Functionalization of graphene and stoichiometric graphene derivatives

(1)

a dissertation submitted to

the institute of materials s ien e and nanote hnology

and the Graduate S hool of engineering and s ien e

of bilkent university

in partial fulfillment of the requirements

for the degree of

do tor of philosophy

By

Hasan ahin

(2)

Prof. Dr. Salim Çra (Advisor)

I ertifythatIhavereadthisthesisandthatinmyopinionitisfullyadequate,

ins opeandinquality,asadissertationforthedegreeofdo torofphilosophy.

Prof. Dr. R. Tugrul Senger

I ertifythatIhavereadthis thesisandthatinmyopinionitisfullyadequate,

(3)

Prof. Dr. Taner Yldrm

I ertifythatIhavereadthisthesisandthatinmyopinionitisfullyadequate,

Asso . Prof. Dr. Dönü³ Tun el

Approved for the Graduate S hool of Engineering and

S ien e:

Prof. Dr. Levent Onural

(4)

Hasan ahin

Ph.D.in Materials S ien e and Nanote hnology

Supervisor: Prof. Dr. Salim Çra

De ember, 2011

Re ent developments in experimental te hniques have made the design and

produ tion of materials at nanos ale possible. In parti ular, graphene has been

thefo usofresear hindiverseeldsowingtohighmobility arriertransportand

other ex eptionalproperties. Overthe past four years experimentalstudies have

demonstrated that hemi al onversion of graphene to its stoi hiometri

deriva-tivesis possiblebyhydrogenation,uorinationand hlorination. The aimof this

thesis is to predi t stable stoi hiometri graphene derivatives and explore their

me hani al, ele troni and magneti properties. Moreover, the fun tionalization

of graphene and its derivatives are a hieved, whereby their physi al properties

aremodiedtoderivenovelmaterials. Our predi tionsrevealingstable2Dsingle

layer onformers, whi h an beusedasnovelnano oetingmaterials,areobtained

from state-of-the art rst-prin iples Density Fun tional al ulations of total

en-ergy, phonons, transitionstate analysis and ab-initiomole ulardynami s.

An extensive theoreti al study on the stability of hydrogenated graphene

(C

n

H), fully hydrogenated graphane i.e graphane (CH), and their quasi one-dimensional nanoribbons is performed. The formation of meshes of

dehydro-genated domains on graphane resulted in geometry spe i magneti stru tures

showing interesting magneti intera tions. Creation of H and CH va an ies,

as well as adsorption of transition metal atoms give rise to signi ant

spin-polarization in graphane nanoribbons. It is shown that as a result of one-sided

ortwo-sided uorinationof grapheneone an obtainnanostru tures with diverse

ele troni and magneti properties. Fullyuorinatedgrapheneoruorographene

CF is a stable, sti and non-magneti semi ondu tor. Additionally, this

on-former of bu kled graphene is fun tionalized by alkali, non-metal, metalloidand

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Adsorption of hlorine to graphene is dramati ally dierent from those of

hydrogen and uorine. While the binding energy of hlorine is signi ant, its

migrationonthe surfa e of perfe t graphene takes pla e almostwithout barrier.

This is ru ial for energy harvesting on graphene surfa e. Energy optimization

andphonon al ulationsindi atethat the hair ongurationof fully hlorinated

graphene ( hlorographene) is energeti ally most favorable and stable. It is a

nonmagneti semi ondu tor with 1.2eV dire tband gap, whi h an betuned by

applieduniform strain.

Graphene by itself an be fun tionalized by reating meshes of va an ies

or adatoms onserving spe i symmetries. Under these ir umstan es linearly

rossingbandsandhen ethemasslessDira Fermionbehavior anbemaintained.

Finally,it isdemonstrated that multilayer, even singlelayer graphene

onsti-tute an ex ellent nanos ale oating, whi h an prevent a rea tive metal surfa e

fromoxidationwithout hangingthesizeandotherphysi alproperties. Graphene

an sti k to at metal surfa es and hinders freeoxygenatom and mole ule from

penetratingtothemetalsurfa e. Singlelayeruorographene an beusedalsofor

the same purposes.

Design of novel nanomaterials, in parti ular biologi al mole ules and

om-plexes using rst-prin iples methods derived from quantum theory indi ates a

new dire tion in theory, whi h promises a produ tive hybridizationwith

experi-mental studies.

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Fluo-Hasan ahin

Malzeme Bilimive Nanoteknoloji, Doktora

Tez Yöneti isi: Prof. Dr. SalimÇra

Aralk, 2011

Deneysel tekniklerdekisongeli³melernanoboyutlumalzemelerindizaynnve

üretimini mümkün klm³tr. Özellikle gran, yüksek hzlarda elektron ta³nm

ve diger üstünözellikleridolaysylaçe³itliara³trmaalanlarnnilgiodag

olmu³-tur. Geçtigimiz dört yl boyun a yaplan deneysel çal³malar granin

hidrojen-lenmesi, orlanmas ve klorlanmas yolu ile stokiyometrik türevlerine kimyasal

dönü³türülebile egini göstermi³tir. Bu tez çal³masnn ama olasgran

türev-lerinin varlgn öngörerek bunlarn mekanik, elektronik ve manyetik

özellik-lerinin ara³trlmasdr. Ayr a, modiye edilmi³ ziksel özelliklere sahip yeni

malzemelereldeetmeyeolanaksaglayan, granvetürevlerinin i³levselle³tirilmesi

mümkündür. Yeni nano kaplama malzemeleri olarak kullanlabile eksaglam iki

boyutlu malzemelerin varlgna i³aret eden sonuçlarmz toplam enerji, fonon,

geçi³ durumu ve moleküler dinamik için üst seviyede temel ilkeler yogunluk

fonksiyoneli kuram kapsamnda yürütülenhesaplardan elde edilmi³tir.

Hidrojenlenmi³ gran (C

n

H), grafan olarak adlandrlan tümüyle hidrojen-lenmi³ gran (CH) ve bunlarn bir boyutlu nano³eritlerinin kapsaml analizi

yaplm³tr. Grafan üzerinde dehidrojene edilmi³ bölgelerin geometrisine bagl

olarakilginç manyetiketkile³melere sahipoldugugösterilmi³tir. Grafan üzerinde

HveCHkusurlarnnolu³turulmasveayr ageçi³metaliatomlarnnbaglanmas

yolu ile grafan nano³eritlerde dikkate deger miktarda spin polarizasyonu saglar.

Ayr a granin tek ve çift yüzeyinin orlanmas sonu u olarak çe³itli elektronik

ve manyetik özelliklere sahip olan nanoyaplarn elde edilebile egi gösterilmi³tir.

Florogran olarak bilinen, tümüyle orlanm³ gran, sglam, gerilmelere kar³

dayankl ve manyetik olmayan bir yariletken malzemedir. Florogranin alkali,

metal olamayan, metaloidvegeçi³ metali ilei³levselle³tirilmi³tir ve her biratom

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Kloratomunungraneabsorbeolmashidrojenveuoratomunagöreoldukça

farkldr. Klor atomunun grane baglanmas çok güçlü olmasnaragmen, yüzey

üzerindeki hareketi neredeyse bariyersizdir. Bu, gran yüzeyinde enerji elde

edilmesi için büyük öneme sahiptir. Enerji optimizasyonlar ve fonon

hesapla-malar, tümüyle klorlanm³ granin (klorogran) hair halinin enerjetik olarak

en ter ih edilebilir ve saglam yap oldugunu göstermi³tir. Klorogran, germe

yoluyla degi³tirilebilir olan 1.2 eV yasak band aralgna sahip ve manyetik

ol-mayan yariletkendir

Gran tek ba³na belli simetrilere uyan yapsal kusurlar ve adsorbe edilmi³

atom gruplarilei³levselle³tirilebilirdir. Bu yaplardalineer olarakçak³an

bant-larn varlgda korunabilmektedir. Hatta belli durumlar için kütlesiz Dira

fer-miyonu davran³ bileelde edilmektedir.

Son olarak, çok tabakal ve hatta tek tabakal gran malzemelerin, reaktif

metal yuzeylerinin boyut ve diger ziksel özelliklerini degi³tirmeksizin koruyan,

mükemmel nano ölçek kaplama malzemeleri oldugu gösterilmi³tir. Gran düz

metal yüzeylere yap³abilir ve serbest oksijen atomunun ve molekülünün metal

yüzeye szmasnaengel olur. Tek tabaka orogran de benzer amaçlar için

kul-lanlabilirdir.

Özelliklebiyolojikmolküllervekomplekslergibiyeninanomalzemelerin

kuan-tum teorisinden türeyen yogunluk fonksiyoneli kuram ile dizayn edilmesi teori

açsndan yenibirdogrultuya i³aret etmektedir.

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Floro-Foremost, I would like to express my sin ere gratitude to my advisor Prof.

Dr. Salim Çra for the ontinuous supportof my Ph.D study and resear h,for

hismotivation,enthusiasm,and immenseknowledge. Iamdeeplygratefultohim

forgivingmethe onden etoexploremy resear hinterestsand theguidan e to

avoid getting lost in my exploration. His guidan e helped me in all the time of

resear h and writingof this thesis.

I amdeeplygratefultomy o-advisorProf. Dr. R.T. Senger, foropeningthe

doors of Bilkent University to me and attra ting my interest to spintroni s and

nanote hnology.

AlsoIthankmyfriendsEnginDurgun,SefaDagandHaldunSevinçlifortheir

valuable friendshipand guidan e.

I alsothank my groupmates: Seymur Cahangirov, Can Ata a, Mehmet T

op-sakaland Ethem Akturk, for the stimulatingdis ussions, for the sleepless nights

we were working together before paper submissions, and for all the fun we have

had inthe lastfouryears.

I would liketo thank my parents forgiving birthto meatthe rst pla e and

supporting me spiritually throughout my life. They are always the ompass of

my life.

Lastly, Iowe mylovingthankstomy wifeeydaHorzumahin. Withouther

en ouragement, patien e and understanding it would have been impossible for

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1 Introdu tion 1

2 Computational Methodology 6

2.1 Density Fun tional Theory . . . 6

2.2 Ex hange-CorrelationPotentials . . . 7

2.2.1 Lo alDensity Approximation . . . 7

2.2.2 GeneralizedGradientApproximation . . . 8

2.3 Cal ulationof Phonon Spe tra . . . 8

2.3.1 Small Displa ement Method . . . 10

2.3.2 Density Fun tional Perturbation Method . . . 11

2.4 Graphene: ComputationalAnalysis . . . 13

2.4.1 Atomi Stru ture . . . 14

2.4.2 Ele troni Stru ture . . . 15

2.4.3 Transport Properties . . . 17

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3.1 Motivation . . . 24

3.2 Graphane: Fully Hydrogenated Graphene . . . 26

3.3 Va an y Formationon Graphane . . . 27

3.3.1 Single-sidedVa an y Domains . . . 27

3.3.2 Double-sided Va an y Domains . . . 29

4 Graphane Nanoribbons 32 4.1 Motivation . . . 32

4.2 Two DimensionalGraphane . . . 34

4.3 Graphane Nanoribbons . . . 37

4.4 Fun tionalizationof Graphane NRs by Adatoms . . . 43

4.5 Va an ies inGraphane NRs . . . 46

4.6 Edge Roughness . . . 49

5 Fluorinated Graphene 52 5.1 Motivation . . . 52

5.2 Stru tures of uorinatedgraphene . . . 54

5.3 Ele troni Stru tures . . . 60

5.4 Elasti Properties of CF . . . 63

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6.1 Motivation . . . 68

6.2 Adsorption of Single Chlorine . . . 72

6.3 Coverage of Graphene by ChlorineAdatoms . . . 75

6.4 StableFully ChlorinationatedGraphene: Chlorographene . . . 80

6.4.1 Stru tural Properties . . . 80

6.4.2 VibrationalProperties and RamanSpe tra . . . 81

6.4.3 Ele troni Properties . . . 83

6.4.4 Me hani alProperties . . . 86

6.4.5 Defe ts. . . 88

7 Graphene Nano oatings 91 7.1 Motivation . . . 92

7.2 Oxidationof Al Surfa e and Graphene . . . 94

7.3 Prote tion of Al Surfa eby Graphene Coating . . . 98

7.3.1 BilayerGraphene Coating . . . 101

7.3.2 Va an y Ee t . . . 103

8 Graphene Nanomeshes 106 8.1 Motivation . . . 107

8.2 Tight Binding Approximationand DFT . . . 109

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2.1 (a) Top view of honey omb stru ture of graphene. Bravais

lat-ti e ve tors for both stru ture are given with

| ~

a

1 | = | ~

a

2 | = a

. Hexagonal unit ell in luding two arbon atoms is delineated by

dashed area. (b) Side view for the sp

2

oordinated arbon atoms

of graphene. ( ) Atomi onguration of zigzag and arm hair

graphene nanoribbons. . . 14

2.2 (a)Ele troni band stru ture ofgraphene obtained by rst

prin i-ples method. High symmetry points and the orbital hara ter of

the bands are delineated. (b) Three dimensional band stru ture,

obtained by tight-bindingapproximation,for valen e and

ondu -tionbandsandtheDira pointslo atingattheKsymmetrypoints.

Nearestneighbor hoppingparameter istaken to be 2.7eV. . . 16

2.3 (Coloronline)Geometryandspin-dependent hargedensityofthe

grapheneake utfrom4-ZGNR.Theedgesaresaturatedwith

hy-drogenatoms. Green (dark) and yellow (light) regionsdenote the

lo almajority spin-typeof the hargedensity. Possibleadsorption

sitesofadatomsarealsolabelled. The lowest energy onguration

for a single vanadium atom is shown in (i). (b) Spin-dependent

transmissionspe traforvarious asesofgraphenefragment. Fermi

level isset tozero. ( ) Lo al density of states (LDOS) isosurfa es

al ulatedfor parti ularenergy values of theup-spin transmission

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dieren e of spin-up (

↑

) and spin-down (

↓

) states for 4-(upper row) and 5-triangle(lower row) graphene akes: Bare,singly-and

doubly-hydrogenated edges. Cal ulated net magneti momentsof

the akes are given in terms of Bohr magneton (

µ

B

). Dieren e harge density of spin-up and spin-down states is shown by red

(dark) and blue (light) isosurfa e, respe tively. Ele trode-devi e

geometry and onvention for forward and reverse bias applied to

triangulargraphene akes(TGFs).(a) Energy level spe tra within

±1

eV range of Fermi level (

E

F

), the HOMO-LUMO gap (

∆

) and isosurfa e of HOMO and LUMO orbitals. (b- ) Cal ulated

I-V urves for hydrogenated and bare triangular graphene akes

(TGFs). Resultsof 4-TGFand 5-TGFare presented, respe tively.

Spin-up (

↑

) and spin-down (

↓

) urrents are shown by red (dark) and blue (light)lines, respe tively. Solid and dashed lines denote

forward and reverse bias al ulations, respe tively. Transmission

urves of spin-up and spin-down under zero bias are also shown

by insets. Transmission spe tra of singly-H and bare TGFs are

plottedup to maximum value of 0.03 and 0.4, respe tively. Fermi

levelsare set tozero. . . 21

2.5 Phonondispersionofthegraphene. TheresultsofSDMnadDFPT

arepresented by solid/blueanddashed/redlines. Atomi motions

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3.1 (Coloronline)(a) Top andside views ofatomi stru tureshowing

ofgraphane primitiveunit ellwithBravais latti eve torsb

1

and b

2

and bu kling of alternating arbon atoms, A and B, in honey- omb stru ture

δ

, bond lengths

d

C−C

and

d

C−H

optimized using

LDA. Large green(light)and smallorange (dark)balls indi ateC

andHatoms,respe tively. (b)Energybandstru tureis al ulated

by using LDA and orre ted using GW

0

(shown by blue lines and orange dots). For graphene, linear band rossing at Dira point

is shown by dashed grey lines. ( ) Cal ulated phonon bands and

density of states DOS proje ted to Cand H atoms. . . 25

3.2 (Color online) Cal ulated magneti state of various domains of

single-sided H-va an ies, where all H atoms atta hed to C atoms

fromuppersideintheunshadedregion(delineatedbydash-dotted

lines)in ludingedges, are removed. The trianglesare spe ied by

∆

s

n

with

n

indi atingthemaximumnumberofCatomsatoneedge and

s

signies the single-sided dehydrogenation. Similar symbols are used also for hexagonal,

H

s

2

and lane

L

s

n

(

n =

4,5) domains. Totalmagneti moment

µ

T

and its omponents

µ

x

,

µ

y

and

µ

z

are given inunits of the Bohrmagneton

µ

B

. Magneti moments onC atomsareshown byred (bla k)arrows. Relo ationsof Hatomsat

theothersideofgraphaneare shown by urlyarrows. Forthesake

of larity

π

-bonds formed after the relo ation of bottom H atoms are indi ated onlyfor

∆

s

4

,

L

s

4

and

L

s

5

stru tures. . . 28

3.3 (Color online) Net magneti moments in Bohr magneton within

the triangular

∆

d

n

, hexagonal

H

d

n

, re tangular

R

d

n

and lane

L

d

n

domains, whi h are delineated by dash-dotted lines and have

n

arbon atoms attheir edges. Here

d

signiesthe double-sided de-hydrogenation. Random shaped domain in luding both one and

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of innite 2D graphane sheet having honey omb stru ture. Two

sublatti esofgraphaneareindi atedbyAandB.Bla k(dark)and

blue(light)balls are for arbonand hydrogen atoms,respe tively.

(b) Charge density ontour plots of diamond and graphane are

shown on a plane passing through C-C-C-C and H-C-C-H bonds,

respe tively. The tetrahedral angle of the diamond

θ

C

= 109.47

o

.

Arrows indi ate the dire tion of in reasing harge density. The

al ulated values of

θ

C

and

θ

H

, namely C-C-C and H-C-C bond angles in graphane respe tively, are given in Table 4.1. Contour

spa ings are 0.0286 e/Å

3

. ( ) The LDA energy band stru ture

where the orbital hara ter of spe i bands is also given. The

bandgap isshaded yellow/gray. . . 35

4.2 (Coloronline)(a)Atomi stru tureofbare arm hairgraphane NR

having

N

=11. The double unit ell of the ribbon is delineated by red/dashedlines with thelatti e onstant

2a

. Large/bla kand small/lightblueballsindi ate arbonandhydrogenatoms. Energy

bandstru ture orrespondingtothearm hairNR and harge

den-sityofsele tedbandsareshowninthepanelsattherighthandside.

(b)Atomi stru tureofbarezigzaggraphaneNRhaving

N

=6with double unit ell delineated by red/dashed lines and with latti e

onstant

2a

. Energy bandstru ture and isosurfa e harge density ofsele ted states orrespondingtozigzagNRareindi ated. Bands

shown by red/dotted lines are derived from edge states. Zero of

energyisset tothe Fermilevel,shown by dash-dottedlines, ofthe

ribbons with H-passivated edges. In spin polarized al ulations

doubleunit ellisused toallowantiferromagneti order alongthe

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4.3 (Coloronline)Totalenergies ofpossible magneti orderingsatthe

edgesofbare zigzaggraphaneribbons. Cal ulationsare performed

indoubleunit elldelineatedbyred/dashedlines. Spinupandspin

downstates areshown by green/darkand grey/lightisosurfa esof

the dieren e hargedensity,

∆ρ

. . . 39

4.4 (Color online) Variation of the energy band gap of H-passivated

zigzagandarm hair NRsofgraphane asafun tion ofwidth given

by

N

. The variationof the bandgap with

N

istted tothe urve given by ontinuous line. (See text) . . . 41

4.5 (Color online) S hemati representations of possible positions of

adatomsonalarge H-passivatedarm hair graphane ribbon.

Posi-tionsofadatomsobtainedafterthestru tureoptimizationthrough

minimizationof total energy and for es exerting onthe atoms are

alsoshown. . . 45

4.6 (Color online) Atomi stru tures orresponding to single-H,

double-sided triangular shaped

∆

2 and

∆

3, double-sided re tan-gular shaped, CH and C

2

H

2

va an ies and resulting dieren e harge

∆ρ = ρ

(↑)

_{− ρ}

(↓)

, surfa es for a bare arm hair graphane

NR. Large/bla k and small/light blue-gray balls indi ate C and

H atoms, respe tively. Only a small part whi h in ludes va an y

region and its nearby atoms, of the arm hair NR with

N

=15 is shown. density. . . 47

4.7 (Color online) Energy band diagram and band proje ted harge

density isosurfa es of various states for bare zigzag NR in luding

edge roughness. The band gap between edge states are shaded

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phonon frequen ies,

Ω

(k) versus wave ve tor, k) of various op-timized C

n

F stru tures al ulated along the symmetry dire tions of BZ. Carbon and uorine atoms are indi ated by bla k (dark)

and blue (light) balls, respe tively. (a) C

2

F Chair stru ture. (b) C

2

FBoat stru ture. ( ) C

4

Fstru ture. Units are Å forstru tural parametersand m

−1

for frequen ies. . . 56

5.2 (Coloronline)(a)Atomi stru ture ofuorographene CF.

a

and

b

arethelatti eve tors(

|a| = |b|

)ofhexagonalstru ture;

d

CC

(

d

CF

) istheC-C (C-F)bond distan e;

δ

isthe bu kling. (b) Phonon fre-quen iesversuswaveve tork,i.e.

Ω

(k)ofoptimizedCF al ulated alongsymmetrydire tions inBZ.( ) Symmetries, frequen iesand

des riptionsofRaman a tivemodes ofCF. (d)Cal ulated Raman

a tive modes of graphene, CH, CF and C

4

F are indi ated on the frequen y axis. Those modes indi ated by "+" are observed

ex-perimentally. ThereisnoexperimentalRamandata inthe shaded

regions. Units are Å for stru tural parameters and m

−1

for

fre-quen ies. . . 58

5.3 (Coloronline)Energybandstru tures ofvariousstableC

n

F stru -turestogetherwiththeorbitalproje teddensitiesofstatesandthe

totaldensitiesofstates(DOS).TheLDAbandgapsareshadedand

thezeroofenergyisset totheFermilevel

E

F

. TotalDOSiss aled to45%. Valen e and ondu tion bandedges afterGW

0

orre tion are indi ated by lled/red ir les. (a) C

2

F hair stru ture. (b) C

2

F Boatstru ture. ( ) C

4

F stru ture. . . 61

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5.4 (Coloronline) (a)Energy band stru ture of CF together with the

orbital proje ted DOS and total densities of states. The LDA

bandgapisshadedandthezero ofenergyissettotheFermilevel,

E

F

. Valen e and ondu tionband edges afterGW

0

orre tion are indi ated by lled/red ir les. (b) Isosurfa es of harge densities

of states orresponds to rst (V1), se ond (V2) valan e and rst

(C1) and se ond (C2) ondu tion bands at the

Γ

- and

K

-points. ( ) Contour plots of the total harge density

ρ

T

and dieren e harge density

∆ρ

in the plane passing through F-C-C-F atoms. Contour spa ings are 0.03 e/Å

3

. . . 62

5.5 (Coloronline)(a)Variationofstrainenergyanditsrstderivative

with respe t to the uniform strain

ǫ

. Orange/gray shaded region indi atestheplasti range. Two riti alstrainsinthe elasti range

are labeled as

ǫ

c1

and

ǫ

c2

. (b) Variation of the band gaps with

ǫ

. LDA and

GW

0

al ulations are arried out using 5x5 super ell havingthe latti e parameter of

0

=5a, and

∆c

isits stret hing . . 64

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gratingalongthe path Top(T)-Bridge(B)-Hollow(H)-Topsites on

a hexagon graphene in a (4x4) super ell. At ea h point on the

energy urve,

x

- and

y

-positions adsorbed Cl atom are xed, its

z

-height,as well as positionsof all Catoms in the (4x4) super ell are optimized by minimizingtotal energy and atomi for es. Zero

ofenergyissettotheenergyofT-site. Thediusionpathwiththe

lowest energy barrier of

Q

=13 meV between two adja ent T-sites are marked with thi k red/ dashed lines. (b) Energy lands apeof

asingle Cladatomadsorbed tographene. Dark (light) olors

rep-resentsthetop(hollow)sites. ( )Potentialenergy ontourplotsof

ClatomadsorbedtotheT-site. The al ulationofjumpfrequen y

of Cl atom

ν

, is obtained from this plot. (d) Band stru ture of a single Cl adsorbed to ea h (4x4) super ell of graphene and

orre-spondingtotal(TDOS) andorbital de omposed(PDOS) densities

ofstates. ThezeroofbandenergyissettotheFermilevel. Up-spin

and down-spinbands are olored with blue and red, respe tively. . 71

6.2 (Color online) The intera tion energy between two Cl atoms

ad-sorbed tothe same sideof a(6x6)super ellof graphene. Thezero

ofenergyis set tothe energy ofCl

2

plus graphene. nn denotes the neatrest neighborin graphene latti e . . . 73

6.3 (Coloronline)Energyband stru tureofasingleClatomadsorbed

toea h (

n

x

n

)super ell of graphene for

n

=2,3,5and 6, whi h or-respond to the one-sided uniform overage

Θ = 1/2n

2

. For

n ≥ 2

the ouplingbetween adja ent adsorbates is not su ient toform

Cl

2

mole ule. Whereas for

n = 1

(or

Θ

=0.5) the oupling is sig-ni ant and form Cl

2

. The units of magneti moments

µ

is Bohr magnetonper(

n

x

n

) super ell. . . 76

(21)

6.4 (Coloronline)(a) Theatomi stru turetwoClatoms adsorbed to

a(4x4)super ellofgraphene. Inthreedierent onguration

illus-trated by top panels, namely ortho top-bottom, para top-bottom

and meta top-bottom two adsorbed Cl atoms are stable.

∆E

in-di atestheir energies relativetothe total energy of the ortho

top-bottom onguration. Double sided adsorption imposes a lo al

bu kling in planar graphene. Three one-sided ongurations,

or-tho top-top, para top- top and meta top-top are not allowed; Cl

atoms annotbeboundtographene,theyratherformCl

2

mole ule. (b) Contour plot of the total harge density of asingle Cl-C bond

and two Cl-C bonds inortho top-bottom onguration. Contours

spa ings between 0.025

e

/Å

3

and 1.0

e

/Å

3

are 0.025

e

/Å

3

. The

Cl-C bonds of ortho top-bottom onguration is reminis ent of

the bonds formed from

sp

3

hybridization, where C atoms are

lo- ally bu kled and bond harges are a umulated between Cl and

Catoms indi atinga ovalent hara ter. Whereasthe singleCl-C

bond isioni with minute lo albu kling of graphene. . . 78

6.5 (Color online) (a) Atomi stru tures of boat, nonbonding hair,

zigzag and arm hair onformations. Large green and small gay

ballsrepresentClandCatoms. (b) Sideviewofnonbonding hair

onformation and its al ulated phonon dispersion urves. Low

frequen yphononmodesshownbyredlinesarerelatedtoadsorbed

Clatoms. Thesemodeshaveimaginaryfrequen iesandhen ethey

are unstable.. . . 79

6.6 (Coloronline) (a) Top, side and tilted views for the atomi

stru -ture of hlorographene layer having hexagonal latti e and

honey- omb stru ture. Carbon and hlorineatoms are indi ated by gray

(dark) and green (light) balls, respe tively. Cal ulated stru tural

parameters are indi ated. (b) Phonon bands of hlorographene

(22)

CCl. The band gap is shaded yellow. The GW

o

orre ted valan e and ondu tion bands are shown by dashed line and red balls.

Thezero ofenergyset tothe Fermilevel

E

F

. (b)Density of states proje ted to various orbitals(PDOS). . . 84

6.8 (Coloronline)(a)Thevariationofthe strainenergyandits

deriva-tive of with applied uniform strain

ǫ

al ulated for CCl, CH and CF. The stru ture of

dE

s

/dǫ

is enhan ed due to the s ale of the plot. After the maxima indi ated by arrows stru tures be ome

unstable and undergoes a plasti deformation. Cal ulations

per-formedin(5x5)super ells. Thestraininonedire tionis

ǫ = ∆c/c

o

.

c

0

is the equilibrium latti e onstant of super ell and

∆c

is its stret hing. (b) Variationof the band gap with uniform strain. . . 87

6.9 (Coloronline)TopPanels: Atomi stru tureofvarioustypesof

de-fe ts, Cl

N

=1-4 and their al ulatedmagneti moments. Middle Panels: (CCl)

N

defe tswith

N

=1-4and their al ulatedmagneti moments. ( ) BottomPanels: Fragmentation ofthe stru tureCCl

havinga single (CCl)

N

hole in the super ell. . . 90

7.1 Oxidation of the Al(111) surfa e: (a) Atomi onguration of an

oxygenatom adsorbed toAl(111) surfa e, whi h isrepresented by

a 4-layerslab. Oxygen and Al atoms are illustrated by small-red

and large-blue balls with numerals indi ating their layer numbers

from the top. Al atoms at the fourth layer are hidden below the

rst layeratoms. (b) Density of states (DOS) proje ted to

s

-and

p

-orbitals of adsorbed O, surfa e and subsurfa e layers of Al(111) slab. Zero of energy isset toFermi energy. . . 93

(23)

7.2 Oxidation of bare graphene: (a) Variation of the energy of

ad-sorbed oxygen atom along T(top)-H(hollow)-B(bridge) site

dire -tionsof ahexagon showing that B-siteisenergeti ally most

favor-able. Starsindi atefavorablepathforthediusionofoxygenatom

onthegraphene surfa e. (b) Oxygen atomadsorbed atthe bridge

siteofa(4x4)super ellofgraphene onsistingof32 arbonatoms.

( )Ele troni energybandstru ture orrespondingto(b)together

with the harge densities of spe i ondu tion and valen e band

states. Upon oxidation the linearly rossing

π

and

π

∗

states of

semimetalli bare graphene are modied and opened a band gap

of 0.58 eV. (d) Density of phonon modes of a pristine graphene

(shaded area) and those of oxygen adsorbed to the bridge site of

the (4x4) super ell of graphene (red line) are al ulated from the

rstprin iples.[98℄Relevantlo alizedphononmodesare indi ated

by insets. . . 95

7.3 Cal ulationofenergybarriersinvariouspathsofO

2

andOmoving fromthe top to bottom side of a suspended graphene, whi hmay

berelevantfortheoxidation: (a)O

2

,whi hmovesperpendi ularto graphenesurfa eand isfor edtopassfromthe toptobottomside

ofgraphene alongaverti allinethrough theholeatthe enter ofa

hexagon. (b)Theenergybarrierforasingleoxygenatomfor edto

pass fromthe top to bottom side of graphene alonga verti al line

through the hole atthe enter of a hexagon. ( ) An oxygen atom

adsorbed at the bridge site above the graphene plane is for ed to

pass to the bottom side. Positions of arbon atoms as wellas the

lateral

x

-and

y

- oordinatesofoxygenareoptimizedforea hvalue of indentation. The path followed in ( ) appears to have lowest

(24)

the top side of single graphene layerto its bottom side and

even-tually adsorbing to Al(111) surfa e underneath. Oxygen atom is

initiallyadsorbedatthebridgesiteabovegraphenesurfa eand

fol-lowsthe pathofminimumenergybarrier

Q

ox

. Variousstagesfrom A(whi h orresponds toanoxygenatom adsorbed tographene at

the bridge site above) through C (i.e. the point of maximum

en-ergywhere Opasses from above tobottompart of graphene) and

E (eventually oxygen adsorbed to Al(111) surfa e) show that the

surfa e of the metal is oxidized if an energy barrier of

Q

ox

=

5.93 eV is over ame. (b) Prote tion of Al(111) surfa e from oxidation

by a graphene bilayer and the variation of energy of oxygen

ad-sorbed at the bridge site of rst graphene layer orresponding to

the stage A. Diusing O has to over ome

Q

ox

=

6.81 eV at B. At Coxygenatomisswit hed tothe bridgesite atthebottomsideof

rst graphene. At E the diusing Oswit hes to se ond graphene

layer and is adsorbed its bridge site. A se ond energy barrier of

Q

′

ox

= 5.20

eV isneeded tobe over omefor Oatom tooxidize the metalsurfa e. . . 100

7.5 (a)Evolutionsofenergeti sandatomi stru turewiththe

indenta-tionof the oxygen atom,whi his initiallyadsorbed atthe edgeof

asingleva an y. (b)Evolutionsofenergeti sandatomi stru ture

oftwooxygenatomsresultingfromthe disso iationofO

2

mole ule atthe edge of asingle va an y and the indentationof one of

oxy-gen atoms from the top to bottom site of a suspended graphene.

( ) The diusion of one of adsorbed oxygen atoms in(b) towards

the Al(111) surfa e resultingin its oxidation. Red, gray and blue

(25)

8.1 Contour plots for band gap. (a) Perfe t graphene. (b) A rystal

havingsquare latti ewith singleatom inthe unit ell. ( ) A

rys-tal having hexagonal latti e with a single atom in the unit ell.

Band rossingo ursalongthe yellow/light ontours onwhi hthe

band gap be omes zero. Band gap takes its maximum value at

brown/dark ontours. . . 110

8.2 Spin-polarized energy band stru ture of a periodi patterns

on-sistingofthe(4x4)super ellsea hhavingasingle(a)hydrogen;(b)

uorine; ( ) oxygen; (d) manganese adatom. (e) A similar

hydro-gen pattern forming the (2x2) super ell on graphene host matrix

allowingsigni ant ouplingbetween adatoms. (f-h)Periodi

pat-terns of two, eight and six hydrogen atoms in the (4x4)graphene

super ell, respe tively. Isosurfa e of hargedensity ofbands

ross-ing near the

K

-point are shown by insets. For the sake of om-parison,linearly rossing

π

and

π

∗

-bandsof perfe t graphene host

matrix are also superimposed in the band stru tures. The zero of

energy is set at the Fermi level

E

F

. The band gaps are shaded in yellow. All the bands presented in BZ orresponding to the (4x4)

super ell. . . 111

8.3 Bandstru turesshowingtheee tsofthe ouplingbetweenvarious

patternsasafun tionoftheirsize andthesizeofmesh(super ell).

Thelinearly rossingof

π

-and

π

∗

-bandsofgraphene areshown by

red/dashedlines. . . 114

8.4 (a) Stru tural parameters for a nanomesh of C

n

hole. (b) Large super ellsofhexagonallatti eea h ontainingsingleholeofC

1

,C

2

, C

4

, C

6

, C

12

, C

24

. The nanomeshes in the third row are obtained by saturating C

6

and C

24

holes by hydrogen and alsoby B and N atomsalternatingly. . . 116

8.5 Bandstru ture of nanomeshes of C

12

forming in the (

n

x

n

) super- ells of graphene with

n

=4...11. . . 117

(26)

and (5x10) graphene super ells. Atomi stru ture of nanomeshes

are given by inset. (d) Semimetalli ele troni stru ture and

iso-surfa e harge densities of valen e (V) and ondu tion (C) bands

(27)

4.1 Comparisonofthe al ulatedquantitiesofgrapheneandgraphane.

Latti e onstant,

a

;C-C bond distan e,

d

C−C

;C-Hbonddistan e,

d

_C−H

; thebu kling,

δ

[seeFig.4.1(a)℄;anglebetweenadja entC-C bonds,

θ

C

[see Fig. 4.1(b)℄; angle between adja ent C-H and C-C bonds,

θ

H

; energy band gap al ulated by LDA,

E

LDA

g

; energy band gap orre ted by GW

0

,

E

GW

0 g

; ohesive energy

E

nm

c

, (

E

m

c

) obtained with respe t to nonmagneti (magneti ) free atom

ener-gies;theC-Hbondenergy,

E

nm

C−H

(

E

m

C−H

)obtained withrespe t to nonmagneti (magneti ) free atom energies; photoele tri

thresh-old(workfun tion),

Φ

; in-plane stiness,

C

and Poison ratio,

ν

. . 37

4.2 Summary of the al ulated quantities for adatoms adsorbed on a

H-passivated arm hair graphane NR.The rst and se ondlines in

ea hrowasso iatedwithagivenadatomadsorbedtoedgesiteand

thesitesnearthe enterofthegraphaneNRs,respe tively.

d

H

,the adatom-nearesthydrogendistan e;

d

C

, thenearestadatom arbon distan e;

E

b

, adatom binding energy;

µ

T

, magneti moment per super ell;

ρ

∗

, ex ess harge on the adatom (where negative sign

indi atesex ess ele trons);

Φ

, photoele tri threshold (work fun -tion);

p

, dipolemoment;

E

i

energies of lo alized states asso iated with adatoms. Lo alized states are measured from the top of the

valen e band. The o upied ones are indi ated by bold numerals

andtheir spin alignmentsare denotedbyeither

↑

or

↓

. Uptorst seven adatom states of

E

i

are shown. . . 42

(28)

graphene stru tures (namely CF, C

2

F hair, C

2

F boat and C

4

F) with those of graphene and CH. Latti e onstant,

a = b

(

a 6= b

for re tangular latti e); C-C bond distan e,

d

CC

(se ond entries with slash dier from the previous one); C-X bond distan e (X

indi ating H (F) atom for CH (CF)),

d

CX

; the bu kling,

δ

; an-glebetween adja entC-C bonds,

α

C

; anglebetween adja ent C-X and C-C bonds,

α

X

; total energy per ell omprising 8 arbon atoms

E

T

; formationenergy per Xatom relativeto graphene,

E

f

; bindingenergy per X atom relative to graphene,

E

b

(the value in parenthesis

E

b

′

ex ludes the X-X oupling);desorption energy,

E

d

(see the text for formal denitions); energy band gap al ulated

by LDA,

E

LDA

g

; energy band gap orre ted by GW

0

,

E

GW

0 g

; pho-toele tri threshold,

Φ

; in-plane stiness,

C

; Poison ratio,

ν

. All materialsare treatedinhexagonallatti e,ex ept C

2

Fboat having re tangularlatti e. . . 57

6.1 Cal ulated values for CX stru tures (where X=H,F,Cl); latti e

onstant(

a

),C-Cdistan e(

d

CC

),C-Xdistan e (

d

CX

),thi knessof the layer (

t

), photoele tri threshold (

Φ

), harge transfer from C to X (

∆ρ

), ohesive energy per unit ell (

E

coh

), formation energy (

E

f

), desorption energy of a single X atom from the CX surfa e (

E

des

), dire tband gap(

E

g

), bandgap orre ted with GW

o

E

GW

o

g

(29)

Introdu tion

The industrialrevolution of the late eighteenth entury had a major impa t on

almost every aspe t of industry, ma hinery and daily life in a manner that very

fewotherdevelopmentsintheworldhave. Subsequentte hnologi alands ienti

developments openedthe way of realizationof novelmaterialsthat are the parts

ofdevi esoperatingatonemillionthofameter. However, abilitytomeasureand

designatmi ro-s alenotonlyledtotheemergen eofmi ros opi devi esbutalso

provide equipments to explore the world at the smaller dimensions. Nowadays,

in the rapidly developing eld of nanote hnology, resear hers are taking ontrol

of atoms and mole ules individually, manipulating them and putting them to

use with anextraordinary degreeof pre ision. Overthe lastde ade, nanos ien e

and emerging nanote hnologies have been dominated by honey omb stru tured

arbon based materials indierent dimensionality, su h as fullerenes, single and

multi walled arbon nanotubes, graphene and its nanoribbons. In parti ular,

graphene, a two dimensional (2D) honey omb stru ture of arbon, has been an

a tive eld of resear hfor lastseven years.

In this thesis we mainly fo us on the use of graphene and graphene-based

hemi alderivativesasnano-s ale oatingmaterialsthat anprote trea tive

sur-fa esunderneathbyposingahighenergybarriersurfa e. Therea tionofmaterial

(30)

tivesurfa es atma ros aleresulted inthe hange ofsize and physi alproperties,

prote tion against orrosion at nanos ale has required new paradigms using

ul-trathinand nano omposite oatings. Graphene, being notonlythe thinnest ever

but alsothe strongest material, has the potential for nano- oating appli ations.

Whensti kstoorgrownonvarioussurfa es,graphene addsonly3-6Åtothesize

of underlyingsampleand formsan ele tri allyand thermally ondu tive oating

on it. Moreover, graphene has ex eptional thermal and hemi al stability; it is

stableup totemperatures higherthan1500 Cunderinertenvironment. Itisalso

stableundermany onditionswhereother substrateswould undergo rapid

hem-i alrea tions endingwith degradation. Withthese propertiesgraphene keep the

promisesof being anex ellent andidate for a nanos ale prote tionlayer against

orrosion.

Although graphene is one of the most me hani allystrong material havinga

wide range of extraordinary properties, pra ti aldevi e appli ations are limited

by it's metalli behavior and tenden y to intera t with environment. Eorts to

manufa tureof hemi allymodiedgraphene ompositeswithtailoredele troni ,

opti al,and hemi alpropertiespresentanex itingnewdire tioningraphene

re-sear h. Inparti ular,bandgapengineeringofgraphenethrough hemi al

modi- ationsu hasoxygenation,andhydrogenationisappealingforele troni

appli a-tionssin eitfa ilitatesthes alablefabri ationofgraphene-baseddevi eswithout

disturbing the strong honey omb latti e. However, due to the omplex atomi

stru ture of grapheneoxidesand thermal instabilitiesof hydrogenated graphenes

even at low temperatures, sear h for the novel graphene-based materials is still

ontinuing. Re ent studies have also revealed that oxygenated, hydrogenated

and uorinated derivatives of graphene having diverse ele troni and me hani

properties an be synthesized. Easy manufa turability, high-quality insulating

behaviorand extraordinary me hani alstrength of uorographene(CF) have

in-spired intense resear h on other halogen de orated graphene derivatives. In this

thesiswealsopresentele troni ,magneti ,me hani andvibrationalpropertiesof

CH,CF and CCl. Itwas shown thatthese derivativesare asstrong asgraphene,

(31)

view,graphene ompositeshavingsaturatedsurfa esarealsogood andidatesfor

hemi allystableand me hani allystrongprote tionlayer. Sin euorographene

isthemost stablederivativeamongtheexperimentallyrealizedmaterialswealso

examine the performan e of CF as a oating material.

The thesis begins with this brief introdu tion, followed by a hapter on the

basi s of density fun tional theory and stability analysis of nanos ale materials

within the framework of ab initio methods. In Chapter III, it is predi ted that

ea hhydrogen va an y reated at the surfa e ofhydrogenatedgraphene, namely

graphane,resultsinalo alunpairedspin. Fordomainsofhydrogenva an iesthe

situationis,however omplexand depends onthe sizeand geometryof domains,

as well as whether the domains are single or double-sided. Owing to the

dif-ferent hara ters of ex hange ouplingin dierent ranges and interplay between

unpairedspinandthebindinggeometryofhydrogen,va an ydomains an attain

sizable net magneti moments.

In Chapter IV, the ele troni and magneti properties of graphane

nanorib-bons are investigated. It is found that zigzag and arm hair graphane

nanorib-bons withH-passivatededges arenonmagneti semi ondu tors. Whilebare

arm- hair nanoribbons are alsononmagneti , adja ent dangling bonds of bare zigzag

nanoribbons have antiferromagneti ordering at the same edge. Band gaps of

theH-passivatedzigzagandarm hairnanoribbonsexponentiallydepend ontheir

width. DetailedanalysisofadsorptionofC, O,Si,Ti,V,Fe,GeandPtatomson

the graphane ribbon surfa e reveal that fun tionalization of graphane

nanorib-bons is possible via these adatoms.

In Chapter V, uorination of graphene are investigated. Analysis of

uori-natedgraphenesshowsthatdierentC

n

Fstru tures an formatdierentlevelof F overage. Cal ulated properties of these stru tures, su h as latti e parameter,

d

CC

distan e, band gap, density of states, work fun tion, in plane stiness

C

, Poisson's ratioand surfa e harge, are shown todepend onthe level of overage.

Perfe t uorographene stru ture is asti, nonmagneti wide band gap

nanoma-terial having substantial surfa e harge, but attains signi ant lo al magneti

(32)

ther fun tionalized by the adsorption of adatoms to other side. It is also noted

that relevant data reported in various experiments do not appear to agree with

the properties al ulated any one of the stable C

n

F stru tures. This nding lets us to on lude that domains of various C

n

F stru tures an form in the ourse of the uorination of graphene. Therefore, the experimental data may ree t

a weighted average of diverse C

n

F stru tures together with extended defe ts in grain boundaries.

In Chapter VI, analysis of the intera tion of hlorine atoms with graphene

and the presen e of possible hllorinated graphene derivatives are presented. It

is found that diering from hydrogen and uorine adatoms, while the binding

of single hlorine atom to graphene is signi ant, its migration on the surfa e

of perfe t graphene takes pla e almost without barrier. It is also found that

strong Cl-Cl oupling in one sided high hlorine overage of graphene hinders

the formation of boat, arm hair and zigzag onguration. While the bonding of

single hlorine atom leads to ioni bondingand negligiblelo al distortionin the

planargraphene, full hlorine overage( hlorographene)whereone hlorineatom

isbonded alternatinglytoea h arbonatomfrom dierentsides givesrise tothe

bu kling and ovalent

sp

3

-bonding. Phonon and ele troni energy stru ture of

hlorographene, astoi hiometri graphene derivative, largely deviate from those

of graphene. Energy optimization and phonon al ulations indi ate that the

hair ongurationoffully hlorinatedgraphene( hloragraphene)isenergeti ally

favorableandstablestru ture. Itisasemi ondu torwith1.2eVdire tbandgap,

whi h an be tuned by applied uniform strain. However, Clva an y defe ts an

lead todisso iations of bound Clatoms athigh temperature.

In Chapter VII, it is demonstrated that ontinuous oating of pristine

graphene on rea tive surfa es an provide for an ex ellent prote tion from

ox-idation of rea tive surfa es at nanos ale. The binding of oxygen atom at low

oordinated arbonatomsisratherhigh, buttheirbarrier topenetratetothe

re-a tivesurfa e under grapheneis low. Therefore dis ontinuitiesingraphene

(33)

ir umventedby oatingofbilayerorpreferablygraphenesheets omprisingafew

graphene layers, whi h provides even more ee tive prote tion. At ma ros ale,

resultsare suggestedthat grapheneadditives animprovethe strength of

antiox-idantpaints. Graphene oating,whi h is thin at the atomi s ale an alsoserve

asa naturalbarrier between environment and solid surfa es for other atoms.

InChapterVIII,fun tionalizationofgrapheneviananomeshesisinvestigated.

Crystal stru ture of graphene and its symmetry properties are investigated by

means of tight-binding model of

π

ele trons. It is predi ted that adatom and holepatternedgraphenenanomeshes anhavemetalli andsemi ondu tor

behav-ior. In parti ular the meshes providing six and threefold rotational symmetires

maintain the Dira fermions. It is also shown that depending on the mesh size

and adsorbate atoms, velo ity of the fermions and the magneti ground state of

the stru ture an be tuned. Zigzag and arm hair edged hole arrays of meshes

an alsobe indierentstates rangingfrom metalli tosemi ondu ting in luding

semimetalli state with the bands rossing linearly atthe Fermi level.

Finally, the ndings on the fun tionalized graphene stru tures and

(34)

Computational Methodology

This hapter of the thesis is devoted to the basi s of omputational many-body

theory, relevant approximate fun tionals and the stability analysis of the

stru -tures via determinationof phononspe tra.

2.1 Density Fun tional Theory

Density fun tionaltheory(DFT) isanextremely su essfulquantum me hani al

approa hmodellingmethodforthedes riptionofgroundstate propertiesof

met-als, semi ondu tors and insulators. With this theory, the properties of a

many-ele tron system an be determinedby usingfun tionals,i.e. fun tionsof another

fun tion,whi hinthis aseisthespatiallydependentele trondensity.[1,2℄Hen e

the namedensity fun tionaltheory omesfromthe use offun tionalsofthe

ele -tron density. DFT is among the most popular and versatile methods available

in ondensed-matter physi s, omputational physi s, and omputational

hem-istry. The su essof DFTnotonlyen ompassesstandardbulkmaterialsbutalso

omplex mole ules.

The main idea of DFT is to des ribe the ground state of an intera ting

(35)

wavefun tion. For su h a system satises the rule of onservation of the

num-ber of ele trons, behavior of the entire many-body system be omes dependent

only on three spatial oordinates. From omputational point of view, DFT

al-lows resear hers to deal with mu h larger and omplex systems by using less

omputationalsour es.

2.2 Ex hange-Correlation Potentials

WhileDFT in prin iple gives a good des ription of ground state properties, the

axa t form of ex hange- orrelation potential that des ribes all ele tron-ele tron

intera tions is not known. Therefore the ee ts of Pauli prin iple and the

Coulomb potentialon ele tron-ele tron intera tions have to be approximated by

the appropriate fun tionalsin terms of the ele tron density. However, there are

well-establishedapproximationstothe ex hange- orrelationpotentialexistwhi h

permitthe al ulationof ertainphysi alquantitiesquitea urately: Lo al

Den-sity Approximationand the Generalized GradientApproximation.

2.2.1 Lo al Density Approximation

Lo aldensityapproximation(LDA)isoneofthemostwidelyusedapproximation

to ex hange- orrelation potential. LDA assumes that ex hange- orrelation

en-ergyisequaltothat ofuniformlydistributed ele trongas atthesame oordinate

and the fun tional isa fun tionof oordinates. Thus the inhomogeneous system

of a mole ularor rystal stru ture is approximated by using lo aldensity of the

homogen ele tron gas.

Foraspin unpolarizedsystem, ex hange- orrelationenergy an be writtenas

E

LDA

xc

(ρ) =

Z

ρ(r)[ǫ

x

+ ǫ

c

]dr

(2.1)

(36)

izedsystem, namelylo alspindensityapproximation(LSDA), an bewrittenby

using spin dependent ele tron density

ρ(r) = ρ

↓

_{(r) + ρ}

↑

_(r)

.

Lots of the ground state properties su h as latti e onstant, stiness, bulk

moduli, band stru ture well approximated LDA. However, while the potentials

of nite systems approximated by LDA de ay exponentially, in the real systems

this de rease is mu h slower ina Coulombi manner.

2.2.2 Generalized Gradient Approximation

For the stru tures having rapidly hanging harge densities, the

ex hange- orrelation energy deviates signi antly from the uniform result and LDA

ap-proximationmay be omepoor. Beyond the LDA, Generalized gradient

approxi-mation(GGA)takesintoa ountthegradientandhigherorderspatialderivatives

of theele tron density to orre tfor this deviation. Ex hange- orrelationenergy

approximated by GGA is inthe form of

E

_xc

GGA

(ρ) =

Z

ρ(r)[ǫ

x

(ρ(r), ~

∇ρ(r)) + ǫ

c

(ρ(r), ~

∇ρ(r))]dr

(2.2)

where

ρ(r)

,

ǫ

x

(ρ(r), ~

∇ρ(r))

and

ǫ

c

(ρ(r), ~

∇ρ(r))

are the ground state harge density, ex hange energy and orrelation energy, respe tively.

2.3 Cal ulation of Phonon Spe tra

In addition to ground state propoerties it is also possible to al ulate a lot of

ex itedstatepropertiesofthestru tures withintheframeworkofDFT.

Determi-nation of vibrational dispersion spe tra of the stru tures is quite important for

the hara terization of predi ted novel materials. Espe ially the al ulation of

(37)

As a starting point of phonon al ulations performed within the framework

of DFT, the Hellmann-Feynman[3℄ theorem, that relatesthe derivatives oftotal

energy and Hamiltonian by a simple parameter, has a great importan e. Sin e

the ground state evolves as a fun tion of ioni motions, the total energy of a

stru ture an be seen as a fun tion of atomi positions. When the parameter

in the Hellman-Feynman theorem is hoosen to be the spatial oordinateof the

nu lei, it is possible to al ulate all the for es in the stru ture. Fundamental

quantities an be formalizedas follows

energy

E = E

tot

(R

i

)

(2.3)

for e ona nu lei lo ating at

R

i

F

i

=

dE

dR

i

(2.4) for e onstants

C

ij

=

dF

dR

j

(2.5)

Byusingthevibrationalpropertiesobtained fromthesimpleformulation

pre-sented above it is also possible to al ulate the diele tri onstants, ee tive

harges, ele tron-phonon intera tions, spe i heat and entropy of the systems.

Inthis thesis, thereare two main omputationalmethod usedfor the al ulation

of phonon spe tra: smalldispla ement method and density fun tional

(38)

For the al ulation of the frequen ies of phonons for any arbitrary hoi e of

q-ve tor in the Brillouin zone smalldispla ement method (SDM) provides a very

simpleandquiteusefulmodelbytreatingthe rystallineandmole ularstru tures

as system of balls onne ted by springs. Propagation of latti e vibration waves

through the stru ture an start with aninitialperturbation onthe atomsand is

mediated by the ele trons. Des riptionof the atomi vibrationsof a rystal ora

mole uleissimplybased onapotentialenergy termexpandedaroundthe atomi

equilibrium positions. As long as the atoms remain lose to their equilibrium

positions,energyofthesystem anbedes ribedbyusingharmoni approximation

asfollows

U

harm

= E

ground

+

1

2 X

Φ

_lsα,l

′

_tβ

R

_lsα

R

_l

′

_tβ

(2.6) where

Φ

lsα,l

′

_tβ

isthe for e onstantmatrix,

R

lsα

isthedispla ementof

s

atom inunit ell

l

, and

α

and

β

are artesian oordinates. Therefore thefor e onstant matrix thatrelates the for eonea hatomi site

F

lsα

toatomi displa ements at neighboringsites

R

l

′

_tβ

as

F

lsα

=

dU

harm

dR

lsα

= −

X

Φ

_lsα,l

′

_tβ

R

_l

′

_tβ

(2.7)

and the for e onstant matrix issimply given as

Φ

_lsα,l

′

_tβ

=

d

2 _U

harm

dR

lsα

dR

l

′

_tβ

(2.8)

It is seen that for es and displa ements are linearly dependent ea hother as

long asthe atomi displa ementsare smallenough.

From the viewpointof omputationalmethodology,itisseen that the atomi

(39)

determined by nding the eigenvalues of the dynami al matrix whose elements are

F

lsα

D =

_√

1 m

s

m

t

X

Φ

_lsα,l

′

_tβ

R

_l

′

_tβ

e

iq·T

(2.9)

It appears that the for e onstant matrix an also be used to determination

ofthe elements ofdynami almatrix forany q-ve torinthe Brillouinzone. Sin e

al ulations are performed for super ells by using periodi boundary onditions,

thesuper ellsmust be hoosenlargeenoughtoprovidethatthe for estake

negli-gible values atthe boundaryof the ell. Formost of the metals,due tothe large

s reening provided by ele trons this onvergen e ondition issatisedfor smaller

super ells.

Intheframeworkofsmalldispla ementmethod,phononfrequen iesofagiven

stru ture an be observed via onstru tion of the for e onstant matrix upon

smalldispla ementof the atomsinthe periodi allyrepeatingsuper ells.[4℄ From

the phonon frequen ies in rystals it also possible to determine various

thermo-dynami quantities; Helmholtz free energy, entropy, spe i heat and internal

energy of the harmoni rystal.

2.3.2 Density Fun tional Perturbation Method

For the al ulation of phonon spe tra of ioni ompounds the onvergen e an

be problemati due to the motions of harged atoms. Sin e the displa ement of

harges reates dipoles that intera ts with long range for es, for eon the

neigh-boringatomsde rease as

r

−3

. Hen e, thelong rangepolarizationelds are

arti- iallysuppressed in super ells resultingina vanishing LO-TO splitting. This is

espe ially due to the wrong LO frequen ies. Su h a slow de rease requires very

extended super ells for the treatment of ioni ompounds within small

displa e-ment method.

(40)

s opi ele tri eld generated by dipols, longitudinal and transversal opti al

phonon bran hes are splitted at gamma point and therefore the so alled

LO-TO splitting an bein luded inDFPT al ulations.

A net dipol moment in the ioni ompound an appear for the small values

wave ve tor

q

i.e. at the vi inity of

Γ

point. A ordingly the non-analyti alpart of the dynami almatrix inthe limitof small

q

an bewritten asfollows

D =

4πe

2 Ω√m

s

m

t

(q · Z

∗

s

)(q · Z

t

∗

)

q · ǫ

∞

_{· q}

(2.10) where

Z

∗

is the Born ee tive harge tensor and

ǫ

∞

is the high frequen y

stati diele tri tensor.

Cal ulationofphononspe traof rystalstru turesintheframeworkofDFPT

is also provide an elegant strategy. A DFPT al ulation starts from the ground

stateresultsobtainedfromthe primitiveunit ellofgivenmaterialand the

prob-lem of dealing with large super ells is avoided. The response to arbitrary and

innitesimallysmall displa ements of the atoms in the unit ell and the hanges

inthe ioni potentialis al ulatedfor ea h

q

by means of perturbationtheory.[5℄ As a result of Hellmann-Feynman theorem perturbations yielding linear order

deviationsinthe ele tron density also ause ase ond orderperturbationintotal

energy. Therefore the linearorder variation inthe ele tron density an be

al u-lated using only unperturbed wavefun tions of the ground state rystal. For the

ase of linearperturbationon hargedensity aused by phononmotion,for e on

ea hatom an be al ulated forany waveve tor

q

without onstru ting extended super ells.

(41)

2.4 Graphene: Computational Analysis

Last two de ades, nanos ien e and emerging nanote hnologies have been

domi-natedby honey omb stru tured arbonbased materialsindierent

dimensional-ity,su hasfullerenes,singleandmultiwalled arbonnanotubes,grapheneandits

ribbons. Inparti ular,graphene, atwodimensional(2D)honey ombstru tureof

arbon, has been ana tiveeldofresear h.[6℄Be ause ofuniquesymmetry,

ele -tron and hole bands of graphene show linear band rossing at the Fermi level[7℄

resultingin amassless Dira Fermionlikebehavior of harge arriers.

The fabri ation of graphene sheets[6 ℄ and observation of their unusual

prop-erties su h as ahalf-integer quantum-Hall ee t have attra ted mu h interest in

ele troni transportpropertiesofthistypeoftwodimensionalgraphiti materials.

Observed gapless energy spe trumand high mobility ele tron transport[8,9, 10℄

are the most remarkable features of graphene. It was shown by tight-binding

al ulations onsidering the

π

bands that in the ele troni energy dispersion of graphene, energy is linearly dependent on the wave ve tor around the Fermi

level[7℄whi h makesit aunique material.

As a result, Kleinparadox, aninteresting result of quantum ele trodynami s

wasexpe ted tobeobserved ingraphene.[6,8℄Moreover, itwasshown that

half-integer quantization of Hall ondu tan e[8, 9, 10℄ an be observed in graphene.

Unusual ele troni and magneti properties of graphene, su h as high arrier

mobility, ambipolaree t, have promised variety of appli ations. In addition to

someearlyworkson rystallineorderinplanarstru tures,possibilityofverylarge

one-atom-thi k two dimensional (2D) rystals with intrinsi ripples is reported

theoreti ally[8, 11℄ and experimentally.[8, 9℄ Not only extended 2D graphene

sheets but alsoquasi-one-dimensional graphene ribbons with arm hair or zigzag

edgeshaveshownunusualele troni ,[12,13℄magneti [14,15℄andquantum

(42)

a

2 a

1 d

Zigzag

Graphene

Nanoribbons

Armchair

Graphene

Nanoribbons

120 o

Figure 2.1: (a) Top view of honey omb stru ture of graphene. Bravais latti e

ve tors for both stru ture are given with

| ~

a

1 | = | ~

a

2 | = a

. Hexagonal unit ell in ludingtwo arbon atoms is delineated by dashed area. (b) Side view for the

sp

2

oordinated arbon atoms of graphene. ( ) Atomi onguration of zigzag

and arm hair graphene nanoribbons.

2.4.1 Atomi Stru ture

Graphene has a 2D hexagonal latti e in whi h C atoms are arranged to form a

planar (PL) honey omb stru ture as shown in Fig.2.1. A ordingly, it has a six

fold rotation axis,

C

6

at the enter of the hexagon, whi h is perpendi ular to the atomi plane. Hexagonal latti e has a two-atom basis in the primitive unit

ell, orresponding to A- and B-sublatti es. That is three alternating atoms of

ea hhexagon belong toone of the twosublatti es. In graphene planar geometry

is assured by the formation of strong

π

-bonding between two nearest neighbor

p

z

-orbitals perpendi ular to the graphene plane. The resulting

π

- and

π

∗

-bands

determine also relevant ele troni properties. In addition, there are strong yet

exible, ovalent

σ

-bonds derived from the planar hybrid

sp

2

orbitals between

adja ent C atoms. Nearest C atoms are separated by 1.42 Å and the magnitude

ofthehexagonalBravaislatti eve toris2.46Å.Briey,theplanar

sp

2

hybridiza-tion and perpendi ular

p

z

orbitals underlieplanargeometry,unusual me hani al strength and ele troni stru ture of graphene.

However, in two of our studies,[20 , 21℄ it is revealed that graphene-like two

(43)

om-stable. We nd that all the binary ompounds ontaining one of the rst row

elements,B,CorNhaveplanarstablestru tures. Ontheotherhand,inthe

hon-ey omb stru tures of Si, Geand other binary ompounds the alternating atoms

of hexagons are bu kled, sin e the stabilityis maintainedby pu kering.

2.4.2 Ele troni Stru ture

Propagation of ele tron waves through the honey omb latti e attributes

ex ep-tional features to graphene.[6 , 8℄ Condu tion of ele trons within one-atom-thi k

layer with minute s attering makes the observation of quantum ee ts possible

even at room temperature.[10℄ In graphene Dira fermions have a high Fermi

velo ity,

v

F

= c/300

. Due to its high arrier mobility, graphene based ballis-ti transistors operating at room temperature have already been fabri ated. In

addition to these unusual ele troni properties of graphene, the observation of

anomalousquantum Hallee t and the possibility ofKlein paradox arefeatures,

whi h attra t the interest of resear hers. Ele troni properties of graphene and

graphene-based stru tures have re ently been reviewed.[11 ℄ Experimental

inves-tigations have reported the observation of half-integer quantum Hall ee t for

arriersingrapheneand possiblemagneto-ele troni devi eappli ations.[9℄Most

oftheuniquepropertiesofgraphenearerelatedtoitsmonolayerlatti estru ture,

linearly rossed

π

bands atFermi level with ele tron-holesymmetry.

Here, we start with a briefdis ussion of hexagonal symmetry and apply

sim-ple tight binding modelof

π

-orbitalsto reveal the ee t of latti e symmetry on the band rossing.[7℄ Graphene has the spa e group P6/mmm and point group

symmetryD

6h

. Atthe-

Γ

point,the groupofthe wave ve tor isisomorphi tothe point group D

6h

. However, irredu ible representation of the wave ve tor point groupturnsintoD

2h

and D

3h

athighsymmetry points

M

and

K

(or

K

′

),

respe -tively. It was shown that the tight binding Hamiltonian with nearest neighbor

hoppingparameter,

t = 2.7

eV

H

₌

X

i

ǫ

_i

c

†

i

c

i

+ t

X

i,j

(c

†

_i

c

_j

_{+ H.c).}

(2.11)

(44)

E

_F

p

_z

p

_x

p

_y

s

-10

-5

0

5 En

er

gy (e

V

)

10 Γ

Μ

Κ

Γ

Κ

*

Ε

gap

Μ

Ε

gap

Γ

Figure 2.2: (a) Ele troni band stru ture of graphene obtained by rst

prin i-ples method. High symmetry points and the orbital hara ter of the bands are

delineated. (b)Three dimensional bandstru ture, obtained by tight-binding

ap-proximation, for valen e and ondu tion bands and the Dira points lo ating at

the K symmetry points. Nearest neighbor hopping parameter is taken to be 2.7

eV.

well approximates the

π

-bands of perfe t graphene.[7 ℄ Here

c

†

i

(

c

i

) is the re-ation (annihilation) operator of a

π

ele tron at the latti e site i. The rst term isthe on-siteenergyof ea h arbonatom andequals toenergyof the

2p

z

orbital. Energyeigenvalues ofgrapheneand othertwohypotheti al rystalhavingsquare

andhexagonallatti eswith singleatominthe ellare al ulated andthe ontour

plotstheirenergybandgapinBZareshowninFig.2.2. Forgraphene,energy

dis-persionislinearatthe vi inityof the

K

-symmetry(Dira ) pointsandthe Fermi velo ity, whi h is linearly dependent to nearest neighbor intera tion parameter,

an be given by the expression

v

F

= 3td/2¯h

.[11℄

Re ently,we showed that thehoney ombstru turewith linearband rossings

at Dira points is also ommon to Si and Ge.[21, 20℄ and binary ompounds

between dierent Group IV elements and Group III-V elements,[20℄ whi h are

stable in either innite periodi form or in nite size, are presented in Fig. ??.

Inthesehexagonallatti estru tures(PLorLB)relevantele troni energybands

aroundtheFermilevelarederivedfrom

π

and

π

∗

(45)

stru tures,su h asgraphene,Siand Ge, thesebands have linear rossingsattwo

in-equivalent

K

- and

K

′

-points of BZ, alled Dira points and hen e they are

semimetalli . Be ause of their lineardispersion of

E

(k), the harge arriersnear the Dira points behave as massless Dira Fermions. By tting the

π

-and

π

∗

-bands at

k

= K + q

to the expression,

E(q) ≃ v

F

¯h|q| + O(q

2 )

(2.12)

andnegle tingthese ondordertermswithrespe tto

q

2

,one anestimatethe

Fermi velo ity for both Siand Geas

v

F

∼ 10

6

m/s. We note that

v

F

al ulated for 2D LB honey omb stru tures of Si and Geare rather high and lose to that

al ulated for graphene using the tight-binding bands. It is also worth noting

thatbe auseofthe ele tron-holesymmetryatK-and

K

′

-pointsofBZ, 2DLB Si

andGeare ambipolarforE(q)=

E

F

± δE

,

δE

being small. Among these,Siand Ge in honey omb stru ture are semimetal and have linear band rossing at the

Fermi level whi h attributes massless Fermion hara ter to harge arriers as in

graphene.

2.4.3 Transport Properties

Inre entexperimentalstudiesgraphenenanoribbons(GNRs)withnarrowwidths

(10-70nm)havebeenrealized.[22℄Liet al reportprodu ingultranarrowribbons

with widths down to a few nanometers.[23℄ In addition to high arrier

mobili-ties that are higher than those in ommer ial sili on wafers, existen e of

width-dependentenergybandgapsmakesthegraphenenanoribbonsapotentiallyuseful

stru ture for various appli ations. The width dependen e of the band gap and

transport properties in quasi-one dimensional narrow GNRs have been studied

theoreti ally.[24, 25, 26, 27, 28, 12℄ Graphene nanoribbons having

r

zigzag rows (

r

-ZGNRs)arepredi tedassemi ondu torshavinganarrowingbandgapwiththe in reasing widthof the ribbon. Arm hair-edged ribbons (AGNRs)are also