Theoretical studies of doped carbon nanotube based materials

LEABHARLANN CHOLAISTE NA TRIONOIDE, BAILE ATHA CLIATH TRINITY COLLEGE LIBRARY DUBLIN OUscoil Atha Cliath The University of Dublin

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T h eo retica l stu d ies o f d op ed

carbon n a n o tu b e-b a sed m aterials

by

Andrew Wall

A thesis subm itted for th e degree

of D octor of Philosophy

School of Physics

D ublin University

D e c la r a tio n

I have not previously su bm itted this thesis as an exercise for any degree.

T h e work in this thesis is mine, except where I specifically refer to published or unpublished work carried out w ith the help of those collaborators whom I have acknowledged in th e text.

T he Library has my permission to lend or copy this work, if requested.

Su m m ary

For th e last fifteen years, carbon nanotubes have piqued th e interest of many in the m aterials research community, due to their superlative individual characteristics. One possible route to their technological exploitation may lie through utilising their interactions with other m aterials. Proposed uses for nanotubes, which depend cru­ cially on their behaviour when exposed to such foreign substances, variously include use as sensors, as interconnects in electronic devices, as non-conventional transistor elem ents, as conductive agents in non-conductive plastics, as hydrogen storage de­ vices and as com ponents in super-tough fibres, capable of being used in light-weight body arm our. In each case of crucial im portance is th e detailed interaction th a t occurs between the nanoscopic carbon cylinders and o th er m aterials.

To this end, we have performed a range of theoretical investigations into the interaction between carbon nanotubes and foreign agents. Among other things, we have proposed a theoretical framework caj)able of modeling the electronic interaction between nanotubes and doping bodies based on the existence of well-known sum rules for interaction-induced changes in the density of states of th e system. Using simple models for the electronic structm 'e of the dopant and the nanotube, we have investigated the effect th a t different dopants have on the electronic stru ctu re of the combined structure. M athem atically transparent, and num erically light-weight, the scheme allows not simply qualitative analysis, but also quantitative. To this end we have utilised the m ethod to perform calculations for th e electronic stru ctu re of nanotubes doped w ith atom s and small molecules. O ur analysis has allowed us to derive a scheme capable of param eterising semi-empirical electronic structure calculations involving many dopants, by using the o u tp u ts of ab-initio calculations for a few foreign objects. O ur m ethod also allows us to show the existence of correlations between the binding energy and the charge transfer th a t takes place between th e parts, when a dopant is brought into contact w ith a nanotube.

A ck n o w led g em en ts

O v er th e last four years I have received im m easu rab le s u p p o rt from so m any people w ith o u t whose help I w ould n o t be w ritin g th is today. In p a rtic u la r, I w ould like to th a n k those people I have been lucky enough to c o lla b o ra te w ith directly: Dr. M auro S. F erreira, Dr. C la u d ia G. R ocha, Dr. A lexandre R. R o cha an d Prof. J o n a th a n N. C olem an. I m ust th a n k my supervisor M auro, for his p a tie n ce a n d his a tte n tio n to m y work. O ne of th e ad vantages of w orking in a relatively sm all g ro u p is th e chance for reg u lar one-on-one in te rac tio n w ith those w ho know far m ore th a n you do. So for a c o n sta n tly open door, an d a c o n sta n tly open m ind, I am g ratefu l. I m u st also give a special th a n k you to C lau d ia, for whose help in p erform ing ab-initio calculations I am especially th an k fu l. T h a n k s as well to D avid K irw an, who pro o fread a po rtio n of th is thesis, and to Felipe, who helped me reason th ro u g h a p a rtic u la rly th o rn y problem tow ards th e end of my thesis. I m ust also th a n k S tefano S anvito and his group, who have provided scientific and co m p u tin g s u p p o rt, as well as som eone to have lunch w ith.

I m u st m ention th e financial sui)port th a t I have received from b o th th e Irish R esearch C ouncil for Science, E ngineering and Technology, w ho aw arded m e an in au g u ral G overnm ent of Irelan d Scholarship u n d er th e E m b a rk in itia tiv e , a n d from Science F o u n d atio n Irelan d , who also funded m e for p a rt of th e d u ra tio n of th e p ro je c t. Suffice to say, th e existence of such funding was a crucial factor in my deciding to continue my studies in Ireland. In d irect a ssistan ce also cam e in th e form of th e co m p lim en tary access to w orld class c o m p u ta tio n a l facilities; for th is I owe th a n k s to th e T rin ity C en tre for H igh-Perform ance C o m p u tin g a n d to th e Irish C e n tre for H igh-E nd C om puting.

inentioi. m ust go to iiiy aunt Eileen M cGlade, who gave me an insider’s glimpse into the acalem ic life, and who always em phasised the imi)ortance of education.

I co uit myself lucky beyond reckoning to have made such good friends over the years. For the love and friendship th a t you have given me, I must thank Ciara, Ronan, Deco, Keith, John and K aren, Enda, Turlough, Leanne, Fiona, Eoin and Aileen^

C ontents

1 I n tr o d u c tio n 1

1.1 Historical p e rs p e c tiv e ... 1

1.2 Geom etry of iia iio tu b e s ... 2

1.3 Intrinsic properties of n a n o tu b e s ... 5

1.3.1 Electronic p r o p e r t i e s ... 5

1.3.2 Mechanical properties ... 7

1.3.3 Therm al P r o p e r tie s ... 8

1.4 Nanotul)es in composite structures ... 9

1.4.1 Utilising mechanical properties ... 9

1.4.2 N anotubes in electronic d e v i c e s ... 11

1.5 The crucial role of c o n t r o l ...12

1.6 T he desire for a general fo rm alism ... 14

1.7 Layout of the t h e s i s ... 17

2 T h e L loyd Form ula M e th o d 19 2.1 M otivation and l a y o u t ... 19

2.2 Canonical form of Lloyd’s f o rm u la ... 20

2.3 Modeling doping via contact p o t e n t i a l s ...24

2.4 Using Lloyd’s formula: a case s t u d y ...27

2.4.1 Linear chain with adsorbed i m p u r i t y ... 27

2.4.3 T he charge neutrality c o n d i t i o n ... 32

2.4.4 Electronic contribution to the binding energy ...34

2.4.5 Charge transfer to adsorbed a t o m ...35

2.4.6 Uses of the fo rm alism ... 36

2.5 A nalytic Investigation of the LFM e q u a tio n s ... 40

2.5.1 Analysis of the charge neutrality c o n d itio n ... 40

2.5.2 Weak binding in the Lloyd formula m e th o d ... 43

2.5.3 Linear Chain with substitutional impurity as a limiting case of the adsorbed o n e ... 46

2.6 C o n c lu s io n s ... 46

3 A p p lic a tio n s o f th e g en eral m e th o d 49 3.1 In tr o d u c tio n ... 49

3.1.1 M o tiv a tio n ...49

3.1.2 Layout of this c h a p t e r ... 53

3.2 Modeling the effect of randomly disi)ersed adatoms on carbon nanotnbes 55 3.2.1 Electronic structure of n a n o t u b e s ...55

3.2.2 Evaluation of tight-binding param eters by the Lloyd formula m ethod ...63

3.2.3 R e s u lts ... 65

3.3 An inverse modeling approach to the gas sensor p r o b l e m ...74

3.3.1 M o tiv a tio n ... 74

3.3.2 C a lc u la tio n s ... 75

3.4 C hapter s im m ia ry ... 79

4 M o d e lin g h e lica lly ord ered p e r tu r b a tio n s o f n a n o tu b es 83 4.1 In tr o d u c tio n ... 83

4.1.1 M o tiv a tio n ...83

4.2 Modeling nanotube polymer interaction via free electron g a s ...87

4.2.1 Overview ... 87

4.2.2 P erturb ed Free Electron Gas in I D ... 89

4.2.3 P erturbed Free Electron Gas in 2 D ... 96

4.3 T ight binding s tu d i e s ...101

4.3.1 Calculational d e t a i l s ...104

4.3.2 Location of van Hove s in g u la ritie s ... 106

4.3.3 W idth dependence of perturbation ... 107

4.3.4 Angular dependence of perturb atio n ... 110

4.3.5 A m plitude dependence of p e r t u r b a t i o n ...I l l 4.3.6 Angular dependence of change in electronic energy ...113

4.4 C hapter Sum m ary ... 114

M ec h a n ism for m ech a n ica l r ein fo r cem en t in n a n o tu b e -p o ly m e r co m ­ p o s ite s 117 5.1 I n tr o d u c tio n ... 117

5.2 M odeling the in t e r a c t io n ... 121

5.2.1 G eom etry of the p ro b le m ...121

5.2.2 N anotube-polym er interaction and the Frenkel-Kontorova m odell22 5.2.3 T h e force equilibrium a p p r o a c h ...123

5.2.4 M ethod of effective p o t e n t i a l s ... 124

5.2.5 D etails of interaction p o te n tia l... 127

5.2.6 Results and d is c u s s io n ... 131

5.2.7 Graphical m ethod for the tem plating f r a c t i o n ... 133

5.2.8 D iam eter dependence of the re in fo rc e m e n t...134

5.2.9 T he effect of c ry s ta llin ity ... 135

6 C o n clu sio n s 139

6.1 Summary of work u n d e r t a k e n ...139

6.1.1 Overview of chapter o n e ...139

6.1.2 Overview of chapter t w o ...140

6.1.3 Overview of chapter t h r e e ... 140

6.1.4 Overview of chapter f o u r ...141

6.1.5 Overview of chapter f i v e ...143

6.2 Possible extensions of this w o r k ...144

A G reen F u n ctio n s 147

B D eriv a tio n o f L lo y d ’s form ula 151

C L lo y d ’s form ula for a lo ca lised p ertu rb a tio n 153

D C a lcu la tio n o f c o m p o s ite G reen fu n ctio n for p ertu rb ed linear chain 157

E A co m m en t on n u m erical im p le m en ta tio n 161

List o f Figures

1.1 G eo m etry for a single grap h en e s h e e t... 2 1.2 Schem atic of geom etry for an arm chair an d an achiral ca rb o n n an o tu b e. 4 1.3 D ensity of sta te s for achiral n a n o tu b e s as calculated w ithin th e

zone-folding approach ... 6

2.1 C hange in th e to ta l density of s ta te s for a linear chain su b je c t to a localized p e rtu rb a tio n p o t e n t i a l ... 23 2.2 A schem atic rep resen tatio n of th e h o s t/ a d so rb an t s y s te m ... 26 2.3 G rap h ical flisplay of tig h t-b in d in g p a ra m e te rs for linear chain and

ad so rb e d ad a to m b o th (a) before a n d (b) after c o n ta c t... 27 2.4 T ig h t-b in d in g b a n d s tru c tu re (a) an d G reen function (b) of th e u n ­

p e rtu rb e d linear chain in th e n earest neighbour a p p ro x im a tio n ... 29 2.5 Schem atic rep re sen ta tio n of th e electronic s tru c tu re of th e

unper-tm 'b ed host / a d so rb a te s y s te m ... 30 2.6 C h a ra c te ristic dependence of change in th e n u m b er of p articles as a

function of th e d ire c t host / a d so rb a te coupling p a ra m e te r for speci­ fied ato m ic ionization p o te n tia l... 38 2.7 C h a ra c te ristic dependence of change in th e n u m b er of p articles as a

function of th e d ire c t host / a d so rb a te coupling p a ra m e te r for a range of a to m ic ionization p o te n tia ls ... 38 2.8 M ap from tig h t-b in d in g p a ra m e te r space to physical observable space

3.1

Schematic of the binding geometry between a (6,6) cart)on nanotnbe

and externally adsorbed hydrogen atom ... 57

3.2 Schematic of the binding geometry between a (6,6) carbon nanotnbe

and adsorbed atom, with preferential binding site at cent re of hexagon. 58

9 ill

, .4^ Green function of a

3.3 Real and imaginary parts of the

(^Ri,A

(6,6) armchair nanotube as calculated using the zone-folding approach. 62

3.4 Folded band structures for a doj^ed (6,6) nanotube with a single H

atom per unit cell, (a) and (c) depict the band structure evaluated

using our approach, while (b) corresponds to the Kohn-Sham DFT

result, (a) and (c) are for perfect / imperfect screening respectively. . 67

3.5 Average conductance as a function of the F'ermi energy for distinct

concentrations of H atoms distributed randomly along a (6,6) carbon

nanotube as calculated via the Kubo formula...73

3.6

A E / A C

graphs (see the text) displaying allowed values for the bind­

ing energy and charge transfer for (a) a Hydrogt'n adatoni and (!:>) a

Lithium adatom, externally bound to a (6,6) N T ... 76

4.1

STM image of PmPV-coated single wall nanolubes displaying

apjmr-ent helical o r d e r in g ... 85

4.2 Schematic diagram of the geometry involved in helical w'rapj)ing of

a nanotube, (a) in three dimensions and (I)) the corresponding two

dimensional representation...88

4.3 Relative energy change per perturbation for a one-dimensional free

electron gas perturbed by a series of ecjnally spaced delta-function

scatterers...95

4.4 Relative energy change per unit length of perturbation for (a) a two-

4.5 S chem atic re[)reseiitation of a u a n o tu b e hehcally w rap p ed by a uiii-fonii one-ciiinensioiial charge d i s t r i b u t i o n ... 103

4.6 (a) T h e DOS for a (4,4) arm chair n a n o tu b e p e rtu rb e d a t a c o n sta n t angle a w rap p ed uniform charge d istrib u tio n , (b) T h e corresponding energy iso-surfaces in reciprocal space for a graphene sheet p e rtu rb e d by a series of equally spaced s trip e s ... 108

4.7 (a) D ependence of th e size of th e induced m ini-gap m \ \ on th e w idth of th e p e rtu rb a tio n for an (18,0) n a n o tu b e coiled by a uniform stripe. (b) W id th -d ep e n d e n c e of th e M u tra n s itio n for th e sam e system . . . 109

4.8 A ngular d ep endence of th e size of th e p e rtu rb a tio n -in d u c e d m ini-gaps m il (a) and of th e M \ \ tra n s itio n (b) for an (18,0) zigzag n a n o tu b e p e rtu rb e d by a uniform coiling p o te n tia l... I l l

4.9 D ependence of th e size of th e p e rtu rb a tio n -in d u c e d m ini-gaps m u (a) a n d of th e 5’n tra n s itio n (b) on th e p e rtu rb a tio n a m p litu d e for various n a n o tu b e s p e rtu rb e d by a uniform coiling p o te n tia l... 113

4.10 A ngular d ep endence of th e relative energy change for helical w rap])ing on a (9,0) zigzag n a n o tu b e , as calculated w ithin th e tig h t-b in d in g m odel. 114

5.1 Ball a n d spring m odel for a single polym er s tra n d coiling a ro u n d a n a n o tu b e a t a c o n sta n t angle 9... 122

5.2 P lo t of two lim itin g cases for th e dependence of th e average in te r­ m onom er se p a ra tio n on th e n a tu ra l bond length for a ball an d spring system in its g ro u n d s ta te in an ex ternal sinusoidal p o t e n t i a l ...126

5.3 A verage inter-m onorner se p ara tio n on th e n a tu ra l bond len g th for a ball an d spring sy stem in its ground s ta te in a section in an e x te rn al hexagonal p o t e n t i a l ...128

5.5 T hree dimensional plot of tlie external hexagonal potential experi­ enced by monomers in the vicinity of a graj)hitic surface... 130 5.6 Fraction of cases T / for which strands coiled at uniformly distributed

angles are tem plated (see the text) by the surface of a nanotube, (a) as a function of n atural inter-m onom er bind length (b) as a function of the nanotube r a d i u s ... 134 5.7 Y oung’s modulus of iianotubc-polym er comi)osites for small loadings

of nanotubes, as per the rule of m ixtures with (purple line) and with­ out (blue line) perfect stress transfer, com pared to experimental find­ ings (crosses) ... 136

E .l Schem atic of the integration contour chosen in the complex plane to ease num erical integration...163

List o f Tables

C hapter 1

In trod u ction

1.1

H isto rica l p e r sp ec tiv e

T h e discovery of the carbon nanotubes is commonly a ttrib u te d to th e Japanese scientist Sumio lijima, who first described them in 1991 [1]. lijirna used state-of- th e -a rt high resolution transm ission electron microscopy to image w hat he referred to as nanoscopic needles of carbon. W hile there is convincing evidence th a t this was not the first tim e th a t nanotubes had been synthesized, w hat is certain is th a t he was the first to give a detailed description of the form and stru ctu re of these rem arkable stru ctiu ’es.

■*1

r '* '.

j r \ r~\ r

\

'r~^.

r

"H

,.

■X

F igure 1.1: Ball an d stick m odels for a grap h en e sheet. D isplayed are th e u n it cell, d elim ited by th e d ashed lines, th e u n it vectors di and a-i-, and th e chiral vector for a (4 ,0 ) n a n o tu b e . T h e two inequivalent sites A and /i are also labelled.

b e st lu b ric an ts we have, while diam ond is b o th th e hardest m aterial an d th e best th erm al c o n d u c to r known. P a rtly by analogy w ith th e known electronic s tru c tu re of g ra p h ite , n a n o tu b e s were quickly pred icted to have rem arkable therm al, mec:hanical, electronic an d electrical pro p erties, which would see them c a ta p u lte d to th e forefront of th e n ascent field w hich h a d th e aim of utilising nan o stru ctu res for technological applications.

Before giving d etails of those p ro p erties th a t excited suc'h early in te rest in these nanoscale s tru c tu re s , it is w orth addressing th e question: ju st w h a t are these carb o n n a n o tu b e s?

1.2

G eo m etry o f n a n o tu b es

a rra n g e d in to a hexagonal honeycom b lattice. Successive layers are spaced by ab o u t 3.5A and a re a ttr a c te d to each o th er th ro u g h a weak van d er W aals interactio n , 'r h e w eakness of th is in te ra c tio n is w h a t allows th e layers to slide over one an o th e r, th e p ro p e rty w hich m akes g ra p h ite such an excellent lu b rican t. A single sheet of g ra p h ite is w h a t is referred to as graphene, a m a te ria l th a t is only recently com ing in to s h a rp focus for th e research com m unity in its own right [3, 4, 5]. In fact, th e g e o m etry of n a n o tu b e s is m ost easily described in term s of th a t of graphene, which is a hexagonal, p la n a r s tru c tu re . A t each vertex resides a carb o n ato m , and th e edges of th e la ttic e c o rresp o n d to stro n g sp^-hybridised chem ical bonds. As carb o n h as four valence electrons, an d it form s th re e bon d s w ith neighbours (each of length a b o u t 1.42A), we m u st account for one e x tra electron p e r atom . T h is e x tra electron goes into a d u m b -b ell sh a p ed o rb ita l p e rp e n d icu la r to th e grap h en e plane, an d th e electronic s tr u c tu r e of th e g rap h en e sheet is essentially d e term in ed by th ese so-called TT-orbitals. G ra p h e n e has a trig o n al B ravais la ttic e sp a n n ed by vectors di and a2 w ith a two a to m basis, th e inequivalent ato m s being labelled A and B. T h e geom etry of a g rap h e n e sh e et is specified in Fig. 1.1. A n im p o rta n t p a ra m e te r is th e length of th ese basis vectors, w hich we will sim ply d e n o te by th e sym bol o; it is th e d istance betw een n e a re st equivalent atom s.

(a)

* » ^ w -» -T a

* f

Figure 1.2: Ball and stick m odels for tw o n an o tu b es. In panel (a) we display th e schem atic for 6 fu n d am e n ta l u n it cells of a m etalhc (7,7) n anotube. T h is achiral n aiiotube displays no helicity, in c o n tra st to th e (6,3) n an o tu b e, displayed in panel (b), which displays a clear helicity.

is uniquely g e n e ra te d by choosing Ch — n a\ + nia-ii and n and rn are called th e chiral :ndices of th e n a n o tu b e . D ue to th e variety of ways in which a grap h en e sheet can b e rolled upon itself, we see th e existence of a p leth o ra of different types of nan o tu b es, each specified by its chiral indices. Theory, and indeed ex p erim ent, show ta a t p ro p ertie s such as th e intrinsic conduc'tance of n a n o tu b e s are governed by their chiral indices. In p a rtic u la r, it is found th a t a n a n o tu b e th a t has n = m is inherei.tly m etallic, a n d such tu b e s are called arm chair nanotubes. T ogether w ith th e zig-zag tu b es, c h a ra c te rise d by rn = 0, we have th e set of achiral n a n o tu b e s. T hese ;nolecules are identical to th e ir m irro r images. All o th er tu b es fall into th e class of achiral tu b es. T h e geo m etry of an achiral (a (7,7) arm chair n a n o tu b e ) and a chiral tu b e (a (6,3) n a n o tu b e ) is d ep icted in Fig. 1.2, w here th e “tw ist” of th e la ttic e is rea d ily a p p a re n t in th e chiral tu b e , b u t non-existent for th e achiral tu b e.

However, typical lengths are in the micro-metre range, giving an extremely large

aspect ratio [7]. Snch a large aspect ratio is what allows us to view the nanotubes as

being one-dimensional structures. As grown, carbon nanotubes tend not to be found

in this pristine form. They can suffer a range of defects, including substitutional

defects (where a carbon atom can be replaced by an atom such as boron), vacancy

defects, and Stone-Wales type defects (where a bond is rotated by 90 degrees in the

plane of the moleciile). In real life, things are further complicated by the propensity

of nanotubes to aggregate together to form bundles.

1.3

Intrinsic p rop erties o f n a n o tu b es

Much research into carbon nanotubes is driven by their remarkable electronic, ther­

mal and mechanical properties. Nanotubes can display high levels of purity, and a

low level of defects, as evidenced by microscopy. W'hile typical defects th at can oc­

cur in nanotubes, including topological defects and vacancy defects, certainly have

an impact on the i)hysical properties, crucially the defected structures still possess

exceptional intrinsic properties. It is worth noting in passing that the amount of

defects in the uanotube is usually determined by how they are grown, be it by laser

ablation, arc-discharge or chemical vapour deposition. Typically, those nanotubes

grown by arc-discharge possess a higher concentration of defects, which can impact

on the as-measured physical properties of the structure. It is w^orth spelling out

some of the most exceptional properties th at as-grown nanotubes possess, which I

now do.

1.3.1

E lectron ic p rop erties

0.6

(b)

0.3: ° -0.3^

Figure 1.3: Density of states per atom for two different acliiral nanotubes as calcu­

lated within the zone-folding approach (see the text), (a) Corresponds to an (8,2)

chiral nanotube, while (b) corresponds to a (6,2) nanotube. The red lines indicate

the distance between the first van Hove singularities above and below the band,

known as A/n transitions in th e case of metallic tubes and S’n transitions in the

case of the semiconducting. As per the zone-folding theory, (a) has

rn — n =

and

is metallic , while (b) has

m. ^ n mod 3 and is semiconducting. The parameter

has the dimensions of energy, and has a value of about 2.7eV.

have been observed experim entally th ro u g h th e investigation of th e local density o f s ta te s th ro u g h th e use of scanning tu n n ellin g m icroscopy [10]. W ith in th e zone- folding ap proach, th e van Hove singularities can b e viewed as th e energies a t which th e lines of allowed reciprocal lattic e vectors a re ta n g e n t to th e a p p ro p ria te iso­ e nergy surface of th e underlying grap h en e sh eet.

It has been show n t h a t carbon n a n o tu b e s c a n display q u antised co n d u ctan ce [11], w hich is d u e to circum ferential confinem ent of th e w avefunction. E lectronic tr a n s p o r t w ith in m etallic carbon n a n o tu b e s is essentially ballistic over distances large co m pared to th e nanoscale, m eaning th a t th e y can carry high c u rre n ts w ith little h e a tin g over such distances, w ith th e p rim a ry source of resistance in th ese tul)es com ing from a c o n ta c t resistance due to th e in terface w ith m acroscopic electrodes. D ue to im perfections tra n s p o rt over longer d ista n c es becom es diffusive. As regards th e m obility for n a n o tu b e s in the diffusive regim e (a q u a n tity th a t d ete rm in e s how easily th e carriers can be induced to d rift by th e a p p licatio n of an electric field), re p o rts in d ic a te m obilities g reater th a n t h a t of any o th e r sem iconductor [12].

1 .3 .2

M e c h a n ic a l p r o p e r tie s

theory says th a t th e Young’s moduhis of a small diam eter SWNT should be of the order of 1000 G P a (which is roughly the in-plane tensile modulus of graphite) [14], b u t should decrease as th e diam eter increases. Nanotubes also have exceptional strength. Com posites m ade with nanotubes have been shown to have exceptional toughness, defined as the to tal energy to break. By way of a reference, it is useful to com pare this value w ith the Young’s modulus of steel, which is commonly cjuoted as being ab o u t 200 GPa.

1 .3 .3

T h e r m a l P r o p e r tie s

1.4

N a n o tu b e s in co m p o site stru ctu res

In th is section, I wiU briefly d etail som e of th e resu lts of investigations perform ed w ith th e aim of in te g ra tin g n a n o tu b e s into useful s tru c tu re s. As we have seen, th eo ry a n d e x p erim en t have show n th a t carb o n n a n o tu b e s are a prom ising c a n d id a te for a n e x t g e n eratio n m ate ria l due to th e ir rem arkable in trin sic tra its . To m axim ise th e u tility of th ese rem arkable s tru c tu re s , th ey m u st however b e com bined w ith o th e r s tru c tu re s . For exam ple, if n a n o tu b e s are to be used as tra n s is to r elem ents, th ey m u st m ake c o n ta c t a t th e tip s of th e tu b e s w ith m acroscopic m etallic electrodes. If a b ad c o n ta c t is m ade, th e in trin sic resistance of th e tra n s is to r can be dw arfed by th e c o n ta c t resistan ce a t th e interface. Since th e in te ra c tio n w ith o th e r su b stan ces can crucially d eterm in e th e usefulness of n a n o tu b e -b a se d devices, it is n a tu ra l to q u estio n how it will be possible to in te g ra te n a n o tu b e s w ith existing m ate ria ls to form com posite stru c tu re s. As a sidenote, in th is w ork, I will often use th e w ord co m posite in its m ost general sense to refer to a com bination of n a n o tu b e s an d o th e r s tru c tu re s , w hich m ain ta in s th e d istin c t m orphology of th e n a n o tu b e s.

1.4 .1

U tilis in g m e c h a n ic a l p r o p e r tie s

typically poor electrical conductors, since they suffer a Peierls distortion which de­ stroys their conductivity. However, the addition of a small quantity of nanotubes can increase the electrical conductivity of th e sample, w ith the resulting increase in the conductivity being normally explained through percolation theory [23].

A nother area where nanotubes promise an imj^rovement over the pure polym er is in the realm of mechanical reinforcement. W hen we think of polymers, we tend to think cf their macroscopic realisation in the form of plastics, so called because of their plasticity. It is not surprising then, th a t polymers tend to form bulk aggregates th a t are mechanically weak. However, it is found th a t the addition of small quantities by volrm e of th e far stronger and stiffer carbon nanotubes to a polymer sample can greatly affect the mechanical properties of the coini)osite material. It is often desired th a t such changes be m ade using iniim te quantities, in order not to adversely affect those properties of the polymer th a t we find desirable (such as malleability). One of the m ajor successes to day in the field is th a t new composite m aterials w ith world-record toughness (defined as the to tal energy a m aterial can absorb before structural failure) have been synthesized, leading to the i)rosi)ect of light-weight protective textiles, which may find application as light-weight bulleti)roof vests [24].

quantifying, the interfacial stress transfer between a polymer / nanotube interface is an open topic which may be crucial for further improvements in th e properties of composites.

W ith regard to th e morphology of com posites it has been reported th a t certain polym ers, such as polyvinal-alcohol (PVA) and Pm PV , crystallize ab out carbon nan­ otubes which are loaded into a polym er m atrix. Transmission electron microscopy reveals a substantial ordered layer of polym er th a t surrounds th e multi-wall nan­ otubes [26]. Such a coating has been linked w ith the observed stren gth increase when Y oung’s modulus experim ents are performed on multi-wall / PVA composites [27], in contrast w ith reports of substantially sm aller strength increases when a polymer which tloes not crystallize, poly(9-vinyl carbazole), is used.

Furtherm ore, when certain polymers bind to carbon nanotubes in a crystalline coating, it is often observed th a t they bind in a helical geom etry [26]. T he he­ lical pitch does not seem to be random preferential coiling angles for different nanotube/i)olym er system have been observed. A simple, classical model based on geom etric considerations [28] sought to explain the existence of spontaneous chiral order in such structures by exam ining the balance between the coiling induced stress in a w rapping molecule and th e surface dependent affinity such molecules have. In­ deed, this fact has been proposed to be a central element of a scheme to separate carbon nanotubes by chirality and electronic properties [29].

1.4.2

N a n o tu b e s in electro n ic d evices

techniques. The basic idea of molecular electronics is to replace the current bulky transistors with small molecules which are capable of eiimlatiug their characteris­ tics. This may provide a possible alternative to the current m initurization schemes employed by the sem iconductor industry. In fact, it has been proposed th a t the underlying properties of nanotubes may allow them to play a role in this budding domain. T he com bination of the existence of sem iconducting nanotubes with their reported high mobility seems to make them ideal candidates for constituents of tra n ­ sistors. Indeed, some of the most exciting early experim ents on nanotubes showed the ability of nanotubes to act as device elements [30], [31].

A nother promising applications for uanotubes is as nanoscale sensing devices. One rem arkable feature of carbon nanotubes (NT) is th a t their conductance is strongly affected by the interaction with certain foreign objects (FO), Atoms, molecules and nanoparticles are some of the objects known to interact strongly with NT, paving their way to being used as nanoscopic sensors. N anotubes have been shown to ex­ perience large changes in their resistance upon exposure to oxygen [32], nitrogen dioxide and am m onia [33]. In this light it may be possible to utilise th e properties of nanotubes to generate selective sensitive nanoscale devices. A recent review of the nascent field of carbon-based electronics has appeared in [34], while a more general treatm en t of th e electronic tran sp o rt properties of nanotubes has recently appeared in [35].

1.5

T h e crucial role o f control

m ust have different diam eters, it is clear th a t many different chiralities m ust be present in th e same tube. Bearing in mind the decisive influence th a t the chiral indices have on the electronic stru ctu re of a nanotube, we see th a t in principle some of the walls of the nanotube will be conducting and some not (under the assum ption th a t the different walls don’t interact with each other). If we take as a rule of thum b th a t one th ird of single walled nanotubes are metallic, we see the difficulty in finding a sem iconducting multi-wall tube.

Even if we are able to restrict ourselves to considering only single walled nan­ otubes, we still run into problems. W hile nanotubes of very small chiral indices are not found since they are energetically unstable, there is still a plethora of different chiral indices th a t are allowed. The problem is th a t for a useful technological device, there is often call for a certain chirality of nanotube. For example, a m etallic nan- otube will not be a useful com ponent where the need is for a sem iconducting one. C urrent m anufacturing m ethods allow only a poor control over the chiral indices of the nanotubes in a sample. While there can be some control over the spread of diam ­ eters th a t are grown in a sample [36], since there exist sem iconducting and metallic tubes w ith sim ilar diam eters, this does not give you control over th e m etallicity of a sample.

causing dam age to th e tubes or irreversibly altering their properties in some way is, as of now, one of the most fundam ental barriers to the connnercial exj)loitation of nanotubes.

On th e other hand, if the nanotube experiences a dram atic dependence on its environm ent, it may be possible to control its properties through controlling w hat is in its vicinity. For example, it has been shown th a t the conductance of a nanotube can be strongly altered by letting oxygen adhere to its surface [32], We will often refer to such an interaction as a doping process, by analogy w ith the conventional definition of th e term . By far the most dram atic success of doping in technology, and in the sciences, is in the addition of foreign species to pure semiconductors to increase the num ber of possible electrical current carriers, a crucial step in the manvifacture of m odern logic devices. Since a sem iconductor is characterised by an energy gap between the conduction and valence electrons, a non-trivial am ount of energy must be added to the m aterial in order to prom ote some electrons to th e conduction band in order to allow an electrical current to flow. However, the introduction of atom s of a different valence into the crystal introduces free carriers th a t can dram atically increase the conductance of the m aterial. If such extrinsic control could be exerted over nanotubes, it could overcome deficiencies in as-grown samples, and j)rovide a path tow ards their integration in technological devices.

1.6

T h e d esire for a general form alism

In the light of th e foregoing discussions, it is clear th a t a theoretical understanding of th e doping process for carbon nanotubes is highly desirable, and the elucidation of a framework capable of accounting for the changes induced is one of th e m ain aims of this work.

conditions a foreign object will bind to a nanotube, and one which investigates the effects th a t this interaction brings to th e electronic properties of the device. Typically, the first step is perform ed by utilising ab-imtio calculations to provide the energy change th a t occurs when the foreign object adheres to the nanotube, while the second considers the conductance of th e stru ctu re in the presence of a finite concentration of dopants. To take into account a disordered array of molecules on the surface of a nanotube, an ensemble average m ust be performed. Since we have to average over a large num ber of configurations com putationally expensive ab-initio evaluations are m iattractive, and so param eterised semi-empirical H am iltonians are often used. Besides from th e large dem and on com putational resources this puts, one reason this approach is dissatisfying is a lack of m athem atical transparency in the first part; the ab-initio calculation is trea ted as a “black-box” , and th e chance to make general statem ents about the binding process can be lost.

in tro d u c tio n of a foreign object n a tu ra lly destroys th e tra n sla tio n a l invariance of th e u n p e rtu rb e d system , th e m ethodology m u st be cap able of calcu latin g q u a n titie s in d ire c t space; however, since we w ould also like to be able to tre a t c e rtain p eri­ odic p e rtu rb a tio n s , it is desired t h a t th e schem e be ex te n d a b le to utilise th e Bloch th eo rem a n d work w ithin a supercell represen tatio n .

Lloyd’s form ula m eth o d (L FM ) provides one c a n d id a te for such a form alism w hich m eets th e c rite ria above. T h is schem e utilises m a th e m a tic a l rules which ex­ press in d u ced changes in th e density of s ta te s in term s of th e p ro p erties of th e isolated su b sy stem s and th e coupling betw een th e p a rts to c alcu late physical p ro p ­ erties of in terest; it will be one of th e m ain tools used in th is thesis. As well as th e fact it is c a p ab le of m eeting th e c rite ria outlin ed above, a n o th e r p o in t in favour of th e L FM is t h a t it utilises G reen F u n ctio n (G F ) techniques. T h e usefulness of G F (see A p p e n d ix A) techniques lies in th e fact th a t th e den sity of s ta te s of a system is in tim a tely re la te d to th e tra c e of th e G F. G reen functions can be used in disordered system s; G F m eth o d s need not rely on th e existence of tra n sla tio n a l sym m etries of th e system . Since th e process of ad d in g a d o p a n t to a carb o n n a n o tu b e breaks tra n s la tio n a l sym m etry, using G F m eth o d s can avoid th e need for a supercell rep re­ se n ta tio n . F u rth e rm o re , w ithin th e L FM , we are free to choose w hichever electronic s tru c tu re m odel is m ost su ite d to th e problem a t hand.

For in sta n c e in th e case of n a n o tu b e s th e functional form for th e zone-folded G reen fu n ctio n m a trix elem ents in real space is well know n [37], so it m ay be a t­ tra c tiv e to work w ith th is choice of basis set. If, however, it becom es clear th a t a m ore so p h istic a te d choice of electronic s tru c tu re m odel is necessary, we are free to use ou r expressions w ith th is m ore so p h istic a te d m odel. In this work, we will work on th e basis t h a t th e d etails of th e isolated su b sy stem s are well known, an d view th e configuration of th e system a fte r th e in teractio n as th e desired unknow n.

s-[)arent schem e w hich allows th e ex tra ctio n of b o th q u a lita tiv e a n d q u a n tita tiv e infoi'- m a tio n a b o u t th e binding process. M oreover, we in ten d to a d o p t a form alism w hich allows th e tre a tm e n t of a diverse range of d o p a n ts w ith o u t m a jo r m odifications. As such we will ap p ly th is m eth o d to a range of d o p a n ts of in te rest, such as ato m s and sm all m olecules, to investigate th e p ro p erties of th e com bined s tru c tu re s so form ed.

1.7

L ayout o f th e th esis

In o rd er to enh an ce th e rea d a b ility of th is work, I will here give d e ta ils of th e layout of th e rem ain d er of th is thesis, which is com posed of six c h a p te rs. T h e n ext c h a p te r will serve to intro d u ce th e m ath e m a tic s an d concepts th a t will be needed to in v estig ate changes induced in th e electronic s tru c tu re of a general system d u e to th e in tro d u c tio n of a co n tact p o ten tia l. T h is gives m e an o p p o rtu n ity to provide a d e ta ile d e x p o sitio n of th e general form alism , which will be th e p rim a ry goal of th e chai)ter. I will th e n use to illu stra te how th e m ath e m a tic a l c la rity of th e expressions p e rm its th e deriv atio n of c e rtain an aly tical results. C h ai)ter T h re e will utilise th e general form alism p resen ted in C h a p te r Tw o, and will be specific to th e case of a to m s and sm all m olecules adhering to th e surface of m etallic n a n o tu b e s. In this c h a p te r, we will show how it is possible to show co rrelations betw een th e charge tra n sfe r betw een th e p a rts and th e electronic c o n trib u tio n to th e b in d in g energy. F u rth e rm o re , we will investigate th e dependence of th e co rre la tio n on th e d etails of th e ato m ic s tru c tu re of th e adsorbed atom . We will show how it is possible to utilise th e afore-m entioned sum rules to e x tra c t d etails a b o u t th e d e ta ile d s tru c tu re of th e coupling betw een th e parts.

structure, irrespective of the underlying chiral order of the host nanotube. We will also perform calculations to investigate how th e changes induced in the electronic stru ctu re of th e system depend on details such as th e stren gth of the coupling perturbatio n, the w idth of th e perturbing strand, and indeed the angle at which the one dimensional charge distribution coils.

C h ap ter 2

T h e L loyd Form ula M eth o d

2.1

M otivation and layout

In order to understand, and ultim ately to predict the physical properties of doped carbon nanotubes, it is desirable to have a theoretical forinalisni based on which these properties can be evaluated. W hen a system w ith known properties is per­ tu rb ed by an external potential, its electronic stru ctu re is affected in a way which is reflecterl by changes in its global density of states. Those physical properties of the system th a t are directly governed by the total density of states will consequently change. Likewise, a num ber of ancillary quantities such as to tal energy, conduc­ tance, and occupation, to name b ut a few, will also be modified. This suggests th a t a study of how doping im pacts on the physical properties of a com posite m aterial should include a description of how th e total density of states changes in response to such a perturbatio n.

relates changes in th e total density of states to m atrix elements of b oth th e G reen function of the isolated system and the pertu rb atio n m atrix itself. It is one goal of this work to adap t this existing formula in such a way th a t suits th e stu dy of doped carbon nanotubes.

For the sake of clarity, it is worth describing the layout of ideas in this chapter. Lloyd’s formula in its general form, which gives inform ation about the global density of states of th e structu re in term s of th e Green function of th e unpertiu’bed system and a p erturbative potential will be introduced in the first section. T his will th en be adapted to take into account the specific geom etry relevant to iianotube doping. We do this by deriving a closed form expression for th e variation in the to ta l density of states appropriate to the interaction between initially disconnected sub-system s th a t are allowed to come into contact. The rem ainder of th e chapter investigates th e consequences of the adapted formula, and indicates some uses to which it can be put. Calculations based on the m athem atically straightforw ard case of a jjerturbed linear chain will be presented for illustration purposes.

2.2

C anonical form o f L loyd ’s form ula

uses include modeling of th e physical properties of disordered alloys [43]. In its canonical form Lloyd’s formula can be w ritten concisely as

Ap { u ) = - ^ ^ I m (T r (lo g (l - g ( E ) V (2.1) E = u )

Here, g{E) is th e single-particle Green function of the to tal H am iltonian before the p ertu rb a tio n is introduced, while 1 is the identity operator. V is th e p ertu rb atio n H am iltonian; it is necessary th a t this be independent of energy. We will refer to 1 — g { E ) V as th e Lloyd m atrix. A formal derivation of this expression is included in A ppendix B of this thesis. The log function maps linear operators into linear operators according to the definition

l o g ( i - j V ' ) = - j ( ' - ^

- 2 ^

- . . .

(2.2)

which echoes th e familiar definition of the Log function via the Laurent series for complex num bers [44].

Taking the trace of th e logarithm for each energy produces a complex num ber which makes th e trace of th e logarithm a complex function of energy. Once we take the im aginary p art, we have a simple real valued function of energy, which we m ust then differentiate. T he external factor of gives th e correct norm alisation of this function, and takes into account spin degeneracy. If spin degeneracy is absent, the external factor is b u t th e m atrix will have twice as many rows and columns. We will also use a com putationally more convenient form of this equation, which takes advantage of the useful relation

Tr (log {A)) = log (det {A)) (2.3)

form of Lloyd’s formula

Ap{lj) = (log(det (^1 - g { E ) V ^ ) ^ (2.4) E = u )

While the logarithm may be regarded as a multivalued function of a complex vari­ able, we will tre a t it as a single-valued function defined on m ultiple Riemami sheets in th e following fashion. Letting det = r {E)e' "^^^\r{E) > 0 ,0 (7 ? ) g M and fixing (t>{E) —> 0 as i? ^ —cxo,

(2.5)

Having presented th e necessary m athem atics to utilise Eq. (2.4), it is worth emphasizing its physical usefulness. One point in its favour is th a t the derivation of Eq. (2.4), as presented in th e A ppendix, includes nm ltiple scattering events of all orders through th e use of the full T m atrix. A nother point in its favour is th a t Lloyd’s formula allows one to derive an expression for the change in th e luiniber of particles in a system, by performing an elem entary integral over energy. Furtherm ore, a knowledge of the change in the density of states allows the calculation of physically relevant quantities such as the electronic contribution to the total energy. A final point in its favour is th a t since the expression is in operator form it is model independent, in so far as th e same expression holds regardless of th e choice of H amiltonian. For the sake of simplicity, we will often use simple models such as th e tight-binding model w ith Eq. (2.4), b ut we are by no means restricted to this approach; if it is found th a t the choice of such a simple H am iltonian is insufficient for accurate calculations, the same expressions will hold w ith whichever H am iltonian is necessary.

2

— L lo y d ’s Form ula result

0

Q .

2

4

6

2

0

2

Energy (t)

Figure 2.1: Change in the to tal density of states for a linear chain, whose Hamil­ tonian is / / = Y ^ j t \ j ) { j ± 1|, subject to a p ertu rb atio n potential V = ^5o|0)(0|5 calculated w ithin the tight-binding model. Different curves refer to different cutoff radii for the brute-force api)roach (see th e text). Here N is tlie distance from the p ertu rb a tio n site at which cutoff occurs. The continuous black curve (as per Lloyd’s formula) agrees w ith the limit —> oo (red circles). In this example Sq = —1.9/.

this approach is th a t the calculation of the com posite Green function is a numerically expensive operation which m ust be done for a large num ber of sites, due to a large cutoff radius. T he need for a large cutoff radius is due to the fact th a t perturbations ten d to have long-ranged effects in low-dimensional systems. This is illustrated in Fig. 2.1, where we display how th e results converge on th e Lloyd formula result in th e case of a diagonal p ertu rb a tio n of a linear chain. W hile not miexpected, it is w orth em phasizing th a t Eq. (2.4), as evidenced by the continuous black curve, reproduces exactly the converged change in density of states, as shown by th e red circles.

cannot then be apphed, and in principle th e evaluation of the eigenvalues and eigen­ states m ust be done in real space for an infinite dimensional m atrix. The stan d ard procedure to get around this in most ab-imtio calculations is to include periodic repetitions of the p ertu rbatio n through the use of a supercell. W hile th e size of th e supercell is chosen as large to reduce the interaction between im purities within adjacent unit cells, tlie exact result is only achieved in th e limit L —> oo.

hi contrast, as shown in A ppendix C, to utilise Lloyd’s formula in the case of a localized perturbation , we need only work with sub-m atrices which involve only those m atrix elements connecting states involved in th e binding. W hen combined with the simplicity w ith which the electronic structure of nanotubes can be described, this m ethod seems to be a promising way of describing the doping process in nanotubes.

2.3

M o d elin g d op in g v ia co n ta ct p o ten tia ls

schem e.

Before th e in te rac tio n , th e electronic ch a ra c te ristic s of isolated su b sy stem s are governed by a to ta l H am ilto n ian /?,. In th e case of an ad so rb a te, th e H am iltonian before in te ra c tio n can be w ritte n as th e sum of th e H a m ilto n ia n s of th e isolated svibsystems:

h = ho + h-A (2.6)

w here refers only to th e host, while refers only to th e ad so rb a te. If th e H ilbert sp ace of s ta te s of su b system O has basis { |o i), I02), • • •} an d t h a t of su b sy stem A has basis { |o i), 10-2), • • •} th en th e H ilb ert space of th e com bined sy ste m is assum ed to have basis { |o i), I02), • • •, |o i), |a2), • ■ •}. Such a schem e is w ell-know n from th e fa­ m iliar linear co m b in atio n of m olecular o rb ita ls m eth o d , in w hich th e aforem entioned s ta te s co rrespond th e m olecular o rb ita ls of th e isolated su b system s. O ften, as far as th e n a n o tu b e is concerned, we will assum e th a t th e set of s ta te s co rresp o n d in g to th e c a rb o n l/;^) s ta te s of graphene will suffice to describe th e electronic s tru c tu re of th e n a n o tu b e , since th ese are th e s ta te s th a t lie closest to th e Ferm i level. W ith reference to Fig. 1.1, we can m ake th e identification { |o i), I02), • • •} = { |/ ? i ,/ I ) , |/ ? i , / i ) , • • •}, w here it is u n d e rsto o d th a t |/ ? i ,/ l ) corresponds to th e pz o rb ita l localized a b o u t th e c a rb o n a to m of ty p e A located in th e supercell a t p o sition R i . F u rth erm o re, unless explicitly s ta te d , we will assum e t h a t all basis s ta te s are o rth o n o rm a l; th is is a reaso n ab ly good app ro x im atio n in th e case carb o n n a n o tu b e s, especially if one is concerned only w ith th e electronic s tru c tu re up to th e Ferm i level.

In th e p a rtic u la r case of a c o n ta c t p o ten tia l, we assum e t h a t only th e s ta te s { |o i), I02), • • •, |o„)} (corresponding to those sites in th e green region of Fig. 2.2) and { k 'i)i 1^2), • • ■, |«m)} (corresponding to those sites in th e p u rp le region of Fig. 2.2) are involved. In o th e r words, we assum e th a t V\oi) — {oi\V = V\a.j) = {a j\ V = 0 if i > n , j > ni. In th is case, we can form ally w rite down a p e rtu rb a tio n H am iltonian of th e form

(a)

_O

(b)

OA

Figure 2.2: Schematic representation of host and adsorbed dopant (a) before and

(b) after a contact interaction. Only matrix elements corresponding to sites in

the vicinity of the contact are perturbed. The green area denotes the region in

sub-system

O where the contact induced perturbation is non-zero, while the purple

region plays the corresponding role for sub-system

A.

Figure 2.3; Linear chain characterized by onsite e and hopping integral

t phis ad­

sorbed single level impurity with characteristic eigenvalue a. Graphical display of

tight-binding parameters both (a) before and (b) after the systems are allowed to

interact. W ithin the efficient screening hypothesis (see the text) only those sites on

the chain nearest to the atomic impurity undergo a correction to their onsite.

2.4

U sin g L loyd ’s formula: a case stu d y

2 .4 .1

L in ear ch a in w ith a d so r b e d im p u r ity

By way of an introduction to Lloyd’s formula, I now present the example of the

linear chain perturbed by an adsorbed single level atomic dopant. This pedagogical

system may serve as a simple model for interaction between carbon nanotubes and

isolated adsorbed impurities. In common with the nanotube, the linear chain is a

one-dimensional system, and in the absence of the inclusion of a Peierls distortion

is metallic (as are approximately one third of nanotubes). One advantage of this

system is th at it has a particularly simple Green function (as do the 7r-bands of

nanotubes) th at allows us a degree of mathematical transparency. A schematic of

the im perturbed system is displayed in Fig. 2.3a, while Fig. 2.3b displays the system

after the perturbation.

o rb ita ls |j ) , se p a ra te d by cornnioii d istan ce a, representing atom s bonded to g eth e r in to a long m olecule. In th e n earest neighbour ap p roxim ation, th e H am iltonian of

th e chain is d e te rm in e d by ju s t two c h a ra c te ristic energies e an d t, and th e to ta l H am ilto n ian is given as

II = E

+ E' d i Xi + i| + liXi - i|).

(2.8)

j j

w here j runs over all th e sites of th e m olecule. T h e eigenvalues of th is H am ilto n ian are o b tain e d using B lo ch ’s theorem ; th e co rresponding b a n d s tru c tu re is displayed in Fig. 2.4a.

As far as th is work is concerned, we tak e as an effective definition of th e G reen function a t energy J? of a system governed by H am ilto n ian h th e expression

g { E ) = run {1{E + trj) - h ) - \ (2.9) 7 ; ^ 0 t

w here ij] is a sm all positive im aginary nu m b er which ensures th a t th e G reen function is well defined.

It can be show n t h a t th e real space G reen function elem ents for th e linear chain are

(2.10)

w here c o s/cq = 9oo{E) = 2 t s i n f c o ( £ )• ^ h e re are two d istin c t values of ko th a t satisfy th e given equation; th e y a re negatives of each other. T h e only one w hich lies inside th e in te g ratio n co n to u r, and hence by C a u ch y ’s theorem will co n trib u te , is th e one w ith a negative im aginary p a rt. T h is ensures th e selection of th e re ta rd e d G reen function, one m a trix elem ent of which is p lo tte d in Fig. 2.4b.

k (a ')

F ig u re 2.4: (a) T ig h t-b in d in g b a n d s tru c tu re of th e u n p e rtu rb e d h n ear chain in th e n e a re st n eig h b o u r ap p ro x im atio n . T h e b a n d w id th is seen to be 4 t, an d is centred a t E = e. (b) R eal and im ag in ary p a rts of th e G reen function (0| g |0) of th e linear chain. T h e G reen function is seen to be p u re im aginary inside th e Ijand.

ionization p o te n tia l {Ui) of th e ato m (th e energy th a t m ust be add ed to th e atom to excite its hig h est energy electron sufficiently to a sc a tte rin g s ta te ). U nder th e assu m p tio n th a t we can identify a conm ion vacm m i level w ith th e host, for w hom th e w ork function (M ') plays th e role equivalent to th e ionization p o ten tia l, we can m ake th e identification t h a t q = IV — Uj. T h e relativ e position of a , E f and th e rest of th ese p a ra m e te rs is schem atically d ep icted in Fig. 2.5. However, to ch aracterize th e ato m ic im p u rity we nm st also give in fo rm atio n a b o u t th e occupancy of th a t level; we d en o te th e n u m b er of p a rticle s on th e site a in th e absence of th e p e rtu rb a tio n

as

Co-2.4 .2

T h e Lloyd Form ula M e th o d

A n in v estigation of th e linear chain plus a d so rb e d single level im p u rity provides th e o p p o rtu n ity to in tro d u c e w h a t will be h e re a fte r referred to as th e L l o y d ’s form ula

sys-E(Y)

Figure 2.5: Schematic representation of the electronic structure of the unperturbed

system. The host is represented by a half-filled band, while the single level is denoted

by a delta function. Here

W is the work function of the host,

Ey its Fermi level and

a is the energy level of the isolated impurity which has ionization potential

Uj. The

relation between the characteristic energies is displayed, which leads to the effective

definition a =

W — Uj.

tem. Combined with the fact that knowledge of the Green function of the composed

system allows direct evaluation of the local density of states at each site, a set

of non-linear equations can be derived th at allows direct evaluation of measiu’able

properties of the system in terms of the elements of the perturbation matrix.

To model the interaction, we allow the initially isolated atom to interact directly

with th at chain atom at location

j —

0. The interaction Hamiltonian displayed in

Eq. 2.7 in this case specialises to

V = ^o|0)(0| + (^a|«)(a| + 'r|a)(0| -I- r*|0)(a|,

(2-11)

where

and r are no longer matrices but now simple scalars. Here

5q

reflects a

change in the electrostatic environment at the site 0;

5a

a corresponding change on

the atomic impurity; and the hopping integral r couples the atomic level directly

to site 0 and reflects the possibility of transfer of particles between the parts. In

assume tliat those sites which are closest to the nearest point of contact should be

affected the m ost, since it is a well known characteristic of a metallic system th a t

the charge density of the m etal can change in such a way as to screen out extraneous

electric fields. W hile this ability is somewhat suppressed in low-dimensional systems,

for the sake of sim plicity we will initially assume th a t the only onsite on the chain

th a t is affected by the interaction is th a t of the site closest to the impurity. This

presum ption will later be referred to as the efficient screening hypothesis] we will

subsequently have to relax this and deal w ith finite-sized screening clouds when we

are considering the case of nanotube-adatom interaction, and wish to com pare with

ab-initio calculations.

In th e case a t hand, th e rank of the p ertu rb atio n m atrix is n = 2. In line with

w'hat has been outlined thus-far, we m ust form the 2 x 2 m atrix of Green functions.

I ’he Green function of the unp ertu rb ed linear chain is given by Ecj. (2.10) while the

Green function of th e bound orbital has a particularly simple form:

^ (2.12)

E + iO+ - Q

T h e cross term s go„ and /7„o both vanish, since the atom ic level is initially dis­

connected from th e host. It is straightforw ard to show th a t in the case of the

linear chain / adsorbed im purity system, the determ inant of the Lloyd m atrix is

1 “ (^0.900

—

kP)-In the light of th e foregoing, I now show how a set of non-linear equations can

be derived th a t show the relation between the param eterisation of the pertu rbation

and some cjuantities th a t depend on changes in the density of states, in the case

th a t there is initially Cq electrons w ith energy a on the doubly degenerate atomic