Modelling of mole ules on sili on surfa esand thin-lm photovoltai absorbersThesis submitted in a ordan e with the requirements ofthe University of Liverpool for the degree of Do tor ofPhilosophyPiotr T. CzekaªaSeptember 30, 2013

Modeling of mole ules on sili on surfa es and thin-lm photovoltai ab-sorbersPIOTR T. CZEKAACopyright © 2013 PIOTR T. CZEKAA

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Prefa eThis thesis overs the work performed by me during my time as a PhD stu-dent in the University of Liverpool in the resear h group of Prof. WernerHofer. My work as a PhD student involved the theoreti al modelling. I havenot parti ipated in the any experimental studies. When dis ussing workother then my own, I learly state it and provide the sour e of presentedinformation.Piotr T. Czekaªa Liverpool September 30, 2013

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Abstra tIn this thesis a range of phenomena related to mole ular adsorption onsili on surfa es is investigated. The majority of the studies are performedin response to experimental results, where, using newly developed meth-ods within the framework of density fun tional theory, we aim to elu idatesome of the underlying physi s as well as test the performan e of the hosenmethodology. The studies over a range of subje ts su h as mole ularly me-diated pinning of surfa e geometry, single mole ular adsorptions, mole ulemigration via exited states and nally an analysis of overage dependentadsorption phenomena, where intera tions between mole ules are mediatedby the surfa e or ena ted via dipole intera tions. The main mole ules of oursimulations were water, ethylene, a etylene, and benzene, as well as halo-genated hydro arbons. We studied pro esses at two dierent surfa es, theSi(111)7×7 surfa e, and the Si(100) (4×2) surfa e.Finally we simulated and hara terized one type of grain boundary ob-served experimentally for a set of photovoltai absorbers (CuInSe2) andkesterite and stannite(Cu2ZnSnSe4 or Cu2ZnSnS4) in order to resolve theopen question of how these grain boundaries inuen e e ien ies of the pho-tovoltai devi e.

ContentsPrefa e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Introdu tion 51.1 Histori al overview . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6I Theoreti al ba kground 82 Introdu tion 92.1 Solid state theory . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Many-body wavefun tions . . . . . . . . . . . . . . . . . . . . 102.2.1 Born-Oppenheimer Approximation . . . . . . . . . . . 112.2.2 Wavefun tions and density fun tionals . . . . . . . . . 123 Density fun tional theory 143.1 Hohenberg and Kohn Theorems . . . . . . . . . . . . . . . . . 143.1.1 Existen e theorem . . . . . . . . . . . . . . . . . . . . 153.1.2 Variational Theorem . . . . . . . . . . . . . . . . . . . 163.2 N, V representability and onstrained sear h . . . . . . . . . 163.3 Self- onsistent Kohn-Sham equations . . . . . . . . . . . . . . 173.4 Solutions to Kohn-Sham equation: Algorithms . . . . . . . . . 193.5 Ex hange and Correlation . . . . . . . . . . . . . . . . . . . . 213.5.1 Lo al density approximation . . . . . . . . . . . . . . . 223.5.2 General gradient approximation . . . . . . . . . . . . . 233.5.3 Spin extension . . . . . . . . . . . . . . . . . . . . . . 234 Dispersion intera tion 254.1 Non-self onsistent method . . . . . . . . . . . . . . . . . . . 264.2 Self onsistent method DFT-D . . . . . . . . . . . . . . . . . 275 Te hni al aspe ts 285.1 Periodi ity: Blo h theorem . . . . . . . . . . . . . . . . . . . 281

5.2 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . 295.3 Basis set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.3.1 Plane wave versus lo al fun tions . . . . . . . . . . . . 295.3.2 Plane wave basis set . . . . . . . . . . . . . . . . . . . 305.4 k-point sampling . . . . . . . . . . . . . . . . . . . . . . . . . 325.5 Ele tron bands and Brillouin zones . . . . . . . . . . . . . . . 325.6 Pseudo-potentials . . . . . . . . . . . . . . . . . . . . . . . . . 335.7 Proje tor augmented waves . . . . . . . . . . . . . . . . . . . 335.8 Delta Self-Consistent Field method : ex ited states . . . . . . 345.9 Computational software . . . . . . . . . . . . . . . . . . . . . 355.9.1 Vienna Ab initio Simulation Pa kage . . . . . . . . . . 356 S anning Tunnelling Mi ros opy 376.1 Ba kground informations . . . . . . . . . . . . . . . . . . . . . 376.2 BSKAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39II Analysed systems 407 Sili on surfa es 417.1 Sili on: hemi al properties . . . . . . . . . . . . . . . . . . . 417.2 Si(111)(7×7) surfa e re onstru tion . . . . . . . . . . . . . . 427.3 Si(100) (4x2) surfa e re onstru tion . . . . . . . . . . . . . 457.3.1 Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . 477.3.2 Defe ts . . . . . . . . . . . . . . . . . . . . . . . . . . 498 Surfa e pinning and water adsorption. 518.1 Water adsorption on Si(100) . . . . . . . . . . . . . . . . . . . 518.1.1 Computational details . . . . . . . . . . . . . . . . . . 538.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 548.2 High overage water on Si(100) . . . . . . . . . . . . . . . . . 578.2.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . 578.2.2 Computational details . . . . . . . . . . . . . . . . . . 578.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 579 Single organi mole ules on sili on surfa es 609.1 Mole ules on surfa es . . . . . . . . . . . . . . . . . . . . . . . 609.2 1,2dibromoethane on Si(111) . . . . . . . . . . . . . . . . . . 619.2.1 Experimental ndings . . . . . . . . . . . . . . . . . . 629.2.2 Computational details . . . . . . . . . . . . . . . . . . 639.2.3 Simulated ongurations . . . . . . . . . . . . . . . . . 652

9.2.4 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . 679.3 Ethylene on Si(100) . . . . . . . . . . . . . . . . . . . . . . . . 729.3.1 Experimental observations . . . . . . . . . . . . . . . . 729.3.2 Theoreti al methods . . . . . . . . . . . . . . . . . . . 759.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 759.3.4 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . 779.4 A etylene on Si(100) . . . . . . . . . . . . . . . . . . . . . . . 809.4.1 Existing experimental and theoreti al data . . . . . . . 809.4.2 Theoreti al methods . . . . . . . . . . . . . . . . . . . 839.4.3 Results and Dis ussion . . . . . . . . . . . . . . . . . . 849.4.4 STM simulations . . . . . . . . . . . . . . . . . . . . . 869.4.5 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . 9010 Towards high overage: benzene on Si(100) 9210.1 Computational details . . . . . . . . . . . . . . . . . . . . . . 9610.2 Single adsorption . . . . . . . . . . . . . . . . . . . . . . . . . 9710.3 Transition from SB to TB . . . . . . . . . . . . . . . . . . . . 9810.4 Adsorption on Cdefe t . . . . . . . . . . . . . . . . . . . . . 10210.5 Line overage . . . . . . . . . . . . . . . . . . . . . . . . . . . 10310.6 Full overage . . . . . . . . . . . . . . . . . . . . . . . . . . . 10610.7 STM simulations . . . . . . . . . . . . . . . . . . . . . . . . . 11010.8 Stru tural analysis . . . . . . . . . . . . . . . . . . . . . . . . 11210.9 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11611 Thinlm photovoltai absorbers 11811.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11811.2 CIGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12111.2.1 Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 12111.2.2 Computational details . . . . . . . . . . . . . . . . . . 12111.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 12211.3 CZTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12311.3.1 Computational details . . . . . . . . . . . . . . . . . . 12311.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 12411.4 Grain boundaries . . . . . . . . . . . . . . . . . . . . . . . . . 12611.4.1 Experimental overview . . . . . . . . . . . . . . . . . . 12711.4.2 Computational setting . . . . . . . . . . . . . . . . . . 13011.4.3 Theory and results . . . . . . . . . . . . . . . . . . . . 13011.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13612 Final summary: Closing remarks 1373

13 Additional material 13913.1 Appendix I : Feynman-Hellmann theorem . . . . . . . . . . . 13913.2 Appendix II : Nudged elasti band . . . . . . . . . . . . . . . 140

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Chapter 1Introdu tion1.1 Histori al overviewOne of the great moments in the development of s ien e was ertainly theinvention of the omputer, whi h not only revolutionized the way we work,but also the way s ien e is ondu ted. In the last de ade parallelizationof omputer pro essing has been the next grand step allowing a ess to omputational power of previously in on eivable magnitude. Parallelizationin high performan e omputing over omes the limits of single pro essorsand gives a ess to methods allowing us to solve problems of quite stunning omplexity.This progress in omputational te hnologies oupled with advan es insoftware engineering opens new routes in the development of most s ienti dis iplines, reforming the s ienti method itself.The su ess of the s ienti method an be attributed to self- orre tion,through whi h s ien e aims at the best tting model. Based on a umulatedempiri al knowledge, through repeated observation of nature, the on lusionsare drawn. These on lusions eventually ontribute to formation of more a - urate s ienti theory. Up until the time omputers be ame widely availableexperiments were the only means of s ienti investigation and theory usedmostly as a tool for interpretation. Due to the nature of experiments, whi hoften are subje t to ne tuning of the apparatus and experimental pro e-dure, onsiderable time is usually required. The omputer simulation allowsone to greatly extend the theoreti al side and to a elerate the feedba k partof the pro ess. It also provides the me hanism, by whi h the a ura y andthe level of detail is greatly improved. Furthermore, if su essful repli ationof the real phenomena is a hieved, the simulation an be used as an inde-pendent resear h tool. In this way the usual order of s ienti pro edures isoften reversed: trial simulations an gauge the onsequen es and potential5

pro esses linked to a parti ular setting, and an thus be used to guide ex-periments. This is happening today in many elds of physi s and hemistry,in hemistry most notably in atalysis, organi and inorgani hemistry, andsurfa e hemistry.Of ourse the range of possible environments and systems a essible to omputer simulations is limited. Even so, there are plenty of problems thatare approa hable and to whi h the questions have not yet been provided.Based on a hosen theory a omputational algorithm is built whi h allowsthe re onstru tion of the experimental set-up or more likely just a sele tedpart of it.This thesis is based on the results obtained from omputational ab-initioquantum me hani al methods, spe i ally on the very su essful densityfun tional theory, whi h is introdu ed in a dedi ated hapter. The theoreti aland omputational aspe ts are introdu ed in the rst part of the thesis.The se ond part is on erned with spe i systems of interest and theobtained results. For the most part of the thesis the systems dis ussed an be ategorized as a part of surfa e s ien e where the range of dierentphenomena are analysed, with the ex eption of the last hapter whi h talksabout bulk stru tures.1.2 OverviewThe presentation en ompasses the following systems: in Chapter 7 thesili on surfa es will be introdu ed to provide the ba kground of the laterpresented results. In Chapter 8 a further analysis of the Si(100) surfa ewill be provided in the ontext of surfa e pinning phenomena by disso i-ated water mole ules. Chapter 9 des ribes adsorption of single mole ulesto determine the basi properties of the investigated systems. The singlemole ule study is very important for understanding the basi hara teristi sof the investigated system. It allows one to simplify the problem to justthe mole ulesurfa e intera tion without having to onsider the intramole -ular intera tions thus greatly redu ing its omplexity. This type of work isne essary before in luding any extra variables present in the higher over-age regime when mole ular intera tions may be ome signi ant. Only afterhaving a well understood foundation, is it reasonable to fo us on the inter-a tions of mole ules. In the next hapter su h high overage systems areinvestigated, rstly with the analysis of benzene adsorption Chapter 10,investigating the surfa e mediated intera tions and, se ondly, with the se -ond part of Chapter 8 investigating high overage water adsorption. In the6

nal hapter, Chapter 11, the subje t of photovoltai thin lms absorbers isintrodu ed, and the analysis of one parti ular grain boundary type is under-taken for a group of thin lm photovoltai absorbers, whi h are the subje tof intense resear h in photovoltai s today.

7

Part ITheoreti al ba kground

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Chapter 2Introdu tion2.1 Solid state theorySolid state physi s and hemistry are governed by the laws of quantum me- hani s (QM). Ideally, it would therefore be preferential to model them byQM alone. However, obtaining a full QM des ription of many-body systemsis not always possible due to the inherent omputational omplexity. This isthe reason why many of the large systems are still des ribed using lassi almodels based mainly ele trostati intera tions. Currently, new and moresophisti ated methodologies and omputational tools are being developed,whi h allow more problems to be ta kled by genuine QM methods.There are two main philosophies in approa hing quantum me hani alproblems. The rst one is alled ab-initio (latin: from the beginning) andit is a purely theoreti al approa h. The input to ab-initio al ulations islimited only to the fundamental onstants and atomi numbers of nu lei. Thea ura y of the results obtained using this method depends on the model's apa ity to orre tly represent the wave fun tion, whi h is indire tly limitedby the omputational power available.The alternatives to this purely theoreti al approa h are semi-empiri almethods, whi h use some empiri al data to either in rease the a ura y of theobtained results, or to simply limit the omputational ost. However, sin ethe results obtained using these methods are experimentally dedu ed, theyare often system sensitive. In my studies I mainly used ab-initio method.However, semi-empiri al methods have been employed to orre t for the dis-persion intera tion. In this hapter I will introdu e the origins and foun-dations of the theories used in ele troni stru ture al ulation leading toDensity Fun tional Theory (DFT), whi h is the ore of all the al ulationsin the thesis. I will also introdu e some te hni al details involved in per-forming DFT al ulations together with extensions to the general theory,9

and more spe ialised topi s su h as, s anning tunnelling mi ros opy, nudgedelasti band, and delta self onsistent eld methods.2.2 Many-body wavefun tionsIn order to fully appre iate the omplexity of a many-body system, it isnatural to rst outline a many-body formulation of the S hrödinger equation(SE). Let us onsider a simple form of the SE, whi h is a time independentrepresentation of a spin-less and non-relativisti many body system, with Matoms and N ele trons. Su h an equation reads:HΨ(r,R) = EΨ(r,R), (2.1)where the wavefun tion Ψ depends on oordinates r of N ele trons andR of M nu lei, r = (r1, r2, .., rN ),R = (R2,R2, ..,RM ). (2.2)The total kineti energy in su h a system is dened as a sum of thekineti energies of all ele trons and all nu lei,

Ekin =

M∑

k=1

P 2k

2Mn+

N∑

i=1

p2i2me

. (2.3)The potential energy is attributed to the ele trostati intera tions be-tween all the harges, whi h yields three possible intera tions: ionion,ele tronele tron, and ele tronion:Epot =

1

2

M∑

k1 6=k2=1

1

4πǫ0

Zk1Zk2e2

|Rk1 −Rk2 |+

1

2

N∑

i1 6=i2=1

1

4πǫ0

e2

|ri1 − ri2 |−

M∑

k=1

N∑

i=1

1

4πǫ0

Zke2

|Rk − ri| . (2.4)In ontrast to two other ontributions the nu lei-ele tron potential energy ontribution is negative due the attra tive nature of this intera tion. Theremay also be additional terms to the Hamiltonian due to the inuen e ofexternal ele tri or magneti elds. These, however, are not onsidered here.The Hamiltonian for the total energy an be expressed as the sum of all omponent Hamiltonians namely two kineti omponents and three potentialenergy omponents.H = Hk,n + Hk,e + Hp,n−n + Hp,n−e + Hp,e−e (2.5)10

H =−M∑

k=1

~2

Mk∇2Rk

−N∑

i=1

~2

2me∇2ri + 1

2

M∑

k1 6=k2=1

1

4πǫ0

Zk1Zk2e2

|Ri1 −Ri2 |

+1

2

N∑

i1 6=i2=1

1

4πǫ0

e2

|ri1 − ri2 | + M∑

k=1

N∑

i=1

1

4πǫ0

Zke2

|Rk − ri| (2.6)Considering the evaluation of the above equations for N ele trons andM atoms it be omes apparent that for only a few atoms one is already fa edwith very large amount of variables. Solutions for realisti systems thus willbe impossible. In order to over ome this issue a variety of approximationsneed to be used.2.2.1 Born-Oppenheimer ApproximationThe Born-Oppenheimer (BO) approximation is one of the most fundamentalapproximations of omputational hemistry. It is based on the re ognition ofthe large dieren e in mass between ele trons and ions and the times-s alesthat are involved in their dynami s. The large mass dieren e is ausingnearly instantaneous rea tion in ele trons while keeping the positions of ionsxed by omparison. Thus, in many ases in ondensed matter problems,the nu lei an be treated adiabati ally i.e. the dynami s of ele troni wave-fun tions an be seen as if it takes pla e in a stati ioni potential energylands ape.This implies the possibility of dividing the ioni and ele troni degreesof freedom, whi h allows their independent treatment,

Ψ(R, r) = ψn(R)ψe(r). (2.7)The ioni kineti energy term an be negle ted, or if ne essary treatedas a small perturbation. This signi antly simplies the system. The rststep in applying the adiabati approximation is to ex lude the ioni kineti energy ontribution from the Hamiltonian. The ele troni wave fun tiondepends parametri ally on the spe i position of the nu lei. Consequently,a SE applying to ele trons alone an be formulated:Heψe = Eeψe (2.8)and the ele troni Hamiltonian is

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He =−N∑

i=1

~2

2me∇2ri + 1

2

N∑

i1 6=i2=1

1

4πǫ0

e2

|ri1 − ri2 |+

M∑

k=1

N∑

i=1

1

4πǫ0

Zke2

|Rk − ri| (2.9)After introdu tion of the BO approximation one an nd the groundstate onguration in a variational minimization of energy,min

⟨ψe|He|ψe

⟩

〈ψe|ψe〉

= E, (2.10)while the optimal ioni oordinates an be found via the negative deriva-tive of the total energy of the systems with respe t to the ioni position RIwhi h provide the for e on the nu lei.

F = − dE

dRI(2.11)Although the BO approximation signi antly simplies many-body prob-lem, all the ele trons are still treated as separate and intera ting entitiesleaving one with a signi ant omputational hallenge. There is no on- eivable way to extend the many-body wavefun tion methodology to anyrealisti ondensed matter problem in luding at least 1001000 ele trons.The BO approximation alone is not su ient, other approximations need tobe present or the many-body wave fun tion needs to be avoided altogether.2.2.2 Wavefun tions and density fun tionalsIn general, in modern studies of ele troni stru ture two dierent routes an be re ognized. On the one hand, there are wavefun tion-based methodsfollowing the work of Hartree, Fo k and Slater. On the other hand there aredensity-based methods derived from work of Thomas, Fermi, and Dira , andlater formulated in a Density Fun tional Theory (DFT) by Kohn and Sham.The wavefun tion approa h, known under a name of HartreeFo k (HF), is alinear ombination of wavefun tions, represented by a set of basis fun tions.The ele tron wave fun tions are anti-symmetri in nature, thus hange theirsign upon ex hange. In order to satisfy the Pauli ex lusion prin iple, theantisymmetri property of the wavefun tion is en apsulated by utilization ofa Slater determinant,

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Ψ(r1, r2, . . . , rN ) =1√N !

∣∣∣∣∣∣∣∣∣∣∣

χ1(r1) χ2(r1) · · · χN (r1)

χ1(r2) χ2(r2) · · · χN (r2)... ... . . . ...χ1(rN ) χ2(rN ) · · · χN (rN )

∣∣∣∣∣∣∣∣∣∣∣

. (2.12)where χN are orthonormal spin orbitals, ea h a produ t of a spa ial or-bital and the spin up or spin down fun tion. In DFT, the ele trons are nolonger expressedf as omplex wavefun tions but as a real ele tron density.This oers substantial omputational advantage over wavefun tion basedmethods. Similarly to HF, DFT has a hieved a great re ognition and hasproven to be valid in a wide range of ondensed matter systems, al ulat-ing binding energies of mole ules and the band stru ture of solids or evenmole ular systems in biology. DFT oers a omplementary perspe tive on aquantum system. It fo uses on quantities in 3dimensional spa e, prin ipallyon the ground-state ele tron density, and provides a simpler and more intu-itive visualization of even larger systems. It so far has proven its reliabilityin many similar systems to the ones investigated here, with limited om-putational ost. For this reason it is the method of hoi e for al ulationsperformed in this thesis .

13

Chapter 3Density fun tional theoryThe initial work on DFT was reported in two publi ations: the rst by WalterKohn and Pierre Hohenberg in 1964 [1 regarding the fundamental theorems.The se ond one by Walter Kohn and Lu Sham in 1965 [2 giving rise to theKohn-Sham (KS) equations. The introdu tion of KS equations gave rise tothe new eld of DFT, whi h together with the on epts of pseudopotentials(Se tion 5.6), super ells (Se tion 5.1) and iterative minimization te hniques(Se tion 3.4) have revolutionized the theory of ondensed matter.In this se tion the basi ideas behind DFT will be introdu ed. It isimportant to remember, however, that DFT is not the rst theory to in ludeele tron density. But in ontrast to its prede essors, su h as the modelproposed by Thomas and Fermi, it is for the rst time meant to be exa t. Itproposes the density as a fundamental entity mapping it uniquely onto thewavefun tion,ψN (r1, r2, ..., rN ) ↔ ρ(r), (3.1)and it is based on the proof that the knowledge of the ground state density ofele trons is su ient to determine all observable properties of an ele troni system. This means that we do not need the full wavefun tion but only theele tron density, and that any observable of the many-body wavefun tion

ψ0,N an be formulated as a fun tional of density ρo(r)O0 = O [ρ0] (3.2)Sin e this is not a trivial on lusion it requires a mathemati al proof ofvalidity.3.1 Hohenberg and Kohn TheoremsThe rst proofs for the above postulates were presented by Hohenberg andKohn in the form of two simple theorems.14

3.1.1 Existen e theoremOne assumes ρ(r) to be a non-degenerate ground state density of N ele tronspla ed in the external potential V1(r), whi h is orresponding to the groundstate wavefun tion with its energy eigenvalue E1:E1 =

⟨ψ1|H1|ψ1

⟩

=⟨ψ1|T + U |ψ1

⟩+⟨ψ1|V |ψ1

⟩

=

ˆ

ψ∗1

(T + U

)ψ1dτ +

ˆ

ψ∗1V1ψ1dτ

=

ˆ

ψ∗1

(T + U

)ψ1dτ +

ˆ

V1n(r)τ. (3.3)Here, T is the kineti energy and U is the ele tronele tron intera tionenergy. They operate independently of a external potential V . If one as-sumes another potential produ ing another ground state wavefun tion ψ2 isrepresented by the same ele tron density ρ(r), then a similar equation forE2 an be formulated:

E2 =⟨ψ2|H2|ψ2

⟩=

ˆ

ψ∗2

(T + U

)ψ2dτ +

ˆ

V2n(r)dτ (3.4)Sin e the ground state wavefun tion is non-degenerate the following anbe dedu ed:ψ1 6= ψ2 (3.5)

E1 <⟨ψ2|H1|ψ2

⟩= E2 +

ˆ

(V1(r)− V2(r))n(r)dτ (3.6)andE2 <

⟨ψ1|H2|ψ1

⟩= E1 +

ˆ

(V2(r)− V1(r))n(r)dτ (3.7)from this it follows that:E1 + E2 < E1 + E2. (3.8)Thus, by the redu tio ad absurdum, the existen e of the se ond potential

V2, giving rise to the same ele tron density is disproved. This proves theexisten e of a unique relation between the external potential and the ele trondensity. This leads to the on lusion that the ground state harge densityprovides a omplete des ription of the system properties via the solution ofa time independent S hrödinger equation. Thus any ground state property15

is a fun tional of the ele tron density. This theorem only proves, that afun tional representation exists, however it does not dene what it is.3.1.2 Variational TheoremIn this theorem the variational energy minimization is dis ussed in terms ofthe positive ele tron density, whi h integrates to the total ele tron numberN in the system.N = N [ρ(r)] = ˆ drρ(r) (3.9)If one assumes the existen e of a ground state ρ0(r), it is the densityof ele trons that determines the total ground state energy E0. We knowthat ρ(r) an be al ulated from the square of the ground state N-ele tronwavefun tion.E0 = E0[ρ(r)] = ⟨

ψ0|H|ψ0

⟩ (3.10)knowing that the energy expe tation value of the ground state wavefun -tion and the energy fun tional of the density obtained from this wavefun tiondenes uniquely (for a non-degenerate ase) the ground state energy of thesystem. Let us propose another wavefun tion ψ1(r) and another orrespond-ing density ρ1(r), whi h by the variation prin iple will ne essary give a higherenergy:E0 = E[ρ0(r)] = ⟨

ψ0(r)|H |ψ0(r)⟩ < ⟨ψ1(r)|H |ψ1(r)⟩ = E[ρ1(r)] = E1(3.11)From this it follows that the total energy of the system an be found by aminimization with respe t to ρ(r), and that the minimum energy orrespondsto the system with the orre t ele tron density ρ0(r)3.2 N, V representability and onstrained sear hDespite the fundamental nature of the above proofs of DFT theory, theHK theorems raise many questions regarding the one-to-one orresponden ebetween the ele tron density and the external potentials. There have beena few generalizations showing that in fa t the HK theorem is only a limited ase. Fortunately, in most al ulations these limits are not an issue. Theone failure of HK formulation whi h is often dis ussed regards the degeneratewavefun tion for whi h one an no longer talk about the uniqueness of theground state expe tation value. 16

This problem has been ta kled by Levy and Lieb (LL) [3, 4. TheHK formulation is based on the assumption that the density ρ(r) is V representable, whi h means that the density is dened by some external po-tential.The LL theorem provided an alternative formulation in whi h insteadof a V representable density the minimization algorithm only required theNrepresentability. By su h a denition they avoided the problem of how anarbitrary density, integrable to number of N ele trons, ould be representedby the ground state of a smooth external potential.Employing this idea it is possible to onstrain the sear h of the groundstate wavefun tion from the full spa e to the sub-spa e limited to all thewavefun tions whi h onstru t the same ele tron density ρ(r). In that waythe minimizations an be performed in two steps instead of one, greatlyde reasing the ost of al ulation.

E0 = minρ

minψ→ρ

[⟨ψ|T + U |ψ

⟩+

ˆ

V (r)ρ(r)dr] (3.12)E0 = minρE [ρ] (3.13)The LL formulation is dened for any density that is derivable from thewavefun tion ψN of N ele trons, therefore it is Nrepresentative. WhileLL eliminates the V representability ondition from the domain of varia-tion, nonetheless the density of the minimum energy solution is still V representable, agreeing with HK even for the degenerate ase.If ψ an be al ulated from ρ and vi e-versa then both fun tions on-tain exa tly the same information. The ground state wavefun tion not onlyreprodu es the ground state density but also minimizes the energy of thesystem.3.3 Self- onsistent Kohn-Sham equationsUp to this point, we know that the density ρ(r) an be used to fully representa many-body system, thus redu ing the many-body problem to a singlebodyproblem. HK and LL have proved that a system an be exa tly des ribedonly by means of ρ(r), but it was still left un lear how this an be done.DFT an be implemented in many ways, from whi h the most widely used isthe Kohn-Sham approa h. Interestingly, this approa h does not ex lusivelywork in terms of the harge density but employs also a spe ial type of wavefun tions alled KohnSham orbitals. As a onsequen e, DFT looks like a17

single parti le theory although many parti le ee ts are still in luded via theso- alled ex hange- orrelation potential.In order to a ommodate the ele tron intera tion the Hamiltonian isH = Ts + Veff (3.14)where Ts is the kineti energy of non-intera ting ele trons, while Veff is anee tive potential of following form,

Veff = Vext + VH + VXC . (3.15)where the Hartree potential is (in atomi units)VH(r) =

ˆ

ρ(r′)|r − r′|d

3r′ (3.16)and the ex hange- orrelation potential isVxc(r) = δExc[ρ(r)]

δρ(r) (3.17)Vext is an external ioni potential. The ex hange- orrelation potential

Vxc is a fun tional of the ele tron density, with the ele tron density denedas follows.ρ(r) =

∑

i

| φKS(r) |2 (3.18)The Kohn-Sham equations are used iteratively. They provide the frame-work for nding the exa t density and energy of the ground state of thesystem.(−~

2∇2/2m+ Veff )φKS(r) = EKSφKS(r) (3.19)The Kohn-Sham equations map many-ele tron systems onto a systemof non-intera ting ele trons moving in the ee tive potential due to all theother ele trons. As shown by Hohenberg and Kohn, DFT is an exa t refor-mulation of the intera tion for a many-ele tron ground state problem but thedi ulty lies in the formulation of the unknown ex hange- orrelation energyfun tional.The Kohn-Sham eigenvalues are not stri tly speaking the energies of thesingle ele tron states but rather the derivatives for the total energy withrespe t to the o upation numbers of these states. DFT al ulations are fo- used on iterative solution of the eigenvalue problem on e an approximate ex-pression for ex hange orrelation energy is given. The minimum energy of thetotal fun tional is the ground state energy and the obtained ele tron density18

is the exa t ground state single parti le density. The ex hange- orrelationpotential is given as the fun tional derivative of ex hange orrelation energyfun tional with respe t to density.The meaning of KS orbitals is often onfused with a standard wave-fun tion representation. It is important to highlight the fa t that the KSeigenvalues are introdu ed as ompletely arti ial obje ts. They are resultsof auxiliary single body equations, even though the eigenfun tions (KS or-bitals) yield the orre t density. The density has a stri tly physi al meaningwhile KS eigenvalues provide only semi-quantitative information of the en-ergy spe trum. Given this nature of KS eigenvalues it is surprising that theyprovide a reasonable approximation to the energy levels.DFT is a very popular method for band stru ture al ulations in solidstate physi s. It has also been shown to be in good agreement with experi-mental studies. It has been also used to perform band stru ture studies inthis thesis. Nonetheless, it is important to remember that the extent of itsvalidity is not yet learly known. DFT also is not a rigorous many-bodytheory but a mean eld theory. The negle t of the derivative dis ontinuityby LDA and GGA fun tionals leads to an underestimation of the band gapwhi h is most severe in metal oxides or strongly orrelated systems.3.4 Solutions to Kohn-Sham equation: AlgorithmsIn the pro ess of solving the KS equations a few essential omponents haveto be taken into a ount for e ient and a urate al ulations.• Introdu tion of ee tive valen e- ore potentials via pseudo-potentials,approximating the ore ele trons• Choi e of periodi boundary onditions or isolated lusters• Choi e of basis set for one-ele tron wave-fun tions: de-lo alized planewaves, or lo alized atomi orbitals• E ient diagonalization of Hamiltonian matrix obtained from the rep-resentation of the one ele tron S hrödinger equation in the basis setIn any numeri al implementation a many-ele tron problem has to besolved iteratively. The reason is the inherent non-linearity of the funda-mental KohnSham equations. While in a single step of the iteration theKohnSham eigenvalues and eigenve tors an be al ulated on the basis ofan existing density distribution, the summation of the harge density over alleigenstates reates a new potential. The nal result an thus not be predi ted19

from one iteration only. The rough stru ture of an ele troni self- onsistentiteration y le is the following [5.1. From an existing distribution of ions and ele tron density ρ0 onstru tthe various omponents of the potential: the external potential Vext(summation over ions, e. g. by Ewald summation), the Hartree po-tential VH (solving the Poisson equation for the harge distribution)and the ex hange orrelation potential Vxc (determined point by pointfrom the density distribution). The sum of all potentials gives theee tive potential in the single-ele tron KohnSham equation:Veff (r) = Vext(r) + VH(r) + Vxc(r) (3.20)2. Solve the orresponding KohnSham equations either by matrix inver-sion if the Hamiltonian is represented in a dis rete basis set, or bysome other method e.g. using trial solutions and a predi tor- orre tors heme:

(− ~

2

2m∇2 + Veff

)ΦKS(r) = EKSΦKS(r) (3.21)3. Cal ulate the total energy from the various ontributions. The to-tal energy in a system is des ribed by a sum over the eigenvalues ofo upied states and some orre tions:

Etot,step =N∑

i

EKS,i − EH [ρ] + Exc [ρ]−ˆ

Vext(r)ρ(r)dr (3.22)4. See if the energy is equal to the previous one. If yes stop the iter-ation. Che k the energy dieren e between the obtained energy andthe energy al ulated of the previous step. If it is smaller then theset threshold value stop the al ulation. The ∆E are usually set smallaround 1meV for good quality al ulations5. If the dieren e is still larger, then sum up all density ontributionsfrom the o upied states and reate a harge density:ρnew (r) = N∑

i

|ΦKS,i(r)|2 (3.23)6. Admix a part of the new harge density to the old harge density ρ0to reate a harge density ρ1: This pro edure is alled mixing. Itis ne essary to retain numeri al stability of the simulation. Several20

s hemes exist to admix a new harge density to the original one. Insome of those s hemes an array of harge densities is kept and mixedin su h a way that the dieren e between the densities gradually goesto zero.ρ1 (r) =Mix [ρ0 (r) , ρnew (r)] (3.24)7. With this density go ba k to step 1.The self onsistent ele troni iteration pro edure is usually embeddedin a y le aimed at a hieving the ground state positions of all ions. Inthis ase the HellmanFeynman for es (see Appendix I for details)onthe ions are al ulated after the ele troni ground state is a hieved, andthe ions moved a ording to the for es a ting on them. Dependingon the system, the ele troni self onsisten y y le in today's odestakes about 1050 iterations. Ioni relaxation depends strongly on theinitial guess for the ioni positions and an take from about 10 to afew hundred y les.3.5 Ex hange and CorrelationDFT oers a way to avoid the enormous omplexity arising from ele tron-ele tron intera tions in many-body systems and substitutes it with the ef-fe tive one ele tron potential, a fun tional of the ele tron density only. Al-though the exa t form of the fun tional is not known, its various approxi-mations have allowed predi tion of many properties of mole ules and solids.In the sear h for the perfe t fun tional a wide variety of approximations hasbeen reated. The fun tionals proposed so far are usually more valid for spe- i systems and tasks, dividing them into separate groups. Some of thesefun tionals are not stri tly ab-initio and in lude empiri al parametrizationsenabling them to obtain results of higher a ura y, or they de rease the ostof the al ulations. While these methods are fairly su essful for very similarsystems they an produ e highly ina urate results for inadequate ones. Agood example of this problem would be a fun tional su h as B3LYP [6, 7,whi h is the most popular fun tional in the quantum hemistry ommu-nity for al ulations of small mole ules, while produ ing inadequate resultsfor systems ontaining a homogeneous ele tron gas. The other approa h todesign the ex hange- orrelation fun tionals is to reprodu e as many exa t onstraints of the many-body system as possible. This approa h aims to21

produ e fun tionals with a mu h more exible range of appli ations, their reation is however not without di ulties [8.Although DFT is an exa t des ription in prin iple, all problems are hid-den in dening the ex hange- orrelation energy fun tional EXC(n). Thegreat su ess of DFT depends on the unexpe ted a ura y of simple ap-proximations. The rst approximation was based on the lo al density ap-proximation (LDA) using the ex hange orrelation energy of a homogeneousele tron gas with the density equal to the lo al density. The LDA has beenvery su essful in the solid state, but does not work well for mole ules wherethere are strong variations in their ele tron density. This weakness was par-tially removed by the onstru tion of gradient orre ted fun tionals su h asGGA and semi-empiri al approximations in luding expli it ex hange su has B3LYP [6, 7, whi h was very su essful in modelling mole ular systems,or HSE03 [9 and HSE06 [10 for solid state systems. The developmentof better and yet omputationally heap fun tionals is still very mu h inprogress.3.5.1 Lo al density approximationThe lo al-density approximation is the simplest method for the des riptionof ex hange orrelation energy [8. In this approximation the ex hange- orrelation energy is obtained by assuming that the energy per ele tron atr, Exc(r), is equal to the energy per ele tron of in homogeneous gas of thedensity equal to the density of ele trons at this the same point, ρrExc[ρ(r)] = ˆ ǫxcρ(r))ρ(r)d3r (3.25)with

εxc(r) = εhomxc [ρ(r)] (3.26)Considering that a homogeneous ele tron gas is far from a real densitydistribution, the LDA approximation has nevertheless been highly su essfulin dealing with a broad range of systems. The reason why the uniformele tron gas has su h a prominent pla e in DFT is that it is the only systemfor whi h we know the form of the ex hange and orrelation energy fun tionalexa tly or at least to very high a ura y.Despite the su ess of LDA, the homogeneous density was learly notsu ient for many systems exhibiting strong ele tron density variation; anew method was needed, if one aimed to approa h more hemi al systems.22

3.5.2 General gradient approximationThis approximation is an extension to the LDA, in whi h the ex hange or-relation energy was based on lo al density only. In this ase the additionallo al information in luded the gradient of density ∇ρ(r). In this ase thefun tional is able to a ount for the nonhomogeneous ele tron density ofthe real system. The name general gradient approximation is reserved forthe group of fun tionals whi h an be dened in the following way:EGGAxc [ρα, ρβ ] =

ˆ

f (ρα, ρβ ,∇ρα,∇ρβ) dr. (3.27)Furthermore, the ex hange orrelation fun tional an be divided intoseparate ex hange and orrelation parts. It is important to realise that inboth ases one is dealing with omplex mathemati al onstru ts whi h havebeen hosen to suit the boundary onditions and to obtain system spe i performan e. One should be aware that in this ase the results of al ulationsare more important than the underling physi s, and it is these results whi hoften determine the mathemati al model. There exist a group of dierentfun tionals whi h operate within a GGA frame-work for whi h ea h ex hangeor orrelation energy an be separately a ounted for. For the ex hangeintera tions, empiri ally based fun tionals an be used su h as PW91 byPerdew et al. [11, and CAM(A), CAM(B) by Handy and o-workers [12. Inaddition there are purely theoreti al fun tionals whi h instead of empiri allyaided parametrizations use rational fun tions of redu ed density gradients.The most prominent of this kind are PBE by Perdew, Burke, and Ernzerhof[13, or its revised form revPBE proposed by Zhang and Wang [14, orre ently its version RPBE [15. From this variety the PBE and RPBE havebeen fun tionals of hoi e in this work.3.5.3 Spin extensionSo far the theorems introdu ed here were formulated for spinless densities;however, there also exists an extended approa h to the DFT in whi h theele tron density is a sum of the two separate densities representing spinupand spindown states. This is essential in a theory of atoms and mole uleswith a net spin dierent from zero and in solids with magneti ordering.n(r) = n(r, ↑) + n(r, ↓) (3.28)Even though from a purely theoreti al point of view the exa t fun tionalwill not depend on the spin densities (as long as the external potential is23

spin independent), the approximations to it will benet from the additionalexibility of having two instead of one variable. In parti ular for an open shellsystem with an unequal number of α (spin up) and β (spin down) ele trons,a fun tional of the two spin densities onsistently leads to more a urateresults. But an improvement is also seen for ertain systems in whi h thenumber of ele trons is even, su h as H2 mole ules at large separation, forwhi h the unrestri ted fun tionals perform signi antly better be ause theyallow symmetry breaking.Just as for the simple spin ompensated situation where the ρα(r) =

ρβ(r) = 12ρ(r) there are related expressions for the ex hange and orrelationenergies per parti le of the uniform ele tron gas hara terised by ρα(r) 6=

ρβ(r), the so alled spin polarized ase.

24

Chapter 4Dispersion intera tionAs dis ussed above the hoi e of the ex hange- orrelation fun tional is veryimportant for the simulation out ome [16. While for large homogeneoussystems, su h as simple metals and semi ondu tors a lo al density approx-imation is appropriate, for inhomogeneous systems like transition metals,ioni rystals, ompound metals, surfa es, interfa es and some hemi al sys-tems semi-lo al approximations su h as a generalized gradient approxima-tion (GGA) work mu h better. These two standard methods are heap andeasily appli able for periodi systems. However, sin e they are not able todes ribe dispersion intera tion properly, they annot be used for modellingsoft matter systems, van der Waals omplexes and biomole ules. In this ase another extension of the method, in luding also non-lo al intera tionsis ne essary. These non-lo al intera tions an be also a de isive ontributionin the determination of the stru ture of highest stability, in parti ular inmetastable hemisorbed systems, su h as the one investigated here. It hasbeen shown that although many abinitio methods are quite su essful inpredi ting bonding energies, they perform rather poorly when estimating thedispersion intera tions.The dispersion intera tion plays a very important role in many self as-sembly pro esses, ranging from simple organi mole ule adsorption to proteinfolding, and its understanding seems to be an essential step on the way toevolve some intuition on how and when the orre tion should be in orpo-rated in the al ulations as not to miss any important phenomena. In gen-eral, dispersion intera tions are des ribed as intera tions between two dipolemoments within the ele tron distribution of atoms or mole ules. The sim-plest model of dispersions is the intera tion of two Drude os illators wherethe instantaneous intera tion between them reates an attra tive intera tionbetween them. The dispersion intera tion an take the form of the series25

expansion Edisp = −C6

R6 − C8

R8 − C10

R10 − ... where higher order terms an benegle ted. The Leonard-Jones potential is often used among hemists todes ribe the intera tion between atoms or harge-neutral mole ules. It is ommonly expressed in the following fashion:VLJ = 4ǫ

[(σr

)12−

(σr

)6] (4.1)where ǫ is the depth of potential well σ is the distan e lose to the parti leat whi h the potential is zero. The R6 term approximates the potential fromthe dispersion for es while R12 orresponds to the repulsive Pauli ex lusionfor e arising from the ele tron loud overlap.It has been shown in re ent studies, utilizing extensions to density-fun tional approa hes for vdW intera tions, that this method is suitableto des ribe adsorption energies, bulk moduli and elasti onstants of layeredmaterials like graphite, boron nitride and MoS2 [17. It has also been on- luded that in most ases the use of post-GGA methods is su ient to obtaina urate results. However, self- onsistent methods an be used in systemsstrongly inuen ed by dispersion intera tions where the VdW would ae tHellmanFeynman for es during ioni relaxation al ulations [18.4.1 Non-self onsistent methodThe non self onsistent method employed in this work is based on the ap-proa h developed by Langreth and Lundqvist where the energy values ob-tained from standard DFT al ulations are updated by the vdW orre ted orrelation energies al ulated separately for lo al and non-lo al intera tions:

ETOT = ESCF − EPBEc + ELDAc + Enlc (4.2)Here, ESCF is the total energy obtained from the DFT/GGA simula-tion, and Enlc is the non-lo al orrelation energy al ulated within the vdW ode. The lo al part obtained using the lo al density approximation ELDAcsubstitutes earlier al ulated EPBEc . Enlc is expressed as the two- entredintegral:Enlc =

1

2

ˆ

ρ(r)φ(r, r′)ρ(r′)d3rd3r′ (4.3)This feature was implemented in VASP by Klimes [1926

4.2 Self onsistent method DFT-DThe self- onsistent method used in a work present in the following hapterswas based on semi-empiri al Grimme orre tions [20, where dispersion or-re tions take the form of C6R−6. Within this method density fun tionaltheory is restri ted to al ulate the short range orrelation energies, whilemedium to long range orrelation energies are des ribed by damped empiri alpotentials given byEdisp = −s6

Nat−1∑

i=1

Nat∑

j=1+1

Cij6 /R6ij ∗ fdmp(Rij) (4.4)Here, Nat is the number of atoms in the system, C6 is the dispersion oe ient for pair of atoms, s6 is a s aling fa tor depending on the fun tionalused, and Rij is the distan e between intera ting atoms. The f represents adamping fun tion of the following form:

fdmp(Rij) = 1/(1 + e−d(Rij/Rij−1)) (4.5)The total energy is then given by:ETOT = EDFT + Edisp (4.6)

27

Chapter 5Te hni al aspe ts5.1 Periodi ity: Blo h theoremPeriodi ity plays a very important role in omputational analysis. The iden-ti ation of the symmetry allows one to minimize the omputational ost byavoiding multiple al ulations of the same problem.Many of the systems in ondensed matter physi s as well as hemistryare periodi (e.g. rystals, wires, tubes, polymers). The symmetry of thesesystems an be used, and their size redu ed to the smallest repeatable unit.Usually this is done by redu tion to the unit- ell or surfa e super- ell size butsometimes its multiples are ne essary to more a urately model the phenom-ena of interest. This type of ase may in lude: isolated mole ules, mole uleson the surfa e, defe ts, doping where one often does not want to model su hunrealisti systems with 20 % doping or defe t on entration. In these asesa larger unit ell may be required to redu e the intera tions between defe ts,mole ules, et ., from the neighbouring periodi images. Although this te h-nique is far from ideal it is ertainly a good approximation for a large enough ell. In the al ulations of isolated mole ules or surfa es high quality resultsare rea hed by introdu ing a large va uum in between the periodi images,thus de reasing the mole ulemole ule or surfa esurfa e image intera tionto a negligible value.In the single-ele tron des ription in a periodi potential V (r) = V (r+R),the Hamiltonian is invariant under the latti e translations. Whi h meansthat it ommutes with the translational operator.

[H, RT

], RT = n1a1 + n2a2 + n1a2 (5.1)

n1, n2, n3 = 1, 2, 3, ... (5.2)28

A ording to Blo h's theorem [5 the wavefun tion an then be writtenas a produ t of a plane wave and the latti e periodi fun tion,ψk = exp(ikr)uk(r) (5.3)where uk(r) has the same periodi ity as the potential, whi h in the re- ipro al spa e leads to:

ψk =∑G Ck+G exp(ir(k+G)) (5.4)where C is a Fourier oe ient and G is re ipro al spa e latti e ve tor.The important on lusion from Blo h's representation is that although thewave fun tion phase is not transitionally invariant the harge density ρ(r) =

ψ(r)ψ(r)∗ is. Therefore, the use of Blo h theorem allows for the periodi representation of systems with (r+R) translational symmetry su h as theele tron density in a periodi latti e potential.5.2 Density of statesThe density of states is the number of states per unit energy per unit volume.Under the periodi boundary ondition the volume of ea h waveve tor pointin three dimentional spa e isδk = (

2π

L)3 (5.5)In the re ipro al spa e the number of states in the shell from k to k+ dkis

dN = 2(L

2π)3intk+dkk dk (5.6)Then for the dispersion relationE(k) = ~

2k2/2m, one an obtain thefollowing density of statesg(E) =

1

L3

dN

dE=

1

2π2(2m

~2)3/2E1/2. (5.7)5.3 Basis set5.3.1 Plane wave versus lo al fun tionsThe basis an be formed from a set of plane waves or some lo alized fun tions.Both of the methods have advantages and disadvantages, depending on thesystem and the resear h interest. Within the plane wave methods it is easyto hange between real and momentum spa e, through utilization of the Fast29

Fourier Transforms (FFT). The kineti and potential energy are then rep-resented diagonally. Plane waves oer very simple onvergen e ontrol andtheir adjustment through a ut-o energy. The Hellmann-Feynman for es [5 an also be easily al ulated with respe t to ioni oordinates and superpo-sition errors from the lo al basis set are avoided. For onvergen e the nodal hara ter of the valen e orbitals needs to be eliminated and ion-ele tron in-tera tions des ribed by pseudo-potentials. While some regard plane wavemethods as an unne essary approximation, some are s epti al about lo al-ized basis sets due to in ompleteness and superposition errors. Lo al basissets have ertain advantages ompared to plane wave basis set, and theyallow analyti al integration of 1/r singularities of Coulomb potentials thusallowing fast al ulation of exa t ex hange.5.3.2 Plane wave basis setThrough the introdu tion of the Blo h theorem the single ele tron wavefun -tion an be expanded in terms of a dis rete basis set. In order to des ribethe system perfe tly, the basis set should be innite. In pra ti e this is ap-proximated. Plane waves are espe ially appropriate for systems with longrange periodi ity. In su h onditions they provide intuitive understandingas well as algorithms for pra ti al al ulations.Be ause many of the ondensed matter systems oer long range peri-odi ity it is natural to express them using periodi fun tions su h as 3-dimensional plane waves. Using the fa t that any periodi fun tion an beexpanded into a omplete set of Fourier omponents an eigenfun tion an bewritten as follows:ψi(r) = ∑

q

ci,q × 1√Ωexp (iq · r) ≡ ∑

q

ci,q × |q〉, (5.8)where ci,q are the expansion oe ients of the wavefun tion in the basisof orthonormal plane waves |q〉, whi h satisfy the orthonormality ondition:δq,q′ =

1

Ω

ˆ

Ωexp

(−iq′ · r) exp (iq · r) ≡ 〈q'|q〉. (5.9)By inserting the eigenfun tion into the KS equation (Equation (3.19))and multiplying by 〈q′| we obtain the KS equation in Fourier spa e,

∑q 〈q′|H|q〉ci,q = εi∑

q

〈q′|q〉ci,q = εici,q′ . (5.10)from that, the plane wave representation of kineti energy is expressedas 30

〈q′| − ~2

2me∇2|q〉 = ~

2

2me|q|2δq,q′ , (5.11)Due to the periodi ity of the rystal and its ee tive potential Veff theonly allowed Fourier omponents are those with the waveve tors in the re- ipro al spa e of the rystal. Veff is expressed as follows.

Veff (r) = ∑

m

Veff (Gm)exp(iGm · r), (5.12)where Gm is a re ipro al latti e ve tor andVeff (G) =

1

Ωcell

ˆ

Ωcell

Veff (r)exp(−iG · r)dr (5.13)with Ωcell representing the volume of a re ipro al unit ell. The matrixelement of the ee tive potential is〈q′|Veff |q〉 = ∑

m

Veff (Gm)δq′−q,Gm(5.14)from whi h it be omes apparent that it an only be zero if the dieren ebetween q and q′ is a re ipro al latti e ve tor Gm. Redening a q to:q = k+Gm,q′ = k+Gm′ , (5.15)the matrix equation an be written,

∑

m′

Hm,m′(k)ci,m′(r) = εi(k)ci,m(k). (5.16)In this way the solutions have been transformed to a set of independentequations for ea h k-point within the Brillouin zone, whereHm,m′(k) = 〈k+Gm|H |k+Gm′〉

=~2

2me|k+Gm|2δm,m′ + Veff (Gm −Gm′), (5.17)is a matrix element of the Hamiltonian. The equations (5.16) and (5.17)are a system of linear equations, whi h an be solved via diagonalization andyield a set of eigenvalues εi(k), one for ea h dened k point. On the basisof these equations the band stru ture an be al ulated where the index irepresents the band number of ea h al ulated k-point value.When one uses plane wave sets it be omes apparent than although thea ura y of the solutions depends on the number of PW basis fun tion, the omponents for small kineti energies are more important that large ones,31

with their weight fa tor approa hing zero for in reasing energies. For thatreason the plane wave basis set an be trun ated. It is thus a viable optionto introdu e an energy ut-o value. Then the number of plane waves in thebasis set is limited,|k+G|2

2< Ecutoff . (5.18)This pro ess allows not only for a dis rete but also a nite representationof the wavefun tion. The use of Ecutoff within the DFT ode allows easy ontrol over regulation of the a ura y optimizing of the onvergen e.5.4 k-point samplingThe onvergen e and a ura y of the al ulation is also dependent on the hoi e of k-points. If the sampling is not su ient to a hieve onvergen e,the grid an be set denser, until the error due to the k point sampling be- omes insigni ant and the al ulation result onverges to a ertain value. Ingeneral the number of k-points required for onvergen e is inversely propor-tional to the size of the unit ell. In DFT al ulations the k-points an be setmanually or a ording to one of the s hemes for the k-point grid generation,with the most ommon one introdu ed by Monkhorst and Pa k [21.5.5 Ele tron bands and Brillouin zonesFor perfe t periodi ity within the latti e one is able to obtain the re ipro allatti e via Fourier transform. The states in the system are now hara terizedby the waveve tor with a spe i eigenenergy or eigenfrequen y. Due to theperiodi boundary onditions ertain energies are forbidden. The obtaineddispersion relation an be divided into bands hara terizing the propertiesof ele trons. The dispersion relation is des ribed within the Brillouin zone hara terizing the periodi nature of a given re ipro al spa e. The solutionof the matrix eigenvalue problem gives rise to a dis rete set of eigenvalues

εi(k) whi h vary with k forming a unique ele troni band stru ture.The presen e of various symmetries for the one-ele tron Hamiltonian re-du es the number of inequivalent k points in the rst Brillouin zone. Forthat reason it is only ne essary to onsider the k-points in the irredu ibleBrillouin zone (IBZ), while the states and energies for other k-points anbe obtained through symmetry operations. Typi ally one visualizes bandsalong the symmetry lines in IBZ. Pra ti ally this of ourse an't be done in32

a ontinuous manner and to obtain a urate band hara terization one hasto a urately enough sample the k-spa e along the dire tions of interest.5.6 Pseudo-potentialsThe idea of pseudo-potentials (PP) is to simply allow heaper omputationalanalysis of many-atom systems without sa ri ing any of the details of hem-i al or physi al systems. It is known that atoms in a hemi al bond engageprimarily due to the intera tion of their valen e ele trons, and that the in-tera tion of the ore ele trons is very small. This allows a greater exibilitywhen des ribing the ore ele trons. The main on ept of PP is to add thetightly bound ore ele trons to the strong potential for the nu lei and in ee treprodu e the ee tive ioni potential a ting on the valen e ele trons only.The pseudo-potentials an be generated beforehand and then applied in theDFT al ulation, thus redu ing the ost. PP have been used both in planewave and lo alized basis methods for quite some time, and although theiruse is very ommon, the pro ess of the produ tion of pseudo-potentials isnot a straightforward one. The pseudo-potentials quality is mat hed againstall-ele tron al ulations. The major di ulty in the use of pseudo-potentialsis due to the non-linearity of ex hange intera tions between valen e and oreele trons. In systems with high valen e- ore ele tron overlaps extensive or-re tions are ne essary. One of the methods of dealing with this problem isto utilize proje tor augmented waves.5.7 Proje tor augmented wavesThe proje tor-augmented wave method, st published by Blo hl [22, is anattempt of reating a method ombining the e ien y of pseudo-potential(PP) methods and the a ura y of full-potential linearised augmented-plane-wave (FLAPW) methods. Unlike PP, PAW a ounts for the nodal featuresof the valen e orbitals and ensures orthogonality between the valen e and ore wavefun tions. In PAW all valen e ele tron wavefun tions ψAEi are onstru ted from pseudo wave fun tions through the linear transformation.|ψAEn 〉 = |ψPSn 〉+

∑

i

(|ϕAEi 〉 − |ϕPSi 〉

)〈pPSi |ψPSi 〉 (5.19)where pseudo wavefun tion ψPSi (of band index n) is a variational quan-tity expanded in the plane waves. ψPSi is identi al to the all ele tron wave-fun tion ψAEn outside of the sphere of a PAW-potential (whi h has a radius33

of about half of the atomi -separation), while inside of the sphere, it is onlya bad approximation to the wavefun tion. The all ele tron partial wavesϕPSn are solutions to the spheri al s alar-relativisti S hrödinger equationfor non-spin polarized atoms while the pseudo (PS) partial waves ϕPSn arenode-less and identi al to the AE partial waves outside of the sphere of thePAW-potential and mat h ontinuously inside of it. The pi is a proje torfun tion whi h is onstru ted su h that 〈pPSi |ϕPSj 〉 = δij and 〈r|pPSi 〉 = 0 forr larger then the radius of the PAW-potential sphere.Within the PAW method the harge density orresponding to the allele tron eigenstate ψAEn , ρ(r) = 〈ψAEn |r〉〈r|ψAEn 〉 is omposed of the three ontributions to ψPSi , ϕPSn and ϕAEn ;

ρ(r) = ρψPS (r) + ρϕAE (r)− ρϕPS(r). (5.20)As dis ussed by Kresse [23 the PAW method gives results that arealmost indistinguishable to the US-PP for materials in whi h the hargedensity losely resembles that of the referen e system within the ore region.However, for the materials with strong ele tro-negativity dieren es or sys-tems with large magneti moments PAW shows superiority over the ultrasoftpseudo-potential (US-PP). For the al ulation in this thesis PAW potentials(as implemented in VASP) have been used due to their a ura y omparableto FLAPW, but under signi antly redu ed ost [23.5.8 Delta Self-Consistent Field method : ex itedstatesAnother area in whi h DFT struggles is for systems not in their ground state.The problem of in luding the ex ited states has been approa hed in dierentways, however, it often suers from high osts or omputational limitations.The most promising methods, be ause of their low ost, ompatible to stan-dard ground state al ulations for tasks su h as the ex ited state mole u-lar energies al ulations are Constrained DFT and the Self- onsistent eldmethod ∆SCF. For the rst one an additional potential is used to a hieve thedesired amount of ele trons in the area. In the ∆SCF s heme, however, theposition of the ele trons is ontrolled by the o upation of the Kohn-Sham(KS) states [24.In this thesis in se tion 9.3, the method used is an extended ∆ Self-Consistent Field method. The DSCF method losely resembles standardDFT; its advantage is that ele trons are not restri ted to o upy the lowest34

possible orbitals. It allows one to pla e one or many ele trons in higher lyingKohnSham orbitals. With this additional freedom it is possible to lo alizethe ex ited state of adsorbed mole ules and perform al ulations from whi hresonan e energies an be estimated. Furthermore, from the al ulated for esit an be dedu ed if the resonan e is responsible for mole ular dynami s.In a tual al ulations the density of a spe ied orbital φs(r) is added tothe total density in ea h step of the self- onsisten y y le [24.ρ(r) =

∑

n

fN−1(T, εn) | φn(r) |2 + | φs(r) |2 (5.21)N is the total number of ele trons and fN−1(T,φ) is the Fermi-Dira distribution of the N-1 ele tron system. φs is our hosen orbital o upied byan ele tron, whi h is taken from the Fermi level to keep the system neutral.To get the orre t band energy φs(r) needs to be expanded in terms of KSorbitals,|φs >=

∑

n

cns|ψn >, cns = 〈ψn|φs〉 (5.22)and the band energy be omes,εs =

∑

n

|cns|2εn. (5.23)5.9 Computational software5.9.1 Vienna Ab initio Simulation Pa kageIn this thesis all the al ulations have been performed mainly using Vi-enna abinitio simulation pa kage (VASP). VASP is based on a Fortran ode is a versatile quantum simulation environment based on the densityfun tional theory. It is a omplex pa kage for performing ab-initio quantum-me hani al simulations using pseudo-potentials or the proje tor-augmentedwave method and a plane wave basis set. In al ulations performed for thisthesis the intera tion between ions and ele trons have been des ribed by theproje ted augmented wave (PAW). PAW method allows for a onsiderableredu tion of the number of plane-waves per atom for transition metals andrst row elements. The size of the basis-set an be kept very small even fortransition metals and rst row elements like C and O.For es and the full stress tensor an be al ulated with VASP and usedto relax atoms into their ground-state. As in any plane wave program, theexe ution time s ales like N3 for some parts of the ode, where N is the35

number of valen e ele trons in the system. In VASP, the pre-fa tors for the ubi parts are almost negligible leading to an e ient s aling with respe tto system size.As for the Brillouin-zone sampling VASP oers a range of meshes ands hemes from expli it kpoint denition to automati mesh determinationproviding a exible hoi e depending on the individual requirements. Theabove mentioned features are only brief outline of the whole pa kage, mu hmore information on VASP an be found on the VASP web site where one an a ess a pra ti al guide as well us other useful resour es on the VASP[30, while the overview of the main methods is provided in the arti le byKresse and Furthmüller [31, 32.

36

Chapter 6S anning TunnellingMi ros opy6.1 Ba kground informationsIn order to obtain atomi s ale resolution with a s anning probe instrument(see Fig. (6.1)), a signi ant hange in probe sample intera tion needs tobe observed within the s ale of atomi units. The only intera tions sensitiveenough to su h small displa ement are hemi al for es and tunneling urrentsand both are giving rise too two important methods, namely the s anningtunneling mi ros ope (STM) and the atomi for e mi ros ope (AFM). Bothof these methods are of tremendous importan e and have revolutionised theeld of surfa e s ien e. In this work, however, we shall only be on ernedwith the rst one. In surfa e s ien e one of the most fundamental problemsis an a urate determination of the surfa e stru ture. To this end, STMhas been the leading method. STM oers high resolution of dire t and realimages making it one of the most favourable tools of surfa e analysis.This is how it works. An atomi ally sharp tip is brought lose to thesurfa e and a voltage is applied. The potential dieren e between surfa eand tip auses the appearan e of a tunnelling urrent, whi h is exponentiallydependent on the tipsurfa e distan e. This makes it possible to a uratelydetermine the topography of the surfa e. The surfa e is s anned line byline and a full surfa e lands ape is obtained. The s anning an be done intwo modes depending on the expe ted topography of the system. Either the urrent or the height of the tip (z oordinate) is kept xed while the othervariable is mapped on the s anned xyplane.Although from these images the interpretation may often seem straight-forward this is not always a ase, and it is important to understand the exa t37

relation between the urrent, the tip and the surfa e of an investigated ma-terial. Depending on the potential dieren e the ele trons parti ipating inthe tunnelling are hosen from ertain "origin" states of the o upied statesin the emitting material to their "destination" states to whi h they tunnel.Thus the availability of states on both sides of the jun tion is a ne essary ondition for tunneling to o ur. From this, it is apparent that tunnelling urrent is a result of the topography of the surfa e; its atomi spe ies, aswell as its lo al ele troni properties.

Figure 6.1: Ilustration of s anning tunnelling mi ros opy operation [27Be ause of the va uum between the tip and the sample in an STM,intera tions of the ele trons an be onsidered small and the problem anbe treated using perturbation models. The tunneling urrent to rst orderperturbation theory isI =

2πe

~

∑

µ,ν

f(Eµ) [1− f(Eν + eV )]× |Mµν |2 δ(Eµ − Eν), (6.1)where f(E) is the Fermi fun tion, V is the applied voltage, Mµν is the tun-neling matrix element between states ψµ of the probe and ψν fo the surfa e,and Eµ is the energy state of the state ψµ in the absen e of tunnelling.With the urrent omputational power the al ulations of the ele troni stru tures of the tip and the surfa e are feasible and the Bardeen method ould by applied. Unfortunately, there is still is not enough informationabout the exa t stru ture of the tip of the probe. For this reason the most ommonly used te hnique of STM simulation is based on a very simple modelproposed by Terso and Hamann in 1983 [28, 29To simplify the Bardeen method Terso and Hamann have restri ted38

the geometry of the tip to one s-like orbital. This simpli ation and the orresponding simplied al ulation of the tunneling matrix element Mµνyields the following expression for the tunneling urrent:I ∝ e2κR

∑

n

|ψn(r0)|2 δ (εn − εF ) = e2κRn(r0, εF ) (6.2)where κ is the inverse de ay length for the tip wave fun tion in va uum, Ris lo al radius of probe's urvature. The de ay length itself depends on thework fun tion φw and the applied bias ∆V a ording to:κ =

√2me|φw −∆V |

~(6.3)where the φw is the work fun tion Equation 6.2 shows that the tunneling urrent is proportional to the lo al density of states of the surfa e, n(r0, εF ).Despite its simpli ity this relation is very often a quite a urate des ription ofSTM images. Its drawba k, however, is that it does only allow for qualitative omparisons.6.2 BSKANAll the STM simulations in this thesis have been performed using bSKAN byW. Hofer. bSKAN is written in modular form and is urrently implementedin Fortran 90. It is an e ient tool for s anning tunnelling mi ros opysimulations, written on the basi of Bardeen treatment of tunneling urrent,and in ludes a Terso-Hamann approximation. For more informations seethe bSKAN guide [34.

39

Part IIAnalysed systems

40

Chapter 7Sili on surfa esThis hapter is aimed as an introdu tion to the sili on surfa es dis ussed inthis thesis. It outlines some basi phenomena observed on the lean surfa esand provides the stru tural informations al ulated in preliminary studies.7.1 Sili on: hemi al propertiesSili on is a tetravalent hemi al element with the symbol Si and atomi num-ber 14. It is less rea tive than its hemi al analogue arbon, the non-metaldire tly above it in the periodi table, but more rea tive than germanium,the element dire tly below in the table. Sili on is the eighth most ommonelement in the universe by mass and se ond most abundant element on earthafter oxygen a ounting for 25.7 % of the Earth's rust by weight. However,sili on very rarely o urs as a free element in nature. It is most ommonlypresent in a form of sili on dioxide (sili a) or sili ates [35.Sili on is widely used in its available forms for building and, in the e-rami industry in the produ tion of por elains and quartz based lime-glass.The other sili on ompounds are sili on arbides and a whole lass of sili on-based polymers alled sili ones. Puried sili on is a base of modern te hnol-ogy, a orner stone of urrent ele troni s and the omputing industry. Asone would expe t sili on also plays its part in the physi s and hemistry oflife, mostly as a tra e element for animals but more importantly in the biol-ogy of plants. The rystalline sili on has relatively high melting and boilingtemperatures of 1687 K and 3538 K, respe tively.Similarly to germanium and arbon, sili on rystallizes in a strong butbrittle diamond ubi stru ture with a latti e parameter of 5.43 Å. Sili on isa metalloid with mu h lower ele tro-negativity than arbon (1.7 omparedto 2.5 on the Allred and Ro how s ale) allowing for many forms of hemi albonding. Like arbon it typi ally forms four bonds but it an also a ept41

additional ele trons and form ve and more bonds [35. Its tetra-bonded hemistry provides opportunities for exible ombinations with other ele-ments and gives rise to stru turally omplex hemistry, thus being the mostlikely andidate for an alternative bio hemistry.Its shell stru ture onsists of 1s, 2s, 2p, and 3s orbitals, whi h are om-pletely lled, and a 3p orbital whi h ontains only two ele trons, adding upto four valen e ele trons. Sili on has a negative temperature oe ient ofresistan e, sin e the number of free harge arriers in reases with tempera-ture. The ele tri al resistan e of single rystal sili on signi antly hangesunder the appli ation of me hani al stress due to the piezoresistive ee t[35. Re ently it has been also dis overed that sili on an exist in anotheranisotropi 2d form alled sili ene [36.The ele troni band stru ture of bulk Si has been widely studied bothexperimentally and theoreti ally [37, 38, 39, 40. The energy of a bandgap at 0C is 1.2 eV while at room temperature it is 1.1 eV [41. Dueto the importan e of sili on in modern industry the study of its propertieshave been of great value. Here, we are on erned with the analysis of itssurfa e properties. The terminated bulk stru ture, when exposed to va uum,usually undergoes some a re onstru tion to redu e its total energy. If there onstru tion pla es the empty states below the Fermi surfa e or o upiedstates above it a band bending is expe ted and ele trons ow from or tothe bulk until the built-up ele tri eld be omes strong enough to reatean equilibrium. This built up eld therefore orresponds dire tly to theband bending. Surfa e dangling bonds an be redu ed by the formation ofdimers or an be saturated by another spe ies su h as hydrogen. The bondhydrogenation auses bonding states to move to the valen e band and emptystates into the ondu tion band whi h attens the band. Even though surfa ere onstru tion minimizes the amount of dangling bonds, the shifted atomi onguration also generates surfa e stress whi h is energeti ally ostly. Forthat reason not all re onstru tions are easy to predi t, and the most stablesolutions often are due to a balan e between bond saturation and surfa estress.7.2 Si(111)(7×7) surfa e re onstru tionThe Si(111) surfa e leavage and asso iated re onstru tion was a topi ofs ienti debate for many years des ribed by a well established theoreti- al model alled dimer adatom sta king fault (DAS). This model was rstpublished by Takayanaki et al. in 1985, and instead of very simple pattern42

arising from the bulk stru ture it proposed a new arrangement of atoms inthe top three layers forming a large (7×7) unit ell, shown in Figure ( 7.1)[42. The DAS model a ounts for many features observed by STM, wherethe most visible atoms orrespond to adatoms and the dark areas surroundedby hexagons of adatoms orrespond to orner holes (Figure 7.2).The trademark of this re onstru tion is the existen e of a 3fold symme-try of faulted and unfaulted top layer atoms with respe t to the orner hole(CH), responsible for very hara teristi triangular patterns visible in STMimages (Fig. 7.2). For the purpose of this work a short ode in python waswritten to generate the re onstru ted Si(111)(7×7) surfa e for given bulkunit- ell latti e parameters, with a number of bulk layers and a hydrogenatedbottom layer. In order to a hieve a (7×7) re onstru tion the rst bilayer ofthe (111) surfa e leavage has to be reorganized, this is done in four dis retesteps:1. Loss of 4 orner atoms resulting in reation of hara teristi deep va- an ies at the apex of the unit ell alled orner holes2. Dimerization of side atoms3. Rearrangement of atoms in one half of a super ell reating a faultedsta king4. Addition of 12 ad-atoms.The obtained re onstru tion is then added to a number of bulk bilay-ers and terminated with hydrogen atoms for saturation of dangling bonds(Fig. 7.2).

43

Figure 7.1: Si(111)(7×7) surfa e re onstru tion model proposed byTakayanaki [42. First three layers of atoms represented as follows: dashedlling ad-atoms, bla k lling rest-atom, white lling re onstu ted bi-layer, dots bulk bi-layer.

Figure 7.2: S anning tuneling mi ros opy image of the Si(111)(7×7) surfa ere onstru tion. Two triangles representing the surfa e super ell where thedierent brightness is due to the periodi fault in top layer sta king. The sixouter bright atoms in ea h triangle represent ad-atoms and the tree dimerones represent rest atoms [4344

Figure 7.3: The top and side view of the (111) surfa e re onstru tion withthe side views presenting the detailed hara teristi of layered arrangementand the nature of the sta king fault. The top layer of a faulted half of asuper ell hanges its sta king sequen e from (bulk: ...AaBbC ) AaBbC to(bulk: ...AaBbC ) AaBbA.7.3 Si(100) (4x2) surfa e re onstru tionThe Si(100) surfa e is the most ommon fa et of a sili on rystal. In are onstru ted surfa e it minimizes its energy by the formation of dimersfrom top layer atoms, so that the number of dangling bonds is redu ed by50%. The surfa e is omposed of Si-Si dimer rows, while dimers in theirenergeti groundstate form a bu kled (4×2) stru ture (Fig 7.4). At roomtemperature, however, the surfa e involves asymmetri bu kling vibrationsof the dimers, in ee t ausing a ip-op motion between opposite bu kling ongurations. The ip-op motion annot be seen dire tly in an STM dueto the low time-resolution, although its result an be inferred from averagedsurfa e images of at dimers, appearing to posses (2×1) symmetry [44. There onstru ted surfa e utilized in the lean surfa e and mole ular adsorbtion45

alulations was also generated using written by python s ript. The pro ess ishowever mu h simpler in omparison to a Si(111)(7×7) re onstru tion, sin eit involves only dimerization, bu kling and bulk surfa e hydrogenation. Theobtained initial atomi arrangements were relaxed by VASP in two stages,rst xing all atoms ex ept hydrogen passivating the bottom layer, and thenkeeping hydrogen together with the two bottom layers xed, while all otherlayers underwent relaxation. The kpoint sampling was limited to Γ pointand energy ut o was set to 350 eV.

Figure 7.4: Si(100) (4×2) re onstru tion. Beige atoms represent the top re- onstru ted layer of sili on atoms, forming rows of dimers bu kled in oppositedire tions. The left row mirrors the right row forming a ctype re onstru -tion. Grey atoms represent se ond, third and fourth sili on layer.The surfa e bu kling is not only hara teristi of Si(100) it has also beenobserved on Ge(100), therefore some of the results highlighted in the follow-ing are transferable between these system. At low temperature the bu klingis stati and an be easily observed in experiments. It takes the most en-ergeti ally favourable symmetry forming a c(4 × 2) super- ell [45. Thisobservation has been also onrmed in theoreti al studies of other dierentpossible phases [46. During the dimerization the harge is transferred fromthe lower to the higher Si atom within the dimer; this asymmetry in lo al harge is also ausing an asymmetry of the atomi arrangement. As shownin the Fig. 7.6, the lo al density of states (LDOS) peak for the up atom(pink line in the gure) in the dimer is in the valen e band, while for thedown atom (blue line in the gure) it is in the ondu tion band. On thesame plot we also in luded the LDOS of atoms in a at dimer onguration(green), obtained from a at dimer unit ell al ulation, whi h shows a tran-sition between the up and down results with the LDOS peak shifted loserto the Fermi level. The dimer up and down bu kling also ae ts the se ondatomi layer, where the atoms hange their position due to the geometry ofthe dimers. The atoms under the up atoms are pulled up, while atoms under46

the down atom are lowered as seen in Fig. 7.5.

Figure 7.5: The illustration of the bu kling inuen e on the the top layerof Si(100)(4x2) re onstru tion, the beige atoms represent the dimer atomswith brighter ones being bu kled up and darker bu kled down. The greyatoms represent rst layer beneath the dimers, while the arrows indi ate thene essery shift of these atoms to a omodate the bu kling.Employing an Ising model Fu at el. have estimated that at room tem-perature the dimers have su ient energy to os illate between the oppositebu kling with a frequen y of about 3× 106Hz [46.7.3.1 PinningAlthough the fast os illations do not allow one to observe this hara teristi c(4×2) zigzag stru ture in room temperature STM images, this is not alwaysthe ase when dealing with defe ts or adsorbates. On many o asions, ithas been observed that dimer os illation an be ae ted and bu kling anbe ome apparent even at room temperature. Depending on the temperatureand the strength of pinning, the observed ee ts an have lo al (just a fewdimers) or mu h broader hara ter responsible for reating domains of inor anti phases (see Fig. 7.7). Controlling and understanding the dierentSi(100) phases is of great signi an e due to its potential to be utilized inappli ations su h as mole ular ele troni s, quantum omputing or quantumopti s [47. Even though the low temperature STM has repeatedly proventhat Si(100) (4×2) is a fa tual ground state phase [45, the phase an beae ted by a spe i mole ular adsorption. In the literature this pro essis referred to as dimer pinning when the bu kling os illation is limited in47

4 2 0 2 4

E-EF [eV]0.00

0.05

0.10

0.15

0.20

0.25

0.30

Part

ial D

oS

FLATDOWNUP

Figure 7.6: Si(100) (4x2) re onstru tion, a omparison between a lo alizeddensity of states for Si atoms bu kled up or down (Gaussian broadening witha width of 0.2 eV).a spe i dire tion. So far, it has been shown that the restri ted bu kling an ontribute to hanged geometry/rea tivity indu tion of surfa e ele tri dipoles and double row in or anti phase relation [47.Previous work has shown that when the symmetry of the surfa e is bro-ken by a defe t or a step edge a regionally xed bu kling may o ur [45, 48.Hossain at el. present no rossdimer row orrelation, thus showing no ev-iden e of a strong intera tion between the rows. The observation regardsmole ular adsorption in whi h the two neighbouring mole ules reate pin-ning of the dimer row. The distan e and the position of mole ules ae tthe result of in phase or out of phase pinning. This ee t is observed at thetemperature of 300K, and the mole ule used is (CH3)2NCH2CH2N(CH3)2

(N,N,N ′, N ′−−tetramethylethylenediamine) (TMEDA) results in visiblepinning ranging 12 dimmers away from adsorption site. The rst introdu edmodel of indu ed dimer pinning was based on an Ising-type model and hasbeen mostly on erned with dimerdimer intera tion rather than mole uledimer intera tions. This has been re ently expanded to mole ules and will befurther dis ussed in the following se tion [49, 50. The relationship betweenthe surfa e and the adsorbate has been proven to be mu h more ompli atedthan initially assumed when not only the geometry but also the hara ter ofthe individual bonds or polar intera tions play a signi ant role. Smith etal. have shown that full knowledge of pinning ould help to determine the48

Figure 7.7: STM images taken by Hossain et al. showing the adsorptionsof small amounts of TMEDA mole ules at 65K, bias −2V and tunnelling urrent 0.2nA. The pinning aused by the mole ule is responsible for reationof two phases: (4x2) and p(2x2). [47mole ular adsorption of a etylalehyde and a etone, in whi h a lear distin -tion ould be derived on the basis of bu kling signatures in between the (2+2) y loaddition or enolate mediated geometries [50. The approa h based onthe Isling model has been shown to be a powerful tool. This method ouldbe used for dierent mole ules and is expe ted to be appli able also on othersurfa es exhibiting bu kling, su h as Ge(100), allowing the determination ofthe pinning strength and its dire tionality.7.3.2 Defe tsThe lean Si (100) surfa e has several types of ommon imperfe tions su has step and point defe ts, all of whi h ould inuen e surfa e properties orlo ally present pro esses. Due to the enormous importan e of Si in urrent49

industry, the issues related to the rystal purity and stru tural perfe tionare of wide interest. Despite detrimental properties of defe ts in some appli- ation, they have been shown to be the a tive site for both bulk and surfa epro esses and an be used as atalyti entres, guides or markers in a selfassembly pro ess. A proper understanding of existing impurities and defe tsis essential for appli ations where impurities are an integral element of thenal material/devi e.In the ase of Si(100), point defe ts have been ategorized into threetypes: A, B and C. The A and B types represent single and double dimerva an ies [51, 52. The origin of the C type defe t was initially ambigu-ous, there were a few dierent models proposed su h as va an ies [53, 54,surfa e or subsurfa e foreign atoms [55, 56 and water adsorption [57. Inthe past de ade a general on lusion has been rea hed and the C defe t hasbeen identied as a rea tion site for a water mole ule. This will be furtherdis ussed in more detail in Se tion 8.1.In order to further investigate the bu kling phenomena a series of dierentstudies were performed investigating the stru tural and energeti hara ter-ization of the system.The systems of hoi e are disso iated water and two of its further tran-sition states observed by both Sobotik and Wars hkow at. el. [58, 59whi h is parti ularly interesting due to the presen e of strong pinning ef-fe ts. In addition to these studies we have performed an ele troni stru tureanalysis to investigate the origin of pinning. Most of the studies have beenperformed on the Si(100) (4×2) surfa e although o asionally a at dimersurfa e Si(100)(2×1) was used.

50

Chapter 8Surfa e pinning and wateradsorption.8.1 Water adsorption on Si(100)Due to the importan e of wet oxidation pro esses for the reation of SiO2layers, studies of water adsorption on Si(100) have re eived onsiderable at-tention. Water has been both appre iated as a rea tant and atalyst, whi hmakes it a key element in oxidation s hemes [35. Moreover it is also widelyre ognized as a surfa e ontaminant due to its high residual on entrationin Ultra High Va uum (UHV) setups, whi h makes it desireble to know itsee t on surfa e properties or surfa e pro esses. Histori ally there have beensome on erns whether the adsorption of water is disso iative, whi h todayare resolved and proven to be the ase. The earlier proposed option wasthe on-dimer (OD) onguration in whi h the mole ule disso iates on twodangling bonds of the same dimer. The STM study of this onguration sug-gested that adsorbed water appears identi al to features previously assignedto single dimer surfa e defe ts [62, 63. The other proposed ongurationwas inter-dimer (ID) with a mole ule disso iating on the nearby danglingbonds of in-line neighbouring dimers whi h is also known under the name ofC-defe t. Subsequent DFT al ulations showed that both of these ongu-rations are stable [64.The water adsorption is a omplished in a two step pro ess in whi h atrst the whole mole ule is adsorbed and then it undergoes disso iation to Hand OH. The pro ess of disso iation an be des ribed by means of transitionstate theory in whi h the total transition rate an be represented as a sumof a quantum tunneling rate and a rate due to thermal u tuations.The motion of the dimer an be des ribed by the same model without51

the tunnelling ee t.A ording to one al ulation of the transition barriers, the transition toeither OD or ID ongurations is almost identi al [65. The minimum energypath (MEP) was investigated with the Climbing Image Nudged Elasti Bandmethod (CINEB with water physisorption above down and up bu kled Siatom). It has been onsistently shown the DOWN Si atom is mediating inadsorption, when the oxygen atta ks on the higher Si atom within the dimerit auses the dimer swit h followed by adsorption [66, 67 The MEP showsthat the oxygen atta k is ee tive from any position with respe t to the Sidimer, onrming the experimentally observed high rates of rea tion [67.The sti king oe ient of water is onstant until saturation, suggesting amobile me hanism for rea tion. Be ause of its high mobility and rea tivity,water an be a signi ant problem when in just one hour under the partialpressure of 10−11mbar of residual water it already reates a 2 % surfa e overage.However 0.24eV higher energeti stability of OD vs ID does not explainthe ve times higher probability of ID adsorption. From the kineti perspe -tive the rea tion barrier is 0.02 eV lower for ID whi h is still quite small toexplain the behaviour. It has been suggested that this inequality in prin iple ould be a ounted for by H atom tunnelling similarly to the one shown forNH3 adsorption [68The subje t of dimer os illation has been analysed in the re ent work byYu, providing some inside on the impa t that dimer os illation may have onthe water disso iation pathways into ID and OD ongurations, and it hasbeen demonstrated that this ee t may even lead to population reversal. Yualso suggested that the dimer's dynami s should be onsidered in analysis ofother polar mole ules on a Si(100) surfa e [69.Although we onsider Yu's point as an important onsideration, we donot agree with his interpretation of the presented data in whi h he attributesall of the observed features to only two distin t pro esses. In fa t, and asshown in earlier work, water disso iation is usually mu h more omplex thanthat [58, 59. It has been spe i ally shown that the elevations of the surfa etemperatures allows a series of transformations to o ur whi h lead to surfa edimer oxidation SiOSi or even hydrogenation of this dimer HSiOSiH, whi h has been observed experimentally [70.

52

8.1.1 Computational detailsIn al ulations presented here a set of dierent unit ells have been used to ompare the dierent possible pinning. Four unit ell ells were generatedusing writen by me python s ript as presented in Fig. 8.1. Ea h unit ell omposes of eight layers of Si atoms saturated from the bottom with Hatoms, the unit ell is extended to nine or ten dimer rows to provide atleast three dimers on ea h side of the adsorbate. The length of ea h is reated in su h a way as to preserve the bu kling at the periodi boundarythus avoiding any unnatural ee ts whi h ould o ur otherwise. Due to asigni ant number of atoms the unit ell in ludes only a single dimer row.Ions were relaxed using PAW-PBE as implemented in VASP until the residualfor es were less than 0.02 eV/Å. The H atom layer and two bottom layers ofSi atoms xed in their bulk positions. The k point sampling was performedwith 1 x 3 x 1 mesh and the energy ut of was set to 350 eV.

Figure 8.1: Four dierent bu kling ongurations for the OH, H interdimeradsorption of disso iated water mole ule on Si(100)(2×1) surfa e; in a) andb) the mole ule breaks the bu kling phase, while in ) and d) the phaseis preserved. The unit ells are always reated to preserve phase upon theperiodi transformation. The beige atoms represent the dimer atoms withbrighter ones being bu kled up and darker bu kled down, the bla k atomsrepresent rst layer beneath the dimers.In order to a ount for the dierent unit ells while omparing totalenergies, the ontribution to the total energy from additional atoms was al ulated. The approximated energy was obtained as a dieren e of total53

OH and HEnergy meVE1 (a) in Figure 8.1 0E2 (b) in Figure 8.1 45E3 ( ) in Figure 8.1 4E4 (d) in Figure 8.1 14Table 8.1: Results of al ulation for adsorption energies of disso iated watermole ule on Si(100) - (4×2) surfa e in omparison with other theoreti alresults.energies from two lean surfa e al ulations, eight and ten dimer divided byto to represent the energy dieren e per extra dimer, whi h was equal to

97.11 eV.∆E =

E10 − E8

2(8.1)The unit ells with even number of dimers were hosen to a ount for the ontinuity of the bu kling phase.8.1.2 ResultsFrom the obtained results presented in the Table 8.1 the most stable ongu-ration predi ted is E1 ( a) in Fig. 8.1), this is in agreement with experiment.Furthermore the swit h of pining from a) to ) is 0.01 eV more likely than a)to d) whi h means that the pinning on the H side is expe ted to be strongerthan on the OH side, this is also observed in experiments (see Fig. 8.2). Ourresults onrm that the phase of bu kling most preferably gets broken andthat the pinning strength on both sides is asymmetri .We have investigated the impa t of the adsorbate on the hange of lo aldensity of states (LDOS) in the vi inity of the adsorbate. By omparing theatom proje ted DOS we have found that the hanges are very minor andare restri ted to the dimer dire tly involved in adsorption and the atomsbelow that dimer. The neighbouring dimer atoms DOS are unae ted, this on lusion applies to all the adsorptions that follow in this thesis unlessstated otherwise. The ex eptions to this are not resulting from mole ulesurfa e harge transfer but from the dipole intera tions between the mole uleand a Si dangling bond, su h as the one dis ussed below.As presented for the H side bu kling in Fig. 8.3 there is no observable hange in the density of states for Cdefe t adsorption ( Fig. 8.1). Fromthe omparison of peaks for UP (o upied states) and DOWN (uno upiedstates) atoms it is evident that the overlap of the olour lines (neighbouring54

Figure 8.2: STM images taken by Wars hkow et al. showing lled and emptystates for −2V and 2V respe tively at urrent 0.3nA of water Cdefe t onSi(100) surfa e. [58

8 6 4 2 0 2 4 6

E-EF [eV]0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

Part

ial D

oS

DOWNUPREF_UP

REF_DOWN

Figure 8.3: LDOS of UP and DOWN dimer Si atoms on Si(100) surfa ewith C-defe t on it. The onsidered atoms are neighbouring to a dimer withH adsorbate (see a) in Fig. 8.1) while REF_UP and REF_DOWN are areferen e up and down atoms far away from the adsorbate. The up bu kledatoms are on the positive axis while down are on the negative one.dimer) and bla k plots (far away referen e dimer) are almost identi al. Forthe OHside Fig. 8.4 however, a slight shift in the density is observed whi h55

an be attributed to the dipole moment of the OH group. The OH dipolemoment is known to be about 1.5 D. Then the potential in the vi inity ofthe dipole will beV (R) ≈ 1

4πε0

qd · RR2

(8.2)where R is a distan e from the entre of the dipole moment, one anestimate that the potential at 4.0 Å will be 0.25 V if the dire tion of the dipolewas exa tly in the dire tion of measurement. In reality the ele tron loudis disperse and the dire tion varies therefore this value represents only thehigher limit. The shift measured in between the UP peak and the REF_UPpeak is about 0.21 V whi h is in good agreement with the approximatedvalue. Furthermore the adsorbed OH group pointing toward the danglingbond has been observed to indu e bu kling up, while the OH group pointingin the opposite dire tion indu es bu kling down [66.

8 6 4 2 0 2 4 6

E-EF [eV]0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

Part

ial D

oS

DOWNUPREF_UP

REF_DOWN

Figure 8.4: LDOS of UP and DOWN dimer Si atoms on Si(100) surfa ewith Cdefe t on it. The onsidered atoms are neighbouring to a dimer withOH adsorbate (see a) in Fig. 8.1) while REF_UP and REF_DOWN are areferen e up and down atoms far away from the adsorbate. The up bu kledatoms are on the positive axis while down are on the negative one.

56

8.2 High overage water on Si(100)8.2.1 Introdu tionAs re ently shown by Gallet and o-workers high overage water adsorptionoers a wide range of patterning. In the STM image presented by them(Fig. 8.5) a set of signatures an be distinguished for a dierent dimer dan-gling bond saturation. The possible ombinations an be attributed to twoH atoms per dimer, two OH groups, or one H and one OH atta hed to thedimer. O asionally, individual dangling bonds are preserved [164.In our observation as well in earlier studies it has been shown that OHgroups have a tenden y to orient themselves orthogonally to the dimer bonddire tion [165. In this se tion we aim at investigating this behaviour inhigh overage ases where many OH groups are able to intera t with ea hother, with H atoms, and with isolated dangling bonds. We have performedstru ture relaxations and from the relaxed geometry simulated the STMimages.8.2.2 Computational detailsWe have re reated the features proposed in the experimental results inFig. 8.5 and reated a unit ell ontaining ea h of the above mentionedsignatures. The ell used is omposed of seven layers of Si atoms hydro-genated from the bottom, with two dimer rows, a total of eight Si dimers.The STM images were performed using TersoHamman approximation asimplemented in bSKAN.8.2.3 ResultsDuring the ioni relaxation, a lear movement of the OH moieties is ob-served. Although this pro ess is very slow and requires over a thousandioni iterations. The general observation for the high overage s enario isthat OH groups preferentially align along the dimer rows if adja ent to an-other OH group. As seen in the stru tural illustration Fig. 8.6 the dipolesalso intera t on the dimer.Analysing the experimental images of single disso iated water (Fig. 8.2)it is evident that it is possible to identify both the OH and H adsorptionsites of the Cdefe t, due to the hara teristi drop shape feature, and theasymmetri dimer pinning. The problem is that the STM feature does notarise from the adsorbate but from an empty dangling bond on the adsorbeddimer. When investigating full adsorption, a majority of dangling bonds are57

Figure 8.5: STM images of high overage of water on Si(100)2×1 by Galletet al. (a) Image (8.5 nm × 4.0 nm) of watersaturated n doped surfa es anned at bias voltageV = −2.3V and I = 0.57nA, showing o upied states.(b) Same as (a) but with enhan ed ontrast to show the alignment (redre tangles) and zigzag (green re tangles) patterns made of HSiSiOHunits. ( ) Line prole through the lo ation marked by the red arrows in (a),en ompassing from left to right, HSiSiOH, HOSiSiH, and HSiSiOH units. (d) H and OH distribution along a dimer row, showing the oexisten e of HSiSiOH, HSiSiH and HOSiSiOH units [164saturated, therefore neither the drop feature nor the pinning are visible.So far no a ount of the OH dire tion has been dis ussed in the exper-imental investigation and the existing high overage STM images. In thisthesis as well as in earlier works it has been observed that the OH bond58

Figure 8.6: The relaxed unit ell of high overage water adsorption onSi(100)-(2×1) surfa e with simulated STM images for the bias voltage ofV = −1.5, −2, 1.5 and 2.0 for LDOS = 1 ×107states/(eVm3).is rotated along the surfa e [166. Here, we want to further investigate theee t. As visible in Fig. 8.5 the feature variation is quite signi ant wherethe brightness of the OH and H features varies onsiderably. From the pre-sented STM images no OH dire tion an be learly determined. This la k ofdenition regarding OH orientation may be aused by the thermal u tua-tion and lari ation of this subje t using a low temperature STM would berequired. When investigating the simulated STM images Fig. 8.6 a similarobservation an be made. In both positive and negative bias images thefeature representing H atoms is onsiderably lower while the OH group'ssignature is varied. For negative biases the height of the features are similarfor all OH singularly o upied dimers while the two doubly o upied dimeroer distin t shape, one is symmetri al while the other is not. The variationbetween the OH signature is even more prominent at positive voltages wherenot just a shape but the height varies onsiderably.59

Chapter 9Single organi mole ules onsili on surfa es9.1 Mole ules on surfa esBefore investigating surfa e hemistry let us rst outline the basis of whatis responsible for the mole ular rea tivity in the rst pla e. How an wepredi t if the ambient mole ule will, or will not atta h to the surfa e. Thesti king probability of Si atoms arriving at the lean Si surfa e at thermalspeed is pra ti ally 1, but this does not have to be the ase for other atomsor mole ules and an also dramati ally hange upon surfa e re onstru tion.In general, the ability of the surfa e to a ept an adsorbate is ontrolledby the lo al ele troni stru ture, the geometry of the surfa e and theirsvibrations of whi h the rst two are a subje t of a following hapter. Inthe rst approximation, the key fa tors are: the ele tronegativity dieren ebetween the rea tion spe ies and the availability of ele trons for the hargetransfer aused by this dieren e. For example the highly ele tronegativeoxygen will prefer a Si atom with o upied dangling orbital lose to theFermi energy Ef , so that the O atom an a quire an ele tron from Si withmaximum energy gain.The hara ter of the rea tion between the surfa e and the adsorbate islikely to hange in the ourse of the adsorption. There is a dependen e of theadsorption rate on the exposure time due to the varying overage, it is wellknown that some adsorbate spe ies may sti k better when agglomerationsin the form of islands are already present on the surfa e. The same appliesfor dierent adsorption rates resulting from adsorption on spe ial sites su has defe ts, dislo ations or ridges whi h are often the rst to be o upied. Inthe ase of island formation the adsorption rate is dependent on the amount60

of island already present on the surfa e [35.In general, the adsorption of mole ular spe ies pro eeds though severalstages. The free mole ule may be ome physisorbed, hemisorbed, or dis-so iatively hemisorbed on the surfa e. These states an most of the timebe separated by energy barriers whi h further permits the determination oftheir transition rates, sti king probabilities and rea tion pathways.Physisorption is the weakest form of adsorption. It is hara terized byla k of true hemi al bond instead the mole ule bonds to a substrate thoughtthe Van der Waals or Coulomb for es. Be ause the mole ule makes no hem-i al bond with the surfa e it an often migrate e iently over the surfa e.The analysis of mole ular physisorption is an important eld of surfa e s i-en e whi h is a driving for e behind self assembled networks on metalli surfa es or mole ular imprinting te hniques when the physisorbed mole ulestemplate the later hemisorbed atomi pattern. Even at large distan es amutual attra tion between the atom and the surfa e exists that arises fromthe intera tions of polarizable solid with the dipole from the quantum me- hani al u tuation of the atomi harge distribution and vi e versa.In ontrast to physisorption, a hemisorbed mole ule does make a hem-i al bond. The mole ule however retains its identity without breaking intoseparate pie es. For some mole ules this adsorption state is not a nal oneand under the right onditions they further undergo disso iation into moi-eties su h as H2O des ribed in se 8.1 .9.2 1,2dibromoethane on Si(111)In this work we aimed to provide some lari ation and an explanation of theunusual behaviour observed experimentally in whi h the hemisorbed, ph-ysisorbed or mixed adsorptions were observed in the one per orner hole(OPCH) pattern, even so there has been another ve empty adsorbtionsited (there is six adatoms per orner hole). The mole ule involved was ph-ysisorbed 1,2dibromoethene whi h gave rise to the bromination of orner-hole (CH) Si atoms. The observed phenomena suggested some me hanismsresponsible for the hara teristi one per orner hole adsorption. The knowl-edge of the underlying atomi pro esses ould be of great value for a better ontrol in selfassembly and nano manufa turing pro ess. Although adsor-bate repulsion has been observed, so far it has been restri ted to metalli surfa es only. On metalli surfa es the mole ular repulsion an be referred todierent pro esses. The presen e of the weak repulsion with for es of around5 meV at 10 Å distan e an be attributed to the standing ele tron waves,61

while stronger intera tions an be attributed to ele trostati intera tion be-tween mole ules that have a quired harge from the surfa e [71, 72, 73.The other reason may be dipolar intera tions of about 20 meV at 15 Å, inthis ase the strength of this intera tion is doubled due to the imagedipoleintera tion [74. In our situation we deal with a semi ondu tor surfa e andthe distan e between the opposite ad-atoms at the orner hole is 13.4 Å. Inre ent works it has been shown that the harge inje tion into the surfa e us-ing the STM tip an result in hopping of hlorine atoms [75, or a non-lo alremoval of hlorobenzene mole ule [76. Non lo al a tivation of a metastableadsorbtion of hydrogen on Si(100)(2×1):H surfa e was also observed [77.9.2.1 Experimental ndingsAs presented by M Nab et al. in STM images [78, the physisorbed mole ulepreferentially atta hes itself to one of the adatoms lo ated by the orner holeFig. 9.1. In general no more then a single mole ule atta hes to the same orner hole, but ex eptional double adsorptions have been also observed. Itis di ult to explain this behaviour from the perspe tive of the ele trostati repulsion of mole ules adsorbed at the same orner hole. The repulsion wouldbe expe ted to be quite weak due to the large diameter of a orner-hole, thusthe observed phenomena would be more likely related to a surfa e mediatedee t.It is expe ted that the single adsorption reates a signi ant hange inthe orner hole environment making se ondary adsorption highly unlikely. In overage studies a threshold for the surfa e saturation has been observed atabout 80 % for both the bromination and pre eding it 1, 2dibromoethenephysisorption further supporting the presen e of signi ant repulsion. Ad-ditionally, the statisti al analysis of the bromination provided eviden e forthe mobile pre ursor. Due to the mobility of the mole ule it is expe ted thatthe minimum energy onguration is rea hed.The experimental data show there was no site sele tivity between ad-sorptions on ad-atoms of the faulted or unfaulted part of the Si surfa e.When physisorbed, 1, 2dibromoethene mole ules undergo a thermally in-du ed hemi al rea tion and brominate the nearby ad-atoms Fig. 9.2. In no ase has any residual part of mole ule been observed after the bromination ofthe surfa e. A ording to experimental results obtained by M Nab and o-workers the bromination of a CH adatom is highly favoured over the middleadatoms. The ratio of rea ted Br atoms o upying the orner-hole adatomsto those o uping the middle adatoms is around 13, making it relatively un-62

likely. For this to happen a strong thermodynami or kineti preferen e isrequired. Here again as in mole ular adsorption, no sele tivity is observedbetween the faulted and unfaulted sites.In experiment of a partially brominated surfa e exposed to the 1, 2dibromoethene, the OPCH pattern was still preserved. In the mixed dBrEand Br there was o asional double o upan y but 93.8 % was still singular.

Figure 9.1: S anning tuneling image of physisorbed dibromoethane mole uleson Si(111)(7×7). The image was taken at 110 K with a bias voltage of+2.5V and a urrent of 0.2 nA [79.9.2.2 Computational detailsThe al ulations were performed using the Vienna Ab initio Simulation Pa k-age (VASP) [80, 32 with Generalized Gradient Approximation (GGA) andPerdue Burke Ernzerhof Proje tor Augmented Wave (PBEPAW) poten-tials. The energy ut o was set to 300 eV, the inter ioni for es wereminimized until the rea hed value was smaller than 0.02 eV/Å with thek point sampling restri ted to the gamma point due to large unit ell size.63

Figure 9.2: S anning tuneling image of hemisorbed Br atoms on Si(111)(7×7) re onstru ted surfa e. The image was taken in 298 K with bias voltage+2.5 V and urrent 0.2 nA [79.The generated unit ell Fig. 7.2 was prepared by at rst xing all the Siatoms and relaxing the H atoms then xing the bottom two layers togetherwith H atoms while the top layers were relaxed. The unit ell ontains 49H atoms and 298 Si atoms and has a size of A1 = (13.539, 23.449, 0.000),A2 = (−13.539, 23.449, 0.000) and A3 = (0.000, 0.000, 32.000) Å.1,2 dibromoethane exists in two forms, as rotational isomers of gau heand trans type (Fig. 9.3) . In the gas phase their relative populations is inproportion of 1 to 9 respe tively [81. I present the stru tural analysis forboth isomers in the Table 9.2.2.Due to very low energy dieren es, both isomers are expe ted to bepresent on the surfa e with their population ratio possibly inuen ed by theintera tions with the surfa e. The al ulated gas-phase energy dieren e is98 meV making the trans isomer more stable due to the lesser strain. In the64

Figure 9.3: Two rotational isomers a) gau he1,2dibromoethane b) trans1,2dibromoethane. trans gaugeangles () ()BrCC 108.8 113.2HCH 110.3 109.8bond length Å ÅBC 1.97 1.97CC 1.51 1.51CH 1.1 1.11Table 9.1: The stru tural hara terization of two rotational isomers a)gau he1,2dibromoethane b) trans1,2dibromoethane.adsorption ongurations the relative stability is expe ted to hange due tothe intera tion with the surfa e and the repulsive vs attra tive intera tionfor hydrogen and bromine atoms.The most prominent feature of the isosurfa e of the potential lands apeare adatoms, while the lower features represent rest atoms. The orner holeatoms reate an indent in whi h the height dieren e between the adatomand the orner hole atom is around 4 Å.9.2.3 Simulated ongurationsIn my analysis I have onsidered a series of physisorption ongurationswhi h are presented in Fig 9.4 9.9. From the experimental images it isvisible that the prole of the o upied orner holes is symmetri al in relationto one of six radial axes ( axis onne ting CH entral atom and CH adatom)thus the adsorbates need to be positioned in an appropriate way to mat hthe image symmetry. Under lose inspe tion all the observed features are hara terized by a dark spot in the pla e of a CHadatom, one an also65

d1 d2 EaÅ Å meVtrans CH adatom and rest-atom, (Fig. 9.4) 3.89 4.00 66gau he CH adatom and rest-atom, (Fig. 9.5) 3.14 3.34 70gau he CH adatom CH interior, (Fig. 9.6) 3.37 3.75 -28gau he CH interior, (Fig. 9.7) 3.29 3.61 94trans two mid adatoms, (Fig. 9.8) 3.37 3.54 122trans CH interior wide, (Fig. 9.9) 3.37 3.54 58Table 9.2: The stru tural analysis of 1,2dibromoethane adsorbates onSi(111)(7×7) together with the relative adsorption energies Ea. The dis-tan es d1 and d2 represent the separation of the ea h Br atom from a losestSi surfa e atom. d EaÅ eVBr faulted CH, (A at (Fig. 9.10) 2.27 2.67Br faulted middle, (B at Fig. 9.10) 2.27 2.67Br unfaulted CH, (C at Fig. 9.10) 2.28 2.62Table 9.3: The stru tural analysis for Br adsorption on Si(111)(7×7): BrSi bond length d and relative adsorption energies Ea.distinguish a slight variation in that some adsorption presents a very darkerfeature and slightly apart from the neighbouring CHadatoms, while othershave an additional dim spot still present at the pla e of CHadatom. It isun lear whether this is due to the experimental impre ision or is in fa t arepresentation of dierent ongurations.In order to elu idate possible ongurations a series of dierent geome-tries were tested. The mole ular adsorption energies were al ulated in ref-eren e to the energy of the trans mole ule 10 Å above the surfa e (middleof the unit ell's va uum region), these are shown in Table 9.2. The samemethodology also applies for the al ulation of adsorption energy of the Bratom (Table 9.2.3). Most of the adsorbed mole ules were positioned in su ha way as to orrelate with the experimental STM image, ie, the mole ulesare pla ed so as to redu e the density of states available to the STM tipabove the adatom and produ e the symmetri al image with respe t to theradial axis. Other ongurations were also tested as ben hmark studies.Four of these ongurations were onsidered, some trans while otherswere gau he. The results of our studies are presented in the Table. 9.2.66

The rst series of geometries are positioned on the radial axis, with therst two ongurations a trans and a gau he mole ule positioned above orner-hole add atoms and dire ted outward Fig. 9.4 and Fig. 9.5. Theadsorption energies are similar for both trans and gau he ongurations at66 and 70 meV respe tively. It is known that the physisorbed mole ule hasa limited impa t on the ele troni stru ture of the surfa e. Therefore itis reasonable to onsider a s enario, in whi h geometri al restri tions aloneprevent se ondary CH adsorption. For that reason we investigated two ge-ometries positioned to the inside of the orner-hole Fig. 9.6, Fig. 9.7. In this ase the CH adatoms interior turned out not to be stable while the adsorp-tion energy of the mole ule within the orner hole is 90 meV. Additionallytwo other on eptually possible ongurations were tested, from whi h onewas adsorbed between mid-adatoms Fig. 9.8 and one was adsorbed in the orner hole, perpendi ular to the symmetry axis Fig. 9.9 Here, surprisingly,the rst turned out to be the most stable onguration with 122 meV, whilethe other was less stable at 58 meV. On the basis of the obtained results, one an on lude that there is a large variation of possible physiorbed geometries,with very similar energeti s. The most stable geometry, however, ontradi tsexperimental observations, in whi h CH adsorption is more stable then midadatoms adsorption.The obtained results therefore are unable to unambiguously onrm thepreferable onguration in the experiments. In on lusion, standard DFTmethodology in whi h van der Waals for es are not in luded turns out not tobe a urate enough in examination of 1, 2dibromethene physisorption onSi(111)(7×7) surfa e. For all physisorptions the distan e of the Br atomto the nearest surfa e atoms is between 3.3 and 4.0 Å as presented in theTable 9.2.Additionally bromine hemisorption was also tested (Table. 9.2.3) and itshows that energeti s are independent of the lo ation of the adatom, whetherit is a mid or CH adatom. The omparison between faulted and unfaultedhalf shows a dieren e of 0.05 eV, showing there is no signi ant sele tivitybetween faulted or unfaulted halfs.9.2.4 Con lusionsIn summary the results presented here show that no single unambiguousgeometry ould be found from the perspe tive of standard DFT al ulations.The stable ongurations obtained are in agreement with the experimentalproles in two ases: the CH adatom and rest atoms ongurations, for both67

trans and gau he mole ule. It seems also possible that the entral CH atom isinvolved in the adsorption, whi h supports a geometri al hindran e s enariodue to hole saturation. As for bromine adsorption, no preferen e betweenthe mid and CH atom is observed supporting the thesis that the sele tivebromination is an ee t of CH sele tive 1, 2dibromoethene physisorption.Due to the omputational expense of these al ulations and a presumedneed for thi ker unit ells the investigation was terminated at this point.Subsequently, in reased omputing power oming to bear on a mu h larger ell was able to pin down the energeti s to subtle hanges in the subsurfa earrangement whi h fully a ounted for the ee t. This work was publishedonly re ently by M. Ebrahimi et al. [82.

Figure 9.4: Top and side view of simulated physisorbed trans1,2dibromoethane above orner-hole adatom and neighbouring entral orner-hole atom. .

68

Figure 9.5: Top and side view of simulated physisorbed gau he1,2dibromoethane between orner-hole adatom and rest atom.

Figure 9.6: Top and side view of simulated physisorbed gau he1,2dibromoethane between orner-hole adatom and neighbouring interior orner-hole atom69

Figure 9.7: Top and side view of simulated physisorbed gau he1,2dibromoethane between orner-hole adatom and neighbouring entral orner-hole atom.

Figure 9.8: Top and side view of simulated physisorbed trans1,2dibromoethane between two mid-adatoms.70

Figure 9.9: Top and side view of simulated physisorbed trans1,2dibromoethane between orner-hole atoms.

Figure 9.10: Top and side view of the three simulated bromine adsorptions:a) faulted orner-hole, b) unfaulted orner-hole ) unfaulted mid-adatom.71

9.3 Ethylene on Si(100)This se tion is on erned with thermal and ele tron stimulated migrationof ethylene mole ules. Adsorbate migration is a fundamental step in manysurfa e phenomena, in luding self-assembly, material growth, phase sepa-ration and heterogeneous atalysis. Migration has been interpreted pre-viously in the work of other laboratories as diusion via sequential ran-dom hopping [83. Real-spa e and even real-time observations of surfa ediusion have been a hieved using s anning tunnelling mi ros opy (STM)[84, 86, 87, 88 and ultrafast laser spe tros opy [88, 89, 90. In all these asesof diusion, the dire tion of motion is random, and it is also short-range ( 5Å), even on smooth metal surfa es [84, 86, 87, 88, 89, 90, 91, 92, 93.In other studies re oil away from surfa es has been studied under thename of DIET (desorption indu ed by ele troni transitions) [94. Thesestudies have provided valuable insights into the dynami s of surfa e diusiondue to thermal, ele troni , or opti al pro esses. Re oil towards surfa es isimplied by the frequent observation in re ent studies of ele tron-indu edsurfa e rea tions [95. Here, however, a dire ted long-range re oil in theplane of the surfa e is analysed, whi h is a genuinely novel phenomenon.The novelty resides not only in the long-range in-plane re oil, but also in theobservation of rea tions at a substantial distan e from the originating events.This ontrasts with frequent observations, in previous work, of lo alized hemi al rea tions [96, 97, 98. The dieren e arises from the fa t that in thepresent instan e the exothermi pro ess at rst yields a mobile physisorbedspe ies, whi h only later rea ts, whereas in the ase of a lo alized rea tion a hemi al bond forms immediately adja ent to the reagent mole ule.9.3.1 Experimental observationsChemisorption of ethylene on Si(100) has been the subje t of extensive ex-perimental studies [98, 100, 101, 102, 103, while theoreti al simulationssupporting the eviden e have been obtained [104, 105. It has been shownthat ethylene atta hes to the surfa e by opening its πbond and formingtwo CSi σbonds. There are two observed ongurations, one where themole ule is adsorbed between two dimers, the more ommon one, where itadsorbs dire tly on top of a single dimer, whi h is about 90 % of the totalethylene population adsorbed at room temperature whi h also agrees withthis studies. This se tion's fo us is to investigate the ause of long-distan emole ular migrations indu ed by the tunneling urrent, for voltages below athreshold of−3V. A ording to experimental STM data by K. R. Harikumar,72

John C. Polanyi and Amir Zabet-Khosousi the mole ule, at rst adsorbeddire tly above the Si dimer, while stimulated migrates for distan es as faras 200 Å. A ording to experimental results this is a one ele tron pro ess,and the migration of the mole ule has a dire tional preferen e, as shown inFigures 9.11 and 9.12.The experimental data have also shown a signi ant temperature depen-den e, as presented in Fig. 9.12. The elevation of the temperature alteredthe out ome of ele tron-stimulated events. The desorption probability risesfrom 5 to 10 and 15 % from 25C to 100 C, to 150 C, respe tively. More im-portantly, the radial and angular distributions hange signi antly. Radiallythe exponential distan e dependen y de ays while the angular distributionbe omes sharper. At 25 C the angular distribution exhibits broad peaks at0 and 63. At 100 C the peaks moved toward larger angles and at 150C the migration was dominant in the 90 dire tion, whi h is along the CCbond of the initially adsorbed onguration.Some of the experimental observations provided by Polanyi et al. were on erned with mole ular migrations as a result of dihalogenation of the Sisurfa e, the onsidered halogens were F , Cl, Br. This pro ess took pla e ata temperature of 25 C. Upon dosing with di-haloethene two new surfa e fea-tures were observed. There have been no features that ould be attributed tothe inta t adsorption of the di-halogen-ethylene (DXE) mole ules. Further-more, it has been re ognized that the separation between the hemisorbedhalogens and the migrated mole ules similarly to the ele tri ally stimulatedethylene mole ule is up to 80 Å distant from the nearest halogen pairs.The work done here is fo used on the analysis of the ex ited states lo- alized in the bond between the mole ule and the surfa e, for the energiesabove the threshold of −3 V. The investigated s enario for the observed phe-nomenon was the possibility of a resonan e of the tunnelling ele tron withthe anti bonding state ausing the SiC bonds to break, while also indu -ing a torque on the mole ule, ausing its rotational ex itation and thus farmigration. The observed long range migration often even surmounts surfa edefe ts, other hemisorbed spe ies or elevated terra es.The observation that re oiling ethylene migrates over distan es as greatas 200 Å surmounting steps and exhibits dire tional preferen e suggests thatthe mole ule moves ballisti ally rather than diusively. Diusive motion in-volves sequential short range hopping in random dire tions, s attering theabsorbates from its initial re oil dire tion. Additionally, on the Si(100) sur-fa e one would expe t the preferred dire tion to be along the dimer row dueto the lower diusion barrier. This however does not orrelate well with73

Figure 9.11: Bias dependent STM images of hemisorbed ethylene. Ele tron-indu ed ethylene migration [106. a) STM image ( +2.0 V, 0.2 nA, 25 C)showing three intra-dimer ( ir les) and one inter-dimer (oval) hemisorbedethylene on Si(100). b) STM image of the same area after an ele tron pulse(−3.0 V, 0.2 nA, 0.5 s). The lo al pulse lo ation is marked by a lightningbolt. A white arrow shows migration of the pulsed ethylene by 41 Å overa few surfa e features. Dashed lines above and below the images indi atethe middle of a dimer rows. ), d) STM images of (−1.7 V, 0.1 nA, 25C)of identi al areas ) before and d) after an ele tron pulse (−3.5 V, 0.1 nA,2 s) on the hemisorbed ethylene. White arrows shows ethylene migrationby 104 Å onto an upper terra e. e) Yield of ele tron indu ed migration as afun tion of surfa e bias. A threshold of −3.0 V obtained assuming a linearthreshold law. f) Rate of ele tion indu ed migration obtained at a bias of−3.2 V as a fun tion of urrent. A linear relation is observed indi ative of asingle ele tron pro ess. Solid lines represent the best linear ts to the data.74

observed data. To a ount for the observation a dierent me hanism hasbeen proposed in whi h the mole ule rotates upon re oil, whi h drives it into artwheel motion perpendi ular to the dire tion of the surfa e.Initially a part of the kineti energy would be stored in rotational motionredu ing the probability of desorption. The distan e of a migration woulddepend on the rate of transfer of this rotational energy to the surfa e. Thisin-elasti ity is expe ted to de rease over distan e until the mole ule is nolonger able to es ape the attra tive potential of the surfa e atoms.In order to support this s enario we have analysed the me hanism bywhi h an initial rotation ould be indu ed. The proposed s enario for rollingof ethylene mole ules would have to be aused during the pro ess of asym-metri bond breaking, whi h would reate a torque on the mole ule andinitiate a rolling motion.9.3.2 Theoreti al methodsTheoreti al simulations of the ground state of ethylene, the ground stateof DFE physisorbed on the surfa e, and the nal state of a rea ted DFEmole ule were performed using VASP [30, 32, 23. A 4 × 4 super ell ofSi(100) was simulated using a slab of eight layers with the bottom layerpassivated by hydrogen. The adsorbate mole ules and three surfa e layerswere relaxed until for es on individual atoms were below 0.02 eV / Å. The uto energy was set to 350 eV.The simulations used generalized gradient approximation (GGA) ex- hange orrelation potentials and proje tor augmented waves to obtain theele troni ground state [107, and typi ally integrated the surfa e Brillouinzone of the system with only the gamma point. Based on obtained ge-ometries, the origin of the torque was explored and the ex ited states of theethylene mole ule were simulated by the Delta Self-Consistent-Field (DSCF)method as implemented in GPAW. The ethylene al ulations in GPAWwhereexe uted by Haiping Lin while NEB al ulations of DFE were exe uted byWerner Hofer. The NEB simulations were undertaken with the VASP odeusing the same setings as spe ied earlier [25.9.3.3 ResultsThe transiently o upied orbital was identied from the pool of uno upiedorbitals in the energy range from HOMO to HOMO+3.5 eV; it is shown inFigure (9.13).Figure (9.14b) shows the results of al ulations for a negatively- harged75

Figure 9.12: Radial and angular distribution of ele tron-indu ed migrationat various temperatures [106. a), b), ) Radial distribution (bin size 15 Åat a) 150 C b) 100 C and ) 25 C. Solid line is an exponential t to thedata. d) Angular distribution at 150 C (solid line) , 100 C (dashed line),50 C (dotted line).

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Figure 9.13: The gure presents a harge density of band 353 whi h was hosen as the main state parti ipating in the pro ess of ele tron stimulatedmigration [106. Its energy is 3.08 eV above the HOMO level, and its shapeis asymmetri al around the mole ule, where the majority of the density islo alized on one of the CSi bond ( al ulated in GPAW). hemisorbed ethylene following a Frank-Condon transition, i.e. without re-laxation of the nu lei. The energy of the ioni state is al ulated to be 3.08 eVabove the ground state, in ex ellent agreement with the observed thresholdfor ele tron-indu ed rea tion. The harge distribution in the ex ited stateshows asymmetri harge lo alizations on the two CSi bonds. The higher harge density on one CSi bond is a ompanied by a larger repulsive for e(0.17 eV/Å) on the C atom, reating a net torque that ould initiate motionalong the dire tion of the CC bond axis. The observed out ome depends,however, on the integral of this and subsequent for es in the negatively- harged state, and also following reversion to the ground potential-energysurfa e.This was onrmed by means of a nudged elasti band al ulation (NEB)for the rea tion path-ways of diuoroethene in the pro ess of asymmetri re oil during uoridation of the Si dimer a) Figure 9.14a.9.3.4 Con lusionsIn on lusion, we demonstrate theoreti al support for the long-range mi-gration of ethyleni mole ules on Si(100), indu ed by six surfa e rea tionsinvolving CC πbond formation. We show the generality of migration for(i) the thermal rea tion of three related dihaloalkanes and (ii) the ele tron-indu ed rea tion of a series of three related hemisorbed alkenes at the same77

Figure 9.14: DFT al ulation of thermal and ele tron indu ed re oil of ethy-lene [106. a) Cal ulated minimum-energy path for the thermal disso iationof DFE on Si(100). Twelve NEB ongurations from the initial state step 0to the nal state (step 11) were al ulated. The presented slides present howthe rea tion pro eeds. White arrows show dierential for es on the arbonatoms. b) is a al ulated harge distribution in the rst ex ited state of hemisorbed ethylene on Si(100). White arrows represent for es on arbonatoms due to a Fran k-Condon transition from the ground to the ex itedstate. 78

surfa e. Ea h migratory event led at its terminus to a diσbound ethylene(90 % intradimer, 10 % interdimer). Migration distan es averaged 29 Åfor the room-temperature surfa e, in reasing to 87 Å for the 150 C surfa eand extending up to 200 Å. This long-range migration o urs despite thewell known roughness of this surfa e that makes possible STM observationof adsorbates on Si(100) at room temperature. The observed migration isenergized by the formation of a CC πbond. For ele tron-indu ed re oilof ethylene the migration is dire ted along the initial CC bond axis in the hemisorbed state. The me hanism for the migration gives eviden e of be-ing rolling, rather than translation, of re oiling physisorbed mole ules a rossthe surfa e, be ause obsta les are surmounted and end-to-end inversion isexhibited. The rotation is indu ed by torque during the re oil of ethylenefollowing surfa e di-halogenation (in three examples examined) or during theele tron-indu ed re oil of the ethyleni mole ules from their surfa e ounter-parts (in three further examples). Eviden e of this torque omes from DFT al ulations performed for both the thermal and ele tron-indu ed rea tions.These ndings open up possibilities for the use of hemi al energy to dire tlong-range adsorbate migration a ross surfa es, and also oer a new meanswith whi h to study adsorbatesurfa e intera tions during the ourse of sur-fa e migration. Dire ted mole ular movement over many tens of Angstromsfollowed by hemi al rea tion oers a means to promote rea tion at a dis-tan e. Additionally, the understanding of long-range migration should assistin appli ations that require substantial movement a ross a surfa e, su h asmole ular nanoma hines [108, 109.

79

9.4 A etylene on Si(100)In this work we investigate a range of possible a etylene geometries on theSi(100) surfa e together STM simulations. So far, this system has re eivedmu h attention be ause of its relevan e to developing te hnologies. In thisrespe t, various hemi al pro esses and mole ular ongurations have beenanalysed due to their importan e for appli ations in mole ular ele troni s, ortowards a omprehensive method of hemi al and physi al fun tionalisation[110. The adsorption of simple hydro arbons su h as a etylene was foundto be essential for analysis of the rst stages in organi lm growth.9.4.1 Existing experimental and theoreti al dataDue to its triple CC bond, the a etylene mole ule on Si is very rea tive. Ithas been on luded that during a etylene hemisorption the stability of thesili on dimer bond is not ae ted [111. It bonds to the Sidimer by hangingthe hybridization of the arbon ele trons from sp3 to sp2. A ording toearly experimental a ounts of a etylene adsorption, a di-σ-bonded mole ulewas believed to be the only stable onguration. However subsequent STMimages and a areful examination have shown that this assumption maynot be orre t. In two arti les by Terborg et al.[112 and by Xu et al.[113 it was laimed that two adsorption ongurations exist, namely a di-σ-bonded (Figure 9.15, and d) bonded to two Si atoms and a tetra-σ-bonded (Figure 9.15, f and g) onguration bonded to four Si atoms. Thisled to the proposition of stable tetra- oordinated ongurations, whi h weshall all the pedestal and "rotated pedestal" (Figure 9.15, f and g). Dueto the un lear interpretation of experimental results, a series of theoreti alstudies have been ondu ted [114, 115, 116, 117, 118, 119, 120, 121. Theresults of these simulations are summarized in Table 9.4. Adsorption energiesof dierent ongurations onsidered in theoreti al studies using dierenttheoreti al methods are onsistently higher for the di-σ than for the tetra-σ onguration. However, despite the agreement of theoreti al studies uponthe energeti s, the interpretation of available STM images is still somewhatun lear [117, 120.Notwithstanding the large number of published results, there are still ontroversial issues to address su h as the appearan e of the mole ule ins anning tunneling mi ros opy (STM) images [118, 122, 123, or the inuen eof van der Waals (vdW) intera tions on the energeti s. VdW intera tionshave only re ently been implemented in density fun tional theory (DFT) al ulations, and it has been demonstrated that they may hange the surfa e80

and adsorbate distan e and therefore have a nonnegligible ee t on the nalresult [124. For these reasons we have de ided to revisit the problem ofa etylene adsorption on Si(100) with the improved theoreti al methods toour disposition today.The Si(100)-(2×1) surfa e is re onstru ted due to the formation of dimersfrom top layer atoms, so that the number of dangling bonds is redu ed by50% (for more details see se tion 7.3. The surfa e is omposed of SiSidimer rows, whi h in their energeti groundstate form a bu kled (4×2)stru ture. At room temperature however, the surfa e involves asymmetri bu kling vibrations of the dimers, in ee t ausing a ip-op motion betweenopposite bu kling ongurations. Due to the low time resolution of an STM,the surfa e is then imaged as a (2×1) re onstru tion [44.(a) (b) ( ) (d) (e) (f) (g)(a) Double bridge 3.19(b) Double rotated bridge 3.72 2.79 2.93 3.35( ) Bridge 2.89 2.63 2.97 3.1 3.2 2.75(d) Rotated bridge 2.53 2.87 2.5 2.63 3 2.45(e) Ring 2.53(f) Pedestal 2.13 1.04 1.2 0.5 1.8 1.3(g) Rotated pedestal 2.04 1.71 2.0 1.4 2.14 2.5 2.15([125) R-bridge Si -ring 3.08Table 9.4: Overview of theoreti al adsorption energies (eV) of single a ety-lene mole ules on Si(100)-(2×1). (a) Lu [125, (b) Morikawa [114,( ) Hofer[115, (d) Mezhenny [120, (e) Cho [116, (f) Kim [118, (g) Sores u [119.The bra keted letters next to the names refer to the onguration in theFigure 9.15. The R-bridge Si-ring onguration is the one proposed by Lu,where in addition to a standard rotated bridge the onguration ontains apair of un oordinated Si-bonds.In order to larify the ontroversial issues we have performed a seriesof theoreti al al ulations. The rst part of our analysis is fo used on theinvestigation of dierent possible ongurations and their energeti stabili-ties. We employed standard DFT, with the addition of post-pro ess vdWsimulations based on the LangrethLundqvist fun tional and also omparedthis set of simulations with DFT enhan ed by semi-empiri al simulations ofvdW intera tions. Subsequently, the most stable ongurations were hosenfor STM simulations. 81

Figure 9.15: Adsorption ongurations of a etylene C2H2 on Si(100)(2×1):(a)double bridge two adja ent bridge adsorption, (b) double rotated bridge two adja ent rotated bridge, ( ) bridge dire tly above single dimer, (d)rotated bridge between two ends of dimers adja ent in the dimer row, (e)ring two a etylene mole ules bonded in the four C ring between two dimers,(f) pedestal four bonded parallel to the Si dimers, (g) rotated pedestal four bonded perpendi ular to Si dimers, (h) twisted bridge diagonallybetween two dimers.82

9.4.2 Theoreti al methodsBe ause of the simpli ity of the system in our investigations we mainlyused non-self- onsistent post-GGA methods to a ount for vdW intera -tions, whi h were, however, ben hmarked against self- onsistent iterationswith semi-empiri al vdW methods.In post-GGA, the vdW fun tional is an approximation for the orrelationenergy alone [17, while the ex hange energy is taken from standard GGA al ulations. However, the right avor of the GGA needs to be appliedin order to avoid an overestimation of the ex hange part of the bindingenergy [126, 127. The DFT method used for the relaxation and for the al ulation of ele troni stru ture in our studies is the Vienna Ab-Initiosimulation pa kage (VASP) [31, 32. In all simulations we employed PBEand RPBE fun tionals for ex hange- orrelation potentials and the proje tor-augmented-wave (PAW) method. The ioni relaxation was performed untilthe Hellman-Feynman for es rea hed values smaller than 0.02 eV/Å. TheSi(100) surfa e, with its bu kled dimer re onstru tion, was mimi ked by a (4×2) super ell. The latti e onstant was 5.46 Å [128. The surfa e slab onsisted of eight layers with a 19 Å, va uum, the last two layers were keptxed during relaxation. The bottom layer was passivated with two hydrogenatoms per sili on atom. The Brillouin zone sampling was limited to thegamma point.Due to the requirement of a urate harge density representations alsofor the ore regions, it was ne essary to use a high resolution Fourier gridin the simulations. To this end the energy uto was set to 600 eV, in this ase the neighbouring grid points of the real spa e grid have a separationof less than 0.1 Å. Based on the obtained harge densities the nal stepin our al ulations was the al ulation of vdW intera tions. This approa hallows adjustment of the resolution depending on the radial and angular oordinates of the atoms whi h minimizes the omputational expense withnegligible hanges in a ura y [18. Energy values obtained from standardDFT al ulations in this step are updated by the vdW orre ted orrelationenergies al ulated a ording to 4.2.The above two-step simulations were repeated for eight dierent ong-urations bridge, rotated bridge, pedestal, rotated pedestal, twisted bridge,double bridge, double rotated bridge and ring shown in Figure (9.15). Inaddition to simulations with non-self- onsistent vdW orre tions, we alsoperformed self- onsistent relaxations using semi-empiri al Grimme orre -tions [20, where dispersion orre tions take the form of C6R−6 as des ribed83

in se tion 4.2.9.4.3 Results and Dis ussionAdsorption energies for single mole ules were obtained with respe t to a ref-eren e system omposed of a lean surfa e and a single mole ule pla ed inthe middle of the va uum. For multiple adsorptions the referen e systemsimilarly ontained one mole ule in the va uum range while the other onewas already adsorbed on the surfa e. All al ulations were performed on a (4×2) bu kled ground state of Si(100). The overage for single and doubleadsorption then orresponds to a overage of 0.125 ML and 0.25 ML, respe -tively. The rst set of adsorption energies is based on non-self- onsistent al ulations and presented in Table 9.5. We nd that the most favourableadsorption onguration for a etylene is the double rotated bridge ongura-tion (Figure 9.15). It is worth pointing out that there is an energy dieren eof 0.36 eV between the bridge and the double bridge far apart (Figure 9.16),whi h ree ts the limited size of our unit ell: even the largest distan ebetween two mole ules in our unit ell does not ompletely de ouple themole ules. Given the limitations of omputational resour es, this annot beprevented. There is also a small dieren e in energy between adja ent andfar apart double bridge ongurations, slightly favouring the more distantsetup, indi ating a small repulsive potential, whi h is due to latti e strain.The pedestal and rotated pedestal ongurations are mu h less stable. Thesame applies to the twisted bridge. These three ongurations have thereforebeen ex luded in the subsequent analysis.

Figure 9.16: Ilustration of the unit ell of Si(100)(2×1) with two a etyleneC2H2 mole ules adsorbed in bridge ongurations above Si dimers lo atedfar apart from ea h other.To gauge the validity of the non-self- onsistent vdW simulations, we alsoperformed al ulations using semi-empiri al but self- onsistent methods forall ongurations with adsorption energies above 2.0 eV. The results of our84

E(eV/mole ule)(a) Double bridge 2.65(b) Double rotated bridge 3.17( ) Bridge 3.08(d) Rotated bridge 2.55(e) Ring from rotated bridge 2.12(f) Pedestal 0.65(g) Rotated pedestal 1.39(h) Twisted bridge 1.16Double bridge far apart 2.72Table 9.5: Results of al ulation for adsorption energies of a etylene onSi(100) using non-self- onsistent vdw-DF. The bra keted letters next to thenames refer to the onguration in the Figure 9.15. al ulations are presented in Table 9.6. Simulations have been performedusing both, RPBE and PBE fun tionals. As seen in Table 9.6 the dier-en e in energies for most ongurations is between 0.2 and 0.3 eV. For thering ongurations we nd slightly higher dieren es of 0.4 eV, presumablybe ause in this ase the mole ule is loser to the surfa e. Comparing self- onsistent and non-self- onsistent methods for the vdW intera tions, we notevery good agreement to within 0.1 eV for most ongurations. However, thebridge turns out to be less stable by 0.26 eV and the ring more stable by0.48 eV (see Table 9.7).We have also analysed the bond length and its dependen e on the methodused. The bonds of interests are shown in Figure 9.17. The values forstandard PBE (geometry used for non-self- onsistent vdw), PBE+vdw andRPBE+vdw are in luded in Table 9.8. The general on lusion here is thatthe use of RPBE in the relaxation pro ess produ es sparser systems byslightly in reasing the bond lengths with respe t to PBE. Comparing PBEand PBE+vdW (non-self- onsistent and self- onsistent geometries) we ndthat all bonds are shortened ex ept the CH bonds. Self- onsistent simu-lations based on Grimme orre tions depend to some extent on the utoradius: they tend to overestimate vdW orre tions for short bondlengths,and to underestimate them for long bondlengths. In our view this is the ase for the bridge and the ring from the rotated bridge, whi h both showthe largest energy dieren e between self- onsistent and non-self onsistenttotal energy values. In prin iple, all parameters in the simulations ould85

be analysed by a method developed by Hanke et al. [129. However, sin ethe ring from the rotated bridge, whi h shows the largest dieren e, is a tu-ally not ba ked by experimental data, we have omitted this analysis in thepresent work.ERPBE EPBE dieren e(eV/mol) (eV/mol) (eV/mol)(a) Double bridge 2.47 2.72 -0.25(b) Double rotated bridge 2.99 3.21 -0.22( ) Bridge 2.56 2.82 -0.26(d) Rotated bridge 2.25 2.57 -0.32(e) Ring 2.20 2.60 -0.40Double bridge far apart 2.46 2.74 -0.28Table 9.6: Comparison of adsorption energies from self- onsistent dispersion orre tion al ulation for PBE and RPBE fun tional DFT-D for dierent ongurations of a etylene on Si(100)(2×1). The bra keted letters next tothe names refer to the onguration in the Figure 9.15.Enon−sc Esc dieren e(eV/mol) (eV/mol) (eV/mol)(a) Double bridge 2.65 2.72 0.07(b) Double rotated bridge 3.17 3.21 0.04( ) Bridge 3.08 2.82 -0.26(d) Rotated bridge 2.55 2.57 0.02(e) Ring 2.12 2.6 0.48Double bridge far apart 2.72 2.74 0.03Table 9.7: Comparison of adsorption energies between self- onsistentand non-self- onsistent method for dierent ongurations of a etylene onSi(100)(2×1) . The bra keted letters next to the names refer to the ong-uration in the Figure 9.15.9.4.4 STM simulationsIn our theoreti al work we aimed at analysing the experimental ndingsof Mezhenny (Figure 9.18), who presents three visually distin t ongura-tions together with their line proles [120. Experimentally, the a etylenemole ules appear as depressions ompared to the non-rea ted dimers, one onguration o upying an area orresponding to a single dimer (Figure 9.18,86

Figure 9.17: An example of super- ell used in al ulations. The atoms in- luded in between dashed lines undergo relaxation, during the self- onsistentvan der Waals al ulations. The labelled bonds the ones onsidered in thedis ussion and in luded in Table 9.8.I) and the other two o upying an area of two adja ent dimers of the samedimer row (Figure 9.18, II and III). The rst approa h to STM simulationsdis ussed here is based on the TersoHamann approximation [130, 131,whi h is restri ted to representations of the lo al density of states (LDOS).Images of our ongurations using the TH method, together with line s ansare presented in Fig. 9.19 for a bias voltage of −1.5 V and in Fig. 9.20 for abias of −1.0 V. All images generated are for an LDOS value of 1×10−8/ eV.The dieren es between ongurations are unambiguous, and the obtainedimages were used to interpret all experimentally observed ongurations.In order to over ome the limitations of a nite unit ell size, whi h gener-ally makes it impossible to nd a urate ontour values from a single simula-tion [115, we used a numeri al interpolation s heme to mimi the transitionfrom a non-rea ted surfa e (at the edges of the unit ell) to a rea ted surfa e(at the position of the mole ule). While adja ent dimers are slightly higher inthe rea ted ontour, they appear un hanged in the interpolated ontour. Wend that a double rotated bridge onguration has a similar shape and lines an (Figure 9.19, b) as feature III, observed experimentally (Figure 9.18).However, for a bias voltage of −1.5 V the depth of the experimental depres-sion is 0.6 Å, while it is 0.25 Å, theoreti ally. But this dieren e is partly87

H-C C-C C-Si Si-SiPBEDouble R-bridge 1.096 1.364 1.924 2.436Bridge 1.094 1.36 1.919 2.365R-bridge 1.096 1.365 1.937 2.465Ring 1.097 1.580(‖) 2.006 2.3621.575(⊥)PBE + vdWDouble R-bridge 1.099 1.362 1.919 2.414Bridge 1.097 1.359 1.911 2.362R-bridge 1.098 1.362 1.925 2.427Ring 1.100 1.579(‖) 1.997 2.3531.574(⊥)RPBE + vdWDouble R-bridge 1.100 1.368 1.928 2.425Bridge 1.099 1.364 1.921 2.367R-bridge 1.100 1.369 1.943 2.449Ring 1.101 1.587(‖) 2.010 2.3611.583(⊥)Table 9.8: Bonds length measurement for dierent ongurations using stan-dard PBE fun tional for relaxation with and without vdW orre tion.due to an elevation of about 0.2 Å, of adja ent dimers in the experiments,whi h are used as referen e heights (graph E-E' on Figure 9.18). It is knownthat bonding of mole ules to sili on an lead to harging of adja ent dimerswhi h then in rease in apparent height [99. In our simulations, the eort toreprodu e also this feature in the experiments would be ome prohibitivelyexpensive, as it requires the use of hybrid fun tionals and a doping of thesili on surfa e. Moreover, it has already been established in the ited paperwhat the origin of this ee t a tually is.The other adsorption onguration with a double dimer footprint (Figure9.18, feature II) whi h shows an asymmetri prole, an be attributed to arotated bridge (Figure 9.19, d). In this ase, there is no in reased height forneighbouring dimers and our lines an in the dire tion a ross the dimers witha depth of 0.3 Å is in good agreement with experimental values. For theperpendi ular dire tion the depth of the depression is 0.4 Å , again in goodagreement with experiments.The last onguration in the experiments (Figure 9.18, feature I), where88

Figure 9.18: The experimental STM images [120 of a etylene C2H2 adsorp-tion on Si(100)-(2×1) at -1.0 V sample bias.the rea ted site is 0.2 Å lower than adja ent dimers, is the bridge ong-uration (Figure 9.19, ), where we nd a feature whi h has no apparentheight ompared to adja ent dimers. This an be understood onsideringthe intrinsi limitation of DFT in simulating the width of the band gap ofsemi ondu tors. It is well known that DFT has the tenden y to underes-timate the band gap, whi h in our al ulations would mean to sample thedensities of states further away from Fermi level than in the experimentals ans. For that reason, onsidering the experimental band gap of 1.1 V andthe DFT band gap of about 0.6 V we de ided to perform simulations also at−1.0 V bias and obtained following results (Figure 9.20).All depressions in this ase be ome deeper. The double rotated bridge 0.4Å , the rotated bridge 0.4 Å and the bridge 0.1 Å , whi h with additionally89

Figure 9.19: Simulated STM images for a density of states of 1 × 10−8/eV and a bias voltage of −1.5 V. The dotted lines on the right representlines ans a ross the images on the left. (a)double bridge adsorption attwo adja ent bridges, (b) double rotated bridge - adsorption at two adja entrotated bridges, ( ) single bridge dire tly above a single dimer, (d) singlerotated bridge between two ends of adja ent dimers in a dimer row, (e)ring two a etylene form a ring of four C atoms between two dimers.in luded perturbation of adja ent dimers provides lose to perfe t agreementwith the experiment. The other two ongurations not onsidered in experi-mental studies have the following proles; the double bridge onguration isseen as a protrusion of 0.1 Å for −1.5 V and a depression of 0.1 Å without onsidering the perturbation of adja ent dimers, and the ring ongurationis of 0.0 Å height for −1.5 V and a 0.1 Å depression for −1 V bias, againwithout onsidering any additional ee ts.9.4.5 Con lusionsWe have al ulated adsorption energies employing two methods, a self and anon-self- onsistent one, a ounting for vdW intera tions for a set of possible ongurations of a etylene on Si(100) surfa e. The obtained results agreedwith the previously published trends without any strong ee ts on the hi-erar hy of the possible adsorption aused by vdW intera tions. Simulationsshowed a good orrelation between self and non-self- onsistent methods ex-90

Figure 9.20: Simulated STM images using TersoHamman approximationfor density of states 10−8/ eV and bias voltage −1.0 V: The dotted linesrepresent lines ans along whi h the prole of the image is presented on thegraphs on the right hand side. (a) double bridge two adja ent bridgeadsorptions, (b) double rotated bridge two adja ent rotated bridge, ( )bridge dire tly above single dimer, (d) rotated bridge between two endsof dimers adja ent in the dimer row, (e) ring to a etylenes bonded in thefour C ring between two dimers. ept for the ring adsorption whi h deviated by 0.48 or 0.78 eV, depending onthe starting onguration. The most stable ongurations have been usedto generate STM images and three of the ongurations (double rotatedbridge, rotated bridge and bridge) were very well mat hed with the exper-imental ndings. The long prevailing problem, whether the double dimerdepression is a single-mole ule adsorption in a tetraσ onguration or atwo-mole ule adsorption with a similar double dimer footprint an now besolved in favour of double adsorptions. Based on the energy al ulationsalone the ring onguration is proposed as another andidate for a stable ge-ometry. However, there is no experimental mat h at present in the literature.Con erning STM simulations we highlighted the importan e of the voltagebias adjustment and the harge agglomeration on the adja ent dimers.91

Chapter 10Towards high overage:benzene on Si(100)In this hapter we investigate some of the fundamental pro esses that maya ompany high overage mole ular adsorption on semi ondu tor surfa es,while revisiting the long debated subje t of benzene adsorption on Si(100).In our work, through al ulations of total energies, transition barriers andsimulations of s anning tunnelling mi ros opy (STM) images, we providesome new insights on both methodology and physi s.The intera tion of a benzene mole ule with a Si(100) surfa e has re eivedgreat attention due to the importan e of benzene as a model system forstudying mole ular adsorption of aromati hydro arbon mole ules. In spiteof the apparent simpli ity of both benzene and the Si(100) surfa e, theground state adsorption geometry of this system has been under debate fornearly two de ades.The adsorption takes pla e at the bu kled Si(100) surfa e (see se tion 7.3)one would expe t orientationdependant rea tivity. As shown in se tions 7.3.1on some o asions, mole ular adsorbates signi antly inuen e or even xthe surfa e bu kling. This Si dimer pinning may, at some range, ae t thesurfa e hara ter and its rea tivity [99, 132. The early experimental ob-servations using nearedge Xray adsorption ne stru ture (NEXAFS) mea-surements, ombined with high resolution ele tron energy loss spe tros opy(HREELS) and thermal desorption spe tros opy (TDS) revealed that ben-zene mole ules are hemisorbed on a Si(100)(2×1) surfa e without mole -ular de omposition and two ongurations an be distinguished [133, 134.Sin e the early experimental a ounts ould not unambiguously determinethe adsorption geometries, various theoreti al models have been proposedand investigated [115, 156, 135, 136, 137, 138, 139, 140, 141, 142.92

It has been agreed for all observed ongurations that the bonding be-tween the C and Si atoms is expe ted to be of σ nature. The geometri- ally possible adsorptions ould be divided into two groups for the benzenemole ule: diσ bonded and tetraσ bonded. In either ase, all of the ongu-rations have been onsidered and investigated in previous theoreti al works.For the diσ bonded mole ule on top of the single dimer, a C1, C2 or C1,C4 pair of C atoms ould be engaged, while a C1, C4 pair alone ould alsobond in between two dimers along or a ross dimer rows, or even diagonallywithin the row (for visualization aid see Fig. 10.1).

Figure 10.1: Si (4×2) surfa e re onstru tion and a benzene mole ule. Thebla k frame represent the super ell and ream larger balls represent bu kledup Si atoms.For the tetraσ bonded ase, C1, C2, C4 and C5 atoms ould engage inbonding, bridging two neighbouring dimers in the dimer row or alternativelya bonding of C1, C2, C3 and C4 atoms ould form a tight bridge onguration,either symmetri al along the row alled tight bridge (TB) or a 90 rotatedversion with respe t to it.On a basis of multiple theoreti al investigations, two most stable on-gurations were determined for whi h the results are presented in Table10.1. In ea h lass, the most stable adsorption ongurations obtained fromrstprin iple al ulations are the standard buttery (SB) and tightbridge(TB), respe tively. As shown in Fig. 10.2, in the SB stru ture ea h benzenemole ule rea ts with the surfa e via a [4+2 y loaddition, that is, the C1and C4 atoms of benzene form two σbonds with two Si atoms of a dimer.The remaining C atoms are tilted up and the mole ule has a C2v symmetry.As a result, the C1 and C4 atoms are sp3hybridized and two πbonds of theadsorbate are retained in the tilted C atoms. The TB stru ture, however,is a [2+2 y loaddition produ t. Ea h benzene mole ule intera ts with twoSi dimers with its C1−4 atoms, whi h are on a plane parallel to the Si sur-fa e. The C5 and C6 are sp2hybridized and tilted away from the surfa e.The adsorbate also shows a C2v symmetry, in whi h the mirror plane is per-93

pendi ular to the [-110 dire tion. Regardless of the pseudopotentials andex hange orrelation fun tional used, the TB onguration is energeti allymore favoured than the SB onguration in all standard DFT al ulations(Table 10.1). Supported by rst-prin iple STM simulations, the SB stru -ture has been assigned to the metastable adsorption state, while the TBstru ture is regarded as the ground state [115, 137. This on lusion oersa logi al explanation for the reation of TB in whi h SB is an intermedi-ate state, less strained and thus more kineti ally favourable, while TB isa nal state whi h is energeti ally more stable. The eviden e supportingthis model an be found in examples of tipindu ed onversion and desorp-tion, demonstrating the possibility of ba k and forth swit hing between thosestates [143, 144, 145.

Figure 10.2: Adsorption geometry of an isolated benzene mole ule on aSi(100) (4×2) surfa e: (a) the SB onguration and (b) the TB stru ture.Although through this model an agreement with some experiments wasrea hed, a fair group of experiments still remain where the TB state is eithernot observed or is a se ondary state. This in onsisten y between experimentsposed new questions and raised doubts about the a ura y of both experi-mental and theoreti al methods. A broad experimental analysis was re entlypresented by Nisbet et al. to whi h we refer the reader for more in depth re-view of experimental data [146. Here, we will mostly fo us on the theoreti alside and on issues that have been omitted in earlier work. One of the latter94

an be that, sin e the gradient orre ted density fun tionals are unable todes ribe dispersive intera tions, van der Waals intera tions between benzenemole ules and the sili on surfa e are usually not in luded [147, 148, 149. Inorder to orre t for this methodologi al de ien y, Johnston et al. revisitedthis system with the van der Waals density fun tional (vdWDF) [140, 141.In their al ulations, the stru tural optimization was performed using thePBE form of GGA and ultrasoft (US) pseudopotentials. The total en-ergy, however, was al ulated to in lude the dispersion intera tions [126. In ontrast to other results, this work has reported that the SB stru ture is aglobal minimum whi h is about 0.08 eV more stable than the TB stru tureregardless of overage. It is important to note that the postGGA totalenergy vdWDF method employed in Ref. [140, 141 does not allow atomi relaxations when the van de Waals for es are al ulated. Taking into a - ount the small dieren e in the binding energies of two adsorption states,further stru tural optimization may play a de isive role in determining theground state adsorption geometry. In order to investigate this further inour work we de ided to use self onsistent DFTD as suggested by Grimmein Ref. [20. Here, the van der Waals intera tions are in luded through theuse of C6R−6 semiempiri al dispersion orre tions where C6 is a oe ientfor xed atom pairs (see Se tion 4.2 for more details). Sin e this methodallows ioni relaxations that also take van der Waals for es into a ount, theobtained geometry of adsorbed stru tures, e.g. on erning bond lengths, ismore a urate than that from standard DFT al ulations. This methodologyhas been employed throughout the work presented here. In Se tion 10.2, wealso tested the redu ed Grimme approa h, in whi h vdW intera tions be-tween substrate atoms are not onsidered and vdW orre tion is limited tothe mole ulesubstrate intera tion.As to this time, it has been pointed out in few publi ations that the dis- repan y between dierent experiments may be related to overage, defe tsor temperature. On one hand, the results are strongly temperature depen-dent via the ee t on the rate of onversion between two states. Therefore,at low enough temperatures only single state o upan y would be expe ted.On the other hand, there is a dieren e in sample preparation between theSTM studies and spe tros opi experiments. The rst usually takes pla eat low defe t density surfa es and lower overages, while for the se ond thedefe t density is not onsidered and the measurements take pla e at high overages. In many publi ations the unexplained reverse relative SB/TBpopulation was observed and it was related to the in rease of overage. Asa result, signi ant mole ulemole ule intera tions were suggested, whi h95

Pseudopotential GGA SB TB Di Ref(eV) (eV) (eV)PAW PW91 1.00 1.25 0.25 [155NC BLYP 2.04 2.10 0.06 [156US PW92 1.12 1.42 0.30 [115US PBE 0.82 1.05 0.23 [157US PBE 0.92 1.19 0.27 [140US RPBE 0.47 0.66 0.19 [140US PW91 0.99 1.24 0.25 [140Table 10.1: Computed adsorption energies (in eV) of the SB and TB ong-urations using dierent pseudopotentials and generalizedgradient approx-imations (GGA). ould ae t the energeti s of the adsorption or the height of the barrier inthe SB to TB transition. This subje t has been for the rst time theoreti allyapproa hed by Lee et al. [139.Here, we expand this dis ussion and also provide omparisons of DFT vsDDFT. In Se tions 10.5 and 10.6 we investigate the energeti s for inlineintera tions and the dire tionality as well as the interrow intera tion. Wealso provide omparison of the isolated vs full overage SB to TB transitionbarriers.In response to re ently published high overage studies, in whi h mixedSB/TB overages were investigated [145, we also have performed a seriesof STM simulations for whi h the results are presented and dis ussed in theSe tion 10.7.Finally, we present the al ulations of the newly proposed geometry forstable benzene adsorption of inter-dimer buttery (IdB) on the Ctype de-fe t [144. We further expanded on this results and perform stru tural anal-ysis of a high overage version of the IdB onguration for whi h we alsopresent simulated STM images in se tion 10.8.10.1 Computational detailsIn this work, all al ulations are arried out using the Vienna Abinitio Simu-lation Pa kage (VASP) [32, 80. The ele tronele tron ex hange orrelationintera tions are des ribed with the PBE form of GGA [13. In order to a - urately a ount for the ionele tron intera tions, the proje tor augmentedwave (PAW) method has been employed [22, 23. The optimized latti e96

onstant obtained from Si bulk al ulations is 5.47 Å, whi h is in agreementwith a previous theoreti al study [128. The mole ule and the ve uppermostlayers of Si atoms are allowed to relax in three dimensions.The energy uto used in all al ulations for the plane waves is 400 eV;the stru tural relaxation, arried out with the onjugate gradient method[150, 151, stops when the for es on ea h relaxed atom are smaller than0.01 eV/Å. At the bottom surfa e of the super ell, all Si atoms are passivatedby two H atoms per Si atom. In all al ulations the va uum range is about23 Å.The values of Eads are determined by the following equation:

Eads = Eref − E , (10.1)where E is the total energy of hemisorbed benzene on a Si(100) (4×2)surfa e, and Eref is the total energy of a referen e system, in whi h thebenzene mole ule is kept about 10 Å above the surfa e.10.2 Single adsorptionIn these al ulations, the sili on surfa e is modelled by a super ell whi h on-tains a 12layer Si(100) (4×2) slab. Ea h Si layer ontains 16 Si atoms andthe surfa e onsists of eight dimers, four per dimer-row. This set up resultsin 0.125 and 0.25 dangling bond saturation for SB and TB respe tively. Forthe te hni al parameters, we note that, due to the large size of the super ell,the rst Brillouin zone an be su iently sampled with the Γ point only.In order to understand the role that van der Waals intera tions play in theadsorption pro ess, three series of DFT al ulations have been performed:(i) standard DFT, (ii) Grimme DFTD and (iii) redu ed Grimme DFTD,in whi h the van der Waals intera tions between Si atoms are ex luded. The al ulated binding energies per mole ule are summarized in Table 10.2.Methods SB TB Dieren e(eV) (eV) (eV)DFT 0.93 1.14 0.21DFTD 1.55 1.84 0.29Redu ed DFTD 1.52 1.77 0.25Table 10.2: Binding energies (in eV) of the SB and TB ongurations al u-lated with dierent methods.As expe ted, both adsorption states are stabilized by taking van der97

Waals intera tions into a ount. A omparison of the binding energies ob-tained from the DFTD and the redu ed DFTD al ulations shows thatthe SiSi van der Waals intera tions do not have a signi ant ee t on Eadsof benzene on Si(100) (4×2) surfa e. The in rease of adsorption energies an be as ribed to surfa emole ule dispersion intera tions. Importantly,the DFTD al ulations indi ate that the TB state is energeti ally morefavoured than the SB state by around 0.29/0.25 eV for DFDD/redu edDFT-D, whi h is onsistent with standard DFT al ulations. This revealsthat the van der Waals intera tions do not hange the relative stability ofthe SB and TB states. In this work, the parameters whi h need to be testedare the free atom radius R0i and dispersion oe ient C6ii for Si, C andH, respe tively. Based on our al ulation, the un ertainty Eads is 0.11 eV.This means that the TB state is more stable than SB state even if the R0iand C6ii vary by 5% for all elements [129.A areful analysis of stru tural relaxations shows that the van der Waalsfor es do not ae t the mole ular stru tures of TB and SB states dramati- ally. The CSi bond lengths are almost un hanged although the van derWaals intera tions among the Si atoms introdu e a ontra tion of the latti e onstant by 1%. A Bader harge analysis also indi ates that the lo al hargeof all mole ular atoms does not vary mu h [152.10.3 Transition from SB to TBOn e it is established that benzene adsorbs on Si(100) (4×2) in a SB on-guration whi h then turns into the more stable TB, it is then possible to al ulate the transition barrier between the two states and hen e the rateof onversion. In order to do that, we have employed the Nudged Elas-ti Band (NEB) method ( al ulations performed by Chiara Panosetti), asimplemented in VASP 5.2.1, proposed by Henkelman and Jónsson [25 tosear h for saddle points, that is, transition states, and minimum energypaths (MEP) between known rea tants and produ ts. The NEB, as a hainofstates method, is more omputationally ostly than a single relaxation,we thus employed a slightly looser onvergen e threshold of 0.02 eV/Å. Therea tion oordinate was sampled using 3 intermediate repli as, for a total of5 images (in luding the initial and nal states), onne ted through a springfor e of onstant k = 5 eV/Å2.The optimized Minimum Energy Path for the onversion of hemisorbedbenzene from the SB onguration to the TB onguration is shown inFig. 10.3. The omputed energy barrier is 0.75 eV. Experimentally, the98

onversion barrier was estimated to be around 1.0 or 0.9 eV in two previousstudies [153, 154 respe tively, both al ulated from the measured rate of onversion R using the Arrhenius equation,R = A · exp−Ea/kT (10.2)assuming the preexponential fa tor A to be 1013.However, several non vdW orre ted theoreti al studies pla e the barrierin a range spanning 0.5 eV (CarParrinello method [156) to 0.87 eV (Gra-dient Proje tor Method [139) to 1.61 eV (Cluster method [138). Hen e,within our model, the onversion barrier lies in the range dened by theoret-i al literature but is underestimated with respe t to the experimental values.However, the geometri features of the transition states are in perfe t agree-ment with the results of Ref. [139. The high value of luster al ulationsindi ates that the stiness of the system, whi h an be inuen ed not onlyby its hemi al omposition and bonding stru ture, but also by its ele troni stru ture and dopants, may play an important role in the absolute valueobtained. Stru tural details of the transition state, ompared to that fromRef. [139, are illustrated in Fig. 10.4 and Table 10.3.We evaluated the ZPE orre tion in the harmoni approximation dis-pla ing the adsorbate degrees of freedom together with four underlying Siatoms by 0.005 Å. The Si atoms in luded in the displa ement are those form-ing the dimer to whi h the buttery benzene is atta hed, and the adja entdimer toward whi h the adsorbate bends along the rea tion path, whi h willbe ome bonded in the tight bridge nal state. The ZPE orre ted lassi ala tivation barrier is 0.67 eV. Using the quantum harmoni partition fun tionwith Wigner's formula [158 in luding the orre tion for tunnelling [159, weobtain an a tivation barrier of 0.70 eV.Furthermore, roughly estimating the van der Waals binding energies ofthe initial state and the transition state, by means of single point al ulationsat the van der Waals optimized geometries, we nd that, subtra ting the vander Waals orre tion, the barrier in reases to 0.79 eV. This is easily explained onsidering that in the transition state the mole ule is loser to the surfa e,more atoms are thus involved in the intera tion; the van der Waals bindingenergy is then larger for the transition states than for the initial state. Thisexplains the lowering of the onversion barrier when the dispersion orre tionis in luded in the al ulation.A summary of all the omputed onversion barriers is shown in Ta-ble 10.4. 99

Method d1 d2Gradient proje tor, no vdW [139 2.45 Å 2.86 ÅCINEB, vdW 2.50 Å 2.91 ÅTable 10.3: Relevant stru tural details of the transition state in the onver-sion of benzene from BS to TB, that is, the lengths of the forming CSibonds ( fr. Fig. 10.4). There is substantial agreement between the GradientProje tor method, whi h does not in lude van der Waals, used in [139 andthe Climbing Image NEB used in our al ulation.

Method Ea Referen eExperimental I 1.00 eV [153Experimental II 0.95 eV [154CarParrinello 0.50 eV [156Gradient Proje tor Method 0.87 eV [139Cluster method 1.61 eV [138CINEB, vdW 0.75 eV CINEB + ZPE, lassi al, vdW 0.67 eV CINEB + Wigner, vdW 0.70 eV CINEB, no vdW (estimated) 0.79 eV Table 10.4: Comparison of onversion barriers from literature with thepresent al ulations. Our results lie in the range dened by previous theo-reti al al ulations, but it underestimates the experimental value.100

Figure 10.3: Computed minimum energy path for the onversion betweenBS and TB states of hemisorbed benzene on Si(100) (4×2) obtained usingClimbing Image Nudged Elasti Band. The onversion barrier ( lassi al) is0.75 eV.

Figure 10.4: Relevant stru tural details of the transition state in the on-version of benzene from BS to TB, that is, the lengths of the forming CSibonds. Numeri al details are given in Table 10.3.It is also important to point out that at room temperature the Si dimersip qui kly up and down. In a theoreti al model this feature is not trivial to onsider; our simulation was arried out on a (4×2) re onstru ted surfa e.While the adsorption energies are not dramati ally ae ted by the hoi eof either re onstru tion, espe ially on erning hemisorption, it is lear thatin a dynami al pro ess involving a rea tion path, the dimer ipping mayalso have a signi ant ee t. This ee t is parti ularly evident when anadsorbed mole ule lo ally pins the surfa e in either onguration (see, forexample, [132), whi h, in this ase, is unknown.Finally, the underestimation of the onversion barrier with respe t to101

the experimental value an also be explained onsidering the limitations ofthe hosen theoreti al setup. It is known that, in general, the employmentof GGAbased fun tionals tends to underestimate a tivation barriers (see,for example, Ref. [160). For the purpose of the present work, where we aremainly interested in investigating overage ee ts, the underestimation of thebarrier is not signi antly hanged at dierent overages, whi h again pointsto a general, rather than a spe i short oming in our theoreti al model. We an thus still ompare a tivation barriers in order to point out the ee t ofsurfa e on entration on the transition from the buttery onguration tothe tight bridge (see Se tion 10.6).10.4 Adsorption on Cdefe tThe lean Si(100) surfa e has several types of ommon imperfe tions, su has step and point defe ts, all of whi h ould have an inuen e on surfa eproperties or rea tivity.The Si(100) point defe ts have been ategorized into three types: type Aand B represent single and double dimer va an y, respe tively [51, 52. Theorigin of the Ctype defe t was not lear from the beginning; dierent modelswere initially proposed, su h as va an ies [53, 54, surfa e or subsurfa e defe tatoms [55, 56 and water adsorption [57. Re ently, a on lusion has beenrea hed and the C defe t has been identied as a rea tion site of a watermole ule ommonly also referred to as interdimer disso iation of water (ID).A ording to re ent observations, the Cdefe t is an a tive site in benzeneadsorptions [144.In our study, we have performed DFT and DFTD al ulations of theadsorption geometry shown in Fig. 10.5. The adsorbates saturate 0.25 of thedangling bonds, two by a C-defe t and two by an inter-dimer buttery(IdB)adsorption geometry. The obtained bond length was 1.50 Å for SiH and1.68, 0.98 Å for SiO and OH, respe tively. Both the OH and H areslightly spread apart in omparison to the Si atoms dire tly below, repelledfrom the axial position by 0.04 Å and 0.05 Å, respe tively. The OH groupis pointing in the dire tion of the neighbouring dangling bond, whi h is on-sistent with the studies of the C-defe t alone (see se tion 8.2. The obtainedadsorption energies of IdB on C-defe t are 1.20 eV and 1.90 eV for DFT andDFTD, respe tively, whi h is also 0.06 eV higher than the TB ongurationfor both methods.These values are slightly smaller than results obtained by luster stud-ies [144, however still supporting the interdimer buttery as the most sta-102

ble benzene adsorption onguration. Similarly to a etylene mole ules onSi(100) (4×2), a se ondary interdimer adsorbent on the same dimer pairhas in reased bonding energy [161. In this ase the rea tivity of the interdimer buttery (IdB) is in reased greatly by ∼ 0.43 eV in omparison to sin-gle IdB adsorption due the disso iated water o upying the other two atomsin bonded dimers (Fig. 10.5). From this observation, it is suggestive to as-sume that some of the in rease in the binding energy results from redu edstrain, due to stabilization of two dimers and de reased surfa e bu kling.This will be further investigated in Se tions 10.6, and 10.8.

Figure 10.5: Adsorption geometry of an isolated benzene mole ule on Cdefe t on Si(100) (4×2) surfa e. A Cdefe t omprises an adsorbed OHgroup and H atom on the nearby dangling bonds of neighbouring dimers,benzene mole ules adsorbed in interdimer buttery onguration.10.5 Line overageIn this se tion, we fo us on the intera tion of a benzene mole ule with itsneighbours. In these al ulations, we in reased the length of the super ell todouble the size used in previous se tions, i.e. to eight dimers in a single row.The Si(100) (4×2) ell used onsisted of 7 layers of Si atoms with 8 dimersper row, thus a total of 16 dimers per unit ell. Ea h Si layer ontains 32 Siatoms. Due to the large size of the super ell, the rst Brillouin zone an besu iently sampled with the Γ point only. The energy uto used was 350eV. All the ongurations dis ussed in this se tion involve adsorptions onthe single dimer-row keeping the other dimer-row uno upied. In order to103

understand the role that van der Waals intera tion plays in the adsorptionpro ess, two DFT al ulations have been performed: (i) standard DFT and(ii) DFTD. The al ulated binding energies per mole ule are summarizedin Table 10.5, while adsorbed geometries are presented in Fig. 10.6. Our aimin performing this al ulation was to provide a more omplete pi ture onhigher overage mole ular adsorption and to investigate the possibilities ofintera tion between mole ules with respe t to energeti s of adsorption. Therst analysis presented here on erns the possibility of dire tional ordering.Two ongurations, labelled with a) and b) in Fig. 10.6, are onsidered.In both a) and b) there are four mole ules o upying eight out of sixteendimers, saturating one dimer row a ounting for a overage of 0.5. In a) allTB mole ules fa e the same dire tion along the line, while in b) one of themole ules is in reversed order. From the adsorption stability it is apparentthat the dire tion does not inuen e the energeti s enough to favour eitherof these adsorptions (see one way and one odd in a Table 10.5).By omparing onguration a) and b) one an see that the TB mole uleis reversed, thus its pre ursor SB mole ule must have o upied a dierentdimer in ea h ase. Earlier studies have shown that the dieren e betweenthese two SB ongurations is only ∼ 0.05 eV whi h also does not supportany dire tionality [139. In all following ongurations investigated in thisse tion there are three benzene mole ules rea ting with eight dimers in onerow as presented in Fig. 10.6 whi h due to mixed SB, TB hara ter resultin a dierent dangling bond overage: 0.313, 0.375, 0.375, 0.188, 0.25 for ),d), e), f) and g), respe tively.Further, we onsidered whether there is any energeti dire tional pref-eren e for SB to TB onversion while surrounded by TB benzenes pointingin a single dire tion ) in Fig 10.6. In d) and e) this SB transforms to TBby ollapsing to the left and righthand side respe tively. The dire tionalpreferen e has been found to be only 0.026/0.030 eV for DFT and DFTDshowing only a slight sidespe i preferen e, whi h is unlikely to have anystatisti al signi an e. Finally, we have performed SB to TB onversion inSB environments, here the obtained onversion energies have been found tobe 0.30/0.39 eV, whi h is ∼ 0.05/0.06 eV more exothermi than the on-version in a TB environment. This suggests that the SB to TB onversionenergy de reases with an in rease in TB population.In stru tural analysis of single phase adsorptions of TB or SB no ee tson angles or bond length were observed that ould be attributed to theintermole ular intera tions. For the TB geometry (bonded with C1, C2,C3 and C4) C2Si and C3Si is 2.01 Å and for C1Si and C4Si is 1.99104

Å with the angle C3C4C5 ∼ 110 and the unbounded wing in luding C5,C6 at 52.5 from the surfa e plane (47 experimental [146). For the SBgeometry C1Si and C4Si are 1.97-1.98 Å depending on the bu kling ofthe neighbouring dimers (1.93 Å experimental [146) while the angle ofthe wing from the surfa e plane is 19.4 (15 experimental [146). All theabove DFTD results were onsistent with single mole ule adsorption. From omparison of DFT to DFTD the CH and CC and CSi for TB bondsstayed un hanged while SiSi and SiC for SB shortened on average by 0.01Å. A stru tural analysis of mixed high overages will be given in Se tion 10.7when we dis uss the relevant STM images.When studying phenomena on a lean Si(100) surfa e it is importantto remember that adsorbates may ae t the bu kling in the neighbouringdimers. Comparing the height of the bu kled dimer atoms of the leansurfa e with the atoms of uno upied dimers on the left and right hand sideof an SB adsorbate in ) in Fig. 10.6, it is observed that the up atoms are0.1 Å lower while the down atoms are 0.05 Å higher, thus attening thebu kling. The same trend is present on uno upied dimers in d), e), f) andg) in Fig. 10.6 where the lowering or raising of atoms are on average around0.1 Å. The phenomenon of dimer attening or pinning, although it does notseem to play as important a role in this parti ular system, will be dis ussedin depth in future publi ations.Methods DFT DFT-D(eV) (eV)one way 0.000 0.000 (a) at Fig. 10.6one odd 0.008 0.001 (b) at Fig. 10.6TB environmentSB→ leftTB 0.268 0.345 ( )→(d) at Fig. 10.6SB→ rightTB 0.242 0.315 ( ) →(e) at Fig.10.6SB environmentSB →TB 0.300 0.389 (f)→(g) at Fig. 10.6Table 10.5: Full line relative adsorption energy of TB oriented in the samedire tion vs one odd TB oriented opposite dire tion. SB to TB onversionenergy in one dire tional TB environment. SB to TB onversion energy inSB environment.

105

Figure 10.6: Adsorption geometries of benzene lines. a) Full line of TB ad-sorbates with the same dire tional orientation, b) Full line of TB adsorbateswith a se ond TB oriented in opposite dire tion to the rest, ), d) and e)three metastable ongurations in dire tional TB environment for SB, left-TB and right-TB respe tively, f), g) two metastable ongurations in SBenvironment, SB and TB respe tively.10.6 Full overageFull overage al ulations have been performed in order to investigate interrow intera tions as suggested by the work of Self et al. [162. We have106

obtained the adsorption energy dieren es related to orrelations betweendimer rows and investigated a few possible phases. The al ulations wereperformed using a Si(100) (4×2) unit ell onsisting of eight dimers. Theadsorption ongurations are shown in Fig. 10.7. Dierent full TB andfull SB overages were investigated. The energy dieren es obtained arepresented with respe t to the most stable onguration TB and dividedby four, to give dieren e per benzene mole ule. The results do not showany meaningful intera tion between benzene adsorbates, suggesting no phasepreferen e in any SB or TB full surfa e overage. The average adsorptionenergy for TB is 0.25/0.3 eV higher than for the SB onguration whi h is lose to the single adsorbate energy dieren e of 0.21/0.29 eV for DFT/DFTD. This shows that with higher overage line adsorption ase in se tion 10.5the average stability of TB over SB in reases by ∼ 0.04/0.02 eV. From thestru tural analysis similarly to the high overage ase no geometri al hanges ould be attributed to mole ulemole ule intera tions. In ea h ase there are4 mole ules per unit ell, diσ bonded ongurations a), ) and g) a ountfor the 0.5 dangling bond overage (whi h is a saturation overage for theSB geometry) and tetraσ bonded onguration b), d), e), f) and h) a ountfor 1.0 dangling bond overage (whi h is a saturation overage for the TB orRTB geometry)After performing the al ulations presented in Se tion 10.4, we have de- ided to onsider the IdB geometry in full overage in whi h double dimero upan y ould in prin iple a t similarly to the C-defe t and stabilize bu k-ling and in rease adsorption energy as shown for double a etylene mole ularadsorption by Czekala at el. [161. In order to do that, the mole ules werearranged in a zigzag pattern as shown in panel g) of Fig. 10.7. The results(Table 10.6) show that, in fa t, the predi tion was orre t and in ompari-son to single adsorption multiple adsorption in reases stability. The averageadsorption energy per IdB vs SB rises from 0.17 to 0.04 eV and from 0.08to 0.09 eV for standard DFT and DFTD al ulations respe tively.Although from an energeti perspe tive, the IdB presen e is mu h moreexpe ted than another diσ onguration, it is still important to point outthat this possibility is somehow limited due to the rst single adsorptionbeing less likely to o ur. So far this high overage geometry has not been onsidered, neither experimentally, nor theoreti ally. But it is possible thatthis phase has been interpreted as SB in spe tros opi studies, as it has asimilar ratio of π and σ bonds. In this ase, STM experiments will allow anunambiguous identi ation.We have also performed the al ulation of another four-sigma bonded TB107

onguration, where the mole ule is rotated by 90 . The intention, with thissimulation, was to show that high overage does not hange the adsorptionenergeti s, unless it saturates both dangling bonds of a dimer without ausingtoo mu h strain, as is the ase for the zigzag-IdD. Rotated TB (RTB) gaveenergies of 0.22 and 0.23 eV for DFT and DFTD respe tively, whi h aresimilar to single adsorption. RTB adsorption has not been identied. Thelikely ause is that it only arises after onversion from IdB, whi h, as noted,is rather unlikely to o ur.Methods DFT DFTD(eV) (eV)TB 0.00 0.00 (b) at Fig. 10.7TBshifted 0.00 (d) at Fig. 10.7TBreversed 0.00 (e) at Fig. 10.7TB rev. and shif. 0.00 (f) at Fig. 10.7SB 0.25 0.31 (a) at Fig. 10.7SBshifted 0.25 0.31 ( ) at Fig. 10.7IdBzigzag 0.22 0.29 (g) at Fig. 10.7RTB 0.22 0.23 (h) at Fig. 10.7Table 10.6: Relative adsorption energies for dierent phases of full overageadsorption of TB and SB onguration and one full overage adsorption ofRTB and zig-zag IDB adsorption. The energies were obtained in referen e tothe lowest energy onguration TB therefore the most stable ones are thosewith the lowest value. The dis ussed ongurations are presented on Fig.10.7.To omplete the pi ture, we investigated the ee t of intermole ular in-tera tions on the onversion barrier between SB and TB, performing a NEB al ulation at full overage. With respe t to the isolated adsorption ase,the dieren e is that the neighbouring benzene may ause steri hindran eto the onversion, as well as inuen e the bu kling of the uno upied dimerwhi h ould in ee t hange the a tivation energy. These al ulations wereperformed with a (4×2) super ell in luding eight dimers in two dimer rows.The use of two dimer rows is ne essary to avoid an artefa t due to the bound-ary onditions. Had we employed a super ell ontaining only one row, thiswould have orresponded to the simultaneous transition of an innite line ofbenzene mole ules from SB to TB. The initial state onsists of four benzenemole ules in the SB onguration, while in the nal state one of the benzenemole ules is onverted to the TB onguration.108

Figure 10.7: Adsorption geometries for full overage of benzene on Si(100) (4×2) surfa e. Panels a) and d) represent standard and shifted SB phaseswhile b), ), e) and f) represent standard, reversed, standardshifted,reverseshifted phases of TBtype adsorption, g) IdBzigzag, h) RTB-typeadsorptionThe omputed onversion barrier at full overage is 0.74 eV whi h is only0.01 eV lower than single adsorption. Hen e, the hypothesis of the populationreversal an not be attributed to high overage mole ular intera tion as theee t on the onversion barrier due to the neighbouring benzene mole ulesis negligible. 109

10.7 STM simulationsIn this se tion we present STM simulations using the TersoHamann ap-proa h [130 as implemented in BSKAN [137 for a bias voltage −1.0 V andlo al density of states (LDOS) value of 1×10−8states/eV In order to providesome additional insight into the STM image interpretations of high overage ases in our analysis, we onsidered groups of mixed SB/TB adsorptions.The adsorbed ongurations together with their STM images are presentedin Fig. 10.8. We hose the geometries in su h a way as to provide a full pi -ture for all the predi ted possibilities. In Fig. 10.8 panel a) the mix onsistsof righthanded TB's (rTB) and SB's. From the prole of the rst threemole ules (rTB, SB, rTB) one an see how the overlap in the lo al densityof states (LDOS) ould make it di ult to distinguish the exa t stru ture.It is important to noti e that, due to the asymmetry of the stru ture, thedepression in between the rst two mole ules is more prominent than inbetween the se ond two. Additionally, due to a slight repulsion from therighthand side the SB is tilted to the left (CSi bond angle in respe t tothe dimer row is 89.3 , and the dieren e in height between opposing C andH atoms is 0.27 and 0.32 Å, respe tively). This an also be observed in theprole of the image. The image of the 3rd and 4th (rTB, rTB) mole ulesin a) shows another asymmetri feature whi h an be used to determine thedire tion of the adsorption. Here, the overlap is mu h weaker than previ-ously and a stronger depression is visible. The gradient of the left mole ule'sprole is lower than the right one. The asymmetry of the feature in betweentwo rTB or two lTB an therefore be used in interpreting the dire tionwhen lower or higher gradient an determine whether they represent the TBside whi h is bonded to the surfa e or tilted away from it. The image of4th and 5th (rTB, uno upied dimer (UD), rTB) an be analysed simi-larly with respe t to the dierent gradients; here, however, the depression ismu h deeper and UD states are visible. In Fig. 10.8 panel b) we analyse thesymmetri al arrangements with 1st, 2nd, 3rd (lTB, SB , rTB) and 4th,5th, 6th (rTB, SB, lTB) mole ules reating two hara teristi images ofnarrow and wide prole.In both ases there is a large overlap of LDOS reating a high prole.However, in the se ond ase the prole is wider due to two maxima fromtilted-up sides of TB adsorptions. The prominent depression in betweenTB and SB is another distinguishable feature, whi h is stronger and furtherapart for a se ond ase. In Fig. 10.8 panel ) we investigate symmetri TBadsorptions with an uno upied dimer in between them (1st, 2nd and 3rd,110

4th mole ules). Here the situation is similar to b), reating narrow and wideversions in the middle; however, UD states exposed in the middle of theprole are slightly more prominent for this ase with a more open prole. InFig. 10.8 panel d) we have simulated only a TB population. Here, anothertwo proles an be distinguished for pairs of 1st, 2nd and 3rd, 4th mole ules(lTB, rTB and rTB, lTB). One an re ognize the strong overlap reatedbetween tilted up from the surfa e parts of the TB's in the rst ase anda wider prole with more prominent middle depression for the se ond pair.Finally, in 10.8 panel e) we have examined an SB only population. Here alsotwo ases an be distinguished with a single and double uno upied dimerin between the mole ules 1st, 2nd and 2nd, 3rd respe tively. All the prolesare in-plane with strong overlap of rst pair and a prominent depression fora se ond pair.

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Figure 10.8: Simulated STM images of high overage mixed SB and TB ad-sorptions of benzene mole ule on Si(100) (4×2) surfa e for -1V bias andisosurfa e of LDOS 10−8states/eV together with the related stru tural rep-resentations.10.8 Stru tural analysisIn this se tion we fo us on a further analysis of IdB stability in the higher overage geometries. We investigated the stru tural hanges that may a - ompany high overage adsorption. It is known that due to dimer bu klingthe oordinates of the atoms in the layer below have to hange, whi h reates112

a more/less pa ked environment (down/up bu kling) ausing the oppositebu kling in the next dimer [49. It has been demonstrated that adsorbedmole ules an have a strong ee t on their surroundings. This in turn ouldlead to a more or less stable adsorption site for the next adsorbate. In Table10.7 we ompare a set of parameters for dierent IdB adsorptions in orderto hara terize the stru tural hanges and their possible relation with ad-sorption ongurations. At the top of the table we present sIdB (singleIdB) adsorption and the referen e to the higher overage ases. The ho-sen parameters are two angles of CSi bond α with respe t to dimer rowand β with respe t to the dimer, and two distan es a(Si,Si) between IdBbonded Si atoms and b(Si,Si) between the two Si atoms in the layer be-low(see Fig. 10.9). The analysed adsorption are nTBIdB (IdB in narrowTB environment) Fig. 10.9(a), wTBIdB (IdB in wide TB environment)Fig. 10.9(b), SBIdB (IdB in SB environment) Fig. 10.9( ) , CIdB (C-defe t adsorption) in Fig. 10.5 and zIdB (zigzag IdB adsorption) in Fig.10.7(g). The omparison of the α shows very little variation, whi h wouldbe expe ted due to the stru tural rigidity in this dire tion. The mole ule isexpe ted to be bent more in the dire tion des ribed by β, where bonds aremore exible. The maximum bend of ∼ 6 is observed in the zIdB asedue to the repulsion of adja ent benzene mole ules. For a and b distan esthe hange is more prominent for the nTB and wTB ases, in whi hthe a distan es are in reased by 0.04 Å due to the strain indu ed by theadja ent tetra-σ-bonded TB mole ules. When these results are omparedto the simulated energies, it an be seen that for the n-TB, w-TB and SB ases the neighbouring mole ules add additional strain to IdB, whi h lowerthe adsorption energy making this onguration even less likely than sIdBby ∼ 0.11 and ∼ 0.17 eV respe tively. We on lude that higher overagedoes not redu e the strain, and the in rease in energeti s of C and z asesshould be attributed to ele troni ee ts mainly, due to the hanges in theele troni stru ture of the dimer.In addition to the stru tural analysis we have also performed STM sim-ulations of the wTB- , nTB- and SBIdB ongurations. In Fig. 10.9 we onsidered two stru tures of IdB with neighbouring symmetri al TB adsorp-tions (lTB, IdB, rTB and rTB, IdB, lTB) and one with neighbouringSB (SB, TB, SB). Similar to what was dis ussed earlier in Fig. 10.8 panelb) there is a lear distin tion between the narrow a) and wide b) prole inFig. 10.9. In a) and b) LDOS of TwB is strongly asymmetri with respe t tothe dimer row with a narrower prole omparing to SB adsorption, thus not reating any strong overlap in the LDOS between neighbouring mole ules. In113

α β a(Si, Si) b(Si, Si)() ()sIdB 77.2/78.7 105.8/114.8 3.77 7.71nTBIdB 77.2 109.8 3.81 7.87wTBIdB 77.2 110.4 3.81 7.87SBIdB 77.4/77.7 113.2/113.9 3.78 7.77CIdB 77.9 110.7 3.77 7.73C-IdB, vdW 77.9 110.4 3.77 7.73zIdB 78.6 116.1 3.76 7.74zIdB, vdW 78.6 116.3 3.77 7.73Table 10.7: Stru tural information for group of dierent geometries: singleadsorption of IdB (sIdB), IdB adsorption in narrow TB environment (nTBIdB) (panel a) in Fig. 10.9), IdB adsorption in wide TB environment(wTBIdB) (panel b) in Fig. 10.9), IdB adsorption with SB environment(SBIdB) (panel ) in Fig. 10.9), IdB adsorption on a C-defe t C-IdB inFig. 10.5, zigzag IdB adsorption (panel a) in Fig. 10.7). Here, α is angle of

CSi bond with respe t to dimer row dire tion, β is an angle of CSi bondwith respe t to the involved dimer (two numbers are present if bu kling stillpersist in underlying stru ture), A is the distan e between both Si atomsinvolved in bonding IdB mole ule, B is a distan e between two pairs of Siatoms in the layer beneath the Si dimer (see Fig. 10.9).

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) the overlap is mu h more prominent with a similar height of the prole forboth SB and IdB thus on ealing the IdB's features slightly; the asymmetryis still visible whi h is a hara teristi signature of the IdB geometry.

Figure 10.9: Simulated STM images of high overage mixed SB/TB withTwB adsorptions of benzene mole ule on Si(100) (4×2) surfa e for −1.0V bias and isosurfa e of LDOS 10−8states/eV together with the relatedstru tural representations.115

10.9 Con lusionsIn summary, our al ulations of the single adsorption geometries agree withthe other results previously obtained using dierent DFT fun tionals. Themost stable adsorption observed is the tight bridge (TB) in Fig 10.2b), whi his 0.21/0.29/0.25 eV more stable than Standard Buttery (SB) (panel a) inFig. 10.2) for DFT, DFTD and redu ed DFTD. The transition barrierobtained is 0.75 or 0.79 eV with and without dispersion orre tion. Overall,Grimme's dispersion orre tion slightly de reases the bond length and on-tributes to stronger bonding and transition barrier lowering. Additionally wealso orre ted the barrier by ZPE ontributions and obtained a barrier heightof 0.67 eV and 0.70 eV in luding the orre tion for tunnelling. Furthermore,for the Cdefe ted surfa e the al ulations of interdimer benzene IdB ad-sorption has onrmed the proposal of another energeti ally stable benzene onguration on the Ctype defe ts, with adsorption energies of 1.2/1.9eVwhi h is 0.27/0.35 eV higher than that of SB and 0.44/0.43 eV higher thanIdB alone, for DFT and DFTD respe tively. In the light of these results and luster al ulations presented at Ref. [144, we onrm the strong eviden efor this onguration on Cdefe ted surfa es. We also highlighted the ee tof in reased adsorption energy due to double dimer adsorption, whi h in this ase has been a hieved by OH and H but is also possible by means of othermole ules, as shown in a double sided intradimer adsorption of a etylenemole ule in Ref. [161. We su essfully tested this idea with high overagezigzag IdB in Fig. 10.7g) and observed the in rease of adsorption energiesto −0.04/0.09 eV in omparison to SB adsorption making it more likely toappear in this phase than as a single adsorption whi h is −0.17/ − 0.08 eV,for DFT and DFTD respe tively.These phenomena an be important not just in hydro arbon mono-layergrowth but also in other self-assembly appli ations su h as for nanos aledevi e manufa turing.An example of this was re ently shown by Bel her atel., where the benzonitrile mole ules were used as nu leation and terminationsites for metalli hain-growth [163.In our high overage analysis of SB and TB states, we have found nosigni ant environmental ee t on the a tivation barrier, energeti s or anydire tional sele tivity in SB to TB onversion. Based on full overage al- ulations, we on lude that TB is still the most stable ongurations. Fora singlestate full overage no phase preferen e has been found, indi atingthe observed phases in Ref. [162 are statiti ally equally probable and theintera tions in between dimer rows are negligible.116

We have also performed a high overage STM analysis providing all thepossible proles expe ted in the mixed and singular SB and TB adsorptionand highlighted a range of hara teristi s that may ease the interpretationof the geometries. We also presented the mixed SB/TB with IdB adsorptionto provide full des ription for another analysed adsorption onguration. Inaddition we provided stru tural analysis of the high overage IdB ongura-tions and on luded, that the neighbouring adsorbates do not stabilize theIdB but instead reates additional strain further de reasing the adsorptionenergy. The gain in IdB stability an be attributed to hange in ele troni stru ture of doubly o upied dimer.No answer to the problem of high SB populations an be given from theperspe tive of any surfa e mediated intera tion. A ording to the above anal-ysis, we on lude that benzene adsorption on Si(100) is expe ted to appearin a SB onguration at rst as it is kineti ally preferable, then, dependingon environmental onditions, it transforms to a TB onguration. It is ex-pe ted that for low overages all adsorbates an undergo transformations.For high overage one would expe t the nal populations to be a mix. Theseresults should not be attributed to mole ulemole ule intera tion, as the to-tal energy and NEB al ulation proves, but should be expe ted as purelygeometri al restri tions, due to the fa t that the SB mole ule only needs oneempty dimer to adsorb, while TB needs two. The adsorption of SB maytherefore be possible while the onversion to TB is restri ted. This, in ee t, auses mixed populations in high overage ases. An additional possible fa -tor ould be IdB adsorption whi h is energeti ally omparable to SB in ahigh overage zigzag phase whi h ould impa t the population ratios in thespe tros opi studies as presenting the same σ vs π signature as SB. Theabove studies were also performed to test the idea of environmental ontrolfor metastable adsorptions. On the basis of our results, we an on ludethat the surfa e mediated environmental intera tions are not su ient foree tive ontrol of the SB/TB onversion.

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Chapter 11Thinlm photovoltai absorbers11.1 Introdu tionDespite oering an alternative energy sour e, PV ells are still far frombeing aordable without government subsidies. This is be ause they arestill above the threshold of 1US $ per Watt oered by the standard fossilfuel-derived ele tri ity. Nevertheless, this is expe ted to hange in the nearfuture, early this year the pri e for German PV ells was estimated around 2$per Watt whi h is half of the pri e from 2010 [175. The rapid rise of ChinesePV manufa turing further in reased ompetition in global PV market, withpri es of Chinese ells already rea hing 1$ per Watt and lower. A ordingto an observed trend, ommonly alled Swanson's law, it is suggested thatthe ost of a photovoltai ell falls by 20 % every time the total globalmanufa turing apa ity doubles. The onstant modernization, automationof large s ale produ tion and re ent innovations in material s ien e are soonexpe ted to make PV ells both environmentally and e onomi ally viable.The onversion of sunlight into energy has been a subje t of interest for afew de ades now, with Si single rystalline PV ells dominating the market,although there are a range of dierent te hnologies now oering alternativeand promising routes to solar energy produ tion. This will open a greatrange of appli ations in the future ( see Fig. 11.1 for a histori al overview).In fa t, sili on is not an ideal PV material, be ause it has an indire tband gap whi h redu es its adsorption oe ient to 103. The impli ation ofthis is that a dire t band-gap material su h GaAs, whi h has an absorption oe ient of 105, an be 100 times thinner than an equivalent Si absorber.[176. As shown in Fig. 11.1 solar ell te hnologies an be divided into single-118

Figure 11.1: Histori al a ounts of PV solar ell te hnologies. [177or multijun tion ells. Furthermore, depending on the material used, they an be lassied into single rystal or thin-lm poly rystalline solar ells; inaddition to these there is a new group of emerging PV te hnologies, su h asdye-sensitized ells or organi quantum dots ells, whi h trade very heapprodu tion ost for still relatively low e ien y [178.This hapter will fo us on one of the available te hnologies, alled thin-lm photovoltai s; whi h in ontrast to the standard Si PV ells are madeof poly rystalline absorbers. These greatly simplify the produ tion pro ess,that an be done by a range of UltraHighVa uum (UHV) and non-UHVmethods.A ording to Powalla et al., thin-lm te hnologies have a high ost-redu tion potential at high produ tion [179. The outstanding issue is the ost and availability of indium and gallium whi h are the main omponentsof the best Cu(In,Ga)(Se, S)2 thinlm solar ells [180. To solve thisproblem, a new quaternary material has been proposed. Cu2ZnSn(S, Se)4,in whi h every element from the III group (In, Ga) is repla ed by a pairof atoms from groups II and IV (spe i ally, Zn and Sn). In that way thematerial is entirely made of abundant elements with the row element pri efor Zn 2$ /kg, Cu 7$ /kg, Sn 20 $ /kg, instead of Ga 300$ /kg, In 550 $ /kg,whi h oers a signi ant ost redu tion [181. These materials have notbeen long used as photovoltai absorbers and the highest e ien y rea hedso far is 9.7 % [182.The work presented here is on erned with both groups of PV absorbers ited above. One of the hosen materials is CuInSe2; an alloy of this has119

been used as a ommer ial PV absorber and is ommonly known as CIGSwhi h represents the family of Cu(In,Ga)(Se, S)2 materials. The alloyingpro ess has been found to allow the ontrol over the poly rystalline stru -ture hara teristi s and phase uniformity, whi h ultimately resulted in higherperforman e of the PV devi e. Other investigated materials are pure stru -tures of the CZTS group of alloys whi h in the longer notations representCu2ZnSn(Se, S)4 ompounds.The basi stru ture of a CIGS or CZTS thin-lm solar ell is depi tedin Figure 11.2.

Figure 11.2: The general stru ture of a thin-lm solar- ell.The most ommon substrate for supporting PVC is soda-lime glass, 13mm thi k and oated on one side with molybdenum (Mo); this serves as ametal ba k onta t. The heterojun tion is formed between the semi ondu -tors CIGS or CZTS and ZnO, buered by a thin layer of CdS or ZnS anda layer of intrinsi ZnO. The CIGS/CZTS is doped p-type from intrinsi defe ts, while the ZnO is doped n-type to a mu h larger extent through thein orporation of aluminium (Al). In CIGS this asymmetri doping ausesthe spa e- harge region to extend mu h further into the ZnO. Mat hed tothis are the layer thi kness's and the band-gaps of the materials. The wideCIGS layer serves as absorber with a band-gap between 1.02 eV (CuInSe2)and 1.65 eV (CuGaSe2). For CZTS band tuning is still an unresolved issue.The light absorption is minimized in the upper layers, by the hoi e of largerband-gaps: e.g. ZnO = 3.2eV and CdS = 2.4eV , ZnS = 3.5eV . The dopedZnO also serves as front onta t for urrent olle tion. Laboratory s aledevi es, typi ally 0.5 m2 large, are provided with a Ni/Al-grid depositedonto the front side to onta t the ZnO.Most of thin-lm photovoltai s have so far been developed empiri ally.The improvement has often been made by trial and error, only some ofthe results were later ba ked up by theoreti al models. This approa h may120

hange in the near future, with rapidly in reasing omputational power andnew s ienti methods. Theoreti al materials s ien e oers a glimpse intothe material world through the quantum lens, allowing quantum design ofthe quantum system.11.2 CIGSThe CIGS ompounds rystallize in the tetragonal halopyrite stru turerepresented by the I 42d (D122d) spa e group (see Fig. 11.3). This groupis an isoele tri analogue of III-V binary ompound semi ondu tors. The

CuInSe2 stru ture ould be dis ussed from the perspe tive of the zin blendstru ture, where ea h anion is oordinated by two dierent pairs of analo-gous ations, Cu and In. This oordination leads to the formation of twodierent ation-anion hemi al bonds with two dierent lengths. Sin e theCu−Se bond is stier than In−Se the Se atoms are slightly displa ed fromthe ideal zin blend positions. The unit ell used is a double of the zin blendin dire tion ontaining eight Se anions, four Cu and four In ations whi his giving rise to slight anisotropy in the dire tion, thus η ≡ c/2a 6= 1.11.2.1 PhasesIn addition to the hal opyrate stru ture, another polymorph has also beenobserved for CIGS lms, CuAu. It has been found that the hal opyratephase is 0.2 meV/atom more stable then the CuAu type stru ture and thatthis value rises to 0.9 meV/atom in ase of CuGaSe2. Polymorphism hasbeen asso iated with the redu ed performan e of the solar ell due to in- reased interfa e related re ombination. In the ase of CIGS it has also beenfound that the CuAu phase ould ause an overall redu tion in the band gap,therefore even further de reasing the solar ell performan es. One way to ountera t the reation of unwanted phases and to in rease the ell e ien yis through alloying of Cu(In,Ga)Se2 [183.11.2.2 Computational detailsFor the optimization of the bulk unit ell, the al ulation have been per-formed using PBE-PAW potentials and GGA fun tional with a ut o energyof 300 eV and 5 × 5 × 3 MonkhorstPa k [21 kmesh sampling as imple-mented in VASP. The ioni relaxation has been performed until the for eper atom has been smaller then 0.01 eV/ Å.

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CuInSe2 Theoreti al Experimental [184Unit ell sizea=b 5.89Å 5.782Å 11.80Å 11.620Å /2a 1.005 1.001Bond lengthsSe-In 2.67ÅSe-Cu 2.45ÅTable 11.1: The stru tural results for ha opyrite stru ture of CuInSe2 omparison of experimental data with optimized stru ture.11.2.3 ResultsThe relaxations have been performed until the latti e onstants (see Fig.11.3) The stru tural parameters obtained are presented in Table ( 11.2.3).The latti e onstants obtained by simulations are in good agreement withexperimental values.

Figure 11.3: Unit ell for CuInSe2 hal opyrite rystallographi stru ture.The dierent atoms are represented by the following olours: brown Cu,grey In, orange Se.122

11.3 CZTSIn omparison to the hal opyrates, the array of the anion atoms is un- hanged in CZTS stru tures, o upying one half of the tetragonal sites.There is however a hange in the oordination of the ations. This is due tothe substitution of the In atom with a pair of Zn and Sn atoms, introdu ingan additional omplexity. For every Se atom there are two Cu and one Znand Sn ation thus a wider range of alternative phases is possible. Thishas been investigated in re ent theoreti al studies [185, 186, 187, 188. Theproposed phases an be hara terized by the symmetry of I 4, I 42m, P 42c,P 421 and P2 spa e group. From the above, the kesterite (I 4) stru ture hasbeen identied both in the experimental and in the theoreti al studies asthe most stable. Additionally, another I 42m phase has been found alledstannite [189, 190, 191, 192. Nevertheless, the exa t CZTS rystal stru -ture identi ation has not been a straight forward one be ause Cu+ andZn2+ are isoele troni thus the X-ray dira tion is unable to distinguish be-tween the two phases and alternative experimental te hniques are required.One su h te hnique is neutron dira tion whi h has been su essfully usedto identify that both sulphite and selenite based materials rystallize in akesterite stru ture. In re ent su h studies although the phase have beenidentied, a disorder between Zn and Cu sites has also been observed indi- ating a possibility for short range ationi displa ement and deviation froma major phase [193. This possibility is supported by the earlier mentionedtheoreti al studies whi h predi t the se ond most stable phase to be P 42c inwhi h there is a slight intermixing of Cu and Zn atoms within the Cu−Znplane.Here, we de ided to investigate the kesterite and stannite phase due tothe signi ant stru tural dieren es and yet still ompetitive energeti s, fordetails see A and B in Fig. 11.4.11.3.1 Computational detailsFor the optimization of the bulk unit ell, the al ulations have been per-formed using PBEPAW potentials and GGA fun tional with the ut ofenergy 300 eV and 5 × 5 × 3 MonkhorstPa k [21 k mesh sampling as im-plemented in VASP. The unit ell size similarly to CIGS is a double of thezin blend in the dire tion ontaining eight anions Se or S, four Cu , twoZn and two Sn ations. The ioni relaxation has been performed until thefor e per atom has been smaller then 0.01 eV/ Å.123

Figure 11.4: Unit ell for Cu2ZnSnS4 or Cu2ZnSnSe4 a) and ) kesterite(I 4) and b) and d) stannite (I 42m) rystallographi stru ture. The dierntatoms are represented by the following olours: brown Cu, grey Zn, blue Sn, orange Se and yellow S11.3.2 ResultsFrom the al ulated stru tural optimization al ulations the lowest totalenergy stru ture were obtained for both kesterite and stannite stru tures.The results are shown in the table 11.3.2. The dieren e between kesteriteand stannite phase is 0.03eV for Cu2ZnSnS4 and 0.15 eV for Cu2ZnSnSe4thus showing that the Se based ompound is mu h less likely to have phasemixing than the S based one.124

Cu2ZnSnS4 Kesterite StanniteTheory Theory Experimental [194Unit ell sizea=b 5.48Å 5.47Å 5.427Å 10.91Å 10.95Å 10.87Å /2a 0.995 1.00 1.001Bond lengthsSeCu 2.35Å 2.34ÅSeZn 2.36Å 2.43ÅSeSn 2.42Å 2.38ÅEnergy 0.00eV 0.03eVCu2ZnSnSe4 Kesterite StanniteTheory Theory Experimental [195Unit ell sizea=b 5.78Å 5.85Å 5.688Å 11.56Å 11.34Å 11.338Å /2a 1.000Å 0.97Å 0.996ÅBond lengthsSeCu 2.44Å 2.464ÅSeZn 2.50Å 2.52ÅSeSn 2.63Å 2.58ÅEnergy 0.00eV 0.16eVTable 11.2: The stru tural hara terization of kesterite and stannite stru -ture for Cu2ZnSnS4 and Cu2ZnSnSe4 materials.

125

11.4 Grain boundariesIn omparison to the single rystalline stru tures with a long range period-i ity in poly rystalline materials the periodi ity is onserved only within thegrains whi h in turn an give rise to quantum onnement of some stateswhile ondu tion for other. The spa e between individual grains alled thegrain boundary (GB) is a transition between grains of a dierent orienta-tion, dierent phase or even dierent omposition. One of the signi ant hallenges of thin-lm te hnology is the full understanding of poly rystallinestru tures and a wide range of ee ts and me hanisms arising from it. One an easily see why the proper understanding of these interfa es is a ne es-sary step for the understanding of the poly rystalline materials. The grainboundaries are known to be sinks for both defe ts and impurities, due tomisalignments and a de reased oordination. One has to onsider that someof the bonds of the interfa e atoms are not satised reating states often lo- alized within the band gap. Considering the above one an see why produ -tion of single rystals whi h are free from any impurities, gaps, dislo ation,and grain boundaries has been a goal in the produ tion of high e ien ysemi ondu tor devi es. In the ase of poly rystalline materials these studiesbe ome even more entral with imperfe tions being a signi ant part of thethe stru ture. In this ase a skilful ontrol over a GB is a way of dening thematerial property. This approa h is known as of grain boundary engineeringand has been proven so far to give improved me hani al and hemi al prop-erties of the materials [196. A ording to the standard perspe tive the GBsin semi ondu tors are viewed as detrimental to the performan e of ele troni devi es. In re ent work on hal opyrites, it has been shown that this maynot always be the ase and GBs in poly rystalline materials ould a tuallyplay a bene ial role in the performan e of photovoltai devi es [197, 198.The interest in grain boundaries has undergone signi ant expansion inthe last 30 years, with origins tra eable ba k to the development of therst poly rystalline photovoltai s. At rst the studies, due the experimentallimitation, were mostly investigating su h properties as resistivity, re om-bination e ien y, and I/V hara teristi s. With the development of moreadvan ed experimental te hniques, detailed studies of GB be ame possible.The major interest in the GB are physi al pro esses like harge trapping andtransport. Modelling of the realisti pi ture is however not straight forwardwith signi ant interfa e diversity and interplay of phenomena aused by amixture of defe ts and impurities.The thin lms are produ ed in a variety of dierent methods. This126

has a dire t impa t on the stru tures reated and the hara teristi s of thegrain boundaries present. The potential for many dierent morphologies andtypes of grain boundaries oexisting within the single sample makes both theexperimental and theoreti al work hallenging.One type of a grain boundry is a twin boundary. It is the in iden e ofhighly symmetri al grain boundaries, it is prevalent in diamond-like semi on-du tors. Twin boundaries are of spe ial interest in the stru tures analysedhere. They represent the low energy boundaries with a minimal number ofdisrupted bonds. All others are more disrupted and hemi ally a tive, thussupporting hemi al diusion and ele tri al re ombination and inhibiting ordisrupting the transport over the boundary. Although the twinning rela-tionship is xed the interfa e plane between the grains is not, oering a fewdierent boundary types. Sin e the twin boundaries are highly symmetri althey an be des ribed by means of tilt, rotation shear or ree tion. While inthe ase of small angle tilt boundaries the number of dislo ations in reaseswith in rease of the angle, there are ertain angles at whi h a latti e pointof both sides of the boundary will be superimposed. Whi h in a ordan e toCoin iden e Site Latti e (CSL) theory an be des ribed by a ratio of pointsin the latti e to the number of points in CSL and is known as the Friedelindex denoted by Σ. In the 1950's the CSL theory was applied extensivelyin the study by Kohn for the twin boundaries of diamond and then in late1980's by Durose in the study of sphalerite [200, 201, 202. The sphaleriteorientation proposed by the model has been onrmed experimentally forCdTe [201, 203, 204. As showed by Hold the mirror plane in sphaleritewould lead to an energeti ally unfavourable boundary of an anti-phase ori-entation, this has been further extended in the re ent work on the sphaleriteand hal opyrite where the CSL was used to identify the allowed twin bound-aries [205. One oherent and ve in oherent boundaries were proposed whilethe oherent one was simulated in further work for a proposed hal opyriteas well as stru turally similar kesterite and stannite stru tures.11.4.1 Experimental overviewThe CIGS materials are a good example for thinlms to rea h an e ien y ompetitive with the single rystalline material [206. The pro ess respon-sible for the unusually high e ien y has been debated for over a de ade(for reviews of the subje t see [198, 197). The most a epted explanationhas been proposed by Zunger and Persson. In their model the valen e bandbending would have been present due the neutral 2VCu+ InCu defe t states.127

There the valen e band oset is aused by the removal of Cu atoms from theboundary interfa e, leading to the formation of a neutral barrier for holesthat prohibits the transport of ele trons. This redu es the probability ofre ombination in the defe ted area. The proposed model was based on theformation of the grain boundary interfa e from two Cu de ient (112) polarsurfa es. In su h a way that the defe ts are lo alized along the two layers ofa (112) interfa e between two rystal grains, where one grain is shifted withrespe t to the other by translation t=0.5b. The formation of this type ofdefe t would also agree with the fa t that the highest e ien y ells werealways produ ed from Cu de ient materials [85, 207. The existen e of theproposed 2VCu + InCu defe t omplex has re ently been been onrmed byexperimental studies whi h have shown that the GB omposition is in fa tCu de ient and In ri h while the grain interior stays un hanged [208. Fromhigh-resolution s anning transmission ele tron mi ros opy (HR-STEM) ex-periments three types of the (112) twin boundary have been distinguishedand are presented in Fig. 11.5.

Figure 11.5: Stru tural representation of twinboundaries and their orre-sponding high resolution high resolution transition mi ros ope and ele tronenergyloss spe trometry of three types of GB identied in the studies per-formed by AbouRas et al. [208. 128

Type I and Type II GB have been identied as Se − Se terminatedlatti e planes with the approximate distan e between the planes of 2.8 Å and2.9 Å respe tively. For both of them a Cu depletion and In enri hment hasbeen observed in omparison to the GB interior. Type III have been iden-tied as Se ation terminated latti e planes with the approximate distan ebetween the planes of 2.0 Å. In this ase no signi ant In enri hment but aslight Cu depletion was also observed.It has has been found that the region of the ompositional hanges isvery lose to the twin boundary plain, ranging in width around 7.08.0 Åand 8.09.0 Å for I and II whi h orrelates well with the thi kness of twoatomi planes.It has been published on few o asions that the variation of a Cu/(In+

Ga) ratio is responsible for the variation in grain size [209, 210, 211,similarly to when the ratio of Cu/(Zn, Sn) is varied for CZTS absorbers[212, 213, 214. On the basis of a series of measurements of random bound-aries, Cu and In signals are always anti- orrelated whi h is also strong ev-iden e that site ex hange In2+Cu and Cu2−In ould be the origin for su h holebarrier and in reased e ien y [208. Although there is a strong orrelationbetween the proposed theoreti al and the experimental omposition of theboundary this an not be said about the stru tures proposed. ComparingFigures 11.5 (experimental GBs) and 11.6b (the theoreti al GB proposedby Zunger) it an be seen that no satisfa tory mat h an be made. Firstlythe type I and II GB are hara terized by Se − Se bonding in experimentsand Se ation in theory, whi h is also ree ted in the dieren e in the GBwidth. Se ondly the hara teristi atomi alignment (blue line in Fig. 11.6b)also diers, for experimental GB type I the atomi lines form a shifted mir-rored stru ture while type II forms shifted in line stru ture. In ontrast tothe theoreti al model the atoms in the bottom and top halfes of the interfa eare in line (blue line Fig. 11.6b). These dieren es prove that GB studiesdo not yet amount to a fully omprehensive model for CIGS poly rystallinematerials.Investigations of the CZTS grain boundary so far have shown two typesof s enario: one with a Cu ri h grain boundary and a width of about 50Å,while the other remains un hanged from the bulk omposition [215, 216.The studies of GB in CZTS have just begun and have not been widelydis ussed in the literature yet. In re ent theoreti al studies, based on thegrain boundary models derived from studies of CdTe, Li et al. omparedCIGS and CZTS grain boundaries for a (114) plane GB. They have foundthat in this parti ular type of GB, dangling bonds are present, whi h in129

the Cu2ZnSnSe4 reate more defe t levels within the band gap than inCuInSe2. These would be expe ted to promote more ele tronhole re om-bination and de rease the overall performan e of the solar- ell.11.4.2 Computational settingThe al ulations have been performed using unit ells relaxed earlier fromwhi h the larger grain boundaries unit ells were reated. The GB unit ells have been relaxed in the z dire tion to a ount for the dierent sta k-ing present in the two GB (one at the middle of the ell and one at the ellboundary). In order to reate a ell in whi h the intera tion between the par-allel boundaries are limited the ell grain interior onsist of nine bilayers (seeFig. 11.6 and Fig. 11.7). The unit ell ontains 144 atoms onserving the bulkstoi hiometry. The unit ell dimensions are 8.33/7.22/61.25 Å for CuInSe2,8.16/7.07/59.97 Å and 8.16/7.06/59.97 Å for Cu2ZnSnSe4 kesterite andstannite, 7.74/6.70/56.88 Å and 7.73/6.70/56.87 Å for Cu2ZnSnS4 kesteriteand stannite respe tively. The k -point sampling has been made with 5×5×3grid. The al ulations have been made with the PBE-PAW potentials withan energy ut o 300 eV. The relaxations have been performed until the for eexerted on the atom was smaller the 0.01 eV/Å.11.4.3 Theory and resultsIn this se tion, a oherent twin boundary within the (112) plane has been onsidered for both CuInSe2 and the CZTS group of materials. This typeof boundary is also present in sphalerite as a (111) twin boundary. Thisis not the ase however for hal oparates in whi h the anti-site o upan yof the latti e is present for one in two of the CSL metal olumns. In this ase the grain boundary reated has a slightly dierent ordering than thegrain (presented in the Fig. 11.6) where for ea h Se atom there are twoCu and two In atoms. It has for every four Se atoms two are oordinatedby 3Cu + 1In and 3In + 1Cu atoms and the two by 2Cu + 2In as for thebulk material. This type of boundary has been hosen to model the type IIIboundary observed by AbouRas et al.(see Fig. 11.5).Similarly as for the hal opyrates the same stru ture of the boundarywould be expe ted in the kesterite or stannite materials whi h have alsobeen investigated in this se tion. For the kesterites, the variation in the oordination similarly to the CIGS, reates dierent oordinations of theanions at the grain boundary where for every four Se or S anions there are3Cu + Zn and Cu + 2Zn + Sn or for the opposite boundary orientation130

Figure 11.6: Unit ell for CuInSe2 hal opyrate grain boundaries. The a)panel presents the (112) twin boundary as proposed by the CSL model byDurose, the grain boundary is highlighted in blue while the grain interior usedin text as referen e atoms are highlighted in red. The blue lines represent thedistin t atomi alignment. The panel b) the side tiled view of the boundaryhighlighted in blue olor in a) is shown. The blue line no. 1 highlights the Seatoms with their usual oordination 2Cu+2In atoms per Se, while the blueline no. 2 highlights the atoms with grain boundary spe i oordination3Cu + In and 3In + Cu per Se atom. The plane ) present the modelled(112) boundary proposed as dis used in text. The blue square highlights theboundary while the line represents the distin t atomi alignment.3Cu + Sn and Cu + 2Sn + Zn . In the ase of stannites the GB orderingis slightly dierent: for every four Se atoms, two are oordinated by 2Cu+

Zn+ Sn and the other two by 2Cu+ 2Zn and 2Cu+ 2Sn.Be ause this type of boundary does not have any defe t or danglingbonds, no trapped defe t states are expe ted to be present within the bandgap [217, 218. In order to analyse the dieren e in the ele troni stru tureat the grain boundary in omparison to the interior the LDOS for the grainboundary layer and a bulk layer have been obtained using the atoms as131

Figure 11.7: Unit ell of (112) plain grain boundary based on the CSL hal opyrate grain boundary model. The panel a) and b) presents GB ofkesterite, while ) and d) stannite type stru tures for Cu2ZnSnSe4 andCu2ZnSnS4. The grain boundary is highlighted in blue while the graininterior used as referen e atoms are highlighted in red.highlighted in blue and red in Fig. 11.6a and Fig. 11.7.The results of this study are shown at Fig. 11.7. Here, the band gap is132

underestimated, therefore an additional 1.0− 1.5 eV (depending on the ma-terial) would have to be added to t experimental band gaps. More a urateband gap al ulations an be obtained with a higher level approximationsu h GGA+U or for hybrid fun tional HSE03, however this has been ex-tensively studied in the literature [219, 185. For the PV appli ation thephoton energy is in the range 1.2 − 3 eV. As shown in the Fig. 11.8 thereis an observable shift of the o upied LDOS loser to the Fermi level forboth hal opyrate and kesterite materials whi h is due to the swit h in theCu oordination of Se where the additional peak in the o upied states is aused by dstates of Cu atoms. The slight shift toward the Fermi level islarger in kesterites than in CuInSe2. For the stannite stru ture the LDOS isun hanged by the hange in oordination from 2Cu+Zn+Sn to 2Cu+2Zn,while 2Cu+2Sn had no signi ant impa t on the ele troni stru ture of theGB. The same hara teristi s are observed independent of the anion type,Se or S. In the Fig. 11.9, the total LDOS are shown for ea h atom. Theyreveal that the majority of the o upied states around the Fermi level areCu states, while Se, In and Sn are largely uno upied. The Zn states donot parti ipate in the ele troni stru ture lose to the Fermi level.

133

Figure 11.8: The omparison of LDOS proje ted on the grain boundaryatoms and the grain interior atoms highlighted by blue and red squares ina) of g. 11.6 and in g. 11.7: a) CuInSe2 hal opyrate, b) CuZnSnSe4kesterite, ) CuZnSnS4kesterite, d) CuZnSnSe4stannite, e) CuZnSnS4stannite. 134

Figure 11.9: The omparison of LDOS proje ted on dierent atomi spe ies:a) CuInSe2 hal opyrate, b) CuZnSnSe4kesterite, ) CuZnSnS4kesterite, d) CuZnSnSe4stannite, e) CuZnSnS4stannite.135

11.5 SummaryThe (112) twin grain boundary has been simulated for CuInSe2 hal opyrateand CuZnSnSe4, CuZnSnSe4 both kesterite and stannite materials. Thisboundary agrees well with one of the experimentally observed GB types forCuInSe2 and is also expe ted to be present in the CZTS materials. Ithas been found that this grain boundary in omparison to the grain interiorhas minor dieren es in the ele troni stru ture. For the stannite materialsno hange has been observed, therefore the GB in this defe t free state isexpe ted to be ina tive. For hal opyrate and even more so for kesterite ashift of states towards the Fermi level was observed, it is expe ted that this ould have just a small ee t on the photovoltai performan e in spe imenswith high GB density.It is predi ted by Zhang et al. that the most important re ombinationdefe t due to its low formation energy and being lo ated 0.34 eV belowthe ondu tion band minimum is InCu. It is, however, also predi ted thatthis defe t would be passivated relatively often due to the formation of thedefe t omplexes 2VCu + InCu [14. In future studies of this boundary thestability of the 2VCu + InCu omplex relative to the grain interior shouldbe onsidered as well as the VCu whi h is the lowest energy defe t in CIS,while alternatively for CZTS there is a larger group of low energy defe ts.In kesterites there are ve low formation energy defe ts from whi h VCu isin se ond pla e after CuZn, then there is additionally ZnSn, VZn and CuSn.The GB lo alization of these defe ts should be studied whi h ould lead tosome interesting results and ould more signi antly ae t the ondu tive hara teristi s of the boundary [14, 220.

136

Chapter 12Final summary: ClosingremarksThis thesis in ludes a broad range of simulations. The work started withthe analysis of mole ular adsorption on sili on, a wide variety of mole ulesadsorbed in varying levels of overage, it tou hed the dynami s of diusionof mole ules along this surfa e in one instan e, and it fo ussed on inter-mole ular intera tions and onformation hanges due to polar intera tions.This part of the work was driven to a large extent by existing experimentaldata and was the ore task of my work in a ollaboration with a group inToronto.The last part of the work, whi h analysed the ele troni properties ofsemi ondu tors at grain boundaries, arose out of a parti ular situation duringmy PhD, the setup of a new institute for renewable energy with an emphasison photovoltai s, and dis ussions with the group of Prof. Ken Durose onthe importan e of grain boundaries for photovoltai e ien ies. As ea hindividual hapter ontained se tions dis ussing the results and summarizingthe main on lusions, I only wish to highlight the main a hievements of thisthesis in the losing remarks. These are, in my view:• Analysis of surfa e pinning: Here we found that asymmetri pin-ning an be a valuable tool to identify the geometry of the adsorbate.In situations where STM proles do not yield a lear identi ation ofthe mole ule, al ulations of relative adsorption energies and pinning an oer an alternative route.• Inter mole ule intera tions on Si(111): Here we found that onlysimulations in luding dispersion intera tions ould orre tly predi tthe preferred adsorption sites. The subsequent in rease in pre ision137

and size of the al ulated system, whi h was performed after this workwas on luded, established that surfa e mediated intera tions via sub-surfa e harge transfer is the de isive ingredient• Adsorption of small mole ules of Si(100): We nd that thedouble-dimer bond onguration is the most favourable, and that thein lusion of vdW intera tions improves the agreement between experi-mental and theoreti al adsorption energies. A tivation barriers for the onversion between single-dimer and double-dimer adsorption for ben-zene are within the range of established values, but 200meV too small ompared to experiments.• High overage benzene: Here, the surfa e mediated intera tions inthis regime play only a limited role. A tivation barriers are largelyunae ted by overage and no signi ant steri hindran e is observed.• Dynami s of diusion: In the study of ethylene on Si(100) we foundthat asymmetri desorption and residual torques on the mole ules areprobably the main ause of the very large diusion lengths observed.• Grain boundaries in photovoltai materials: In the study of (112)twin boundaries the model proposed by experimenters has been vali-dated by its total energy. It was also found that the anion oordinationat the grain boundary of hal opyrite and kesterite is the main driverfor an in rease of the LDOS near the bandgap. From a devi e point ofview this indi ates that su h grain boundaries have only a minor ee ton the e ien y of su h solar ells.It goes without saying that work, in parti ular in the last eld, is stillin omplete and one would wish that it ould be ontinued with a prospe tof a full understanding of fun tioning mole ular ele troni s and photovoltai devi es.

138

Chapter 13Additional material13.1 Appendix I : Feynman-Hellmann theoremOn e the ground state of the system is rea hed through a self onsistentiteration y le, KS orbitals an be used to determine the properties of thesystem. They an also be used to al ulate the net for e experien ed by theions in a not fully relaxed atomi arrangement. The al ulation of the for esis the end step of the self onsistent DFT y le at whi h the al ulatedfor e is used in the pro ess of stati minimization or mole ular dynami .As des ribed earlier, this approa h is possible due to BO approximation(see Se ition 2.2.1). In VASP, the ioni for es are determined on the basisof Feynman-Hellmann theorem, i.e. the for es experien ed by the ions aredened as the derivative of the generalized free energy. The fun tional Fdepends on the KS orbitals φ, the partial o upan ies and the ioni positionsr. The set of the orbitals is denoted Φ and the set of partial o upan ies asf. The ele troni ground state is dened by the variational properties of thefree energy

0 = δF [Φ, f,R] (13.1)for an arbitrary variation of the equation it an be written as∂F

∂Φδφ+

∂F

∂fδf = 0. (13.2)For arbitrary variation this is true only if both of the partial derivativesare equal to zero leading to the system of equations whi h determines Φ andf at the ele troni ground state.

Force =dF [Φ, f,R]

dR=∂F

∂Φ

∂Φ

∂R+∂F

∂f

∂f

∂R+∂F

∂R(13.3)139

where at the ground state the rst two terms are equal to zero thussimplifying the equation to,Force =

dF [Φ, f,R]

dR=∂F

∂R(13.4)The for e is obtained by al ulating the partial derivative of the freeenergy while keeping the wavefun tion and the partial o upan ies xed attheir respe tive ground state values.13.2 Appendix II : Nudged elasti bandCommon questions onsidering rea tions, whether they are in the gas phaseor on a surfa e, are the pro ess of bonding, the transition between startingand nal onguration, and the energy ne essary to allow for the transition.The nudged elasti band method is one of the te hniques dealing with theseissues. In its appli ation it is on erned with determining the minimumenergy path (MEP) of the rea tion. In the nudged elasti band (NEB)method a hain of intermediate positions is generated between two stable ongurations (before and after rea tion). The generated sets of oordinatesare then updated in a ordan e to the rst derivative of energy, the minimumenergy path rosses the saddle points dening the transition barrier.Henkelman and Jónsson in Ref. [25 proposed a method for the sear h ofsaddle points, that is, transition states, and minimum energy paths (MEP)between known rea tants and produ ts. The method requires no prior knowl-edge of the transition state as it works by optimizing a number of interme-diate images, initially hosen as appropriate guesses (typi ally, by linear in-terpolation of the oordinates between the initial and the nal state), alongthe rea tion path. Ea h image onverges to the lowest possible energy whilemaintaining equal spa ing to neighbouring images. This onstrained opti-mization is a hieved by adding spring for es along the band between imagesand by proje ting out the omponent of the for e due to the potential per-pendi ular to the band. An elasti band with N+1 images an be denotedby R0,R1...RN where R0 and RN are xed (initial and nal state). Thetotal for e a ting on an image is:

Fi = Fs

i |‖ −∇E(Ri) |⊥ (13.5)whereFs

i |‖= k(| Ri+1 −Ri | − | Ri −Ri−1 |) (13.6)140

and∇E(Ri) |⊥= ∇E(Ri)−∇E(Ri) · τi (13.7)where k is the spring onstant and τi is the tangent unit ve tor at image

i. The Climbing Image NEB or CINEB [26 is an improvement of the NEBmethod, within whi h, after a few iterations, the image imax with the highestenergy is identied as the transition state and hen e driven up to the saddlepoint by maximizing its energy along the band while minimizing it in allother dire tions. When this image onverges, it will be at the exa t saddlepoint. The for e on this image is not given by Eqn. (13.5) but byFimax

= −∇E(Rimax) + 2∇E(Rimax

) |‖ , (13.8)that is, the image does not feel the spring for es along the band; instead,the true for e a ting upon this image along the tangent is inverted.

141

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A knowledgementI would like to express my gratitude and a knowledgement to all the peoplethat made this postgraduate period of my life smooth and enjoyable, andsupported me by their s ienti knowledge or through their friendship and ompany so I ould prevail and omplete this thesis.• First I would like to thank Prof. Werner Hofer for giving me theopportunity to do this PhD. I thank him for his supervision over theyears and a lots of freedom and ontrol that he gave me over my re-sear h proje t. It taught me a lot on how to manage myself and mys ienti obje tives in order to be ome an independent resear her.• Spe ial a knowledgements to my group members Chiara Panosetti,Alejandro San hez Ron o, Dr. Haiping Lin as well as to Dr.Adolfo Fuentes, who oered me their help and support and withwhom I had many fruitful dis ussions.• I would also like to thank Dr. Felix Hanke, Dr. Matthew Dyer, LizCo klin, Joshua Elliott, John Sharp, S ott Devoy, EmilianoPoli, Dr. Ivan S ivetti, Dr. Jonas Bjork, Dr Georey Thomasand Christoper Collins for their kind advi es or proofreading someof my work.• I gratefully a knowledge my ollaborators for the dis ussions and veryvaluable insights to experimental side of my proje t. Thanks to theProf. John Polanyi and Dr. Ian M Nab from Toronto, Dr. AmirZabet from Columbia University, Prof. Ken Durose from Universityof Liverpool and Dr. Budhika Mendis from Durham University.• I also a knowledge Dr. Andris Gulans from Berlin and well as Dr.Daniel Bel her from University of New astle, Australia for their ad-vi e on the theoreti al side.• I give my spe ial thanks to Dr. Olalla Nam LorenzoCarballa forher time and support in di ult moments and her pre ious words ofen ouragement.• Many thanks to the eorts of Cli Addison and Dave Love for theirwork in taking are of the omputer luster in Liverpool and for theirhelpful assistan e.

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• I would also like to thank my parents for their onstant and un ondi-tional love whi h keeps me strong in all ir umstan es and to my un leRoman and my aunt Alinka for their support and belief in me.• Finally, I would like to thank all my friends who are always with meno matter how far they are.

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Publi ations and ontributions at onferen esPublished• Dire ted long-range mole ular migration energized by surfa e rea tion.K. R. Harikumar, J. C. Polanyi, A. Zabet-Khosousi, P. Czekala*, H.Lin* and W. A. Hofer*Nature Chemistry 3, 400408 (2011).• A etylene adsorption on sili on (100)(4 x 2) revisited.P. T. Czekala, H. Lin, W. A. Hofer and A. Gulans*,Surfa e S ien e 605 (1516) , pp. 13411346 (2011).• Van der Waals orre ted DFT study of high overage benzene adsorp-tions on Si(100)(4 x 2) surfa e and STM simulations.P. T. Czekala, C. Panosetti, H. Lin and W. A. Hofer,Surfa e S ien e 621, 15 (2013).To be published• Stru tural and ele troni hara terization of (112) twin boundary in hal opyrite, kesterite and stannite materials.P. T Czekala, W. A Hofer and K. Durose• Wet oxidation and mole ule mediated Si(100) (4x2) surfa e pinningP. T. Czekala and W. A. Hofer• Dipolar group fun tionization for selfassembled mole ular line swit hon HSi(100).P. T. Czekala and W. A. HoferConferen es, Workshops and Meetings• High overage adsorption of benzene on Si(100) van der Waals ex-tended studies.Postgraduate talks 2012, University of Liverpool, (oral presentation)• A etylene adsorption on sili on (100)-(4 x 2) revisitedPostgraduate online poster presentation, 2012 (poster presentation)• A etylene adsorption on sili on (100)-(4 x 2) revisitedCIFAR Nanoele troni s Meeting, Ban, CanadaNovember 16, 2010 November 18, 2010, (poster presentation)157

• DFT study of a etylene adsorption on Si(100) surfa e with onsider-ation of van der Waals intera tions.ECOSS27, (European Conferen e of Surfa e S ien e), Groningen,Netherlands, August 29 September 3 2010 (oral presentation)• Ab Initio Ele tro hemistry Workshop, CECAM-HQ-EPFL, Lausanne,SwitzerlandJuly 12, 2010 July 14, 2010• Materials Studio Workshop,University of Liverpool, UK12th May 2010,• ECOSS26 (European Conferen e of Surfa e S ien e) Parma (2009)

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