 with figures 3-18 and

3-19 that the band width is p r e ~ e ~ e d when we switch fiom an infiaite chain to a

M t e chain, and the positions of the edges remain the same. Only the fatures at

the edges ofthe band are diftkrent (the LWS goes to infinity at the edge of a band

for a bulk state and it goes to zero at the edge of a band for a d a c e state). We

can easily repeat our analysis of a pure substance for a substance made of pure A

material (energy of negative two arbitrary energy units instead of the B sites of

energy negative three arbitrary energy units) we see the same effects in figwes 3-

20 and 3-21 when compared to figure 3-3. This repeat of our resuits tells us that

the same positioning of band edges we observe in figures 3-18 and 3-19 is not an

accident but appears to hold true as a general rule (which is what we would expect

given that we have solved the system analytically and predicted this). This is a

reason to be coafidat in our method of calculating the LDOS as it matches the

results that we predict by solving the system. The other thing it is important to

consider is whether or not the pealry structure due to imdicient b f d e n h g

affects the position ofthe edges of the energy band. We see in figures 3-22 and 3-

23 that it does not. The independence of the band edges fkom broadening effects

will become important in section 4.3 when we discuss self-similarity as some of the

early structures will tend to be a bit pesky. Since the band edges are not effected

by making the "resolutionn too sharp, we can have confidence in increased

resolution not dfixthg the positions of the eigen-values (which is what we would

expect after a i l as the pealq structure is a result of sampling too fbely as opposed

to any difliculties with the eigen-values themselves)). So when we calculate the

LDOS for a binary atomic chain in which the atoms are arranged in a F'ibonacci

sequence, we only have to question whether or not the edges of the band have the

correct fatures. We can be confident in the positions of the eigen-values despite

using the LDOS aad despite any peaky structure due to the close match between

our r d t s and the resuhs predicted by the analytic solutions.

We are emmining our finite chains at the d a c e of the material (Le. we

are examining one end of the chain). The result of this is that we will have a

surface state and we will evaluate the W S b r n the mfiace state- Since we need

to choose some site to evaluate the L W S at, it seems a logical d&t choice as it

is guaranteed to be unique.

3.7 Firmres for Chapter 3

Firmre 3-1 : A graph of the analytic expression for the LDOS for a semi-hhite

pure lattice. The chain is of entirely B type atoms (site energy is -3). The energy

range is fkom -5.1 to 0.1 arbitrary units. The interaction energy is 1 energy unit.

Fimre 3-2: A graph of the andytic expression for the LDOS fir a semi-idkite

pure lattice. The chain is of entirely B type atoms (site energy is -2). The energy

range is &om -5.1 to 0.1 arbitrary units. The interaction energy is 1 energy unit.

w e 3-3: A graph of the analytic expression for the LDOS for an *te pure

lattice. The chain is of entirely B type atoms (site energy is -3). The energy range

is from -5.1 to 0.1 arbitrary units. The interaction energy is 1 energy unit.

Fiaure 3 4 : A graph of the analytic expression for the LDOS for an infinite pme

lattice. The chain is of entirely B type atoms (site energy is -2). The energy range

is from -5.1 to 0.1 arbitrary units. The interaction energy is 1 energy unit.

Figure 3-5: LDOS of an infinite pure lattice- This chain is of entirely B type

atoms (with site energy -3). The energy range of the real axis is -5.1 to 0.1

arbitrary energy units. The range of the yoaxis (broadening term a) is 0 to 0.025.

The interaction energy is 1 energy unit.

Firmre 3.6: LDOS of an infinite pure lattice at two difEerent broadening. This

chain is composed eatirely of type atoms with site energy -3 and interaction

energy 1 . The top figure is a slice at broadening o=0.00125. The bottom figure is

a slice at broadening m=0.0125. These results are consistent with the analytic

results we saw in Figure 3-3.

Fimrre 3 -7: LDOS of an infinite pure lattice. This chain is composed entirely of A

type atoms (site energy -2 energy units) and has interaction energy of 1 energy

unit. The range of the x-axis is @om -5.1 to 0.1 arbitrary energy units. The range

of the y-axis (broadening term a) is 0 to 0.025.

Figure 3-8: LDOS of a periodic (BABABABA..) infinite lattiee. We take €he

LDOS Eoom a B type site. The A sites have an energy of -2 energy units while the

B sites have an energy of -7 energy units. The energy range of the real axis is from

-9.1 to 0.1 energy units. The top figure has a y-axis ranging f?om 0 to 0.025

representing the broadening, a. The bottom figure is a slice at a=0.00125. The

interaction energy is 1 energy unit.

Firmre 3-9: LDOS of a periodic (BABABABA..) infinite lattice. We take the

LDOS from a A type site. The A sites have an energy of -2 energy units while the

B sites have an energy of -7 energy units. The energy range of the red axis is tiom

-9.1 to 0.1 energy units. The top figure has a y-axis ranging from 0 to 0.025

representing the broadening, a. The bottom figure is a slice at (u4.00125. The

interaction energy is 1 energy unit.

Fiaure 3- 10: LDOS of a periodic (BABABABA..) infinite lattice. We take the

LDOS tiom a B type site. The A sites have an energy of-2 energy units while the

B sites have an energy of-3 energy units. The energy range of the real axis is -9.1

to 0.1 energy units. The top figure has a y-axis ranging from 0 to 0.025

representing the broadening, a. The bottom figure is a slice at a=0.00125. The

interaction energy is 1 energy unit.

Figure 3- 1 1 : LDOS of a periodic (BABABABA..) infinite lattice. We take the

LDOS fiom a B type site. The A sites have an energy of -2 energy units while the

B sites have an energy of -2.2 energy units. The energy range of the real axis is

60m -9.1 to 0.1 energy units. The top figure has a y-axis ranging ftom 0 to 0.025

representing the broadening, a. The bottom figure is a slice at (u-0.00125. The

interaction energy is I energy unit.

Do,

Figure 3-12: LDOS of an semi-infinite pure lattice. This chain is of entirely B

type atoms (with site energy -7). The length of the chain is that of the 19th

Fibonacci number (6765 sites) - where we begin our count of Fibonacci numbers

from 0. The energy range of the real axis is -9.1 to 0.1 arbitrary energy units. The

range of the y-axis (broadening term a) is 0 to 0.025. The interaction energy is 1

energy unit.

Figure 3- 13: LDOS of an semi-infinite pure lattice. This chain is of eutireiy B

type atoms (with site energy -7). The Length of the chain is that of the 10th

Fibonacci number (89 sites) - where we begin our count of Fibonacci numbers

&om 0. The energy range of the real axis is -9.1 to 0.1 arbitrary energy units. The

range of the y-axis (broadening term a) is 0 to 0.25. The interaction energy is 1

energy unit.

Fiwe 3-14: LDOS of an semi-infinite pure lattice. This chain is of &iy B

type atoms (with site energy -7). The length of the chain is that of the 10th

Fibonacci number (89 sites) - where we begin our cwnt of Fibonacci numbers

fiom 0. The energy range of the real axis is -9.1 to 0.1 arbitrary energy units. The

range of the y-axis (broadening term a) is 0 to 0.0025. The interaction energy is 1

energy unit.

Firmre 3-1 5: LDOS of a periodic (BABABABA..) finite lattice. We take the

LDOS £tom a B type site. The A sites have an energy of -2 energy units whiie the

B sites have an energy of -3 energy units. The lattice length is twice the 19th

Fibonacci number (13,530 sites). The energy range of the real axis is from -5.1 to

0.1 energy units. The top figure has a y-axis ranging fkom 0 to 0.025 representing

the broadening, a. The bottom figure is a slice at 0=0.00 125. The interaction

energy is 1 energy unit-

Firmre 3- 16: LDOS of a periodic (BABABABA..) finite lattice. We take the

LDOS &om a B type site. The A sites have an energy of -2 energy units while the

B sites have an energy of -3 energy units. The lanice length is twice the 10th

Fibonacci number (1 78 sites). The energy range of the real axis is &om -5.1 to 0.1

energy units. The top figure has a y-axis ranging fiom 0 to 0.025 representing the

broadening, o. The interaction energy is 1 energy wit.

Figure 3-17: LDOS of a periodic (BABABABA..) finite lattice. We take the

LDOS from a B type site. The A sites have an eaergy of -2 energy units wdde the

B sites have an energy of 3 energy units. The lattice length is twice the 10th

Fibonacci number (1 78 sites). The energy range ofthe real axis is from -5.1 to 0.1

energy units. The top figure is a slice at a=0.00125. The bottom figure is a slice

at 0=0.0125. The interaction energy is 1 energy unit.

Figure 3-18: LDOS of an semi-idbite pure lattice. This c h is of entirely B

type atoms (with site energy -3). The length of the chaia is that of the 19th

Fibonacci number (6765 sites) - where we begin our count of Fibonacci numbers

fiom 0. The energy range of the real axis is -5.1 to 0.1 arbitrary energy units. The

range of the y-axis (broadening term a) is 0 to 0.0025. The interaction energy is 1

energy unit.

F i m 3-19: LDOS of an semi-infinite pure la*. This chain is of eatirely B

type atoms (with site energy -3). The length of the chain is that of the 19th

Fibonacci number (6765 sites) - where we begin our count of Fibonacci numbers

Grom 0. The energy range of the real axis is -5.1 to 0.1 arbitrary energy units. This

is a slice across the y-axis at a=0.00125. The interaction energy is 1 energy unit.

Compare to the graph of the 8naIytic solution in Figure 3- 1.

Figure 3-20: LDOS of an semi-infinite pure iattice. This chain is of entirely B

type atoms (with site energy -2). The length of the chain is that of the 19th

Fibonacci number (6765 sites) - where we begin our count of Fibonacci numbers

fiom 0. The energy range of the real axis is -5.1 to 0.1 arbitrary energy units. The

range of the y-axis (broadening term a) is 0 to 0.0025. The interaction energy is 1

energy unit.

Figure 3-2 1 : LDOS of an semi-infinite pure lattice. This chain is of entirely B

type atoms (with site energy -2). The length of the chain is that of the 19th

Fibonacci number (6765 sites) - where we begin our count of Fibonacci numbers

from 0. The energy range of the real axis is -5.1 to 0.1 arbitmy energy units. This

is a slice across the y-axis at a~O.00125. The interaction energy is I energy unit.

Compare to the graph of the analytic solution in Figure 3-2.

F i w e 3-22: LDOS of an semi-infinite pure lattice. This chain is of entirely B

type atoms (with site energy -2). The length of the chain is that of the 10th

Fibonacci number (89 sites) - where we begin our count of Fibonacci numbers

fiom 0. The energy range of the real axis is -5.1 to 0.1 arbitrary energy units. The

range of the y-axis (broadening term a) is 0 to 0.0025. The interaction energy is 1

energy unit.

Figure 3-23: LDOS of an semi-infinite pun lattice. This chain is of entirely B

type atoms (with site energy -2). The length of the chain is that of the 10th

Fibonacci number (89 sites) - where we begin our count of Fibonacci numbers

from 0. The energy range of the red axis is -5.1 to 0.1 arbitrary energy units. This

is a slice across the y axis at ao=0.00125. The interaction energy is 1 energy unit.

This warns us that the LDOS can appear to be "artificially" peaky ifwe do not use

a sufficient number of sites in our calculation.

Chapter 4 : The Local Density of States for Fibonacci Sequence Structures

4.1 Band Edge Effects for a Fi'bonacci Sequence Structure

When dealing with a binary atomic chin amnged in the Fibonacci

sequence, we have a set of unique problems that we did not have when dealing

with a periodic sequence of atoms. Because the Fibonacci sequence never repeats

itself(even in the case where it is infinaely long), every single site in a binary chain

arranged in the Fibonacci sequence is unique. In the case of a periodic infinite

binary chain (see figure 3-6) at a B &e site, we get two distinct bands. The B

band (the lower band) wiU show singularities at the edges due to conm%utions

firom B sites that are "far away". If we alter our system to a semi-idkite (the

chain extends away only in one direction) binary periodic chain (see figure 3-1 1).

then we no longer get the edge singdarities due to "fju away" sites (as the infinite

number of sites comniutiag to the total are no longer present). Instead we get the

lower band forming a bowl shape (while the shape of the suppressed upper band

remains relativefy unchanged).

When we generate a finite Fibonacci chain of atoms we are dealing with a

very dif]Fient case. Because of the disorder present, we no longer need to worry

about the influence of "fm away" sites as they are masked by scattering. This

means that we would never have the baud edge singularities of the periodic case,

as we no longer get the a d a t i o n of small contniutions fiom an infinite

number of sites (see the analytic solutions and the discussion in chapter 3). This

makes the finite chain an even better model for the Fibonacci sequence binary alloy

than for the periodic binary alloy; which is fortunate, since the techniques used to

simulate a infinite periodic chain won't work with the Fibonacci sequence alloy.

However, we will also get A and B bands in the F t ' b o d case as we see in Figure

4-1 where we have a high degree of separation between the bands by sating the B-

site energy to negative seven arbitrary energy units (defined such that the

interaction energy, b, is equal to one) and the A-site energy to negative two

arbitrary energy units. W e begin a on B-site and that leads to the lower energy

band (the B baad) being more detailed while the upper energy band is suppressed.

If we reverse tbis aml start with an A-site, we get figure 4-2 in which the upper

band (higher energy) is detailed and the lower baad (lower energy) is suppressed.

If we take a cross section of figure 4-2, we get figure 4-3. Here we can see that

the upper band is split into three distinct parts while the lower band is one single,

thin band. This Mows the pattern we saw with periodic binary alloys in chapter

3; the type of site we measure our LDOS Eom has it's band emphasized while the

other band is suppressed. This much has not changed 6om the periodic case.

Now let us discuss what will happen if we were to impose the periodic

boundary condition on our system (so the fm left band atom interacts with the f a

right hand atom). If we begin on a B-site, we get figure 4-4 in which the lower

band is split into three parts and the upper band is now a single thin line. If we

compare this with figure 4-1 we see no apparent dafereme between the system

with the periodic boundary condition and the o w without it.. We can do the same

for a chain that begins on an A-site and the result is figure 4-5. Here, once again,

we see no discernible effect compared to the case without a periodic boundary

condition (figure 4-2). What this means is that the baud edge singularities will not

appear in the Fibnacci chain Lack of periodicity in the sequence is enough to

destroy it.

This brings us to an important conclusion: the results of a long, but finite,

chain are nearly indistinguishable &om an in6inite chain for a Fibonacci sequence

binary alloy. Band edge singularities are not important in the case of a Fibonacci

sequence structure case and so the LDOS of a Fibonacci sequence structwe should

be compared to that of a Semi-infinite chain (i.e. ody one side extends outwards).

Another conclusion impIied by these r d t s is that we only have to worry about

the structure in one of the two bands. The result of figures 4-1 and 4-2 is to

d e m o m e that switching the beginning site win only result in a "mirro~g" of

the bands; the structure of the dominant band will be the same whder it is the

upper or the lower band. We can also gain confidence in the fhct that the peaky

structure of the LDOS of a Fibonacci sequence is not artificial because it remains

despite f k z y levels of resolution; it is a red feeture of the band. Accepting that

the band structure ofthe LDOS is supposed to be peaky means that the features of

a band of a F~boaecci sequence structure are quite Merent fiom that of the

smooth semi-infinite periodic chain. They are characteristics due to the

introduction of the Fibonacci Seqllence and are not completely irregular or due to

having too smal l of a broadening term. W e will examine the nature and

importance ofthe peaky structure ofthe LDOS firrther in the succeeding sections.

4.2 Resolution Effects on a Fibonacci Chain Structure

One of the major speculations about a binary atornic chain with the atoms

arranged in a Fibonacci sequence is whether taking it's LDOS will generate a

fiactal. As we know, one of the central properties of a fkactal is that it should be

se@similar. What this means is that if we decrease the broadening a, we should

observe the fine structure of the peaks repeating the gross stnrcture we observe

when o is large. Let us consider a system of 6765 sites (enough that we can

negfect any problems with trying to resolve too finely). To begin with, we will

consider our system to have a binding energy at the A site of negative two

arbitrary energy units aad a energy at the B site of negative three arbitmy

energy units. Here we choose the interaction energy, V, to be equal to one unit (it

sfales the other energies in this system). This system is not the split band case but

the LDOS ofa F ~ h c c i sequence atomic chain evaluated at a B site. It will still

show an energy gap because the spread of the A-band is suppressed. We can

represent the LDOS of this system usiug a three dimensional graph (figure 4-6). A

more 6wly resolved three dimensional graph is shown in figure 4-7.

To d e the behavior of the LDOS more closely, we can take slices

across the y axis (the imagbry axis in the sense of complex munbers actually with

the x-axis being the real lint since we r e d that the LDOS is the imaginary portion

of the fimction G) and see exactly what is happening as we reduce the broadening.

If we examine figure 4-8, we see that there is little to no fine detail present. It is

possible to distinguish the LDOS for the A (binding energy of -2) and B (binding

energy of-3) sites, but not ciearly. If we increase the resolution by decreasing our

value of o fiuther we get figure 4-9. Here we note that the structure has become

much sharper and better resolved. The distinction between the contribution due to

A sites and the contriiiution due to B sites has become clearer. When we halve the

value of a, for our next look at the LDOS we get figure 4-1 0. Here the separation

of the A and B conm%Ution, is clear. More interesting, the peaks seen in figures 4-

8 and 4-9 are splitting up into muItipIe peaks.

This trend of single peaks splitting into multiple peaks is clearly continued

in figure 4-10 in which we can see a clear gap in the LDOS between about -2.0

and -I -6 arbitrary energy units. In figure 4-1 1, this trend towards higher detail at

lower energies continues. Among the B contrrbution (which, if you recall eorn

chapter 3, the LDOS is well defined where B sites colttribute to it when the binary

atomic chain begins on a B site), the LDOS has ctearly broken into three parts.

This same feature of peak splitting is present, but not as obvious, for the

conmcbution &om the A sites. In figure 4-12 we see the pattern of peak splitting

continue and we see the beginniag of the three groups of B site contnions

breaking into three groups themselves. Since the idea behind broadening is that it

represents a "blurringw of the energy spectnrm (or the spectral lines of the energy

spectrum in this case), one would expect to see this sort of minute structural

change when focusing in more c i d y on the spectnun This is, however, the

weakest evidence that we have for the fkctal effect because the fine detail is often

lost due to limitations in the software displaying the results. In the next d o n we

will deal with this problem by actually shrinLing the x-axis (changing the energy

scale) to see if fine detail showing u a l effects does in &ct exist without Nnning

into problems displaying the increasing amount ofdetail.

4.3 Cantor Set E n m S~ectrum

The previous section only hinted at the possicbility of there being a Cantor

set behavior to the energy spectrum. We saw how the LDOS slowly began to

develop a fine structure that suggested that it might possess self-similarity (groups

of three pattern beghhg to repeat itself in the groups of three seen in the B site

conmiution to the W S ) . However, in order to c1aim that the energy spectrum

is seKsimilar we require a much clearer demonstration of the pattern of peaks

splitting into three parts. So what we are going to go is to slowly restrict the

energy range of the LDOS. By "zooming in" on individual peaks, we plan to show

that the peaks themselves break up into groups ofthree. We then focus on mother

peak and h e it and so forth uatil we show this pattern over many repetitiom.

We cannot show this pattem ofpeak splitting indefinitely as this would fequire an

infinite number of sites. However, if we can show this pattern occurring a number

of times then we have clear evidence of the beginnings of a sef-similar spectrum.

In figure 4-16 we see a M energy spectrum for 317,811 sites. It is

necessary to use as many sites as possible in order to enable the rmadmum ~nmber

of amgdications before we no longer see the self-simiiar f w e s (remember, we

will only see these fatwes indefinitely with an idhite number of sites). In figure

6 1 3 we see that each bad of the LDOS bas roughly three separate peaks

associated with it. The B band is clearer as we have evaluated the LDOS at the

first Fibonacci sequence site wbich is a B site.. In figure 4-14, we have fbcused in

on the region that represents the contribution to the LDOS fiom the B sites- The

three distinct peaks in the LDOS for this band are clearer- In figure 4-15 we focus

on the right hand cluster and magnify it and obsene that the three distinct peaks

reappear again when we rescale the energy axis to contain only this peak (this is

what we mean by focusing in). W e continue focusing in figures 416,417 and 4-

18. By the time we reach figure 4-1 8 we have focused in 5 times and the group of

three peaks of the LDOS is still present in our graph (although it is beghing to

break down here). Here we used the m e degree of broadening for all of these

figures (a=O.OOIO). The problem is that as we restrict the energy scale further and

fkther, each graph contains fewer sites. As a result, we will quickly run into the

problems associated with not enough sites to generate a clear w e e To get a

better degree of resolution, we would need to generate more sites (with

diminishing returns in terms of our ability to achieve further rescalings of the

energy axis). The problem of finite computers is paramount here and makes it

impossible to continue this exercise indefinitely.

The other way to demonstrate seif-shddy better is to inmeme y as we

go back through figures 4-17 to 4-14 successivelyly By doing this, we plan to

"adjust our fbcus* for the swath of the energy spectrum which we are examining-

In figure 4-19, we have increased the broadening and increased the energy range to

that of figwe 4-17. In fxgure 4-20, we have increased the energy range to that of

figure 4-16 while also increasing the broadening. In figure 4-2 1 we have increased

the energy range to that of figure 4-1 5 M e once again increasing the broadening-

In figure 4-22 and figure 4-23, we are looking at the energy range of figure 4-14

(the fidl range of the B-band) with increased broadening. The truly remarkable

thing about this series offigures (4-18 to 4-23) is how similar the LDOS is in al l of

these figures. The shape of the LDOS remains very close to the shape shown in

figure 4-18 all the way back up the energy range to figure 4-14 with very little

variation In other words, we can recover the eatire curve almost perfectly from

part of the m e . This is the clearest demonstration of seKsimilarit y that we have

seen yet-

Now let us consider the section of the energy spectrum that represents the

A sites of the energy spectrum In figure 4-24 we see the energy specttum for the

A sites. We it by fausing on an individual peak in figures 4-25 to 4-29.

In these figures we have used the same parameters and broadening that we used in

figures 4-14 to 4-18. It is instructive to compare figures at the same degree of

magnification (example: figures 4-17 and 4-28 are an especially clear case) to see

how similar the form of the LDOS is. We could also vary the broadening function

o to bring the series of figures to look like figure 4-29 as we did for the B band.

W e could continue this examination by showing many more examples of this

behavior when focusing in on the curve but we are limited by the demands of

space.

What this examidon ofthe LDOS has clearly shown is that self-similarity

is present in that the general shape of the LDOS remains constant though many

rnagdications. Note, wen the gross proportions remain constant 0.e. the right

hand cluster tends to be broadest while the left hand duster tends to be murowest).

Also note that the curves tend to look similar even when taken eom very different

por&ions ofthe specmun (as seen with our A and B clusters above). Self-similarity

occurs when a portion of an energy spectnun contains the entire spectrum and we

most definitely have this here.

4-4 The Devil's Staircase

The last thing to discuss about the notion of ~e~similarity is the integral

density of states (IDOS). The lDOS is a counting density of states in which we

coum the number of eigen-values as we ascend in energy ftom the beghning to the

end of our energy range. The integral density of states is the number of eigen-

states below the energy value in question. We no longer need to worry about the

distortions based on whether the LGF is taken at an A site or a B site as we are

considering only the position of the eigen-values. Unfortunately, tb W s us to

considering relatively few sites as the effort in minting the eigm-values by hand

quickly becomes prohibitively time consuming.

Ifwe have a Cantor Set energy spectrum what we would expect to see is a

structure known as the Devil's Staircase. It is called the Devil's Staircase because

between each *stepn of the staircase is another entire staircase. This pattern of

each step containing the entire steircese continues on indefMe1y for an idbite

system. The obmation tbat the staircase contimed to get dkite1y small led to

the analogy of a "stairway to heUn in which the step continue downwards

indefinitely- The steps of this "steirway to hello get smaller and smaller until they

can no longer be resobed by mere mortals but are the stuff of "angels and

demons". The analogy explains the name, but, unfortunately, doesn't do much to

clarify the physics!

In figure 430 we see the f U spectnun for a energy spectrum of 233 sites

obtained by counting the eigen-values in narrow strips and summing the total. It

has the fatures we would expect (we see the basic staircase structure the name led

us to expect). We immediately focus in (by restricting the energy range) on the

lower energy band in figure 4-3 1 and note that we can still see the beghmhgs of a

Devil's Staircase form (each of the three rising portion of the curve also have two

steps in them). We know from our work in Chapter 3 and earlier in this chapter

that the band corresponding to the site we begin our chin with (typically a B-site)

is better defined than the band corresponding to the other element in the binary

chain Examining figure 4-3 1, we note tbat around each of the two largest flat

steps the fegtures of the curve appear to be similar. However, we can see the

beginning of this imbedded staircase phenomema which is a sure sign of the

Cantor Set being present (Schroeder, 1991). It is, however, the weakest evidence

we have of the Cantor set being present as we m o t easily compute cases with

enough sites to show fine detail. StilL, it is one more support for our central

hypothesis that a binary chain arranged in a Fibonacci sequence has an energy

spectrum that is a Cantor Set. While we have no actual proof of this assertion,

based on this chapter we have extreme1y good reasons to infer it &om the presence

of self-similarity and the marmer in which the energy spectrum behaves when

magnified.

That we would end up gettiag a Devil's Staircase structure is no big

surprise to us given that the integral density of states is lDOS = DOS (E' )& '

and so the seKsimilar nature of the energy spectnrm we observed in section 4-3

would logically be repeated here as well. Examining the DeviI's Staircase is just

one more way of checking our r d t s to see that they are consistent and this

examination has not shown any flaw in the logic of our conclusions. W e have self-

similarity and some form of Cantor set present.

4.5 Figures for Cha~ter 4

Figure 4-1: The LDOS for Semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -7 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 19th Fibonacci

number (6765) - we start our count of Fibonacci numbers at zero. The energy

range is from -9.1 to 0.1. The broadening, a, ranges fkom 0 to 0.025.

Firmre 4-2: The LDOS for semi-infhite binary chain arranged in a Fibonacci

sequence beginning with an A site. The site energy of the A site is -2 energy units

and the site energy of the B site is -7 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 19th Fibonacci

number (6765) - we start our count of Fibonacci numbers at zero. The energy

range is fiom -9.1 to 0. I . The broadening, a, ranges fiom 0 to 0.025.

Firmre 4-3: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with an A site. The site energy of the A site is -2 energy units

and the site energy of the B site is -7 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 19th Fibonacci

number (6765) - we start our count of Fibonacci numbers at zero. The energy

range is fiom -9.1 to 0.1. The broadening is 04.00 125.

m r e 4-41 The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -7 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 19th Fibonacci

number (6765) - we start o w count of Fibonacci numbers at zero. The energy

range is £?om -9.1 to 0.1. In the top figure, the broadening (y-axis) ranges from 0

to 0.025. In the bottom figure. the broadening is 0 ~ 0 . 0 0 125.

F i m 4-5: The LDOS for Semi-infinite binary c h i a m g e d in a FiIbomcci

sequence beginning with an A site. The site energy of the A site is -2 energy units

and the site energy of the B site is -7 energy units. The interaction energy is I

energy unit. The number of atoms in the chain is equal to the 19th Fibonacci

number (6765) - we start our count of Fibonacci numbers at zero. The energy

range is fkom -9.1 to 0.1. In the top figure, the broadening &-axis) ranges from 0

to 0.025. In the bottom figure. the broadening is a=O.OO 125.

Fimrre 4-6: The LDOS for semi-infinite b h u y chakr arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 19th Fibonacci

number (6765) - we start our count of Fibonacci numbers at zero. The energy

range is from -5.1 to 0.1. The broadening (y-axis) ranges fiom 0.00 125 to 0.025.

Firmre 47: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site en- of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 19th Fibonacci

number (6765) - we start our wunt of Fibonacci numbers at zero. The energy

range is @om -5.1 to 0.1. The broadening (y-axis) ranges from 0.000125 to

0,0025

w e 4-8: The LDOS for semi-infinite binary chain arranged in a F ~ h a c c i

sequence begimdag with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. Tbe interaction energy is 1

energy unit. The number of atoms in the cbain is equal to the 19th Fibonacci

number (6765) - we start our count of Fibonacci numbers at zero. The energy

range is from -5.1 to 0.1. The broadening is cu=0.175.

F i w e 4-9: The LDOS for semi-idrite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is 3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 19th Fibonacci

number (6765) - we start our count of Fibonacci numbers at zero. The energy

range is from -5.1 to 0. I . The broadening is 0 4 . 1 125.

Firmre 4-10: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 19th Fibonacci

number (6765) - we start our count of Fibonacci numbers at zero. The energy

range is from -5.1 to 0.1. The broadening is a=O.OS. Note that the bands are

finally becoming distinct.

Figure 4-1 1: The LDOS for Semi-infinite binary chain manged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is L

energy unit. The number of atoms in the chain is equal to the 19th Fibonacci

number (6765) - we start our count of Fibonacci numbers at zero. The energy

range is tiom -5.1 to 0.1. The broadening is a=0.0125. Band separation is even

clearer in this figure.

-re 4-12: The LWS for Semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 19th Fibonacci

number (6765) - we start our count of Fibo~cci numbers at zero. The energy

range is fiom -5.1 to 0.1. The broadening is o4.005.

Figure 4-13: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 17,811) - we start our count of Fibonacci numbers at zero. The energy

range is fiom -5.1 to 0.1. The broadening is o=0.0010. This is the entire energy

spectrum for this atomic chain.

Fimre 4-14: The LDOS for semi-idbite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 17.8 1 1) - we start our count of Fibonacci numbers at zero. The energy

range is from -5.1 to -2.0. The broadening is o 4 . W 10. This is the entire energy

spectrum for the contributions fiom B sites (the B band).

Figure 4-15: The LDOS for semi-infinite binary chain arrauged in a Fi'bonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 17,811) - we start our count of Fibonacci numbers at zero. The energy

rangeisfiom-3.3 to-2.0. The broadeningiso=0.0010. We beginto focusinon

one of the clusters of peaks.

Fiaure 4-16: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 17,811) - we start our count of Fibonacci numbers at zero. The energy

rangeisffom-3.2 to-2.8. The broadeningisa4.0010.

Figure 4-17: The LDOS for semi-infinite binary chain arranged in a Fibopacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 17,811) - we start our count of Fibonacci numbers at zero. The energy

range is eom -3.0 to -2.85. The broadening is ~4.00 10.

F i p e 4-18: The LDOS for semi-idbite binary chain amnged in a Fibonacci

sequence begiMing with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 17,8 1 1) - we start our count of Fibonacci numbers at zero. The energy

range is from -2.98 to -2.93. The broadening is o=0.0010. We appear to be

reaching the stage where the peaks begin to blur too much to focus in any fbther.

F i w e 4-19: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site eaergy of the A site is -2 energy un i ts

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 17,8 1 1) - we start our count of Fibnacci numbers at zero. The energy

range is fiom -3.0 to -2.85. The broadening is ~=0.0020. Note the similarity

with figure 4- 1 8.

Firmre 4-20: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 l7,8 11) - we start our count of Fibonacci numbers at zero. The energy

range is from -3.2 to -2.8. The broadening is ~ 0 . 0 0 6 0 . The similarity between

this figure and figure 4-1 8 is evident.

Fime 4-21: The LDOS for semi-idbite binary chain arranged in a Fiknacci

sequence beginning with a B site. Tbe site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 l7,8I 1) - we start our count of F~hnacci numbers at zero. The energy

range is tiom -3.3 to -2.0. The broadening is a=0.020. We still have a structure

that appears similar to that seen in figure 4-18.

Firmre 4-22: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 l7,8 1 1) - we start our wmt of hhnacci numbers at zero. The energy

range is fiom -5.1 to -2.0. The broadening is w=0.070. The similarity with figure

4- 18 persists, but now with the entire B band!

Fimue 4-23: The LDOS for semi-idhite binary chain arranged in a Fiboaacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 l7,8 1 1) - we start our count of Fibonacci numbers at zero. The energy

range is from -5.1 to -2.0. The broadening is o=O.OSO. This is the complete B

band with a slightly sharper broadening than in figure 4-22.

Figure 4-24: The ll)OS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy ofthe A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 17,8 11) - we start our count of Fibonacci numbers at zero. The energy

range is fiom -2.0 to 0. The broadening is o=0.0010. This is the entire A band

from figure 4-13. It shows the same group of three structure that we saw in the B

band.

Figure 4-25: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 l?,8 1 I) - we start our count of Fibonacci numbers at zero. The energy

range is 6om - 1.7 to - 1.1. The broadening is (u=0.00 10.

w e 4-26: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 l7,8 1 1) - we start our count of Fibonacci numbers at zero. The energy

range is from -1 -25 to - 1.1. The broadening is (u=O.OOlO.

F i m e 4-27: The LDOS for semi-infinite binary chain manged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 17,8 1 1) - we start our count of Fibonacci numbers at zero. The energy

rangeisfrorn-1.25 to-1.20. Thebroadeningis(u4.0010.

Fipure 4-28: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 17,811) - we start our count of Fibonacci numbers at zero. The energy

range is fiom -1 -2221 to -1.200. The broadeaing is ~ 0 . 0 0 10.

F i ~ e 4-29: The LDOS for semi-infinite binary chain arranged in a Fibonacci

sequence beginning with a B site. The site energy of the A site is -2 energy units

and the site energy of the B site is -3 energy units. The interaction energy is 1

energy unit. The number of atoms in the chain is equal to the 27th Fibonacci

number (3 17,811) - we start our count of Fibonacci numbers at zero. The energy

range is fiom -1 -2221 to -1.214. The broadening is ao=0.0010. This appears to

the limit of how far we can focus in on the A band. However, it is very interesting

that, in this size of a system, it possesses the same seif-similarity traits as the B

band does.

m e 4-30: The integral DOS for a semi-infinite binary chain arranged in a

Fibonacci sequence. The site euergy of the A atoms is -1 energy unit, the site

energy of the B atoms is -2 energy units and the interaction energy is 1. The

number of atoms in the cbain is equal to the 12th Fibonacci number (233 sites).

The energy range is from -4.0 to 0.5. This is the entire energy spectrum.

Firmre 4-31: The integral DOS for a semi-infinite binary chain arranged in a

Fibonacci sequence. The site energy of the A atoms is -1 energy unit, the site

energy of the B atoms is -2 energy units and the interaction energy is 1. The

number of atoms in the chain is equal to the 12th Fibooacci number (233 sites).

The energy range is from 4.0 to 1.2. This is just the B band. The structure above

is called the Devil's Staircase-

Chapter 5 : Conclusions

5.1 Conclusions

The god of this project was to examine sequence effects on the energy

spectrum of a one dimensional b i chain of atoms (which of course could also

be used to model a series of tbin films). We chose to examine a binary system

which used the Fibonacci sequence to determine the sequencing of the two

Werent types of atoms (or materials in a thin film structure). The literature had

suggested that introducing this type of sequencing led to a system that behaved

differently than the well known types of systems @we, periodic and disordered).

The excitement of exploring this novel type of atomic arraLlgernent was only

enhanced by the subtle hints that the novel type of eigen-value specttum that we

were looking at was linked to the much larger field of fiactals by the promise of

possi1y seeing self-similar energy spectra (which, as we know, would mean we

could see the same gross structure of the energy spectrum repeated in a portion of

the energy spectrum).

Another reason to have looked at this model is the nwnber of types of

systems that this research can ultimately be applied to. Whether we examine a

superlattice, a series of dielectric waveguides, a one dimensional atomic chain or a

cut across a Penrose tiled surface our model will describe what happens to the

energy spectrum when the individual elements are aperiodicly arranged. The

choice of the Fibonacci sequence as our particular aperiodic sequence was justified

by claims in the literature that it led to an exotic eigenvalue spectrum that

corresponded to a novel type of material that is neither crystalline nor arnphorous.

The existence of these "quasi-crystals" was in fkct the primary reason for interest

in this field-

In this thesis we developed a method of evaluating the LWS expressed in

the form of a continued haion- While this form ofthe LDOS bad been discussed

before, we did a thorough examination of how it differed fiom tbe DOS for a wide

variety of systems- For the Fibonacci sequence? we bther simpwed the

caldation by malring use of h c t i o d s and the intrinsic properties of the

Fibonacci sequence to vastly shnplifil calculations. This sirnplIfican was introduced

in order to prevent problems ftom occurring due to the numerical instability of

doing calculations over a large number of computations (of the order of lo5 or

greater which would greatly m@jr even small errors due to unavoidable

problems like rounding off numb). W1th tbis scheme we could easily handle a

chain of F27= 3 17,8 1 1 atoms with only 27 steps instead of 3 l7,8 1 1 steps. This is

equivalent to solving a matrix of size 3 l7,8 1 1 by 3 l7,8 1 1 in only 27 steps!. The

calculation of a matrix of this size is a feat which would be extremely difficult to

duplicate using wnventionai techniques and the redts would probably be poor

due to numerical instability (even round-oE error becomes important over this

many iterations). Our preliminary studies of idkite and semi-infinite pure,

periodic and binary alloys, using the LDOS scheme? allowed us to extract

meaningful results &om the numerical calculation. Based on the success of these

results compared to the analytic solutions availible, we extended the LDOS

method to examining the Fibonacci chain.

The important wnclusion and the one which any reader must be weU

familiar with by this point is the self-similarity present in these systems. This was

the central goal of the entire thesis: to show that we have self-similar behavior

occurring when atoms are ordered in a Fibonacci seq~euce. When examhhg an

eigmalue spectnun for this system we have &om that the spectrum is seK

W a r in several Merent ways. The "finger-like" behavior of the peaks as we

increased o, was an early sign that the behavior of the system was diffkent than

that of the periodic and pure systems. It was cleady shown by zooming in on

clusters of peaks and seeing the same grouping within the clusters as we see for

the energy spectnun as a whole (which shows that we have seEsimilarity present).

The ~e~similarity property became especially apparent when we varied both the

broadening and the energy range and got identical shapes for the LDOS as we

cycied through a large number of magnifications. The change in broadening was

needed to compensate for not having a truly infinite Cantor set. A Cantor set is a

self-sirnilar hctal ~tructure and is actually the most basic example of a fi-actai;

having the energy spectrum behave in this maMer shows that it is a hctal. The

clustering of the eigen-vdues in groups of three was demonstrated to be sew

similar- W e clearly observed that each individual cluster in the group of three

dusters also contained a group of three clusters when we magnitied it. This is the

confirmation of the presence ofa t'ractal that we have been searching for. This has

not been absolutely proven, but inferring it fkom out numerical experiments seems

eminently reasonable.

This is an exotic and interesting result which seems to confirm the general

consensus of the literature: that something strange happens when we mange a

binary chain of atoms in a Fibonacci sequence that is qyite different b m the pure,

periodic or the disordered case. One should also carefidly note that this result

arises entirely due to sequence effcts and that the property of ~e~similarity was

not assumed in any of our basic asmptions (expect for the sequencing of the

atoms which is seKsimhr)). This provides evidence that sequence effects are

extremely importam to these sorts of systems and that certain types of orderiag

can cornpIetely aher the behavior of the system This is good evidence for the

source of- behavior in a system described by bear equations being brought

into the system by sequence effects.

So we have shown what we set out to demonstrate in this thesis_ We do

have a significant alteration of the energy spectnm due entirely to sequence effects

of the elements in a b i chain. The energy spectrum it& seems to show the

properties of a Cantor set (it is self-similar). This enables us to iafer that it is

actually a fractal energy spectrum. And that is quite interesting indeed!

5.2 Directions for Further Studv

This is a new field and it is quite nifficult to easily spell out all ofthe things

which one could do in order to extend these r d t s . One wdd extend the system

of a one dimensional aperiodic binary chain to a two dimeasioaal system

(corresponding to a Pemose T i S h c e ) or a three dimensional system (a real

quasicrystal) to verify which properties of the energy spectrum will be changed,

enhanced and repressed by extending dimensionality. Or one wdd expand the

model 1 have presented by removing one or more of the simplifjing assumptions

stated in section 2.7 that are assumed as part of the Tight Binding Model (i.e. use a

more complicated and possibly more realistic model). The astute reader could also

wonder whether these novel properties exist in other types of aperiodic sequences

(The-Morse and Period Doubling) and seek to do detailed calculations on these

sequences.

APPENDIX I

The program Gmap89 is an illustration of the method used to directly

calculate the Local Green's Function in this thesis. The program generates the

entire continued fiaction form of the Local Green's Function and evaluates it. In

this early form ofthe program, we were still using density and contour plots to

show the form ofthe Local Green's F d o n (which we call g in this appendix). K

you examine these plots you wiU note the group of three feature for the B sites

(which are the ones that we start with in our standard case). The A conmbution is

too blurred to make this group of three feature out clearly.

This program was written for Mathematics version 2.2. Later uses of this

program typically focused on taking a slice at a specific value of a and examining

the resultant curve. The program in the form included in this appendix is geared

towards evaluating 89 atomic sites. It can be extended to more sites but going

beyond 233, the program becomes too slow for pradcai work. For reasons of

calculation efiiciency we stick to only evalua~g a chain of length equai to a

Fibonacci number. Apologies in advance if the work in the appendices are a little

rough as they represent working equations more than a finished product.

Map of local g e Introduction {GmapfM)

In this program, the density map and the contour map of g are plotted. The calculation is by direct evaluation of the mtinued fraction fwm d g, without analytically simplified. If the continued fiaction is first put into a fraction of hrvo

polynomials, the expressions are too complicated for Mathematics to handle if the continued fraction is longer than 15 or so. It is far too slow compared to the direct nurnenmencal evaluation of the g.

rn The Fibonacci sequence generation

In this Program, another way of generating the Fibonacci sequence is introduced. It is based on the formula of writing the Hull function:

where f is the Fibonacci number, the Golden Mean, -61 8.... , Floor rounds the number "down" to the nearest integer, and k is an integer. If Vk is 1, it is a 8 site and if Vk is zero, it is an A site. Go to the end of the lines in dark , is., Table ..... }I, and press shift-enter. The result is shown in light prints following.

In this calculation, we shall choose A= - 3, B= - 2 intearall as 1 -

and the interadion (the overlap

w Direct numerical calculation of g

The normal RecursionLirnit is too small for this calculation. Expand it.

We calculate on 89 sites : Fn =89 , n =10.

Gma@

DensityPlot [ -fm[g[89,u,w] ] , {u, -5.1,0.5), {w,0.0001,1.0), PlotPoints->{150,25), Mesh->False]

-5 -4 -3 -2 -1 0

-ContourGraphics-

Comment

In the above calculation of 89 site chain, it took about 5 minutes for each plot. The RecusionLimit has to be increased to 500. This number increases exponentially with N, the number of sites. Thus it is not a practical way in calculating chains of thousands of sites. It does have the advantage that information at each site is available.

APPENDIX TI

This appendix presents the method used to determine the local Green's

function for very long chains. The program bas been adapted considerably for

looking at much larger chains and most of the figures in this thesis have beea slices

at a specific value of a. This section also presents a second discussion of the

theory behind calculations for very large chains.

The program here is calcul~g the 18th Ftbnacci number (6765). In this

thesis we have often gone much higher (to the 27th F~bnacci number 3 17,4 17) at

considerable expense in computational efficiemcy. When comparing to the work

done in &on 2.8, the reader should realize that this document uses g for the

Local Green's Function while section 2.8 uses G. Otherwise, the notation is the

same.

Local g for a Fibonacci sequence March 1 4,1995.

w Calculate g for a Fibonacci sequence. Introduction.

We make use of the fad that in a Fibonacci sequence, eacht subsqence is the sum of the previous Wt:

B. BA BA8 BABBA BABBABAB BABBABABBABBA

....................................................................... P I

The local go at the end of a finite art off Fibonacci chain is of the form:

where a = [ complex energy E - site energy at A] and b = that at 6. The complex energy E is represented in the following as x + I'y. For the DOS evaluation, y=>O. We take the interaction, the overlap integral, to be I. If we call g i [z., a

functional of z, as the Fl sequence:B, we see that g2 for BA is

gl [z J = l/(b-2); g2=g 1 [I /a].

.................................................................................................. P I

Similarly, for BAB

and so on.

So we can generate the Fibonacci sequence of Fn site by only n steps. However, can we evaluate g at only n steps instead of unfolding everything ending up in Fn steps? Indeed we can, if we use Mathematica, (or Fortran).

Let gb=l l(b-2); and ga=l/(a-2). Each step of [1] for a subsequence of [0] is

g 1 =l l(b-z) , with z=0; gZ=ll(b-il(a-z) , with z=O, i.e. gZ=gl[ga] ; g3=g2[g1]; g4=g3[g2]] ... with z=O.

Mathematica can evalute such function of a fundion type of expression by use the functional expressions g[z J. However, it we use everywhere such expresions, it means that the numerical evalution is left to the end. Then in the calculation, each process unfolds thus for a chain of 6000 sites it unfolds 6000 times, way beyond the recursion limit of Mathematica. Besides, it did not take advantage of the Fibonacci sequence property except in labelling and in generating the sequence.

The real advantge takes place if we evaluate numerically at each step of gk, k=l ... n, but we also need to be expressed them in terms o f t Then we have only to evaluate n steps instead of Fn steps. How to do this is shown belaw. While in fundionals like g[z J the evaluation is suspended, in Ui=..[z] evaluation is taken place, but then we must express the result as a functional v [ z . before we can go to the next step, so that the function in the next step can be evaluated Ui. To get the result out, it must again be expressed in terms of z as a functional g[zJ for the whole chain of finite length. The last site is 1la or 1lb depending on the site. n i s is done by putting g[z=O].

In this way the full advantage of the local g approach is exhibited.

Let E=x+l y; A= -2 ; B= -3; and the interaction be 1. For n=19, or 6765 units, it took about 5 mins.

Calculation

Let E=x+l y; A= -2 ; B= -3; and the interaction be 1. For n=19, or 6765 units, it took about 5 mins.

t=Tablelden[irj],{j,21),Ci,npp)1;

Lis tP lo t3D [t]

ListDensityPlot [t]

Comments In the above example calculation, the energy range was from -3.88 to 3.80, (y range is also small). You compare this result with one with the range -5.1 to 0.1 (y ranges from 0 to 10 for 80 divisions, but we only evaluate 21 points, i.e. 20 divisions). You will see obvious similarities in the DOS.

Self-similarity should be more dramatic l you express the y-axis in log scale (try to avoid log 0 which is - infinity!) , and if the number of points in the ydirection is increased from 21 to at least 80. Of course you enlarge and shring the axis scale and so on,

The purpose of this appendix is to illustrate exactly how we went about

taking slices across the i m . axis. The calculation shown here is the actual

one used to compute figure 4-13 in the thesis (and extended to cover that entire

series of figures).

The only innovation here is to set the imapinanr axis to specific number

instead of permitting it to vary across a range. The resultant graph is easier to see

as it is two rather than three dimensiod. However, note that the method of

calcdating the Local Green's Function is identical to that used in the three

dimensional case,

Local g for a Specific Broadening Local g at a specific level of Broadening

In Appendix II we examined the general three dimensional map of the locai Green's function. However, mostof the work in this thesis involves looking at a specific value of w (the broadening) as it is much easier to see fine detail this way. This program is an adaption of ?he program in Appendix I1 in which we only examine a single slice of the local Green's function.

h[29]:=

m Comments Up to this point the program is basically identical. However, note

that we are computing for a far larger chain and that we don't calculate anywhere near as far out into the imaginary plane. This is because we no longer have to calculate any further than our specific cut. We don't go far out because we plan to leave the broadening constant as we repeatly focus in on the peaks. The sharper the resolution we begin with, the better the final result after we "zoom in" on the peaks five or six times.

Out&'39]=

-Graphics-

@I Conclusions This is an example of how the programs in the previous two appendices

were actually adapted for use in this thesis. The cutting of a specific slice across the imaginary plane gives a much clearer view of the peak structure (compare this to the contour and density plots we previously examined).The only major hazards are toavoid scaling the broadening too small (and this over-emphasizing the peaky structure) or too large (and thus drowning out important details).

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