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ELECTRONIC AND STRUCTURAL PROPERTIES OF FUNCTIONAL
NANOSTRUCTURES

By
TENG YANG

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Physics and Astronomy
2009

ABSTRACT

ELECTRONIC AND STRUCTURAL PROPERTIES OF FUNCTIONAL
NANOSTRUCTURES

By

Teng Yang

In this Thesis, I present a study of electronic and structural properties of functional
nanostructures such as M 0351; I y nanowires, self-assembled monolayer on top of metal-
lic surfaces and structural changes induced in graphite by photo excitations. M 033;]?!
nanowires, which can be easily synthesized in one step, show many advantages over
conventional carbon nanotubes in molecular electronics and many other applications.
But how to self-assemble them into desired pattern for practical electronic network?
Self-assembled monolayers of polymers on metallic surfaces may help to guide pat-
tern formation of some nanomaterials such as M 03:1: I y nanowires. I have investigated
the physical properties of these nanoscale wires and microscopic self-assembly mech-
anisms of patterns by total energy calculations combined with molecular dynamics
simulations and structure optimization.

First, I studied the stability of novel Molybdenum chalcohalide nanowires, a can-
didate for molecular electronics applications.

Next, I investigated the self-assembly of nanoparticles into ordered arrays with the
aid of a template. Such templates, I showed, can be formed by polymer adsorption
on surfaces such as highly ordered pyrolytic graphite and Ag(111).

Finally, I studied the physical origin of of structural changes induced in graphite

by light in form of a femtosecond laser pulse.

To my father and mother,
younger sisters Hsia and Caiping

and black-belted Barbara da Fonseca Magnani.

iii

ACKNOWLEDGMENTS

Countless number of people contribute to one Ph.D Thesis and this one is no
exception. I am so grateful to many people — my advisor, my Guidance Committee,
my friends, my family and others. This Thesis can’t be possible without their strong
support and encouragement, I want to take a full use of this part to thank them.

First of all I thank my advisor Prof. David Tomanek. He has done so much that
an excellent advisor does for his student. He initiated many fruitful discussions with
me and showed me how to communicate more efficiently with research colleagues. He
showed me hand by hand how important it is to make a successful presentation and
how to make me successful. He supported me to attend domestically and internation-
ally as many workshops, colloquia and conferences as possible and introduced me to
almost all the top scientists in my field. During these years, I have also learned from
his great passion to scientific research, his unique way to inspire students to think,
and his sincere concern to human beings, for instance, on the ethnical controversy
of poverty and bio-fuel. And far more than that, Prof. Tomanek is like my second
father who puts so many touching concerns to my life and my future. I would say
that I would not have grown up so fast and be in this position without his continuous
support and guidance.

I want to show my gratitude to the members of my Guidance Committee: Pro-
fessor Vladimir Zelevinsky, Professor Chong—Yu Ruan, Professor Carlo Piermarocchi,
Professor C. P. Yuan.

I feel fortunate to collaborating with Prof. Savas Berber, Prof. Gotthard Seifert,
Prof. Glen P. Miller and Prof. Karsten Pohl. A special thank-you goes to Prof.
Savas Berber and Gotthard Seifert. Savas Berber taught me in an exhaustive way

on how SIESTA code works and tolerated me and my trivially naive questions. The

iv

productive exchange of viewpoints and discussions with him has been a treasure to
me. I want to thank Prof. Seifert for his great encouragement to me.

Among many friends, Jerry McAllister takes a special position. He has been a loyal
listener to” Science Friday” in National Public Radio program every Friday afternoon,
always had a great perspective on science-related issues and most importantly never
hesitated to share it with me. I still clearly remember how excited he was when
talking to me on the phone about his ”buckyball”, in which he accidentally burned
his sweet potato. Moreover he has been a wonderful English tutor to me. I also
want to say a special ”thank you” to Anne Serotkin for her encouragement and her
wonderful book gift ”a beautiful mind”. I thank Daniel Enderich for his great effort
to polish my writing. I thank Kiseok Chang and Kristsada Kittimanapun for many
scientific discussions in our BPS-4240 office. Many friends added a lot of enthusiastic
and beautiful memories to my MSU life: they are Sharron Xiaohong Ye, Nini Xian
Zhang, Jiwu Liu, the Yorks, Cathy Yunfang Cui, Yueya Sun, Fangye Sun.

I acknowledge financial support from the National Science Foundation Nanoscale
Interdisciplinary Research Team (NSF NIRT) and Nanoscale Science and Engineering
Center for High-Rate Nanomanufacturing (NSEC) program. I appreciate the efforts
that Prof. Tomének, Prof. S. D. Mahanti and our secretary, Debbie Barratt, made
to my application for the Dissertation Completion Fellowship.

Finally I thank my family — my parents and my sisters for putting up with me.

Table of Contents

LIST OF TABLES ix
LIST OF FIGURES x
1 Introduction 1
1.1 Overview .................................. 1
1.1.1 Transition metal chalcohalide nanowires ............ 1

1.1.2 Controlled self-assembly of molecules as nano—template . . . . 4

1.1.3 Photo-induced structural transformations ............ 6

1.2 Outline of the dissertation ............ » ............ 7
Density Functional Theory for ab initio simulations 10
2.1 Density Flmctional Theory and Kohn-Sham equations ......... 12
2.1.1 Hohenberg-Kohn theorem .................... 12

2.1.2 Kohn-Sham equations ...................... 13

2.2 Solving the set of Kohn-Sham equations ................ 15
2.2.1 Self-consistency process overview ................ 15

2.2.2 Norm-conserving ab initio pseudopotentials .......... 16

2.2.3 Orbital basis ........................... 19

2.2.4 Hamiltonian matrix elements .................. 20

2.3 Total energy ................................ 25

Interplay between structure and magnetism in M0128919 nanowires 29

3.1 Introduction ................................ 29
3.2 Structural and Electronic Properties .................. 31
3.2.1 Structural Properties ....................... 31
3.2.2 Electronic structure ........................ 34
3.3 Summary ................................. 39
Unique structural and transport properties of molybdenum chalco-
halide nanowires 40

vi

4.1 Introduction ................................ 40

4.2 Structural and Electronic Properties .................. 42
4.2.1 Structural and Chemical Stability ................ 42
4.2.2 Dispersability ........................... 45
4.2.3 Electronic Structure ....................... 46

4.3 Summary ................................. 49

Compositional ordering and quantum transport in M0639 _ 3:11? nanowires
51

5.1 Introduction ................................ 51
5.2 Theory ................................... 53
5.3 Results ................................... 55
5.3.1 Optimum functionalization of the building blocks in nanowires 55
5.3.2 Connection between building blocks in nanowires ....... 57
5.3.3 Elastic properties of nanowires .................. 57
5.3.4 Compositional ordering and phase separation in nanowires . . 59
5.3.5 Quantum transport in nanowires ................ 64
5.4 Summary and conclusions ........................ 68

Self-assembly of long chain alkanes and their derivatives on graphite 69

6.1 Introduction ................................ 69
6.2 Computational details .......................... 71
6.3 Experimental details ........................... 72
6.4 Results ................................... 72
6.4.1 Free-standing long-chain alkanes ................. 72
6.4.2 Structure and stability of long-chain alkane monolayers . . . . 73
6.4.3 Bonding nature of long-chain alkanes on graphite ....... 77
6.4.4 STM images of long-chain alkane monolayers on graphite . . . 78

6.4.5 Structure and stability of long—chain alkane derivatives on graphite 82
6.4.6 Domain formation in self-assembled monolayers of long-chain

alkane derivatives on graphite .................. 84

6.4.7 STM images of monolayers of long-chain alkane derivatives on
graphite .............................. 85
6.5 Discussion ................................. 88
6.6 Summary and conclusions ........................ 90

Molecular self-assembly of functionalized fullerenes on a metal sur-
face 92

7.1 Introduction ................................ 92

vii

7.2 Computational details .......................... 98

7.3 The hierarchy of interactions ....................... 99
7.4 Summary and conclusions ........................ 103
8 Optically induced transient structures in graphite 104
8.1 Introduction ................................ 104
8.2 Theory ................................... 111
8.3 Summary and conclusions ........................ 114
LIST OF REFERENCES 115

viii

List of Tables

2.1 The self-consistency procedure. ..................... 16

6.1 Relative stability of finite-width domains of C6H130H in self-assembled
monolayers on graphene. The structure labels correspond to the ad-
sorption arrangements of Figs. 6.9(a)-(g). The adsorption energy Ead
of C6H13OH is given by Ead = Et0t(C6H13OH array/graphene)-—-Et0t (isolated
C6H13OH in vacuum)—Et0t(graphene). ................ 83

ix

1.1

1.2

3.1

3.2

List of Figures

Schematic structure of M03ns3n + 2 clusters assembled to infinite
nanowires. (a) Chevrel phase of M06S8. (b) End—on view of nanowires

forming triangular lattices, stabilized by intercalated counter-ions (e.g.
lithium). .................................

(a)- ((1) Schematic depiction of a controlled assembly of nanOparticles
on a reusable nano-template (green), followed by transfer onto an
unpatterned substrate (red). ......................

Structural properties of M012S919 nanowires. (a) Atomic arrangement
within a unit cell. Planes normal to the wire axis are denoted by A,
A’, B, C, D. (b) Binding energy as a function of the lattice constant a.
(c) Optimized inter-plane distances as a function of a. (d) Optimum
values of the Mo-S-Mo bond angle 6 as a function of a. Values in the
two sulphur bridges per unit cell are distinguished by different symbols.
Dashed lines in (b—d) are guides to the eye. ..............

Electronic band structure E (k) of M01 8919 nanowires with lattice
constants (a) a = 11.0 A, (b) a = 12.3%, and (c) a = 13.8 A. Solid
lines represent majority and dashed lines minority spin bands. Bands
crossing the Fermi level are emphasized by heavy lines. Densities of
states of nanowires with (d) a. = 11.0 A, (e) a = 12.3 A, and (f)
a = 13.8 A, convoluted with 0.01 eV wide Gaussians. The Fermi level
lies at EF = 0. ..............................

30

3.3

4.1

4.2

4.3

Contour plots depicting the charge distribution in a M012S919 nanowire

at a = 11.0 A. (a) Total charge density contour at p = 0.05 el/a03. (b)
Difference charge density Ap(r), indicating regions of charge depletion
and excess with respect to the superposition of isolated atoms. (0)

p = 1.0x10"3 el/aO3 charge density contour associated with states
close to the Fermi level, Ep — 0.05 eV< E < Ep + 0.05 eV. (d) Differ-
ence spin density pT(r) — p l (r), showing regions of excess majority spin
T. Contour plots (b) and (d) are depicted in a plane, which contains
the wire axis. ...............................

(a) Schematic structure of a Mo6S6 _ x113 nanowire in side and end-
on view. (b) Energy gain AE(x) with respect to the average binding
energy and (c) energy AH(:c) of the substitution reaction in Eq. (4.1)
leading to the formation of a Mo6S6 __ ng; nanowire as a function of
composition. Among the data points for all structural isomers (o),
the most stable structures are identified by the solid circles (o). ((1)
Deviation from the equilibrium binding energy E0 as a function of the
relative unit cell size a/aeq, for a segment of the “magic” M06S412
nanowire containing N Mo atoms, and for an N -atom segment of the
(13, 13) carbon nanotube. ........................

(a) Structure of M068412 nanowires arranged on a simple hexagonal
lattice in a plane normal to the wire axes. (b) Contour plot of the
nanowire binding energy in this lattice as a function of the wire ori-
entation (,0 and separation d. The energy is given in eV per formula
unit. ....................................

Electronic properties of a M0686 _ $155 nanowire (a-f) in comparison to
a (13, 13) carbon nanotube (g-i). Displayed is the band structure E (k)
of the M0656 _ $13; nanowires in (a,d), their density of states (DOS)
in (b,e), and quantum conductance G (E) in units of the conductance
quantum Go (c,f). The corresponding quantities for the (13,13) car-
bon nanotube are shown in (g—i). E = 0, given by the dashed lines,
corresponds to the Fermi level in (a,b,d,e,g,h), and to zero source—drain
voltage in (c,f,i). The dotted lines show the position of E F in externally
gated / doped chalcohalide nanowires resembling carbon nanotubes.

37

43

45

48

5.1

5.2

5.3

5.4

5.5

5.6

Optimized atomic arrangements within finite building blocks of M06S9 _ $13;
nanowires. The structures are labelled according to the number of io-
dine atoms, and structural isomers with the same stoichiometry are
distinguished by lowercase Latin characters. The finite clusters have
been terminated by S atoms on both sides along the wire direction, in-
dicated by the arrows. The most stable isomers for each stoichiometry
are underlined and marked in red. ................... 53

Schematic arrangement of sulfur and iodine ligand atoms in the struc-
tures of Fig. 5.1, along with the binding energ values Eb with respect
to isolated atoms. The most stable isomers for each stoichiometry are
underlined and marked in red. ..................... 56

Optimized structure of two isomers with the same stoichiometry M06 S613
representing the connection between functionalized M06 building blocks
in nanowires by iodine (left) and sulfur (right) bridges. ........ 57

Equilibrium structure and axial rigidity in (a) M012S919 and (b) M012S1119
nanowires. The main difference between the compounds is an addi-
tional S atom in (b) inserted inside each of the S3 bridges in (a). E
denotes the total energy relative to the optimum structure and a is the
lattice constant. Dashed lines are guides to the eye, connecting the
data points. Quadratic fits to the data points near the equilibrium,
related to the axial rigidity, are given by the solid lines. ....... 58

Optimized atomic arrangement in double unit cell of infinitely long
M06S9 _ $13; nanowires for 03:636. Only the most stable isomers for
each stoichiometry are shown and labelled according to the number of
iodine atoms :1: per formula unit. The nanowire direction is indicated
by the arrows. .............................. 60

Mixing and phase separation tendency in M06S9 __ $13; nanowires. (a)
Binding energy Eb as a function of the iodine concentration x. The
data points refer to structures depicted in Fig. 5.5. The solid line, ob-
tained by connecting the data points at a: = 0 and a: = 6, depicts the
expectation value < Eb >. The dashed line is a guide to the eye. (b)
Energy gain AEb(x) with respect to < Eb >. The energy gain asso—
ciated with phase separation into domains with iodine concentrations
3:1 and x2 is presented in contour plots for systems with < :5 >= 3.0
in (c) and < 3: >= 4.5 in (d). Quantitative results are only valid for
integer values of 11:1 and 9:2, indicated by the grid. The contours in
(c) and (d), given in units of eV/ unit cell, are guides to the eye. Dark
areas represent particularly stable stoichiometries. .......... 61

5.7

5.8

5.9

6.1

6.2

Energy cost associated with grain boundaries in M06 S9 _ $11- nanowires
with < 3: >= 4.5, consisting of domains A with x = 3 and domains B
with :r: = 6. (a) Reference structure, with a single boundary separat-
ing domains A and B. (b) Double unit cells of A and B alternate along
the nanowire. (0) Individual unit cells of A and B alternate along the
nanowire. The energy differences AE are given for two formula units
of M012S919, containing 60 atoms. ...................

63

Quantum conductance of an isolated, infinitely long M06S4. 514. 5 nanowire.

(a) Definition of the scattering and the lead regions in an atomistic
model of the nanowire. (b) Quantum conductance G of the nanowire
in units of the conductance quantum GO, calculated at zero bias using
a non-equilibrium Green function formalism. .............

Quantum conductance of an isolated, finite M06S4. 514. 5 nanowire sand-
wiched between Au(111) surfaces. (a) Definition of the scattering and
the lead regions in an atomistic model. The left and right leads are
semi-infinite gold surfaces, indicated by the dashed lines. The scatter-
ing region contains the nanowire segment connected to topmost layers
of the Au(111) surface at both sides. (b) Quantum conductance G of
the system in units of the conductance quantum GO, calculated at zero
bias using a non-equilibrium Green function formalism. .......

(a) Equilibrium geometry and (b) electronic band structure of an in-
finitely long polyethylene chain. Wave functions of the states at k = I‘
corresponding to (c) the top of the valence band and (d) the bottom
of the conduction band are shown in the same perspective as used
in (a). The wave function contours are presented for the amplitude

Itbl = 0.04 a63/2 and superposed with the atomic positions. The
wave function phase is distinguished by color. .............

Interactions between two infinitely long polyethylene chains in (ac)
co—planar (P) and (d—f) stacked (S) arrangements. Interaction energies
are plotted in (c) and (f) as a function of the inter-chain distance d for
polymers in vacuum, depicted in (a) and (d), and polymers adsorbed on
graphene, depicted in (b) and (e). Energy values per primitive unit cell
of the polyethylene chain pair in vacuum are indicated by empty trian-

gles and those for chains chemisorbed on graphene by empty squares
in (c) and (f). ..............................

xiii

67

73

6.3

6.4

6.5

6.6

6.7

6.8

Alignment of polyethylene (PE) chains in different orientation on graphene.
Polyethylene may lie parallel to graphene in the P orientation or nor-

mal to graphene in the S orientation. (a) Definition of armchair and
zigzag directions on graphene. (b) PE(P) along the armchair direction,

(c) PE(P) along the zigzag direction, and (d) PE(S) along the zigzag
direction, viewed from the side or the top. ............... 76

Nature of bonding of polyethylene in P orientation, lying parallel to
graphene. The polyethylene adsorption geometry on graphene, shown
(a) in side view and (b) in top view. Contour plots of (c) the total
charge density p P E / gra(r) and (d) the charge density difference, de-

fined as Ap(r) = p P E / gm(r) — pp E(r) — pgm(r). The plane of the
contour plots is normal to the polymer axis and is indicated by the
dashed line in (a) and (b). ....................... 78

Nature of bonding of polyethylene in S orientation, standing normal to
graphene. The polyethylene adsorption geometry on graphene, shown
(a) in side view and (b) in top view. Contour plots of (c) the total
charge density p P E / gm(r) and (d) the charge density difference, de-

fined as Ap(r) = p P E / gm(r) — p P E(r) - pgm(r). The plane of the
contour plots is normal to the polymer axis and is indicated by the
dashed line in (a) and (b). ....................... 79

(a) Density of states (DOS) of polyethylene (PE) monolayers in P- and
S-orientation on graphene, compared to the DOS of pristine graphene.
The DOS has been convoluted by 0.2 eV. (b) Difference density of
states ADOS(PE(P)/gra)= DOS (PE(P) / gra)—DOS (graphene) for P-
oriented PE. (c) Difference density of states ADOS(PE(S)/gra) =
DOS (PE(S) / gra)—DOS (graphene) for S-oriented PE. ........ 80

Scanning tunneling microscopy (STM) images of self-assembled alkane
monolayers on graphene at the bias voltage Vbz'as = 2.0 V (sam-
ple negative). (a) Calculated STM image of a polyethylene (PE)
monolayer on graphene, corresponding to a constant charge density
10er = 10‘ e/ a . The image is superimposed with the model of
the PE structure. (b) ST M image of an n-docosane monolayer on
graphene, observed at the constant tunneling current of 215 pA. . . . 80

Highest occupied (HOMO) and lowest unoccupied (LUMO) molecular
orbitals of polyethylene chains terminated by (a) an OH group and (b)
a COOH group. Charge density contours are superposed with ball-
and-stick models of the molecules. ................... 82

6.9

6.10

6.11

7.1

7.2

Adsorption arrangements in finite-width domains of straight chain al-
cohols in self-assembled monolayers on graphene. Common to all struc-
tures is the P-orientation of the alkyl chains and their alignment along
the zigzag direction on graphene. Adjacent chains within one domain
can be parallel, with OH groups pointing in the same direction, or
antiparallel, with OH groups pointing in opposite directions. .....

Calculated STM images of alkane derivatives forming self-assembled
monolayers on graphene. (a) C6H13OH at the bias voltage Vbz'as =
1.6 V, (b) C6H13COOH at Vbz'as = 1.3 V (sample negative). Charge
density contours for ,0er = 10"6 e/ag are superimposed with struc-
tural models. The y—axis is aligned along the zigzag direction of the
underlying graphene. ..........................

STM images (a,c) and structural models (b,d) of alkane derivatives
forming self-assembled monolayers (SAMS) on graphite. (a) Observed
and (b) model structure of l-docosanol (022H45OH) on graphite. (c)
Observed and ((1) model structure of 1-docosanoic acid (C22H45COOH)
on graphite. The STM image (a) was obtained using the bias voltage
1.6 V (sample negative) at the constant tunneling current of 30 pA,
the STM image (c) was obtained using 1.3 V (sample negative) bias
and a 25 pA current. Domains are labeled by numbers. The size of in-
dividual molecules is indicated by a black rectangle in each panel. The
OH and COOH terminal groups are emphasized in red in (b) and (d),
and the domain boundaries are indicated by the heavy dashed lines.

Self-assembled monolayer of F-C60 on Ag(111). (a) Equilibrium struc-
ture of an isolated F-C60 molecule; carbon atoms are shown in green,
oxygens in red, and hydrogens in grey. (b) STM constant current im-
age of F'C60 SAM (295 K, —1 V sample bias, 0.4 nA tunneling current)
also showing the geometry of the unit cell of the molecular structure.
(0) Island of pristine C60 on Ag(111) imaged in identical conditions
(295 K, —1 V, and 0.4 nA). ........ ‘ ...............

F-C60 STM signature. Wavefunctions of (a) the HOMO and (b) the
LUMO of the free F-C60 molecule, superposed with the atomic struc-
ture. (c) Calculated contours of constant local density of states for an
F-C60 molecule on Ag(111) representing constant current STM images
under experimental conditions. (d) 3D representation of the STM data
showing the corrugation of the F-C60 SAM (295 K, —1 V sample bias,
0.4 nA tunneling current). ........................

83

87

94

95

7.3

7.4

8.1

8.2

Large scale STM image of F—C60 SAM on 2 ML Ag/Ru(0001). The
terrace edges of the Ag(111) substrate, corresponding to one of its
compact directions, are marked by black dotted lines. The F'C6O self-
assemble in domains with six different orientations, shown schemati-
cally in the bottom diagram. We distinguish two domain groups, char-
acterized by the solid bright green and dashed dark blue lines. Each
group contains three types of domains with distinct orientations, sep-
arated by 60°. Any domain of one group is related to a domain of the
other group by reflection symmetry on a compact direction of Ag(111).
The sharpest angle between any domain direction and a Ag(111) com-
pact direction is 14°. ...........................

Equilibrium configurations of free and adsorbed F-C60 molecule: (a)
top view in the I-shape configuration and (b) top view in the V-shape
configuration. (0) Detail of the F'C6O assembling in the optimum
geometry. ((1) Adsorbtion geometries of the F-C60 SAM resulting from
the proposed model. Besides the backbone of the F-C60 molecules, I
also depict the envelope associated with the Van der Waals radii of the
atoms. The primitive unit cell depicted by the black lines, containing
two F-C60 molecules, agrees with the unit cell shown in Fig. 7.1(b).

Ultrafast Electron Crystallography (UEC) of highly oriented pyrolytic
graphite. (a) The UEC pump—probe setup. (b) The layered structure of
graphite. (c) Ground state diffraction pattern of graphite. ((1) Ground
state layer density distribution function (LDF), obtained via a Fourier
transform of the central streak pattern in (c). .............

(a) Decay of the diffraction intensity. (b) Change in mean-squared
atomic displacements perpendicular to the graphite planes, estimated
from the Debye—Waller analysis of the intensity drops in (a). (c) Ef-
fective surface potential V3 at the HOPG surface as a function of the
applied fluence F. (d) Decay of the effective surface potential with fit
to the drift-diffusion recombination model (see text). (e) Contraction
of the interlayer spacing Ad (left axis) and the corresponding increase
in the effective surface potential (right axis) in the far-from equilibrium
regime. ..................................

97

100

106

8.3 Signature of sp3-like bonding in HOPG. (a) Molecular interference pat-

8.4

tern M (s) and the corresponding LDF curves at selected time stances
following strong photo-excitation. The transient peak at 1.9 A in the
LDF indicates formation of an interlayer bond. (b) Ratio of peak in-
tensities obtained from the diffraction patterns. The enhancement of
the (0,0, 12) with respect to the (0,0, 10) peak cannot be explained by
thermal vibrations and indicates a transient structure consistent with
sp3 bonding. (0) Ratio of peak intensities at R = 1.9 A and 3.35 A in
the LDF. .................................

(Color online) Electronic structure changes during photo-excitations in
graphite. (a) Schematic view of the structure. (b) Electronic density of
states (solid line) and the Fermi-Dirac distribution of graphite at T = 0
(dashed line) and kBT = 1.0 eV (dotted line). (c) Total pseudo-charge
density p(r) and (d) change in the pseudo-charge density Ap(r) due to
an electronic temperature increase to kBT = 1.0 eV. The plane used
in (c) and (d) intersects the graphene layers along the dashed lines in

xvii

110

Chapter 1

Introduction

1.1 Overview

1.1.1 Transition metal chalcohalide nanowires

One of the major challenges in the emerging field of molecular electronics is
to identify conducting nanostructures with desired electronic properties, which are
stable and easy to manipulate. Due to their high stability and favorable electronic
properties [1], carbon nanotubes [2] have been discussed extensively as promising
candidates for molecular electronics applications. A major drawback is our current
inability to synthesize or to isolate nanotubes with a given diameter and chiral an-
gle, which determine their metallic or semiconducting nature. Moreover, single-wall
carbon nanotubes are hard to isolate from stable bundles, which form spontaneously
during synthesis [3]. In this respect, recently synthesized Mo chalcohalide nanowires
[4], such as MoxSyIz, seem to offer several advantages by combining uniform metallic
behavior, atomic-scale perfection and easy dispersability [5, 6].

Chalcohalides of molybdenum and other transition metals are known to form
stable, intriguing 2- and 1-dimensiona1 structures [7] with an unusual combination of

electronic prOperties [8] including good conductance, superconductivity, magnetism,

 

 

 

 

 

 

 

 

 

 

 

 

 

      

Counter Ions (e.g. Ll)

Figure 1.1: Schematic structure of M°3ns3n + 2 clusters assembled to infinite
nanowires. (a) Chevrel phase of M06S8. (b) End-on view of nanowires forming
triangular lattices, stabilized by intercalated counter-ions (e.g. lithium).

and nonlinear polarizability. These layered or filamentous substances are known as
catalysts [9] and, to a much larger degree, as excellent solid lubricants [10, 11, 9]. Their
potential to become unique building blocks of nano-devices has barely been noticed so
far [9], in stark contrast to popular carbon nanotubes [12]. Recent progress in the syn-
thesis of MoxSyIz nanowires [13] suggests that these monodisperse, self-supporting
nanostructures may nicely complement carbon nanotubes by avoiding their short-
comings such as strong dependence of conductivity on the nanotube structure and
difficulty to separate bundled tubes [12].

Chalcogenide compounds containing Mo and S have been studied for a long
time [7, 14]. Whereas the best known allotropes, including MoS2, are insulating and
form layered compounds, more interesting structures often occur at lower sulfur con-
centrations. Well known are Chevrel phases, characterized by cluster compounds with
M0658 subunits, which may be finite clusters or needle-like quasi-1D systems [15],
shown in Fig 1.1(a). All these interesting structures necessitate the presence of metal
counter-ions, depicted in Fig 1.1(b), for their synthesis. Besides providing structural
stability, the main role of the counter-ions is to transfer electrons into the Chalcogenide
substructures [16, 17], thereby promoting formation of an ionic crystal [18]. In their

most stable electronic configuration, many of these compounds contain (M06S6)2_

building blocks [14].

I found it intriguing to study the possibility of stabilizing Mo—based nanowires
by substituting the divalent sulfur by a monovalent halogen atom (i.e. iodine) with a
similar electronegativity, thus avoiding the need for metal counter-ions. In this way,
a desirable electronic configuration could be preserved while maintaining the covalent
character of the system and avoiding transformation to an ionic crystal.

Combining sub—nanometer diameter with structural stability and interesting elec-
tronic properties, transition metal Chalcogenide nanowires have been discussed as a
potentially viable alternative to carbon nanotubes [19] for many applications. M06
S9 _ $19; nanowires, in particular, have been studied extensively both experimen-
tally [20, 21, 22, 6, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] and theoret-
ically [35, 36, 37]. In contrast to other nanowires and nanotubes, M06S9 _ x133
nanowires are believed to be metallic, independent of their structure [35, 36, 21],
and can be synthesized [38], dispersed [6, 5] and functionalized [23] in a straight-
forward manner. Nano—mechanical studies indicate a very low shear modulus of the
bundled nanowires [26], suggesting self-lubricating properties and potential applica-
tions in nano-tribology [39]. Sulfur terminators act as ”alligator clips”, providing
optimum contact to gold leads for molecular electronics applications.

In spite of these many studies, several questions about Mo6S9 _ 221-73 nanowires
still await a definitive answer. There is a consensus that the skeleton of these systems
consists of Mo octahedra, which are functionalized by S and I adsorbates and con-
nected by bridges containing S or I atoms. Still, there is an open controversy about the
atomic arrangement in nanowires with a: = 4.5 and a: = 6, which have been observed
as stable compounds. X—Ray Diffraction (XRD) [21] and Extended X—ray Absorption
Fine Structure (EXAFS) [40] studies agree in the identification of Mo octahedra as
stable building blocks of the nanowires, but disagree about the precise positions of
sulfur and iodine. In particular, both iodine [21, 40] and sulfur [33, 35, 36] have

been proposed as constituents of the bridges connecting the Mo octahedral building

blocks. Additional uncertainty exists about whether the connecting bridges contain

three [41, 35] or four [26, 40] atoms.

1.1.2 Controlled self-assembly of molecules as nano-template

Controlled self-assembly of molecules to form ordered nanoscale patterns is an
area of active research. These superstructures often exhibit unusual properties [42]
and are finding numerous applications in nanotechnology, including use as nanotem-
plates [43]. Nanotemplates enable directed assembly of nanoscale objects and allow
for the transfer of those patterned objects onto a substrate. For example, as nanopar-
ticles can be selectively attached to either molecular or polymeric species, nanotem-
plates derived from self-assembled molecules or polymers provide an efficient means
to prepare ordered arrays of uniformly spaced nanoparticles. N anotemplates of this
type permit small feature sizes and high assembly rates that are unrivaled by current
lithographic techniques including electron-beam nanolithography and scanning probe
methods such as dip-pen nanolithography [44] and field-assisted nanolithography [45].

Self-assembled monolayers (SAMs) of molecular and polymeric species on highly
oriented pyrolytic graphite (HOPG) have attracted much attention, since the forma-
tion of various adsorption patterns can be monitored using scanning tunneling mi-
croscopy (STM) methods [46, 47, 48, 49]. Although many patterns have been reported
even for alkane films on graphite, the energetic grounds for the optimum SAM struc-
ture have never been investigated. Since the interplay between adsorbate—adsorbate
and adsorbatesubstrate interactions ultimately determines the equilibrium geometry
of the molecular assemblies, it should be possible to induce specific changes in the
SAM superstructures by modifying the terminal functional groups of the molecules in
a predetermined manner. Unfortunately, published STM data of SAMs on graphite
do not provide a quantitative interpretation of the resulting superstructures and there-

fore do not allow for predictions concerning the effect of specific functional groups on

nanowires

 

(a) nano- template

If‘m'fh {1.1-‘30....ms'1y

A“

 

     

 

 

 

 

 

Figure 1.2: (a)-(d) Schematic depiction of a controlled assembly of nanoparticles on a

reusable nano-template (green), followed by transfer onto an unpatterned substrate
(red).

the corresponding self-assembled superstructure.

Ordered arrays of molecules or nanoparticles such as C60 fullerenes have a great
number of practical applications, ranging from sensors and biological interfaces [50]
to organic electronics [51], photovoltaics, and novel computational methods such as
quantum dot cellular automata and, with endohedral fullerenes, spin-based quan-
tum computers [52, 53]. Generally speaking, self-assembled monolayers of molecules
(SAMs) form as a result of a delicate balance between competing molecule—substrate
and intermolecular interactions [54]. Therefore, to control such self-assembly pro-
cesses and grow molecular arrays of designed geometries, it is mandatory to under-

stand how this balance of interactions reflects onto the SAM’s final structure [55].

1.1.3 Photo-induced structural transformations

There is growing interest in displacing atoms in materials by photo-excitations
[56]. Observations of transient structures thus formed offer a glimpse into the trans-
formation pathways between different structures. Carbon, with its propensity to. form
a wide range of bonding networks (31), spz, .9193), is ideal to study the dynamics of
bond formation and rupture. Of particular interest is the conversion of rhombohe—
dral graphite to diamond [57, 58, 59, 60, 61]. Whereas pioneering ultrafast optical
studies of graphite have provided evidence of photo-induced melting [62, 63] and gen-
eration of coherent phonons [64, 65] by observing changes in the electronic properties,
direct determination of lattice structural dynamics proves to be difficult, especially
in the far-from—equilibrium regime. X—ray diffraction has been successful in observ-
ing nonthermal structural changes in semiconductors [66], relying primarily on the
integrated Bragg intensities. However, direct observation of atomic motion in nanos-
tructures with low atomic number, such as carbon, is very challenging and has not
yet been achieved. Much more promising is electron diffraction, with its five orders

of magnitude enhanced scattering cross-section and advances allowing experimental

observations on the femtosecond time scale [67, 68, 69, 70], offers a new window
into the realm of photo-excited structural dynamics, with resolution down to 51 nm

[67, 70, 71].

1.2 Outline of the dissertation

This PhD Thesis contains 9 Chapters, including this introductory chapter. In
Chapter 2 I present a detailed description of the computational methods used in this
Thesis.

In Chapters 3 to 5, I present a study of various physical properties of a new kind
of molybdenum chalcohalide nanowires, which are considered powerful candidates for
molecular electronics applications.

In Chapter 3, I investigate the equilibrium geometry and electronic structure of
M012S919 nanowires. The skeleton of these unusually stable nanowires consists of
rigid, functionalized Mo octahedra, connected by flexible, bi—stable sulphur bridges.
Interestingly, this structural flexibility translates into a capability to stretch up to
z20% at almost no energy cost. My results indicate that the nanowires change from
conductors to narrow-gap magnetic semiconductors in one of their structural isomers.

In Chapter 4, I combine ab initio density functional and quantum transport cal-
culations based on the nonequilibrium Green’s function formalism to compare struc-
tural, electronic, and transport properties of M0636 _ x1113 nanowires with carbon
nanotubes. I find systems with .7: = 2 to be particularly stable and rigid, with their
electronic structure and conductance close to that of metallic (13, 13) single-wall car-
bon nanotubes. M0686 _ 231$ nanowires are conductive irrespective of their structure,
more easily separable than carbon nanotubes, and capable of forming ideal contacts
to Au leads through thio-groups.

Chapter 5 covers the compositional ordering and quantum transport study in

M06S9 _ 331$ nanowires in complement to Chapter 3. I find nanowires with x = 3
to be particularly stable. Nanowires with other compositions are likely to phase
separate into iodine-rich and iodine-depleted segments, some of which should have
the stoichiometry M06S6I3. My transport calculations, based on the nonequilibrium
Green’s function formalism, indicate that the nanowires are metallic independent
of composition and exhibit a quantum conductance of G = 3G0, with the three
conductance channels involving the S3 bridges.

Self—assembly of nanomaterials into ordered arrays with the aid of a template is
important to form desired networks for molecular electronics applications. In Chap-
ters 6 to 7, I present results for molecule and polymer adsorption on metallic surfaces
such as HOPG and Ag(111).

Chapter 6 is devoted to the self-assembly of long chain alkanes and related al-
cohol and carboxylic acid molecules on graphite. I studied it using ab initio total
energy calculations and compare my results to scanning tunneling microscopy (STM)
measurements. For each system, I identified the optimum adsorption geometry and
could thus explain the energetic origin of the domain formation observed in the ST M
images. My results for the hierarchy of adsorbate-adsorbate and adsorbatesubstrate
interactions provide a quantitative basis to understand the ordering of long chain
alkanes in self-assembled monolayers and ways to modify it using alcohol and acid
functional groups.

Combining my computational results with experimental studies performed by
others, I present in Chapter 7 a theoretical study of the self-assembly of C60 molecules
functionalized with long chain alkanes on the (111) surface of silver. I find that
the conformation of the functionalized C60 molecule changes upon adsorption on
Ag(111) and that the unit cell size in the self-assembled monolayer is determined
by the interactions between the functional groups. I show that C60 molecules can

be assembled in ordered two-dimensional arrays with intermolecular distances much

larger than those in compact C60 layers. This introduces a novel way to control the
adsorption pattern by appropriate chemical functionalization.

Finally, in Chapter 8 I address the stability of particular templates and study
the possibility of structural changes induced in graphite by a femtosecond laser pulse.
Experimental results suggest that a variety of interesting phenomena occur following
irradiation by light. At moderate fluences of $21 mJ/cm2, lattice vibrations are
observed to thermalize on a time scale of ~8 ps. At higher fiuences approaching
the damage threshold, lattice vibration amplitudes saturate. Following a marked
initial contraction, graphite is driven non-thermally into a transient state with sp3-like
character, forming interlayer bonds. Using ab initio density functional calculations,
I trace the governing mechanism back to. electronic structure changes following the
photo-excitation and find that the modified force field in the excited state and the

Coulomb stress are the main forces driving this structural change.

Chapter 2

Density Functional Theory for ab

initio simulations

In order to get insight into the intriguing electronic and structural properties
of nanostructures, we need to solve the many-body Schrédinger equation for those

systems,
H (D = E<I>,

where the many-body wavefunction <1) depends on electronic and ionic coordinates.

The Hamiltonian is written as

 

”‘25; +239— M+ZUronW RI) +23% -‘r'| +2 M. (2.1)
275]” .7 1¢J|RI'RJ|
The first two terms are kinetic energies of electrons and nuclei; the last 3 terms are
electron-ion, electron-electron and ion-ion interactions, respectively. r", denotes the
electron coordinates and E1 the nuclei positions in the lattice.
In systems with many electrons, it is impossible to solve equation (2.1) exactly.

In view of this difficult problem, several approximations have been developed and

10

applied successfully to a large number of solid-state systems [72]. In the ground
state, these include the adiabatic (or Born-Oppenheimer) approximation [73], which
has long been used to separate the nuclear motion from electron motion, since the
electrons are four orders of magnitude lighter than the ions. Other approximations
include the Hartree-Fock (HF) self-consistent method [74, 75] and the popular Density
Functional Theory (DFT) [76]. The basic advantage of DFT over HF is the treatment
of both electron exchange and correlation in DFT. The exchange-correlation energy
is exact in selected systems and approximate in others. Infinite periodic solids are
treated as ideal crystals with translational symmetry. Using these methods, we can
map the unsolvable many-body problem approximately onto a single-particle problem.

DFT would not be very successful in studying electronic and structural properties
of atoms, molecules, solids, surface and interfaces without the Kohn-Sham approach
to obtain a workable solution [77]. The Kohn-Sham approach maps a many-body
problem onto an independent-particle problem by splitting off the difficult many-body
terms into an exchange-correlation functional of the electron density. I will start with
a brief introduction of the basic idea behind the Kohn—Sham approach, discuss how it
transforms interacting electrons to non—interacting quasi-electrons and finally include
selected details about how it is implemented into popular computer codes like SIESTA
[78, 79, 80] (with a localized basis) and Quantum-Espresso [81] (with a plane-wave
basis) to obtain the electron density, total energy, and other derived quantities such

as forces.

11

2.1 Density Functional Theory and Kohn—Sham equa-

tions

2.1.1 Hohenberg-Kohn theorem

The essence of the Density Functional Theory is that in the electronic ground
state, the total energy of a system of interacting electrons is a unique functional of the
ground state electron density n00") only. According to the Hohenberg—Kohn theorem,
the external potential Vert (7") determines uniquely the ground state density 7100’) of
the interacting electrons. For any external potential Vert (1"), we can write the total
energ as a universal functional E [n] in terms of the density n(r"), which is globally
minimized only by the exact ground state density n00"). The global minimum value

of this functional gives the ground state energy as

EHKln] = T[n] + Eintln] + [d3TVext(fln(fl + E”, (2.2)

where EU is the energy of ions, which are treated classically in the following.

It follows that once the kinetic energy T[n] of the interacting particles and inter-
action energy Eint [n] (equal to the repulsive Hartree energy and energy terms associ-
ated with exchange and correlation interactions) are known, the ground state charge
density n(r"') and energy can be determined by minimizing the total energy (2.2) with
respect to 71.0") The most challenging of the energy terms is Eintlnla which is known
exactly only in the hydrogen atom and the uniform electron gas. In other systems, the
Eintlnl functional is typically determined numerically using various approximations,

to be discussed in the following.

12

2.1.2 Kohn—Sham equations

The ground state charge density determines the total energy and derived prop-
erties of the many-body system. The Kohn-Sham approach is to identify a system
of non-interacting quasi-electrons, the ground state density of which is equal to that
of the original system of interacting electrons. Different from the energy functional

(2.2), the ground state energy functional of this system is rewritten in the form

EKSlnl = TOInl + Eint [n] + [dBTVextlflnlfl + EII

___ T0[n] + E Hartree [n] + E$c[n] + / d3rVe$t(F)n(r") + E I I. (2.3)

The system of N independent particles (quasi-electrons) in a given potential
Veg/.150") can be described using a Schrodinger equation, which yields the wavefunctions

WU") of the quasi-electrons. Then, the total electron density is given by

N 2
”(70: 2 M00] - (2-4)

i=1

With the quasi-electron wavefunctions identified, the kinetic energy T0 of the quasi-

electrons (which is different from the electronic kinetic energy T) is

N
To = --;- .2 (w. [V2] 212,-) (2.5)

and the Coulomb energy of the interacting quasi-electrons (Hartree energy) is given
by
1 ,n 7' n F ’
EH artr‘ee = 5 / d3TdBT M (2-6)

[F—F’l '

The remaining term, describing the exchange and correlation energy of these particles,

13

can be rewritten as a functional of n that needs to be determined, as
EIECInl = Tlnl + Eintlnl _ TOInl — EHartreelnl (2'7)

The most common way to solve the Kohn—Sham independent quasi-particle prob-
lem in the electronic ground state is to perform variational minimization of the energy
functional (2.3) with respect to the charge density 71(7‘). In the energy functional (2.3),
most terms (i.e. Ere, E Hartr‘ 68) are functionals of the electron density, except T0 in
Eq.(2.5), which is a functional of the wavefunctions. Variational minimization with
respect to orthonormal wavefunctions (< wilt/J]- >2 6U) can be performed using the

method of Lagrange multipliers, yielding

5 [EKSIR] - )‘i(< t/Jz'ltbz' > -1)] = 0. (2.8)

This leads to a set of equations that, together with Eq.(2.4), are called the Kohn-Sham

equations,

‘% V2 +Veffm] 1M?) = Az-w.(e- (2-9)

Here, Ve f f(r"') can be viewed as the effective potential, which depends implicitly on

71(1") and is given by

(SE (SE
Veff(fl = VextU-f) + W + ‘6—Tf2 = 6513250?) + VHartreg<fl + V$C(fl' (2'10)

The Kohn—Sham eigenvalues A2- are not to be confused with the energies of electrons
[82].

The attraction of the Kohn—Sham approach lies in the way of separating off
the independent quasi-particle kinetic energy, the Hartree energy, and the interaction
with the external potential Vezt(f'); these form the largest fraction of the total energy

and can be determined in a straightforward way. The remaining exchange-correlation

14

functional Exc[n] is rather small and can be approximated by a local function of
n(r"') and its gradients. As mentioned earlier, the DFT treatment of correlation is
approximate, but still superior to its complete neglect in the Hartree-Fock approach.
Once a proper approximation of Exc[n] is identified, the Kohn—Sham equations can
be solved self-consistently, yielding n(f) and the total energy of the many-body sys-
tem. Widely used methods to describe the exchange-correlation functional are the
Local Density Approximation (LDA) [77] and Generalized Gradient Approximation
(GGA) [83]. LDA is exact in the uniform electron gas and justified by the finding
that the exchange-correlation energy per electron exc(r’) varies slowly in an electron
gas of slowly varying density, where it is well represented by a local function. The
proper parametrization of this local function can be obtained using more microscopic

quantum Monte Carlo calculations [84, 85, 86] for a uniform electron gas.

2.2 Solving the set of Kohn—Sham equations

In the following, I provide a step-by-step explanation of solving self-consistently
the set of Kohn—Sham equations, as implemented in the SIESTA code [78, 79, 80, 87]
that uses a localized basis of atomic orbitals, and the Quantum-Espresso code [81]

that uses a plane-wave basis for a given system.

2.2.1 Self-consistency process overview

The self-consistency procedure can be summarized within a few essential steps as
indicated in Table 2.1. Once the initial electronic charge density nanfi') is determined,
most commonly as a superposition of atomic orbitals, the single-particle effective
potential Ve f f can be constructed by using approximations such as LDA. Next, we
need to decide if we wish to expand the system wave function in a plane-wave basis

or a localized atomic orbital basis. A finite system is solved conveniently in real

15

 

Step 1:
Get an initial 72.67%")
i
Step 2:
Construct VK S
Ve f f = Vert + VHartree + V170
l

 

 

 

 

 

 

 

 

Step 3:
Expand 112,-(77’) in local orbitals Q50")
and solve Hefffpifi‘) = Eitbz'U-f)

1
Step 4:

Calculate new noutfi) = Z,- Ir/Jz-(r‘fl2
Obtain difference 5n = noun?) — nm(F)
Is 672. _<_ tolerance?

 

 

 

 

 

 

 

 

 

 

l
. No. . Yes.
Construct 72:7; 1 = (mg-mt + (1 — (1)722” Self-consistency has been reached.
Go back to Step 2 Calculate the total energy.

 

 

Table 2.1: The self-consistency procedure.

space, whereas in an infinite periodic system, we will benefit computationally from
using reciprocal space to expand the Bloch states and to diagonalize the Kohn-Sham

matrix. To accelerate convergence, only fraction 0: of the output charge density nzout
is typically mixed with the input charge density 72]", with a S, 5%. Once we find
that the output charge density is nearly equal to the input density, which typically
requires several itineraries, we have reached self-consistency and can determine the
total energy and related quantities. In the next sub-section, I will introduce an

important simplification that will reduce the computational effort significantly.

2.2.2 Norm-conserving ab initio pseudopotentials

Besides the Born-Oppenheimer approximation for the separation of the electronic
and ionic motion and the Local Density or Generalized Gradient Approximation for

the exchange and correlation energy terms, we introduce pseudopotentials as a useful

16

concept to separate the interesting valence electrons from the relatively inert core
electrons.

The core electrons play a negligible role in the physical properties of solids. Con-
sequently, it is convenient to combine the strong nuclear Coulomb potential and the
potential of the tightly bound inert core electrons into one effective pseudopotential
acting on valence electrons. Parameter-free pseudopotentials can be generated from
atomic calculations and used without any modifications to describe valence electrons
in molecules or solids. While not unique, pseudopotentials need to accurately repre-
sent the scattering of valence electrons from the ion cores. Ab initio pseudopotentials
are typically non-local and energy dependent (i.e. they depend on the quasi-electron
wave function and energy eigenvalue). Norm-conserving pseudopotentials [88] gener-
ate valence electron pseudo—wavefunctions with the same norm (total charge inside a
finite—size sphere) as all-electron atomic wave functions outside a cutoff radius. Pseu-
dopotentials should furthermore be transferrable, i.e. they should reproduce well the
DFT energy eigenvalues in atoms with different electronic configurations. Unlike the
standard Hamann—Schluter—Chiang [88] or Troullier-Martins [89] pseudopotentials,
”ultrasoft” pseudopotentials [90] are also energy dependent.

Most calculations in this Thesis are performed using LDA. I construct norm-
conserving non-local Troullier-Martins pseudopotentials [88, 89] using the Ceperley-
Alder exchange-correlation functional [91] in LDA. As in Hamann—Schluter-Chiang
pseudopotentials of the same type, non-locality is expressed in terms of a dependence
of the pseudopotential on the orbital angular momentum l. The generation of a
pseudopotential involves the following steps.

(1) Valence electron eigenvalues El and eigenfunctions 451(7‘) are obtained by
performing an all-electron calculation for an atom in the ground state and selected
excited or charged states, using a given exchange-correlation density functional.

(2) For each angular momentum l, a pseudo-wavefunction (ifs (7") with no radial

17

nodes is generated such that the norm Q(r =f0 d3r' ’¢lIDS*(F ’)¢PS(F ’) of the
pseudo—wavefunction 45113307) equals that of the all-electron wavefunction (”(7") for
r > rc. In general, reducing the value of TC leads to more ”transferable” [82], but
deeper pseudopotentials, associated with less smooth pseudo—wavefunctions [72].

(3) Inverting the Kohn-Sham equations, leads to a total ” screened” pseudopo-

 

 

tential,
d2 PS
V _ r12 l(l+1) 32¢: (7‘)
lvtotal(’")‘5‘2me 27‘2 ’ We)

The ionic pseudopotential is then obtained by subtracting the Hartree term Vflfftgrtreeh)

and exchange—correlation term V555 (7") from Vl total, yielding

vim = vi, mm — Vfifmem — noise).

As will be shown later on, it is convenient to decompose Vl(r) into a l—independent

local term and a non-local part,

V10) = Vlocal(r) + 6V1”)

Vlocalo') includes all the long-range effects of the Coulomb potential, and (SI/[(7‘) is a
non-local l-dependent pseudopotential.

(4) A convenient nonlocal form, proposed by Kleinman and Bylander (KB) [92],
reduces the computational effort by requiring only products of projection operators,

‘ “KB
VNL = Vlocalo‘) + VNL ’

‘ |6Vr (infer >< «rm, ”avian

=22

l=0m= —r <f’zm Paw“ )I‘ferS>

 

18

The V155 operator is computationally rather inexpensive as 1713,53}, is typically small.

I usually use a 15,5}, = lmar + 1 [87] in SIESTA calculations, where $5,? is the
atomic pseudo-wavefunction and lmar is the maximum angular momentum used in

the basis.

2.2.3 Orbital basis

In a periodic solid with an infinite number of electrons and ions, we use Bloch’s
theorem to expand electronic eigenfunctions in plane waves on the grid of reciprocal

lattice vectors G as
, = z‘E-r’ ,té-r’ = q ~r(é+i€)-F
$2,120?) 6 Zci,Ge :ci,k + Ge , (2.11)
G

where i denotes the state and If the Bloch wave vector.

The Fourier representation of 2,0,, k is dominated by components with a low wave
number I: + G. Therefore, it is common practice to ignore high-frequency Fourier
components, setting Ci, If + (3 = 0 if E = (fi.2/2m)/|lfi+G.|2 > Ecutoff' The value of
Ecuto f f determines the accuracy of the total energy. Since pseudopotentials suppress
nodes in atomic waveftmctions, also the electronic waveftmctions in a solid are rather
smooth and require a low value of Ecuto f f for convergence.

The size of the plane wave basis set decided by the kinetic energy cutoff Ecuto f f
determines the size of Hamilton matrix. Finally the problem of dealing with an
infinite number of electronic wave functions has been transformed into calculating a
finite number of electronic wave functions at an infinite number of k-points within
the finite Brillouin zone [82]. In most instances, sampling the Brillouin zone by a
finite set of k—points instead of performing an integration over the Brillouin zone is

the most convenient procedure.

Distinct from codes using a plane wave basis set, SIESTA has a lower requirement

19

on ”smoothness” and uses localized numerical atomic orbitals (NAOs) as a basis.
NAOs are strictly confined in space, but non-orthogonal. Still, the resulting Kohn-
Sham and overlap matrices are sparse, enabling linear scaling of the computations.
The level of confinement is defined by a quantity called ”energy shift”, corresponding
to the energy increase associated with the localization of the orbital. The radial
and angular flexibility of atomic orbitals can be increased by increasing the basis
size (i.e. by changing what is called a single-C to a double-C and triple-C basis,
with increasing number of basis functions, and by adding polarization orbitals with
a higher angular momentum [93]). A basis set can be optimized by varying the basis
size, the confinement range and the shape of the basis functions. I found the NAOs
basis sets with double-C polarized orbitals (DZP) to provide dependable results for
most systems studied here. Still, a localized basis set is always inferior to a plane-
wave basis that is free of prejudice. Selecting an atomic orbital basis causes a degree
of arbitrariness in terms of location of the basis functions and introduces basis-set
superposition-errors (BSSE) due to the non-orthogonality of the basis. To limit those
errors, we use the counterpoise method [94] and compare our results to more involved

calculations using plane-wave basis sets.

2.2.4 Hamiltonian matrix elements

The standard Kohn-Sham one-electron operator may be written as

HKS = T'0 + Z: V1550?" Rf) +2: Vlocalfi; Ii) + VHartree(f) + V3002), (2'12)
2' i
where T0 is the kinetic energy operator, R2- is the position of ion i, V151? and Vlocal
are the nonlocal and local pseudopotentials of ion i, respectively, and VHar‘treeU?)
and ch(f') are the total Hartree and exchange-correlation potentials.

The Kohn—Sham Hamiltonian H K S has eigenfunctions girl-(F) with corresponding

20

eigenvalues 52-. We can expand viz-(7") in the orbital basis (”(7") as

«(t-(r) = Zahara. (2.13)
it

Then, finding eigenfunctions and eigenvalues of H K S is reduced to diagonalizing
the matrix (1:! #V — eiS'uy). In the following, I describe the way to determine this
matrix.

In a crystal, the potential Ve f f('F) is periodic and can be described as a sum of

Fourier components,

1

V. f m = 2V. f nememn ° 7". (2.14)
n

where

 

Vary-(Cf) = Q 1 A Veff(f)e—iG . Fdf" (2.15)

cell cell

and where chll is the volume of the primitive unit cell. When plane-wave basis
states [6' > (or qu (F) = efé.’ 7?) are used to expand 2122(7"), the matrix elements for the

potential become
<(f'IV effmlq >=ZV eff(Gn)5 q" -é’5n' (2-16)

If we define (j’ = I: + Gm and cf ’ = If + Gmr, we can rewrite the Kohn—Sham matrix

as
Z (Hm, m,(‘~‘) — 5,(E)5m, m.) C, mm?) = 0, (2.17)
ml
where
s n2 .. c
Hm, TIT/(k) 2771.6 ml + Veff(Gm — Gml). (2.18)

        

The reciprocal space representation makes the calculation of the matrix elements

equivalent to determining the Fourier components of the effective potential. The

21

kinetic energy terms contribute only diagonal elements to the Kohn—Sham matrix.
In SIESTA calculations, we distinguish in the calculation of matrix elements of
the Kohn-Sham one-particle Hamiltonian (2.12) between elements that involve either

two—center integrals or grid integrals, as described in the following.

2-center integrals

The matrix elements of the overlap matrix Spu=< ¢ul¢u > are given by

swat) a / «rpm/(r— am. (2.19)

The orbital wavefunctions (by and a“ can be expanded in spherical harmonics as

lmax l
are) = Z Z af‘mvmme), (2.20)
l = 0177. = —l
where
7r 27r
affine): f0 smear/0 d903’12n(9,<p)¢p(r.9,90)- (2.21)

Using the Fourier transform 45])(7‘) = $5 f ¢#(E)eik ' 77d]: leads to an alter-
(27f)

nate expression in reciprocal space,
5M1?) = / anaemia—“i ' Rdic'. (2.22)
We can furthermore make use of expanding a plane wave as

.-. -. m A A
elk ' R = Z 47rz'ljl(kR)Yl’;n(k)Ylm(R), (2.23)
z = 0

22

where jl(kr) is the Bessel function, and expand

lmax l
MI?) = Z Z b’lL‘m(k)}’lm(I}), (2.24)
l = 0m = —l
with
bflm(k) = £(—i)l/O rzdrjl(kr)a7m(r). (2.25)

Substituting expressions in Eq. (2.23) and (2.24) into Eq. (2.22) leads to

Sui/(1i): 2Z2 El: 2 Gfigmr’lvmsllumwlrmwa) 13mm), (2.26)
l=0m=—l l,,m,,,l,,m,,
with
chm“ ”m” =/ dQYZLm (my, m (n )i%(n), (2.27)
and
S[“m“’l"m”(rz) =4m‘ln —.,z [tr/000 k2dkjl(kR)b*'uk (k )bl m (k). (2.28)

lumu

. l . .
The Gaunt coefficrents G 1:an , WhJCh express angular momentum conservation

rules, can be calculated by recursion from Clebsch—Gordan coefficients and be stored
once and for all [95]. The overlap coefficients S§”m“’ lum" (R) depend on the inter-
nuclear distance R and vanish for R > (rfiutoff + rgmoff), where Tzutoff denotes
the confinement of the basis wavefunction ¢u(r").

The kinetic operator matrix elements T(E) can be evaluated similarly to S (E),

except for including an additional factor k2 / 2 within integral (2.28), as

lm ,lumu . _ _ 1
T,” # (R)=47rzll‘ l 1.]: 21.4.1ij (new (ab’l’m (k) (2.29)

23

The angular integral in Eq. (2.27) is performed by a Gaussian quadrature and
the radial integrals in Eq. (2.25), (2.28) and (2.29) are calculated using fast Fourier
transform (FFT) along the radial direction.

Grid integrals

Matrix elements of the remaining three terms in the Kohn-Sham Hamiltonian (2.12)
can be calculated on a real-space grid. The grid spacing is controlled by the en-
ergy cutoff in the plane-wave expansion of the potential, called ”MashCutoff” [78] in

SIESTA code. We use this real-space grid to evaluate matrix elements of the potential

V“) = Z Vlocal(77_ dz') + VHart'reeU’) 'l' V5500? (2'30)
2'

To eliminate the large computational cost associated with the long range of the
local pseudopotential 171060“? — 1352-), we rearrange the potential V’ by introducing

neutral-atom potentials [96] as

 

- r , ~_ g.
VNA(F— R.) = Vzocafi- R.) + / d2“ ”“03”.” z). (2.31)

This new potential is constructed by populating the localized basis functions
with appropriate valence charges to cancel the pseudonuclear charge inside the radius
rcuto f f' Consequently, VN A (7’) vanishes for r > Tcuto f f’ which greatly reduces
the number of non-zero matrix elements [96]. The difference 6720") between the self-
consistent electron density no") and ”atom (7") reflects formation of covalent bonds
and is determined by a ”Hartree-like” potential (ll/Hartree“), obtained by solving

the Poisson equation

1

2
6710;) = _4—71'8 V 6VHGT‘tT68(F)' (2.32)

24

Using the above expressions, the potential V'(f') can be rewritten as
V '(F) = Z VNA(F— R‘,) + (Si/Hammem + chm. (2.33)
2'

Following the initial guess of the charge density n(7‘~‘) on each grid point 1", we
evaluate 6VHamee(f') by solving the Poisson equation with the help of an FFT. We
then evaluate V9300") using LDA or GGA and determine Zz’ VNA(7"' — E). The sum
of these terms yields the desired value of V '(F) on the grid. To obtain the matrix
element H #V =< ¢p|H K glqbu > of the Kohn—Sham Hamiltonian, we sum up the
contributions (1);) (1")V '(f‘) (by (7")/_\.7"3 over the unit cell size. We need to point out that
the G = 0 component is not determined using FFT when determining potentials
such as the Hartree potential in periodic charge neutral systems, but rather using
sum rules. Charge neutrality must be enforced to avoid divergence of the G = 0

component.

2.3 Total energy

Once we determine the eigenvalues and eigenfunctions of the Kohn—Sham Hamil-

tonian, we may continue with the evaluation of the total energy,
Etotal = T[n] + Eext + EHartree + E330 + E11' (2.34)

Here, E I I describes inter-ion interaction

 

E11=§Z Z '* zmzm.

fl .. ~, (2.35)
§.m§’,m’ |R+Tm _ RI +77“,

 

where R is the Bravais lattice vector, 7'" is the atomic positions within the unit cell

and Zm is the valence charge of ion at position if [72, 97].

25

Using Eq.(2.9) we evaluate the kinetic energy functional as

Tlnl = 22' (WW ‘* VKSI¢2>~ (2-36)

Combining Eqs.(2.36) and (2.34), we obtain the total energy expression
1 3 3.0710972 ’) 3
EtOtal = Zéi—§ d T‘T‘d TTr—TT'l' d T[€$c(fl—V$c(fl]n(fl+EII. (2.37)
’i

where Exc = f d3re$c(f')n(7‘).
For an infinite periodic system, we will again benefit from performing the inte-

gration in reciprocal space, using Fourier transforms of the quantities of interest. We

find

 

)+ QC :[Exc(0)- -V$C(é)ln*(é) + EII’

‘ QC
Etotal = 252' — ? Zn
2 G 750

(2.38)

where QC is the volume of the unit cell. Here, we have excluded the G = 0 components,
as they would cause divergence in the numerical evaluation of Etotal° Physically, the
divergence in the Hartree energy is compensated by a divergence in E I I and external

potential energy. These diverging terms, given by

 

 

 

1 87rn(G ZmZml
Etotal,dz'v= Glim QC 2 GS? szexda) ”(G) +% Z d: R + _. _ R.’ - .. 1
—-> O 2R m R’m , Tm Tm!
(2.39)

are treated separately.

First, we note that the 1127332452) and n(C-vl) terms can be expanded close to G = O

_87me
9002

 

2.3;,(6‘): + am + 0(6), (2.40)

26

where Zm and am are expansion coefficients and

72(6) = Egg—’3 + [302 + 0(6). (2.41)

Substituting the expressions in Eqs.(2.40) and (2.41) back into Eq.(2.39), and drop-

ping negligible higher-order terms, we get for the G = 0 contribution to Etotal

 

. 1 8wn(G) 87r 2m Zm ..
E - = hm Q - — + a n(G)
total, dzv G _) 0 C [2 02 QCG2 2 m]

 

 

 

R, m R’, m’ m
This expression can be further simplified to
1 ZmZ . 1 82(2 2...)2
Etotal,dz'v = Zamzzm’+§ Z Z: ’7 ~ _. TI] .. "—5 9722 '
m m’ R,mR’,m’ R+Tm—R_Tm’ C

 

(2.43)
The last two terms include the diverging Hartree, external potential and inter-ion
Coulomb repulsion energies. They are treated using a technique due to Ewald [98],
which accelerates convergence by treating a charge neutral sub-system in reciprocal

space and performing the remainder in real space using the identity

 

 

 

(2.44)

2 —o —o —o
+2_7r Z/n d3: e—E‘gei(R '- R’) ' G

_1_
90 _. 0 1103'
G

Ewald formulates the charge neutral sub-system by superposing a virtual com-

pensative Gaussian charge distribution of width 77 at each ion, which renders the

27

system charge neutral. This convergent energy sum is evaluated in reciprocal space.
The virtual charge distribution is subsequently subtracted and evaluated in real space
with accelerated convergence for a finite value of 77. Ewald’s final expression for the

total energy of the ions is [99]

1 erfc(77:r) 277
Ell—=5 Z’ZmZmI;T——fi6m,ml
m m

 

"2
GI
471’ 1 -L‘2' .. _. ~ 7r
+— _. e 4'! cos R—R’ -G ——, 2.45
0c 2:: [G]2 K ) ] 7729c ( )
0750

independent of n. In practice, the value of 77 is varied to minimized computational

effort both in real and in reciprocal space.

28

Chapter 3

Interplay between structure and

magnetism in M0128919 nanowires

The following discussion is closely related to a publication by Teng Yang, Shinya
Okano, Savas Berber, and David Tomanek, Phys. Rev. Lett. 96, 125502 (2006) [35].

3.1 Introduction

One of the major challenges in the emerging field of molecular electronics is
to identify conducting nanostructures with desired electronic properties, which are
stable and easy to manipulate. Due to their high stability and favorable electronic
properties [1], carbon nanotubes [2] have been discussed extensively as promising
candidates for molecular electronics applications. A major drawback is our current
inability to synthesize or to isolate nanotubes with a given diameter and chiral an-
gle, which determine their metallic or semiconducting nature. Moreover, single-wall
carbon nanotubes are hard to isolate from stable bundles, which form spontaneously
during synthesis [3]. In this respect, recently synthesized nanowires based on Mo chal-
cohalides, such as Moleglg, seem to offer several advantages by combining uniform

metallic behavior, atomic-scale perfection and easy dispersability [5, 6].

29

 

 

MA)
A
D.
O >
> 0'
‘fl
\\
El \
\\

 

 

 

 

 

 

 

 

 

3-151 .
(b) 8 (d) 160 - ‘n o .

"E Q I, r e A -

§_152 I \\ 1‘ l ’1? 140 I s r

w \ I l [ I

g \\ 7 g, 120

0-153 . \\ ‘ B I 0 MO .

\ , 100 I A

a ‘ A m" ° ! ‘

U) \ A -

E 10’ ‘ 80 3* «

E—154 L g ,1 i I

“I so

 

 

 

 

 

910113131415 910113131415
a() a()

Figure 3.1: Structural properties of M012S919 nanowires. (a) Atomic arrangement
within a unit cell. Planes normal to the wire axis are denoted by A, A’, B, C, D.
(b) Binding energy as a function of the lattice constant a. (c) Optimized inter-plane
distances as a function of a. (d) Optimum values of the Mo—S-Mo bond angle 6 as
a function of a. Values in the two sulphur bridges per unit cell are distinguished by
different symbols. Dashed lines in (b—d) are guides to the eye.

J

Using ab initio density functional calculations, I study the suitability of M0123919
nanowires as potential building blocks of electronic nanocircuits. In terms of binding
energy per atom, I find these nanowires to be almost as stable as carbon nanotubes.
The nanowire skeleton consists of rigid, functionalized Mo octahedra, connected by
flexible, bi—stable sulphur bridges. I find this structural flexibility to translate into
an unusual capability to stretch up to 220% at virtually no energy cost, as a nanos-
tructured counterpart of an accordion. My calculations suggest that the nanowires
change from conductors to narrow-gap magnetic semiconductors in one of their struc—

tural isomers.

To gain insight into structural and electronic properties of these unusual systems,

30

I optimized the geometry of infinite M0128919 nanowires as well as selected structural
building blocks using density functional theory (DFT). I used the Perdew—Zunger
[100] form of the exchange-correlation functional in the local density approximation
(LDA) to DFT, as implemented in the SIESTA code [87]. The behavior of valence
electrons was described by norm-conserving Troullier-Martins pseudOpotentials [89]
with partial core corrections in the Kleinman-Bylander factorized form [92]. I used
a double-zeta basis, including initially unoccupied Mo5p orbitals. I arranged the
nanowires on a tetragonal lattice, with one side of the unit cell given by the lattice
constant a. The other sides of the unit cell were taken to be 19.7 A long, about twice
the nanowire diameter, to limit inter-wire interaction. I sampled the rather short
Brillouin zone of these 1D structures by 6 k-points [101]. The charge density and
potentials were determined on a real-space grid with a mesh cutoff energy of 150 Ry,
which was sufficient to achieve a total energy convergence of better than 2 meV / atom
during the self-consistency iterations.

Magnetic ordering in the nanostructures was investigated using the Local Spin
Density Approximation (LSDA) within the SIESTA code [87]. Even though LSDA
may be considered less rigorous and dependable than LDA, I use it here as a system-
atic way to estimate net magnetic moments and exchange splitting in 1D nanowires

and finite segments of these systems.

3.2 Structural and Electronic Properties

3.2.1 Structural Properties

With 30 atoms per unit cell, corresponding to 84 degrees of freedom, global
structure optimization is a formidable task. The initial experimental structure de—
termination, based on Atomic-Resolution 'Ifansmission Electron Microscopy (TEM)

and X—ray diffraction data [41], provided me with a useful idea of the equilibrium

31

structure. In spite of the significant energy gain of 50 eV with respect to the ex-
perimental structure during the initial structure optimization, the calculated atomic
arrangement within the M0128919 unit cell, depicted in Fig. 3.1(a), was found to lie
very close to the experimental structure.

I found it useful to assign atoms to planes normal to the wire axis. I label these
atomic layers A, A’, B, C, and D in Fig. 3.1(a). Layers A and A’ are structurally
identical in terms of atomic arrangement, but rotated by 180° with respect to each
other. Mo octahedra, decorated by S and I atoms, form the structural motif of the
AA’ and CD bilayers. As I discuss later on, these bilayers can be thought of as rigid
blocks, connected by bridges formed of three S atoms, denoted as layer B.

My results for the binding energy of the system as a function of the lattice
constant a. are summarized in Fig. 3.1(b). For lattice constants other than the exper-
imental value [41, 102] aexpt = 11.97 A, the initial structures used in the optimization
were based on uniformly expanded or compressed nanowires, subject to random dis-
tortions, or structures optimized using semiempirical force fields. For a given value
of the lattice constant, I considered a structure as optimized when different starting
geometries resulted in the same structural arrangement.

The cohesive energy of the 30-atom unit cell, displayed in Fig. 3.1(b), translates
into a formidable average binding energy exceeding 5 eV per atom. I found that inclu-
sion of Mo5p orbitals provided a net stabilization of the system, but did not modify
much the interatomic forces or equilibrium geometries. As indicated by my LSDA
calculations, additional structure stabilization occurred due to magnetic ordering for
selected geometries.

Most intriguing in these results is the presence of multiple structural minima.
Since structures with optimum lattice constants a = 11.0 A, 12.3 A and 13.8 A are
very close in energy, a distribution of bonding configurations along the chain axis

will likely occur in order to maximize configurational entropy, resulting in the loss

32

of long-range order. Identifying structures with very similar energies in a range of
lattice constants also implies the possibility to stretch the nanowire by 20% at virtu-
ally no energy cost, similar to an accordion. The remarkable ability of M06ley to
expand easily by nearly 30% has been previously noted in Atomic Force Microscope
experiments on closely related systems [103]. This uncommon flexibility of an inor-
ganic nanowire may explain in retrospect, why the observed lattice constant [41, 102]
an,” = 11.97 A may differ from the structural minirna in Fig. 3.1(b).

Trying to understand the origin of the unusual elastic behavior of M0128919
nanowires, I first investigated the inter-layer spacings as a function of the lattice con-
stant. Analysis of my data, presented in Fig. 3.1(c), suggests that interatomic spacing
within AA’ and CD bilayers changes very little with increasing lattice constant. The
motif of the bilayers is formed of six Mo atoms in near-octahedral arrangement, with
Mo—Mo bond lengths close to 2.6 A within the layers and 3.0 A along the wire axis.
In the following, I consider these decorated octahedra as rather rigid building blocks
of the nanowires. I found the decoration order by I and S atoms to be non-random,
since an interchange of these atoms within the unit cell raised the total energy by
more than 0.5 eV.

Most importantly, results in Fig. 3.1(c) suggested that virtually all structural
changes are accommodated by the sulfur bridges, which form the A’-D and C-A
connections. I found that most changes occurred in the Mo-S-Mo bond angle 8,
whereas the Mo—S bond lengths remained nearly constant. My results for O as a
function of the lattice constant (1, shown in Fig. 3.1(d), suggest that the expectation
values of 8 do not change uniformly, but rather group around 90° and 150°. In the
following, I will call the corresponding structural arrangements “short” and “long”
bridges. Similar metastable structures have also been reported for 1D wires of group
IV elements [104]. In the case of isolated sulphur chains and S3 molecules, I have

also observed a preference for similar bond angles as in Fig. 3.1(d). In analogy with

33

similar preferential bond angles found in different allotropes of C and Si, I associate
the short bridge with sp3 and the long bridge with sp2 hybridization.

In the case of M0128919 nanowires, I find two such sulphur bridges in each unit
cell. Local minima in the cohesive energy, plotted in Fig. 3.1(b), are found at the
lattice constants a = 11.0 A, 12.3 A, and 13.8 A. The fact that only three minima
are found, and that a = 12.3 A is close to the average of the other two lattice
constants, supports my conclusion that there are only two types of S-mediated bonds
connecting the blocks, which are decoupled within the unit cell. The three local
minima in Fig. 3.1(b) thus correspond to a short-short, short-long (or long-short),
and a long-long bridge configuration.

To verify that the optimum structures are not influenced by the constraints of
a periodic lattice with a fixed unit cell size, I independently optimized the structure
of the Mo—based building blocks with the proper decoration by S and I atoms, with
sulphur bridge trimers attached at both sides. With the exception of the bridge
atoms, which changed their positions, I found the atomic arrangement in M0686161
modelling the AA’ bilayer, and M068913, modelling the CD bilayer, to lie close to
globally optimized nanowire structures, discussed in Fig. 3.1.

Comparing the binding energies of the bilayer clusters, Ecoh(AA’) = —81.0 eV
and Ecoh(CD) = —89.2 eV, I conclude that the binding of sulphur atoms to the
cluster is significantly stronger than that of iodine atoms. I also found that the
isolated CD cluster acquires a net magnetic moment p(CD) = 1.00 #0. As I discuss
in the following, not only finite nanostructures with dangling bonds, but also infinite

nanowires may develop a net magnetic moment.

3.2.2 Electronic structure

I found the infinite M0128919 nanowires to generally behave as conductors in the

lattice constant range 9 A< a < 15 A, with the exception of the metastable structure

34

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

r x '0 1020304050

 

 

 

 

'p x '010 20 ’30 4o 50
k DOS (states/9V)

) - . ) ‘

r S T A
“J / - C .
3

-o.5 ~d§+~J -------

F X 0 10 20 3O 40 50
k DOS (states/9V)

 

 

 

 

 

 

 

 

 

 

Figure 3.2: Electronic band structure E(k) of M012S919 nanowires with lattice con-
stants (a) a. = 11.0 A, (b) a = 12.3 A, and (c) a = 13.8 A. Solid lines represent
majority and dashed lines minority spin bands. Bands crossing the Fermi level are
emphasized by heavy lines. Densities of states of nanowires with (d) a = 11.0 A, (e)
a = 12.3 A, and (f) a = 13.8 A, convoluted with 0.01 eV wide Gaussians. The Fermi
level lies at EF = 0.

35

at azll A discussed below. In Fig. 3.2 I present results for the electronic structure at
a = 11.0 A, 12.3 A, and 13.8 A, corresponding to the equilibrium structures indicated
in Fig. 3.1(b). The interactions along the nanowire, causing band dispersion, depend
crucially on the hybridization near the sulphur bridges. As discussed earlier, changing
the lattice constant does not affect interatomic distances, but rather modifies the
Mo—S-Mo bond angle 8, thus changing the hybridization of directional orbitals. An
increase in band dispersion, caused by increased hybridization, may be counter-acted
by band repulsion in complex systems.

The interplay between lattice constant and band structure is depicted in Figs. 3.2(a-
c). My results in Figs. 3.2(b—c) suggest that the metallic character of the wires de-
rives from a rather dispersionless band, which crosses the Fermi level. As shown
in Fig. 3.2(a), this band becomes very flat and even changes its slope at az11.0 A
due to the changed hybridization associated with Mo—S-Mo bond bending. Conse-
quently, the density of states develops a peak at Ep, which subsequently splits due
to a magnetic instability, as shown in Fig. 3.2(d). This splitting opens up a fun-
damental [gap of 73 meV, accompanied by a net magnetic moment of 1.00 #0 per
unit cell. This magnetic moment is obtained using the LSDA densities of states as
p = [5:50 dE(nT (E) — nl(E)). At other lattice constants, the dispersion of this
partly filled band increases, as seen in Figs. 3.2(b) and (c). Consequently, as seen in
Figs. 3.2(e) and (f), the density of states at the Fermi level is lower, suppressing the
magnetic instability.

A more detailed discussion of the electronic structure in the system with a net
magnetic moment, at a = 11.0 A, is presented in Fig. 3.3. The charge distribution in
the M012S919 nanowire, depicted in Fig. 3.3(a), is very similar to the superposition of
atomic charge densities. This is better seen in Fig. 3.3(b), which displays the charge
redistribution within the system with respect to isolated atoms, given by Ap(r) =

p(r) — Spatom (r) As expected based on electronegativity differences, I observe a

36

0.003
0.002

0.001
0'91 91
0(e/ao 3)

—o. 001

—o.002

-0.003

 

Figure 3. 3: Contour plots depicting the charge distribution in a N103128919 nanowire
at a = 11. 0 A (a) Total charge density contour at p- — 0. 05 el/ao. (b) Difference
charge density Ap(r), indicating regions of charge depletion and excess with respect to
the superposition of isolated atoms. (c) p = 1.0x 10‘3 el/aO3 charge density contour
associated with states close to the Fermi level, Ep — 0.05 eV< E < EF + 0.05 eV.
(d) Difference spin density pT(r) — Pl (r), showing regions of excess majority spin T.
Contour plots (b) and (d) are depicted in a plane, which contains the wire axis.

37

small amount of charge, stemming mostly from Mo4d orbitals, to be transferred to
the region within the Mo octahedra, in particular to Mo-Mo bonds and sulphur atoms.
The decorating iodine atoms do not experience a change in their charge distribution
due to their presence in the nanowire structure. More interesting than information
about all populated levels is the spatial distribution of states close to the Fermi level,
which form the valence and conduction band. My results, shown in Fig. 3.3(0), suggest
that these states are rather delocalized. Together with the band structure results of
Fig. 3.2, I conclude that the nanowires considered here should be conducting at room
temperature, in agreement with experimental observations [5, 6].

Spatially resolved information about the magnetic structure of the nanowire at
a = 11.0 A is presented in Fig. 3.3(d). Associating up—spin with the majority and
down-spin with the minority states, the plotted quantity pT (r) — p 1 (r) allows us to iden-
tify spatial regions with a net magnetic moment. Based on the results in Fig. 3.3(d)
and a set of 3D spin density contours, not shown here, I find the majority spin to
be localized mostly on the bridging sulphur atoms, in their non-bonding 2p orbitals
normal to the wire axis, and to a smaller degree on adjacent Mo atoms.

Since magnetism occurs in selected structural isomers of the infinite M0128919
nanowires as well as in isolated building blocks, I expect the possibility of modifying
the magnetic structure by locally stretching or expanding the nanowires. Since such
structural changes may be induced locally at negligible energy cost, various mag-
netic patterns could possibly be obtained by mechanical manipulation of M0123919
nanowires, possibly using an Atomic Force Microscope. Since magnetic ordering will
likely affect spin transport in these systems, M0128919 nanowires may be viewed as

spin valves, which could change their state by applying local stress.

38

3.3 Summary

I note that the overall metallic behavior of M0128919 nanowires is rather unusual
among Chalcogenide structures. Significant interest has been devoted in the past to
the ability of layered semiconductors, such as M082, to form fullerene- and nanotube—
like structures [105, 106, 107]. Similar to their bulk counterparts, M082 nanotubes
were found to be semiconducting, unless intercalated with iodine to enhance conduc-
tance [108, 109]. MOSQ-based fullerenes and nanotubes, often containing iodine, were
noted in particular for their favorable tribological properties [105, 107, 39]. At other
stoichiometries, such as M06 S316 or Mo6S4I5, self-assembled nanowires have been re-
ported to form with either semiconducting or semi-metallic behavior [20, 21]. In the
particular M0128919 stoichiometry studied here, the rich behavior of molybdenum
chalcogenides is complemented by new properties, such as high structural flexibility,
metal-semiconductor transition, and magnetism. I propose to use these systems as

unique building blocks in hierarchically assembled nanostructured electronic circuits.

39

Chapter 4

Unique structural and transport
properties of molybdenum

chalcohalide nanowires

The following discussion is closely related to a publication by Igor Popov, Teng
Yang, Savas Berber, Gotthard Seifert and David Tomanek, Phys. Rev. Lett. 99,
085503 (2007) [110].

4.1 Introduction

Chalcohalides of molybdenum and other transition metals are known to form
stable, intriguing 2— and 1-dimensional structures [7] with an unusual combination of
electronic properties [8] including good conductance, superconductivity, magnetism,
and nonlinear polarizability. These layered or filamentous substances are known as
catalysts [9] and, to a much larger degree, as excellent solid lubricants [10, 11, 9].
Their potential to become unique building blocks of nano-devices has barely been
noticed so far [9], in stark contrast to popular carbon nanotubes [12]. Recent progress

in the synthesis of MoxSylz nanowires [13] suggests that these monodisperse, self-

40

supporting nanostructures may nicely complement carbon nanotubes by avoiding
their shortcomings such as a strong dependence of conductivity on the nanotube
structure and difficulty to separate bundled tubes [12].

Here I combine ab initio density functional theory [76, 77] and quantum transport
calculations based on the nonequilibrium Green’s function formalism [111, 112] to
compare structural, electronic, and transport properties of M0636 _ $13; nanowires
(NWs) to those of carbon nanotubes (CNTs). I find systems with x = 2 to be
particularly stable and rigid, with their electronic structure and conductance close to
that of metallic (13,13) single-wall carbon nanotubes. M0686 __ 1.1x nanowiras are
conductive irrespective of their structure and capable of forming ideal contacts to
Au leads through thio—groups. Due to the weak inter-wire interaction, M0686 _ 2:113
systems should be more easily separable than carbon nanotubes.

As mentioned before, Chalcogenide compounds containing Mo and S have been
studied for a long time [7, 14]. Whereas the best known allotropes, including M082,
are insulating and form layered compounds, more interesting structures often occur
at lower sulfur concentrations. Well known are Chevrel phases, characterized as
cluster compounds with M0688 subunits, furthermore finite clusters with a similar
structure, and needle-like quasi-1D compounds [15]. All these interesting structures
necessitate the presence of metal counter-ions for their synthesis. Besides providing
structural stability, the main role of the counter-ions is to transfer electrons into
the Chalcogenide substructures [16, 17], thereby forming an ionic crystal [18]. In their
most stable electronic configuration, many of these compounds contained (M0686)2_
building blocks[14].

The idea behind this work is to study the possibility of stabilizing Mo-based
nanowires by substituting the divalent sulfur by a monovalent halogen (I) with a
similar electronegativity. In this way, the “magic” electronic configuration could be

preserved, while maintaining a covalent character of the system and avoiding the for-

41

mation of an ionic crystal, where electron correlations would dominate the electronic
structure. In the following, I study the properties of M05S6 _ x133 nanowires, where

iodine was used to substitute for sulfur.

4.2 Structural and Electronic Properties

4.2.1 Structural and Chemical Stability

To gain insight into structural and electronic properties of the proposed sys-
tems, I optimized the geometry of infinite M0636 _ $11; nanowires for x = 0 — 6
using density functional theory (DFT). I used the Perdew-Zunger [100] form of the
exchange-correlation functional in the local density approximation (LDA) to DF T,
as implemented in the SIESTA code [87]. The behavior of valence electrons was de-
scribed by norm-conserving Troullier-Martins pseudopotentials [89] with partial core
corrections. I used a double-zeta basis, including initially unoccupied Mo5p orbitals.

The numerical results were obtained for the primitive unit cell of length a, con-
taining 12 atoms, depicted in Fig. 4.1(a). For selected structures, I also compared my
results to those for a double unit cell with 24-atoms. To describe isolated nanowires
while using periodic boundary conditions, I arranged them on a tetragonal lattice
with a large inter-wire separation of 20 A. I sampled the rather short Brillouin zone
of these 1D structures by at least 8 k-points. The charge density and potentials were
determined on a real-space grid with a mesh cutoff energr of 150 Ry, which was suf-
ficient to achieve a total energy convergence of better than 2 meV/ atom during the
self-consistency iterations.

To accelerate the global structure optimization of the nanowires with 36—7 2 de-
grees of freedom per unit cell, I first explored the configurational space and pre-
optimized the systems using the faster density functional based tight binding (DFTB)

method [113, 114], which had been used successfully to describe the deformation of

42

 

(a ) wire direction

 

 

 

0.0 (I ------------------------- A.
9 ,,
3 \\ ’8’
g '0.5 - \\ O” .
.s , 9
a I
a a 8 I
E I
g -10 - \\ ’fi’ _
u, (a,

_15 I l I 1 1

0 1 2 3 4 5 6
Iodine content x

 

 

 

 

 

 

 

 

 

 

. . . 6:.
9 0.4 " . MOSS4|2 NW 4;." -
g :-, + (13.13) ONT ,1,-
2 '1. If'
+\ 1,.
>3 0 2 - - \ . .-
-.\ I,
[ll +3} ’f
E 00 — W -
0 1 2 3 4 5 6 0.9 1 1.1
Iodine content x a/aeq

Figure 4.1: (a) Schematic structure of a M0686 _ 31x nanowire in side and end-on
view. (b) Energy gain AE(;1:) with respect to the average binding energy and (c)
energy AH($) of the substitution reaction in Eq. (4.1) leading to the formation of a
M0636 _ 331$ nanowire as a function of composition. Among the data points for all
structural isomers (o), the most stable structures are identified by the solid circles (0).
(d) Deviation from the equilibrium binding energy E0 as a function of the relative
unit cell size a/aeq, for a segment of the “magic” Mo6S4I2 nanowire containing N
Mo atoms, and for an N -atom segment of the ( 13, 13) carbon nanotube.

M082 layers to tubular structures [115].

The optimized structure of M0686 _ $13; nanowires, shown in Fig. 4.1(a), consists
of a Mo backbone decorated by S and I ligands. The Mo core structure is formed by
Mo trimers of alternating orientation forming a chain. There is one M06 octahedron
per unit cell, surrounded by (6 — r) sulfur atoms and r iodine atoms. For each value
of x I performed an exhaustive isomer search and a global structure optimization.
In general, I find the Mo—Mo bond length to increase from z2.6 A when close to
sulfur to #27 A when close to to the larger iodine. Introduction of iodine also

causes an increase of the equilibrium lattice constant from aeq (M0636) = 4.34 A to

aeq(M0616) = 4.55 A.

43

I find M0686 __ $15,; nanowires to be rather stable, with an average binding energy
per atom ranging from 5.0 eV in M0616 to 6.3 eV in Mo6S6 with respect to isolated
atoms. The nanowires thus rival the stability of graphite with 7.3 eV/ atom, and the
slightly less stable carbon nanotubes Changing the iodine content a: in M0686 _ 331:1:
nanowires, I could naively expect the binding energy E to vary linearly between that
of M0636 and M0616, showing no dependence on the particular structural isomer. In
reality, deviations from this linear behavior [116], depicted as AB in Fig. 4.1(b), are
substantial, suggesting that the stability of the various structural isomers at a par-
ticular composition varies by up to a large fraction of an eV. Focussing on the most
stable isomers, I find a general tendency to selectively stabilize particular chalcohalide
stoichiometries, such as M068412. As I will show below, this “magic” composition op-
timizes the electronic configuration of the building blocks to agree with the optimum
charge state identified above using heuristic arguments.

Among the many structural isomers of M0686 _ $123, I generally found structures
with the largest separation between iodine atoms to be most stable. When increasing
the variational freedom in the arrangement of iodine atoms by doubling the unit cell
size, I identified geometries that further stabilize the structures by providing energy
gain per unit cell ranging from 0.09 eV for a: = 1 to 0.29 eV for :1: = 3.

The selectivity of a possible synthesis pathway becomes apparent, when studying
the reaction energy AH of the substitution reaction

1‘

M0636 + 2

12 fl M0636 _ $13; + 288 , (4.1)

depicted in Fig. 4.1(c). Clearly, substitution of sulfur in Mo6S6 by gas-phase iodine
is only exothermic for the “magic” iodine content :1: = 2.
In the following, I will compare various properties of M0686 _ $12: nanowires to

those of carbon nanotubes. As will become clear later on, the electronic structure of

44

  

 

 

 

 

(a) (W ' ' '
.- «993.75 1
_ 0.25

,a ,4 ‘0 0.00

. 3’10 0_ +0375
‘9: 0.00
d 0M0 "0’ . -o.25 ‘
O S -20 . ' .0375 -

‘ ,a O | 8.8 930 9.2 9.4 9.5 9.8

d (A)

Figure 4.2: (a) Structure of M068412 nanowires arranged on a simple hexagonal
lattice in a plane normal to the wire axes. (b) Contour plot of the nanowire binding
energy in this lattice as a function of the wire orientation go and separation d. The
energy is given in eV per formula unit.

the chalcohalide nanowires matches well that of a conducting (13, 13) carbon nanotube
with a diameter of 17.6 A. The equilibrium unit cell size aeq(13, 13) = 2.46 A of the
armchair nanotube is about half the value of the “magic” nanowire, aeq(Mo6S412) =
4.45 A.

The axial stiffness of a M068412 nanowire in comparison to that of a (13,13)
carbon nanotube can be inferred from Fig. 4.1(d). I feel that in a fair comparison
between the different systems, the energy investment upon axial strain should be
normalized by the number of Mo backbone atoms in the nanowire and number of C
atoms in the nanotube. Inspection of my results in Fig. 4.1(d) suggests that the high
axial stiffness, based on the above definition, is nearly the same in the two systems.
Thus, the M068412 nanowire differs significantly in its rigidity from the accordion-like
behavior identified recently in the “floppy” M06S4_5I4_5 nanowire [35].

4.2.2 Dispersability

Contrasting with the high stability and axial stiffness of the M06S412 nanowires

is their lateral inter-wire interaction, discussed in Fig. 4.2. I calculated the binding

45

energy of straight nanowires on a simple hexagonal lattice with respect to isolated
nanowires using DFTB, augmented by Van der Waals interactions [117]. My results
for the binding energy as a function of the inter-wire separation d and the wire orien-
tation (p are presented in Fig. 4.2(b). I find the binding energy to be generally weak
and strongly anisotropic. The most stable arrangement occurs at an inter-wire sepa-
ration d = 9.3 A. The binding energ in this geometry corresponds to 0.1 eV for a 1 A
long nanowire segment and equals that of a corresponding segment of bundled (10, 10)
CNTs [118]. In realistic bundles, individual nanowires and nanotubes are likely to
be twisted rather than being perfectly straight over long distances [119]. Then, their
effective interaction should be closer to an average over all possible orientations.
Due to its anisotropy, seen in Fig. 4.2(b), the effective attraction between M068412
nanowires is strongly reduced or changes to repulsion when averaging over 4p, whereas
the rather isotropic interaction between nanotubes does not affect their strong bind-
ing [118]. This explains the observation that bundled chalcohalide nanowires are

much easier to separate than bundled carbon nanotubes.

4.2.3 Electronic Structure

Among the different properties of M0686 _ x133 nanowires, I find their electronic
structure to be most intriguing. In Fig. 4.3 I compare the band structure E(k), the
density of states (DOS), and ballistic conductance of isolated M0686 _ x13; nanowires
to those of an isolated (n, n) single-wall carbon nanotube. Independent of compo-
sition, I find all M0686 _ 931$ nanowires to be metallic or semi-metallic. This is
appealing in view of the fact that carbon nanotubes may be conducting or semicon-
ducting, depending on their chiral index (n, m) [12].

In Fig. 4.3(a-f), the electronic properties of the M068412 and the M0686 nanowires
can be compared side-by side. External gating or doping can furthermore be used

to shift the Fermi level of the chalcohalide NWs, as indicated by the dotted lines

46

in Fig. 4.3. Similar to conducting (71,12) carbon nanotubes, I find an energy range
with a constant DOS also near the Fermi energy of the gated chalcohalide NWs, sug-
gesting high electron mobility, which is flanked by a pair of van Hove singularities.
As can be seen by comparing Figs. 4.3(e) and (h), the energy separation of the van
Hove singularities in the M0686 nanowire is best matched by the metallic (13,13)
carbon nanotube, which I consider here. For the sake of easy comparison with carbon
nanotubes, I will focus on externally gated / doped chalcohalide N Ws in the following.

The band dispersion of gated M0686 _ $13; nanowires and the (13,13) carbon
nanotube are compared in the left panels of Fig. 4.3. In all three systems considered
I can identify nearly free-electron bands in the vicinity of EF. Whereas the constant
DOS near Ep in the (13,13) carbon nanotube, shown in Fig. 4.3(h), derives from
bands of C21),r character, a very similar constant DOS of M0636 in Fig. 4.3(e) de-
rives from 02 bands [14] with predominant Mo4da character. In general, I expect the
stability of any system to increase when populating bonding or depopulating anti-
bonding states. In M0686 __ 31x nanowires, the optimum stabilization is achieved by
filling the Mo4d-derived 02 band up to the folding point at X.

As suggested by comparing Fig. 4.3(a,d), (b,e), I find that the main effect of
changing the composition of M0686 _ $13; nanowires by iodine substitution prior to
gating is to electronically dope the system by shifting the Fermi level. This finding
agrees qualitatively with that in LixMOGS6 nanowires [18], where the most stable
composition a: = 2 has been associated with the “magic” (M0686)2_ complexes. As
suggested above, stabilization of the system should depend on the oxidation state of
the Mo backbone. In that case, initial withdrawal of 12 electrons from M06 by divalent
sulfur ligands, followed by adding two electrons to M06 in the “magic” (M0686)2_
complexes, should be equivalent to withdrawing only 10 electrons from M06 in the first
place, by substituting two divalent sulfurs by monovalent iodines. Further increase in

the concentration of Li or I should increase the population of the antibonding levels

47

E (eV)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l I 4 1

 

 

 

 

1 4L

 

 

0

20
DOS (arb. units)

 

 

 

 

 

 

 

 

 

DOS (arb. units)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8
G/Go

Figure 4.3: Electronic properties of a M0686 _ $13; nanowire (a—f) in comparison

to a (13,13) carbon nanotube (g—i).

Displayed is the band structure E(k) of the

M0686 _ $13; nanowires in (a,d), their density of states (DOS) in (b,e), and quantum
conductance G (E) in units of the conductance quantum Go (c,f). The corresponding
quantities for the (13,13) carbon nanotube are shown in (g—i). E = 0, given by the
dashed lines, corresponds to the Fermi level in (a,b,d,e,g,h), and to zero source-drain
voltage in (c,f,i). The dotted lines show the position of EF in externally gated / doped
chalcohalide nanowires resembling carbon nanotubes.

48

and thus destabilize the system. I find this reasoning confirmed by the observed
unusual stability of Li2M0686 and Mo6S4IQ, both of which should display a similar
electronic configuration of the Mo backbone.

As seen in Fig. 4.3(a,d), substituting sulfur by iodine atoms in M068412 not only
shifts the Fermi level, but also opens a narrow band gap, while introducing a new
band above E F. This nearly free-electron band of predominantly Mo4d character is
also observed in the Li doped M0686 system [18] and caused by locally changing the
crystal potential along the chains of I or Li atoms in the crystal. I find the position
of this band to depend sensitively on the lattice constant, suggesting the possibility
to modify the electronic structure by axial deformation.

Addressing the usefulness of M0686 _ x117 nanowires as ballistic conductors, I
calculated their quantum conductance G as a function of the carrier injection energy
E using the nonequilibrium Green’s function approach [111, 112]. The conductance
results for a: = 2 and a: = 0, presented in Fig. 4.3(e) and (f), are compared to those
for a (13,13) carbon nanotube in Fig. 4.3(i). In the gated chalcohalide NWs, the
Mo4d character of the states near EF suggests that conduction involves mostly the
Mo backbone and not the ligands.

Since all these systems are metallic, with a constant density of states near Ep,
I observe similarities in the conductance spectra of gated M0685 _ $13 nanowires
and metallic carbon nanotubes. A major advantage of M0686 _ 31x nanowires is the
natural termination of finite segments by sulfur atoms, which are known to bind to

Au electrodes as thio—groups in a well-defined way.

4.3 Summary

In conclusion, I combined ab initio density functional and quantum transport cal-

culations to compare structural, electronic, and transport properties of M06S6 _ 331$

49

nanowires with carbon nanotubes. I find that the M068412 system may form partic-
ularly stable, free-standing quasi-1D nanowires with electronic structure and conduc-
tance close to that of metallic (13,13) single-wall carbon nanotubes. M0686 _ x135
nanowires have favorable properties in comparison to carbon nanotubes by being con-
ductive irrespective of their structure, more easily separable, and capable of forming

ideal contacts to Au leads through thio—groups.

50

Chapter 5

Compositional ordering and
quantum transport in M0689 _ 3315,;

nanowires

The following discussion is closely related to a publication by Teng Yang, Savas

Berber and David Toma’nek, Phys. Rev. B 77, 165426 (2008) [120].

5.1 Introduction

Combining sub-nanometer diameter with structural stability and interesting elec-
tronic properties, transition metal Chalcogenide nanowires have been discussed as a
potentially viable alternative to carbon nanotubes [19] for many applications. In
particular, M0689 _ $15,; nanowires have been studied extensively both experimen-
tally [20, 21, 22, 6, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] and theoret-
ically [35, 36, 37]. In contrast to other nanowires and nanotubes, M0689 _ 31110
are believed to be metallic independent of their structure [35, 36, 21] and can be
synthesized [38], dispersed [6, 5], and functionalized [23] in a straight-forward man-

ner. Nano-mechanical studies indicate a very low shear modulus of the bundled

51

nanowires [26], suggesting self-lubricating properties and potential applications in
nano-tribology [39]. Sulfur terminators act as ”alligator clips”, providing optimum
contact to gold leads for molecular electronics applications.

In spite of the numerous studies, several questions about M0689 _ x113 nanowires
still await a definitive answer. There is a consensus that the skeleton of these systems
consists of Mo octahedra, which are functionalized by S and I adsorbates and con-
nected by bridges containing S or I atoms. Still, there is an open controversy about the
atomic arrangement in nanowires with r = 4.5 and a: = 6, which have been observed
as stable compounds. X-Ray Diffraction (XRD) [21] and Extended X-ray Absorption
Fine Structure (EXAFS) [40] studies agree in the identification of Mo octahedra as
stable building blocks of the nanowires, but disagree about the precise positions of
sulfur and iodine. In particular, both iodine [21, 40] and sulfur [33, 35, 36] have been
proposed as constituents of the bridges connecting the Mo octahedral building blocks.-
Additional uncertainty exists, whether the connecting bridges contain three [41, 35]
or four [26, 40] atoms.

Obviously, the atomic structure plays a key role in determining the stability
and conductance of the nanowires. [110] In spite of apparent interest in the topic,
no conductance data are available that would illustrate the potential of M0689 _ $13;
nanowires, sandwiched between gold leads, for molecular electronics applications. Be-
sides available structural results for nanowires with :r = 4.5 and a: = 6 stoichiometries,
which were assumed to be homogeneous, nothing is known about structural properties
of nanowires with other compositions, which may phase separate into iodine-rich and
iodine-depleted segments. Lack of precise structural information also offers a plau-
sible explanation for the apparent discrepancy between the measured and calculated
Young’s modulus of M0689 _ $11: nanowires [26] with :1: = 4.5 and a: = 6.

To address these open points, I combined ab initio density functional theory [76,

77] and quantum transport calculations based on the nonequilibrium Green’s function

52

  
   

l

9 b 0 ‘

“(28> 42—h) |(20)l

   

C

 

 

     

. do 9 ’
I I(3a) 1(35) m ] l 1(4a) 1(45) M21 1
0M0 O | 0 S

 

 

 

 

 

Figure 5.1: Optimized atomic arrangements within finite building blocks of
M0689 _ x133 nanowires. The structures are labelled according to the number of
iodine atoms, and structural isomers with the same stoichiometry are distinguished
by lowercase Latin characters. The finite clusters have been terminated by S atoms
on both sides along the wire direction, indicated by the arrows. The most stable
isomers for each stoichiometry are underlined and marked in red.

formalism [111, 112] to determine stability, equilibrium structure, electronic, and
transport properties of M0689 __ x1117 nanowires in the full composition range. I found
the M068613 nanowire to be particularly stable and identify stoichiometries, where
separation into two phases, one of them being :1: = 3, is energetically favorable. Sulfur-
terminated wire segments are known to form optimum contacts to gold leads. I find
that sulfur forms the most stable bridges connecting the functionalized Mo octahedra
and determine the relative stability of bridges containing three or four sulfur atoms.

Finally, I determine the conductance of M0689 _ x11: nanowires with a: = 3 and

r = 4.5, sandwiched between two gold leads.

5.2 Theory

To determine the optimum geometry, relative stability and ground-state elec-
tronic structure of the nanowires, I used the density functional theory (DFT) [76, 77],
as implemented in the SIESTA code [87]. I used the Perdew—Zunger [100] form

53

of the exchange-correlation functional in the local density approximation (LDA) to
DFT. The behavior of valence electrons was described by norm-conserving Troullier-
Martins pseudopotentials [89] with partial core corrections in the Kleinman-Bylander
factorized form [92]. I used a double-zeta basis, including initially unoccupied M05p
orbitals. I studied the possibility of spin polarization using the Local Spin Density
Approximation (LSDA) and found all the nanowires investigated here to be nonmag-
netic.

To test the usefulness of these systems as ballistic conductors, I calculated the
quantum conductance G of M0689 _ x1173 nanowires sandwiched between two Au(111)
surfaces as a function of the carrier injection energy E. My calculations were per-
formed using the nonequilibrium Green’s fimction approach [111], as implemented in
the TRAN-SIESTAC code [112], and the localized basis set described above.

My studies focus both on the infinite nanowires and their building blocks, de-
picted in Fig. 5.1. Infinite nanowires are represented by a periodic arrangement of
unit cells of length a containing two formula units of the compound. To describe
isolated nanowires while using periodic boundary conditions, I arranged them on a
tetragonal lattice with a large inter-wire separation of 30 A. I sampled the rather
short Brillouin zone of these 1D structures by 6 k-points. The charge density and
potentials were determined on a real-space grid with a mesh cutoff energy of 150 Ry,
which was sufficient to achieve a total energy convergence of better than 2 meV / atom

during the self-consistency iterations.

54

5.3 Results

5.3.1 Optimum functionalization of the building blocks in

nanowires

To get a better understanding of the optimum arrangement of iodine and sul-
fur ligands on the M06 octahedral building blocks in the nanowires, I optimized the
structure of all possible ligand arrangements in M06812 _ $11; systems. To avoid
complications caused by stoichiometry-dependent unit cell sizes, I first considered fi-
nite nanowire segments. The wire segments were terminated on both sides by sulfur
trimers as reference ligands, which act as bridge connectors in the infinite nanowire.
Such chalcohalide clusters have been observed in cluster compounds, with a partic-
ular oxidation state stabilized by counter-ions [121]. Even though I consider charge
neutral wire segments, the electron count on the M06 octahedra may differ from the
infinite nanowire due to the presence of ligands on both sides, causing minor struc-
tural and energetic changes. My results for the equilibrium structure of these finite
building blocks with the composition 03x36 are shown in Fig. 5.1. For the sake of
fair comparison, I oriented the M06 clusters along the nanowire direction, indicated
by the arrows, and grouped visually the different structural isomers with the same
composition.

The choice of using sulfur reference ligands is inspired by the results of the com-
bined scanning transmission electron microscopy (STEM) and X—ray photoelectron
spectroscopy (XPS) study of M068316 nanowires [33], suggesting that all bridges
connecting M06 octahedra, covered by iodine ligands, contain only sulfur atoms.

To interpret my results, I represent the structures of Fig. 5.1 schematically in
Fig. 5.2. In the representation of Fig. 5.2, I first lay down the molybdenum octahedra
onto one of their faces, to be considered the projection plane. Next, I consider the

position of ligands decorating all faces except those parallel to the projection plane.

55

   
 

 

E, = -95.836 av E, -.- 493.940 eV E = -92.069 eV E, = .92.120 av §s= -91.954 av]

O Iodine
O Sulfur

 

 

 

= -86.536 eV Efi-aesss eV E, = -86.646 efl E, = -83.709 eV E, .-. 60.950 ev

 

Figure 5.2: Schematic arrangement of sulfur and iodine ligand atoms in the structures
of Fig. 5.1, along with the binding energy values Eb with respect to isolated atoms.
The most stable isomers for each stoichiometry are underlined and marked in red.

Of the six ligands, two triplets will form two equilateral triangles, rotated by 180°
with respect to each other, in planes parallel to the projection plane. These triangles,
with the corners corresponding to ligand positions, are depicted in Fig. 5.2. Clearly,
structures with a: = 0, l, 5, 6 iodine atoms have only one structural isomer, labeled
as I(:L') in Fig. 5.1 and Fig. 5.2. Structures with x = 2,3, 4 iodine atoms have three
structural isomers each. My binding energy results in Fig. 5.2 suggest that the relative
stability of these structural isomers can differ up to @035 eV. I find that the most
stable isomer for each stoichiometry, marked by an underlined red label, also has
the highest symmetry. Since the inter-iodine distance is maximized in the favored
structures, energy stabilization of the symmetric structures can be understood in

terms of minimizing Coulomb repulsion.

56

   

0M0 CI 08

Figure 5.3: Optimized structure of two isomers with the same stoichiometry M06S6I3
representing the connection between functionalized M06 building blocks in nanowires
by iodine (left) and sulfur (right) bridges.

5.3.2 Connection between building blocks in nanowires

In previous studies, both sulfur [33, 35, 36] and iodine [21, 40] have been pos-
tulated to form connecting bridges between functionalized M06 building blocks in
M0689 _ 551$ nanowires. Furthermore, the stability of sulfur— and iodine—based con-
nections has been claimed to be nearly the same [36]. To identify the nature of en-
ergetically preferred bridges, I considered two structural isomers of infinite M065613
nanowires, one with iodine and the other with sulfur bridges connecting the M06'
based building blocks. A unit cell of the optimized structures, containing 30 atoms,
is shown in Fig. 5.3. I performed a global structure optimization, including the re-
laxation of the unit cell size, and found the nanowire connected with sulfur bridges
to be energetically more stable by 0.65 eV per 30 atoms, providing further support

for the presence of sulfur bridges.

5.3.3 Elastic properties of nanowires

In bundles of nanowires, the Young’s modulus can be determined from the
stress/strain ratio in the wire direction. Experimental observations [26] of high
Young’s modulus values near 420 GPa in M0123919 and M01256112 suggest that

these nanowires should be very rigid in the axial direction. To explain this high

57

 

 

 

 

 

 

 

 

 

 

 

 

 

3-
A 9
3. 32
w w
1<L
1o 11 12 13 14 15 011 in, 13 14
a(A) a(A)

Figure 5.4: Equilibrium structure and axial rigidity in (a) M0128919 and (b)
M012Slllg nanowires. The main difference between the compounds is an additional
S atom in (b) inserted inside each of the S3 bridges in (a). E denotes the total energy
relative to the optimum structure and a is the lattice constant. Dashed lines are
guides to the eye, connecting the data points. Quadratic fits to the data points near
the equilibrium, related to the axial rigidity, are given by the solid lines.

rigidity, the bi—stable S3 bridges [41, 35], connecting the Mo-based building blocks,
were postulated to contain one extra atom in the middle [26, 40]. To understand the
difference between S3 and S4 bridges in terms of axial rigidity, I determined the sta-
bility of M0128919 and M01281119 as a function of the lattice constant. Optimized
equilibrium structures and binding energies of these systems are presented in Fig. 5.4.
As seen in Fig. 5.4(a), owing to the bi—stability of the two S3 bridges, the E(a) graph
exhibits three minima, the most stable of them corresponding to one short and one
long sulfur bridge. In the finite lattice constant range between the different minima,
stretching and compression of the nanowire occurs at minimum energy cost. This
is no longer true, however, when considering additional compression of a completely

compressed or additional stretching of a completely extended nanowire. A quadratic

58

fit of the E (a) plot near the minima, depicted by the solid lines in Fig. 5.4, suggests
that there is little difference in the spring constant k = 82 E / 80.2 of the nanowires rep-
resented in Figs. 5.4(a) and 5.4(b). Using the quadratic fits of Fig. 5.4, I can calculate
the Young’s modulus as Y = k(aeq/A), where aeq is the equilibrium lattice constant
and A is the cross section of the nanowire. The cross-section area per nanowire in an
infinite bundle, where the inter-wire distance has been optimized [110] to be 9.4 A,
is A = 76.5 A2. This allows us to determine the Young’s modulus of M0128919 with
$3 bridges as Y = 99 GPa, and that of M01281119 with S4 bridges as Y = 109 GPa.
Thus, I conclude that the central atom in the S4 bridges does not necessarily increase
the axial rigidity of the nanowires.

My results are well within the wide range of reported theoretical and experimental
values of the Young’s modulus in M012SzIy nanowires, with calculated data being
generally much softer than observed values [26, 36, 122]. The calculated value [26]
Y = 45 GPa in M01286112 nanowires is much smaller than other reported theoretical
values [36] Y = 82 GPa and Y = 94 GPa in this system, which contains bi-stable
S3 bridge connectors, and the Y = 114 GPa value in M01288112 nanowires with
S4 bridges [36]. An even larger value Y = 320 GPa has been predicted for M0686
nanowires [122]. Comparing all available data, I conclude that the Young’s modulus

may depend sensitively on the stoichiometry.

5.3.4 Compositional ordering and phase separation in nanowires

Since the synthesis of M0689 _ x1513 nanowires occurs in a single-step reaction di-
rectly from the elements [38], I explore the intriguing possibility of forming nanowires
with an arbitrary iodine concentration. To account for phase separation, which may
be expected in this case, I consider coexistence of domains with different composi-
tions. In the simplest scenario, I consider domains with only two different composi—

tions, identified by the iodine concentration 3:1 and 102. My objective is to determine

59

 

 

 

Figure 5.5: Optimized atomic arrangement in double unit cell of infinitely long
M0689 _ (cl-’73 nanowires for 03:036. Only the most stable isomers for each stoi-
chiometry are shown and labelled according to the number of iodine atoms a: per
formula unit. The nanowire direction is indicated by the arrows.

the most stable domain structure for M0689 _ 2:13 nanowires with an arbitrary value
of the average iodine concentration < a: >.

Before addressing inhomogeneous nanowires, I first identify the most stable struc-
ture of homogeneous nanowires. I identified the optimum geometry and binding en-
ergy of all nanowires formed with any of the building blocks depicted in Fig. 5.1 and
Fig. 5.2, considering the possibility of parallel and antiparallel alignment of asym-
metric building blocks along the nanowire, and also the possibility of the bi-stable
sulfur bridges being short or long. As suggested earlier, the most stable structural
isomers in the infinite systems, depicted in Fig. 5.5, may deviate from those for the
finite building blocks. The binding energy Eb of the most stable structural isomer,
given per unit cell with respect to isolated atoms, is indicated by the data points in
Fig. 5.6(a). The straight solid line, connecting the data points for structures with
no iodine and with no sulfur ligands, represents the expectation value < Eb > in
case that there is no energy preference for specific ligand arrangements. Similar to
alloys [123], deviations from the straight line for 0 < a: < 6 indicate a tendency to

mix or to phase separate.

60

(

Figure 5.6: Mixing and phase separation tendency in M0689 _ $155 nanowires. (a)
Binding energy Eb as a function of the iodine concentration at. The data points refer
to structures depicted in Fig. 5.5. The solid line, obtained by connecting the data
points at a: = 0 and :c = 6, depicts the expectation value < Eb >. The dashed line is
a guide to the eye. (b) Energy gain AEb(x) with respect to < Eb >. The energy gain
associated with phase separation into domains with iodine concentrations 2:1 and :62
is presented in contour plots for systems with < m >= 3.0 in (c) and < x >= 4.5 in
(d). Quantitative results are only valid for integer values of 1:1 and :52, indicated by
the grid. The contours in (c) and ((1), given in units of eV/ unit cell, are guides to the

)3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A

AE, (eV/unit cell) 0'

  
 

V
_n

O

I
M

 

‘~
‘~

 

I
_.
'

 

 

 

 

eye. Dark areas represent particularly stable stoichiometries.

61

To better distinguish, which structures are more stable than what would be ex-
pected for a mixture of noninteracting ligands arranged randomly along the nanowire,
I show in Fig. 5.6(b) the energy difference between the optimized uniform systems
and the expectation value < Eb >. The higher-than-average stability of nanowires
with a: = 3 suggests the corresponding stoichiometry to likely be present in one of
the components in case of phase separation. On the other hand, I are unlikely to
encounter dominant phases with a: = 1 or x = 5.

Next, I consider a nanowire with the average composition M0689_ < a: >I< :1: >,
separated into domain 1 of length L1 with the composition M0639 _ $1155, and a
second domain 2 of length L2 with the composition M0689 _ 3321132- The lengths of
the domains are trivially related by L131 + L2x2 = (L1 + L2) < a: >. On the other
hand, provided that 51:15 < :1: > _<_x2 or 1:23 < x > $231, the relative segment lengths
are given by Ll/L = (< a: > —:I:1)/(2:2 — 11:1) and L2/L = (502— < a: >)/(a:2 — 3:1)
in a nanowire of length L = L1 + L2. Even though both 331 and x2 are integers,
I find it useful to visualize stability islands using non-integer values of 2:1, 9:2 and
interpolated energy values presented in Fig. 5.6(b). The allowable concentration range
($1,222) is shown by shaded areas in Fig. 5.6(c) for < 2: >= 3.0 and in Fig. 5.6(d)
for < 2: >= 4.5. These figures show the contour plot AE($1, 2:2), corresponding to
the energy difference between the expectation value of the binding energy for < a: >
and the weighted average of Eb(:1:1) and Eb(a:2). For a given value of < r >, the pair
of integers (:31, :52), corresponding to the lowest value of AE, identifies the preferred
concentrations and relative lengths of the two phases. Inspecting Figs. 5.6(c) and
5.6(d) for dark areas, corresponding to particularly stable structures, I find that at
least one of the two phases occurs with a: = 3.

In my simple energy estimates underlying Figs. 5.6(c) and 5.6(d), I ignored the
energy associated with forming domain wall boundaries between regions with differ-

ent compositions. In Fig. 5.7, I investigate the energy cost of domain wall boundaries

62

(a) JAE = 0.00 eV

H.

 

 

 

 

 

Figure 5.7: Energy cost associated with grain boundaries in M0689 __ x155 nanowires
with < 3: >= 4.5, consisting of domains A with :1: = 3 and domains B with x = 6. (a)
Reference structure, with a single boundary separating domains A and B. (b) Double
unit cells of A and B alternate along the nanowire. (0) Individual unit cells of A and
B alternate along the nanowire. The energy differences AE are given for two formula
units of Molgsglg, containing 60 atoms.

63

in M0689 _ $13; nanowires with < :1: >= 4.5, addressed in Fig. 5.6(d). Those re—
sults suggest that the nanowire will be most stable when separating into domains A
with the composition MOGS6I3 and domains B with the composition M068316. To
maintain the average stoichiometry, the size of A and B domains must be equal.

The reference structure of an infinite M0684. 514. 5 nanowire, depicted in Fig. 5.7(a),
contains a single domain of phase A, separated by a domain wall from a single do-
main of phase B. As suggested in Fig. 5.4(a) and Fig. 5.5, the most stable structures
of phases A and B, labeled A00 and Boo in Fig. 5.7(a), exhibit an alternating se
quence of short and long sulfur bridges. Figure 5.4(a) also suggests that in phase A,
the orientation of the I(3c) building blocks, defined in Fig. 5.1, should alternate, as
indicated by the arrows in Fig. 5.7 (a).

The same average stoichiometry M0684. 514. 5 can be achieved by periodically
alternating double unit cells of domains A and B, labeled A2 and B2 in Fig. 5.7(b).
The decrease in stability with respect to the structure of Fig. 5.7 (a), reflected in
the positive value of the energy cost AE, indicates that the creation of additional
domain walls is energetically unfavorable. This finding is further supported by an
even larger value of AE in the structure of Fig. 5.7(c), where the number of domain
walls has doubled with respect to Fig. 5.7(b). The value of AE for the system of
Fig. 5.7(c) is not twice the value for the system of Fig. 5.7(b), since the local atomic
arrangement near the (A1,B1) domain boundary differs from that near the (A2,B2)

domain boundary.

5.3.5 Quantum transport in nanowires

One of the most attractive features of M0689 _ 2:193 nanowires is the prasence of
S3-bridges, known to form well-defined, stable bonds to Au surfaces. I calculate quan-
tum conductance in nanowires with the M0684. 5I4_ 5 stoichiometry, since Mo6S3I6

nanowires do not attach readily to Au electrodes [124]. In the following, I will consider

64

 

 

 

 

  

 

 

Right Lead

 

 

 

. . All

0
E(eV)

 

 

_L

Figure 5.8: Quantum conductance of an isolated, infinitely long M06S4'5I4_5
nanowire. (a) Definition of the scattering and the lead regions in an atomistic model
of the nanowire. (b) Quantum conductance G of the nanowire in units of the conduc-
tance quantum G0, calculated at zero bias using a non-equilibrium Green function
formalism.

the optimized, frozen structure of infinitely long and of finite nanowires, sandwiched
between gold leads, and determine the quantum conductance of this system within
the Landauer-Biittiker formalism.

In the first study, I consider an infinitely long Mo6S4.5l4'5 nanowire, shown
in Fig. 5.8(a). I treat one unit cell as the scattering region and define the rest of
the nanowire as the semi-infinite leads. The quantum conductance G of the infinite
nanowire in units of the conductance quantum G0 is depicted in Fig. 5.8(b). The
incident energy E of the carriers is with respect to the Fermi level of the leads,

which in this case refers to the entire nanowire. G (E) has a nonzero value at E = 0,

suggesting metallic conductivity at very small bias values. Its maximum value of 300,

65

corresponding to 3 conduction channels in the nanowire, implies that each atom of the
S3 bridge contributes one conduction channel. The predicted zero conductance value
just above E = 0 suggests that n-doping should lead to semiconducting behavior.
This explains the observed conductivity drop in nanowires with iodine impurities [25].

Next I study the quantum conductance of a finite M0684.5I4.5 nanowire seg-
ment sandwiched between Au(111) surfaces as “ideal” leads, shown in Fig. 5.9(a). I
selected Au(111) leads, since gold forms stable covalent bonds with sulfur, and since
the atomic structure at the Au(111) surface is very similar to that of the M06 octa-
hedra. This allows for a well-defined connection between the nanowires and the gold
leads by S3 bridges, causing negligible structural changes near the wire-lead interface.
Besides the nanowire segment, I included also the topmost layers of the Au(111) sur-
face at both sides in the scattering region, as indicated in Fig. 5.9(a), to correctly
represent changes in the electronic structure in the contact region. Closer inspection
of the charge density of the composite system suggested that changes in the electronic
structure are very small, and that including only 2-3 topmost layers of Au(111) in
the scattering region should represent the electronic properties and conductance of
the system adequately.

The quantum conductance G (E) of the system in units of the conductance quan-
tum GO is presented in Fig. 5.9(b). Here, incident energy E of the carriers is given
with respect to the Fermi level of the gold leads. The energy dependence of G sug-
gest a good metallic conductance and closely resembles my results for the infinite
nanowire, depicted in Fig. 5.8(b). The reduction from the maximum value G = 3G0
in the infinite nanowire, caused by the reflection at the nanowire-gold contacts, is
relatively small, confirming the high quality of contacts between sulfur-terminated
nanostructures and gold. The fact that the dip in the conductance of the infinite
nanowire and the sandwiched nanowire segment occurs at the' same energy just above

E = 0 suggests that the contact with gold does not change the Fermi level of the

66

 

 

 

 

o l l l

0
E (eV)

 

Figure 5.9: Quantum conductance of an isolated, finite M0654. 5I4_ 5 nanowire sand-
wiched between Au(111) surfaces. (a) Definition of the scattering and the lead regions
in an atomistic model. The left and right leads are semi—infinite gold surfaces, in-
dicated by the dashed lines. The scattering region contains the nanowire segment
connected to topmost layers of the Au(111) surface at both sides. (b) Quantum con—
ductance G of the system in units of the conductance quantum G0, calculated at zero
bias using a non-equilibrium Green function formalism.

67

nanowire significantly. The filling of the conductance dip above E = 0 in Fig. 5.9(b)
is caused by the wavefunctions of the leads, which extend into the nanowire and
overlap, thus increasing the conductance of the short nanowire segment.

My conductance calculations confirm the initial expectation that sulfur atoms
form not only stable bridges between functionalized M06 octahedra, but also robust
and electronically transparent contacts to gold leads. Whereas the conductance of
nanowires is generally dominated by contact regions, I find the conductance reduction
at the interface between M0684. 514. 5 nanowires and gold to be very small. This is
likely to hold true also for other M0689 _ $13; nanowires with a different stoichiome-
try, as long as the Mo-based building blocks are connected by sulfur bridges. Also the
delocalization of conduction electrons within the M06 octahedra should contribute to
an excellent electrical conductance behavior of the nanowires [110], independent of

the stoichiometry.

5.4 Summary and conclusions

I used ab initio calculations to study compositional ordering and quantum trans-
port in M0689 _ $13; nanowires. The skeleton of these nanowires consists of Mo
octahedra, which are functionalized by S and I adsorbates and connected by flexible
S3 bridges. Optimum geometries and relative stabilities at different compositions
are determined using density functional theory. I find nanowires with as = 3 to be
particularly stable. Nanowires with other compositions are likely to phase separate
into iodine—rich and iodine-depleted segments, some of which should have the stoi-
chiometry M0686I3. My transport calculations, based on the nonequilibrium Green’s
function formalism, indicate that the nanowires are metallic independent of compo—
sition and exhibit a quantum conductance of G = 3G0, with the three conductance

channels involving the S3 bridges.

68

Chapter 6

Self-assembly of long chain alkanes

and their derivatives on graphite

The following discussion is closely related to a publication by Teng Yang, Savas
Berber, Jun—Fu Liu, Glen P. Miller, and David Tomanek, J. Chem. Phys. 128, 124709
(2008) [125]. This study is a collaborative project with an experimental group at the
University of New Hampshire. Even though my contribution is limited to calculations
in the theoretical part, results of the entire study are reproduced in the following as an
example, how a Theory-Experiment collaboration can help to understand nanoscale

phenomena.

6. 1 Introduction

The controlled self-assembly of molecules to form ordered nanoscale patterns is
an area of active research. These superstructures often exhibit unusual properties [42]
and are finding numerous applications in nanotechnology, including use as nanotem-
plates [43]. Nanotemplates enable directed assembly of nanoscale objects and allow
for the transfer of those patterned objects to receiving substrates. For example, as

nanoparticles can be selectively attached to either molecular or polymeric species,

69

nanotemplates derived from self-assembled molecules or polymers provide an efficient
means to prepare ordered arrays of uniformly spaced nanoparticles. Nanotemplates
of this type permit feature sizes and feature densities that are unrivaled by current
lithographic techniques including electron-beam nanolithography and scanning probe
methods like dip-pen nanolithography [44] and field-assisted nanolithography [45].

Self-assembled monolayers (SAMs) of molecular and polymeric species on highly
oriented pyrolytic graphite (HOPG) have attracted much attention, since the forma-
tion of various adsorption patterns can be monitored using scanning tunneling mi-
croscopy (STM) methods [46, 47, 48, 49]. Although many patterns have been reported
even for alkane films on graphite, the energetic grounds for the optimum SAM struc-
ture have never been investigated. Since the interplay between adsorbate—adsorbate
and adsorbatesubstrate interactions ultimately determines the equilibrium geometry
of the molecular assemblies, it should be possible to induce specific changes in the
SAM superstructures by modifying the terminal functional groups of the molecules in
a pre—determined manner. Unfortunately, published STM data of SAMs on graphite
do not provide a quantitative interpretation of the resulting superstructures and there-
fore do not allow for predictions concerning the effect of specific functional groups on
the corresponding self-assembled superstructure.

Here I study the self-assembly of long chain alkanes and their derivatives on
graphite using a combination of scanning tunneling microscopy (STM) measurements
with ab initio electronic structure and total energr calculations. I identify the op-
timum adsorption geometry of the molecules and explain the energetic origin of the
domain morphology observed by STM. My electronic structure calculations allow
a quantitative interpretation of the STM images. Calculated total energy results
provide a quantitative basis for judging the ordering tendency of alkane chains in
self-assembled monolayers and ways to modify the ordering using alcohol (OH) and

carboxylic acid (COOH) functional groups.

70

6.2 Computational details

To obtain fundamental insight into the origin of different adsorption patterns of
long chain alkanes and their derivatives on graphite, I determined the equilibrium
geometry, binding energy and electronic structure of these systems under different
conditions. My calculations are based on the ab initio density functional theory
(DFT) within the local density approximation (LDA) [126]. I used the Perdew-
Zunger [100] parameterized exchange-correlation functional, as implemented in the
SIESTA code [87], and a double-C polarized basis localized at the atomic sites. I
used the counterpoise method [94] to avoid basis-set superposition errors (BSSE)
introduced by the localized basis. The valence electrons were described by norm-
conserving Troullier-Martins pseudopotentials [89] in the Kleinman-Bylander factor-
ized form [92]. This computational approach is known to correctly describe the equi-
librium geometries and energies associated with the alkane-alkane and alkane-HOPG
interaction, as well as the strength of alcohol-alcohol and acid-acid hydrogen bonds.

Since adsorption of alkane chains on graphite and graphene should be very sim-
ilar, I performed my calculations on infinite graphene monolayers using periodic
boundary conditions. In the superlattice geometry modeling isolated monolayers,
graphene layers were separated by 20 A in the normal direction and represented by
orthorhombic unit cells containing 4 carbon atoms. The supercells were sampled
by a 32x32x1 k—point mesh. When modeling finite alkane chains on graphene, I
used 3x5 graphene supercells with 60 C atoms and sampled the Brillouin zone by a
11 x7x 1 k—point mesh. The self-consistent charge density was obtained using a real-
space grid with a mesh cutoff energy of 250 Ry, sufficient to achieve a total energy

convergence of better than 0.05 meV/ atom during the self-consistency iterations.

71

6.3 Experimental details

As a counterpart to the theoretical results, my experimental colleagues observed
overlayers of alkanes and their derivatives on graphite using scanning tunneling mi-
croscopy (STM). Saturated solutions of n-docosane (99%, Aldrich, Milwaukee, WI),
l-docosanol (97%, Aldrich, Milwaukee, WI), and 1-docosanoic acid (99%, Aldrich,
Milwaukee, WI) were prepared in 1-phenyloctane (99.9%, Aldrich, Milwaukee, WI).
A small drop of the solutions was spread onto a freshly cleaved surface of HOPG
(ZYA grade, MikcroMasch, Wilsonville, OR). Scanning tunneling microscope imag-
ing was performed under ambient conditions (21° — 23°C) using a Nanoscope IIIa
STM (Digital Instruments, Santa Barbara, CA) with platinum/ iridium tips (Digi-
tal Instruments, Santa Barbara, CA). The STM was calibrated using freshly cleaved
HOPG. Images were captured in situ in constant current mode at the HOPG~liquid
interface, while integral and proportional gains were in the range 2-5. The raw images

were filtered to remove high frequency noise.

6.4 Results

6.4.1 Free-standing long-chain alkanes

The calculated equilibrium geometry and electronic structure of free-standing
polyethylene, representing an alkane chain, is shown in Figs. 6.1(a—b). To repre-
sent isolated polymers in a superlattice geometry, I arranged them on a tetragonal
lattice with a large inter-chain separation of 20 A and sampled the rather short Bril-
louin zone of these 1D structures by 16 k—points. The optimized structure of free-
standing polyethylene, with the carbon backbone forming a zigzag chain, is shown in
Fig. 6.1(a). The primitive unit cell of length a. = 2.55 A contains two C and four H

atoms. The LDA band structure of the infinite chain, depicted in Fig. 6.1(b), sug-

72

 

 

E (eV)
1!“

 

 

 

 

-1o.o 5
r x
k

 

Figure 6.1: (a) Equilibrium geometry and (b) electronic band structure of an infinitely
long polyethylene chain. Wave functions of the states at k = 1" corresponding to (c)
the top of the valence band and (d) the bottom of the conduction band are shown
in the same perspective as used in (a). The wave function contours are presented for

the amplitude |¢| = 0.04 a63/2 and superposed with the atomic positions. The wave
function phase is distinguished by color.

gests that polyethylene is a wide—gap insulator. The wave functions associated with
the top of the valence band at I‘ are shown in Fig. 6.1(c), those of the conduction
band bottom at I‘ are shown in Fig. 6.1(d). The spatial distribution of these frontier
orbitals is useful for judging the nature of polymer—polymer and polymer—graphene

interactions.

6.4.2 Structure and stability of long-chain alkane monolayers

For long chain alkanes on graphite, controversy exists concerning whether the
carbon skeletal plane should lie parallel [127], perpendicular [49], or at an z30° an-
gle [128] to the graphite basal plane in the SAM superstructure. To help solve this
controversy, I compared the total energies of the different structures and show my re-
sults in Fig. 6.2. I distinguish the co-planar orientation (P), where the zigzag carbon

backbones of the alkanes or polyethylene, separated by the distance d, all lie in the

73

 

 

(a) W ' MW (C)°'2_ ' ' 'p‘eanuoee'a' ]

PE in vacuum A
Side view (P) Side view (P) 0-1 .'

P-orientatlon

    

 

 

 

 

 

]d ~ - [d '_
-‘\«\.L-- 02[---.m__.___
Top view (P) Top view (p) 3.5 4 d (A) 4.5 5
(COM WW (0” , - sigmoid-s:
01-. PEinvacuum A .'
Side view (8) Side view (3) 9 S-orlentation ;

rrrqrr wmqr LU ,
rrrrrr d ffigfl-fi‘rb d —O'1.'
W 1} ji- _,,.

v
Top view (8) Top View (8) 3 3.5

 

 

 

 

Figure 6.2: Interactions between two infinitely long polyethylene chains in (a-c) co-
planar (P) and (d-f) stacked (S) arrangements. Interaction energies are plotted in (c)
and (f) as a function of the inter-chain distance d for polymers in vacuum, depicted
in (a) and (d), and polymers adsorbed on graphene, depicted in (b) and (e). Energy
values per primitive unit cell of the polyethylene chain pair in vacuum are indicated
by empty triangles and those for chains chemisorbed on graphene by empty squares

in (c) and (f).

same plane, from the stacked orientation (S), where all alkane or polyethylene chains
are axially rotated by 90° and the zigzag carbon backbone planes are separated by d
in the normal direction. On graphene, the P orientation of polyethylene in vacuum,
depicted in Fig. 6.2(a), translates to that of polyethylene chains lying down parallel
to graphene, as shown in Fig. 6.2(b). The energetic comparison in Fig. 6.2(c) suggests
that the graphene substrate plays only a minor role in the inter-chain interaction. In
particular, the equilibrium inter-chain distance of 4.26 A in the co-planar orientation

is nearly the same in vacuum and on graphene.

74

Similarly, the S orientation of polyethylene in vacuum, depicted in Fig. 6.2(d),
translates to that of polyethylene chains adsorbed with their zigzag planes normal to
the graphene substrate, as indicated in Fig. 6.2(e). Again, the equilibrium inter-chain
distance of 3.50 A in this geometry is very similar on graphene and in vacuum, as
seen in Fig. 6.2(f). Comparing energy values in Figs. 6.2(c) and 6.2(f), I find that the
inter-chain attraction leads to an energi gain of z0.1 eV per C2H4 unit with respect
to separated polymers, both in the co—planar (P) and the stacked (S) orientation. The
small energy differences in the inter-chain interaction are introduced by the graphene
substrate and can be attributed to inequivalent adsorption sites on graphene. These
are likely to be modified in the case of incommensurate overlayers, which are known
to form Moiré patterns in STM images [49].

Considering a C2H4 unit of an isolated polyethylene chain, adsorption on graphene
leads to an energy gain of 0.09 eV in the S orientation and 0.12 eV in the more stable
P orientation. In view of the polyethylene—substrate attraction that is similar to the
inter-chain attraction, I expect the formation of self-assembled monolayers forming
both S- and P-domains on graphene. Due to their higher stability, P-domains should
prevail unless the system is subject to external constraints including lateral pressure.

Having established the optimum polymer arrangement within the polyethylene
layers, I next address the preferential orientation of the polymer chains with respect to
the graphene substrate. Different representative alignments for the two orientations
are depicted in Fig. 6.3. The optimized adsorption geometry of polyethylene in the
preferential parallel (P) orientation, lying down on graphene along the armchair di-
rection, is depicted in Fig. 6.3(b), and the corresponding zigzag alignment is shown in
Fig. 6.3(c). I should note that in the optimum adsorption geometry of P-oriented PE
aligned along the zigzag direction on graphene, the plane of the PE carbon backbone
is tilted by 25° with respect to graphene, which will have important consequences

for the STM images of alkane chains.

75

 

 

 

 

Top view (P) po view (8)

Figure 6.3: Alignment of polyethylene (PE) chains in different orientation on
graphene. Polyethylene may lie parallel to graphene in the P orientation or nor-
mal to graphene in the S orientation. (a) Definition of armchair and zigzag directions
on graphene. (b) PE(P) along the armchair direction, (c) PE(P) along the zigzag
direction, and (d) PE(S) along the zigzag direction, viewed from the side or the top.
Energetically, I find zigzag aligned PE in Fig. 6.3(c) to be favored by 0.08 eV per
02H4 unit with respect to armchair aligned PE. The zigzag alignment represents the

most favorable structure, since axial displacements of the chain reduce its binding

76

energy by no more than 0.02 eV per C2H4 unit, suggesting that even misaligned
or displaced chains prefer the zigzag direction. As mentioned in the discussion of
my results in Fig. 6.2, zigzag aligned polyethylene lying down parallel to graphene
in Fig. 6.3(c) is energetically favored by 0.03 eV per C2H4 unit over polyethylene
standing up normal to graphene in Fig. 6.3(d). Based on the hierarchy of interactions,
I expect coexistence of polyethylene chains in P and S orientation, with zigzag and
armchair alignment on the graphene substrate. The largest domains are expected for
the stable P-oriented PE in zigzag alignment, the smallest domains for S-oriented PE

in armchair alignment.

6.4.3 Bonding nature of long-chain alkanes on graphite

The chemical origin of the adsorbate interaction with the graphene substrate
is addressed in Fig. 6.4 for P-oriented polyethylene and in Fig. 6.5 for S—oriented
polyethylene. The height of the polyethylene axis center above graphene is h = 3.4 A
for P-oriented polyethylene and h = 3.8 A for S-oriented polyethylene. This relatively
large adsorbate-substrate separation is consistent with the relatively small adsorption
energy. The total charge densities in Figs. 6.4(c) and 6.5(c), and in particular the
difference charge densities in Figs. 6.4(d) and 6.5(d) indicate that there is essentially
no net charge transfer between adsorbate and substrate. Small, but nonzero values
of the charge density difference in Figs. 6.4(d) and 6.5(d) are indicative of weak
covalent bonds. The moderate charge density accumulation in the region between
PE and graphene, observed in Fig. 6.4(d), explains the slightly stronger adsorption
bond in the P orientation. The weaker adsorption bond in the S orientation may be

traced back to a moderate charge density depletion in the same region, observed in

Fig. 6.5(d).

77

 

(C) P-orientation p(r) (eI/ao3)
0.20
0.15
0.10
0.05
0.00

 

 

 

 

Ap<r> (el/aoal
0.0005

0.0000
-0.0005

-0.0010

Figure 6.4: Nature of bonding of polyethylene in P orientation, lying parallel to
graphene. The polyethylene adsorption geometry on graphene, shown (a) in side view
and (b) in top view. Contour plots of (c) the total charge density p P E / gr (1 (r) and (d)

the charge density difference, defined as Ap(r) = p P E / gra(r) — p P E (r) — pgm(r).
The plane of the contour plots is normal to the polymer axis and is indicated by the
dashed line in (a) and (b).

6.4.4 STM images of long-chain alkane monolayers on graphite

To verify my results for the preferential adsorption geometry of polyethylene or
finite alkane chains on graphene, I compare the optimized structures to STM images
of the corresponding systems. Since scanning tunneling microscopy only images elec-
tronic states in the energy range Ep — eVbias < E < Ep, where Vbz‘as is the bias
voltage and Ep is the Fermi level, I must further inspect the electronic structure of
polyethylene on graphene to interpret the images.

Results for the electronic density of states (DOS) of polyethylene on graphene

78

 

(c) S-orientation p(r)(e|/aoa)

& 0.20
1.. 9 0.15

.. e . . 0.10
m m 0.05

0.00

 

 

 

Apm (GI/303)
0.0005

0.0000

-0.0005

 

Top view (8)

-0.0010

 

Figure 6.5: Nature of bonding of polyethylene in S orientation, standing normal to
graphene. The polyethylene adsorption geometry on graphene, shown (a) in side View
and (b) in top view. Contour plots of (c) the total charge density p P E / gr a(r) and (d)

the charge density difference, defined as Ap(r) = p P E / gra(r) — p P E (r) — pgm(r).
The plane of the contour plots is normal to the polymer axis and is indicated by the
dashed line in (a) and (b).

are presented in Fig. 6.6. In Fig. 6.6(a) I compare the DOS of pristine graphene to
polyethylene/graphene in P- and S-orientation. As seen in Fig. 6.1(b), polyethylene is
a wide-gap insulator with a fundamental gap exceeding 7.5 eV and becomes visible to
the STM only in the energy range, where the electronic states of PE—covered graphene
display a significant PE character. I expect a non-vanishing polyethylene character at
those energies, where the DOS of PE on graphene differs from the pristine graphene
substrate. This can be best judged by inspecting the difference density of states
ADOS for P-oriented PE in Fig. 6.6(b) and that of S—oriented PE in Fig. 6.6(c).

79

These results suggest that no signal related to polyethylene should occur at bias
voltages Vans-SL4 V when imaging the occupied states.

Calculated and observed STM images of self-assembled alkane monolayers on
graphene are shown in Fig. 6.7. STM images obtained in the constant current mode

display changes of the tip height h as a function of the lateral tip position (x, y). The

 

 

 

N
I'O

    

 

   

5 ' l . l - 1 .
--. IPE(P)/gra PE S/ ra
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'g —-_graphene ’ E 1 i g 1 i
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0 -1 _

 

 

 

 

 

 

 

 

 

. . I - 1 . n
-10 -5 EF 5 10 —10 —5 EF 5 1O —10 —5 EF 5 10
E(eV) E(eV) E(eV)

Figure 6.6: (a) Density of states (DOS) of polyethylene (PE) monolayers in P- and
S—orientation on graphene, compared to the DOS of pristine graphene. The DOS
has been convoluted by 0.2 eV. (b) Difference density of states ADOS(PE(P)/gra)=
DOS (PE(P) / gra)—DOS (graphene) for P—oriented PE. (0) Difference density of states
ADOS(PE(S)/gra) = DOS(PE(S)/gra)—DOS(graphene) for S-oriented PE.

2.0

0.0

 

Figure 6.7: Scanning tunneling microscopy (STM) images of self-assembled alkane
monolayers on graphene at the bias voltage Vbz’as = 2.0 V (sample negative). (a)
Calculated STM image of a polyethylene (PE) monolayer on graphene, corresponding
to a constant charge density pSTM = 10‘6 e/ag. The image is superimposed with the
model of the PE structure. (b) STM image of an n—docosane monolayer on graphene,
observed at the constant tunneling current of 215 pA.

80

corresponding calculated images display the height h(a:, 3;) corresponding to a given
value of the charge density psrrM that is associated with states in the range Ep —
eVbz'as < E < EF. The calculated STM image in Fig. 6.7(a), obtained for Vbz'as =
2.0 V and pSTM = 10‘6 e / a8, can be directly compared to STM images obtained in the
constant-current mode at the same bias voltage. Superposed to the calculated image
in Fig. 6.7(a) is the underlying structure of P-oriented polyethylene chains, aligned
along the zigzag direction of graphene and separated by 4.26 A, commensurate with
the underlying graphene layer. Comparison between the calculated STM image and
the underlying polymer structure indicates that the bright spots in the image closely
correlate with the position of H atoms of the alkane chains. As mentioned above, due
to the small horizontal tilt of polyethylene on graphene, the hydrogen atoms on the
right-hand side of the polymer in Fig. 6.7 (a) are closer to graphene by 0.2 A and thus
could naively be expected to be observed at a slightly smaller height than those on
the left—hand side of the chains. At the finite bias voltage used in the STM, however,
the height contours are not directly correlated with atomic positions. In the present
case, the small height difference of 0.2 A between the hydrogen atoms translates
into a much larger apparent height difference of 0.6 A in STM images observed at
Vbz' as = 2.0 V.

Figure 6.7(b) shows STM data obtained when imaging a solution of n-docosane
in 1-phenyloctane on graphite at Vbz'as = 2.0 V (sample negative), at the constant
tunneling current of 215 pA. The STM image, obtained using a tip submersed in
the solution, shows a well-formed self-assembled monolayer of n-docosane, closely

resembling my calculated image of Fig. 6.7(a) showing atomic resolution.

81

 

HOMO HOMO

Figure 6.8: Highest occupied (HOMO) and lowest unoccupied (LUMO) molecular
orbitals of polyethylene chains terminated by (a) an OH group and (b) a COOH
group. Charge density contours are superposed with ball-and-stick models of the
molecules.

    
 

 
 

6.4.5 Structure and stability of long-chain alkane derivatives
on graphite

Next, I studied the effect of alcohol and carboxylic acid functional groups on
the adsorption geometry and the corresponding STM images. Whereas the frontier
orbitals of polyethylene or unmodified alkane chains extend along the entire chain, as
seen in Fig. 6.1(c) and 6.1(d), this is generally not the case with molecules possessing
terminal functional groups. Since frontier orbitals are very important for understand-
ing the interaction and optimum alignment of molecules, I plot the charge density
associated with the highest occupied (HOMO) and lowest unoccupied (LUMO) molec-
ular orbitals of polyethylene chains terminated by an OH group in Fig. 6.8(a) and
those of COOH-terminated polyethylene in Fig. 6.8(b).

These functional groups introduce new localized states between the top of the
valence and bottom of the conduction band of polyethylene, which become frontier
states and reduce the fundamental gap. Inspection of Fig. 6.8 suggests that the

frontier states are localized near the functional groups.

82

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6.9: Adsorption arrangements in finite-width domains of straight chain al-
cohols in self-assembled monolayers on graphene. Common to all structures is the
P-orientation of the alkyl chains and their alignment along the zigzag direction on
graphene. Adjacent chains within one domain can be parallel, with OH groups point-
ing in the same direction, or antiparallel, with OH groups pointing in opposite direc-
tions.

Possible adsorption arrangements of long chain alcohols within self—assembled

monolayers on graphene are displayed in Fig. 6.9. The relative stability of the different

Structure E a d

a -1.00 eV
(b) -0.95 eV
(c) -0.98 eV
(d) -1.45 eV
(e) -1.03 eV
(f) -1.20 eV
(g) -1.60 eV

Table 6.1: Relative stability of finite-width domains of CGH13OH in self-assembled
monolayers on graphene. The structure labels correspond to the adsorption ar-
rangements of Figs. 6.9(a)—(g). The adsorption energy Ead of C6H13OH is
given by Ead = Et0t(CGH13OH array/graphene)—Et0t(isolated C6H13OH in
vacuum) —Etot (graphene).

83

arrangements depends upon whether the alcohol functions are arranged head-to—head
or head-to-tail within a domain, and also upon the nature of the domain boundary.
Relative stabilities of these systems are compared in Table 6.1 for the arrangements

displayed in Fig. 6.9.

6.4.6 Domain formation in self-assembled monolayers of long-

chain alkane derivatives on graphite

Within an isolated domain, which contains head-to-head parallel or head-to—tail
antiparallel OH-terminated alkanes, possible adsorption arrangements are displayed
in Figs. 6.9(a)-(c). The corresponding adsorption energy values in Table 6.1 suggest
that the parallel arrangement of Fig. 6.9(a), with all alcohol ends aligning head-to-
head along a domain edge normal to the chains, is most stable. The staggered parallel
arrangement of Fig. 6.9(b), with the domain edge at a 30° angle to the chains, is least
stable. The antiparallel arrangement of Fig. 6.9(c), with the alcohol ends aligning
head-to—tail along a domain edge normal to the chains, is not as stable as the parallel
head-to-head arrangement of Fig. 6.9(a). Nevertheless, since it contains a shorter
domain boundary than the staggered arrangement of Fig. 6.9(b), the arrangement of
Fig. 6.9(c) lies energetically between those of Fig. 6.9(a) and Fig. 6.9(b).

Possible boundaries between domains of long chain alcohols are considered in
Figs. 6.9(d)-(g). All these arrangements are energetically preferred over those for a
single domain with the energetically taxing open edge. In long chain alcohols, the
OH bond direction forms a 60° angle to the axis of the alkyl chain. Among the many
ways to assemble long chain alcohols into a monolayer superstructure, arrangements
that from H-bonds at the domain boundary are energetically preferred [129]. The
chain directions within the SAMs are furthermore constrained by the symmetry of
the underlying graphene lattice, which allows lamellar directions in adjacent domains

to be either parallel or to form a 120° angle.

84

Figures 6.9(d)-(e) depict head-to—head and tail-to—tail alignment of long chain
alcohols at the boundary of adjacent domains, in which all long chain alkyls are
aligned parallel to one another. The head-to—head alignment provides additional
hydrogen bonding at the interface, thus stabilizing the structure of Fig. 6.9(d) over
that of Fig. 6.9(c). In the arrangement of Fig. 6.9(f), the parallel alignment of all long
chain alcohols in adjacent domains is maintained, but the domain boundary is rotated
by 30° with respect to the alkyl axis. Even though hydrogen bonding stabilizes this
arrangement over that of Fig. 6.9(e), the longer boundary length makes it energetically
unfavorable relative to the arrangement of Fig. 6.9(d).

The symmetry of the underlying graphene lattice also allows long chain alcohols
of two different domains to meet at a 120° angle, as shown in Fig. 6.9(g). Since the
arrangement of the terminal OH groups at the domain boundary in Fig. 6.9(g) is most
favorable among all the arrangements discussed here, I expect it to occur frequently
in the self-assembled monolayers.

My findings for self-assembled monolayers of alkane derivatives with COOH func-
tional groups are qualitatively similar to those discussed in Fig. 6.9. However, the
energy to separate a pair of facing carboxylic acid molecules (so-called carboxylic acid
dimers), AEb = —1.72 eV, is more than three times larger than the separation energy

AEb = —0.47 eV of a pair of facing alcohol molecules.

6.4.7 STM images of monolayers of long-chain alkane deriva-
tives on graphite

To help identify the location of OH and COOH functional groups in self-assembled
monolayers of long chain alcohols and carboxylic acids, I calculated the correspond-
ing STM images of C6H13OH and C6H13COOH assemblies on graphene. My re-
sults, shown in Fig. 6.10, suggest that the OH terminal group should appear at a

larger apparent height, whereas the COOH group should appear at a smaller ap-

85

parent height than the alkane chain in STM images obtained in the constant cur—
rent mode. Figure 6.11 displays STM images and corresponding structural mod-
els of self-assembled monolayers of 1-docosanol (022H45OH) and 1—docosanoic acid
(C22H45COOH) molecules on graphite. According to Fig. 6.7, I associate the lamel—
lar structure with parallel alkane chains. For illustration purposes, the length of fully
stretched (all s-trans conformations) docosanol (l = 29.5 A) and docosanoic acid
(l = 30.8 A) molecules is indicated in the figure. The STM images in Figs. 6.11(a)
and 6.11(c) suggest the presence of stripe-shaped domains that are split in the middle
into sub-domains spanned by parallel arrays of the alkane derivatives.

According to my calculations, the equilibrium separation between docosanol
chains in the S—orientation is 3.50 A, incommensurate with the graphene lattice, which
agrees well with the inter-chain separation in domains (1) and (2) of Fig. 6.11(a).
Similarly, the predicted equilibrium separation of P-oriented docosanol chains of
4.26 A agrees well with the observed inter-chain separation in domains (3) and (4)
of Fig. 6.11(a). My energy results in Table 6.1 suggest that hydrogen bonds stabilize
the interface between adjacent domains of head-to—head aligned long chain alcohols.

      

0 37.0 A 5.0
4.0
3.0
g; 2.0
' 1.0
0.0
0.0 ' " ’* NA)
0.0 X(A) 37.0 A

Figure 6.10: Calculated STM images of alkane derivatives forming self-assembled
monolayers on graphene. (a) C6H13OH at the bias voltage Vbias = 1.6 V, (b)
C6H13COOH at Vbz'as = 1.3 V (sample negative). Charge density contours for
pSTM = 10—6 e/ag are superimposed with structural models. The y—axis is aligned
along the zigzag direction of the underlying graphene.

86

As a plausible working hypothesis, I will assume that the sub-domains in Fig. 6.11(a)
contain OH groups at both sides of the interface, with hydrogen bonds acting as a
stabilizing element of the semi—rigid domain backbone. This assumption is confirmed

by the observation of the largest apparent height in the STM image of Fig. 6.11(a) in

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Figure 6.11: STM images (a,c) and structural models (b,d) of alkane derivatives
forming self—assembled monolayers (SAMs) on graphite. (a) Observed and (b) model
structure of 1-docosanol (C22H45OH) on graphite. (0) Observed and (d) model
structure of 1-docosanoic acid (C22H45COOH) on graphite. The STM image (a) was
obtained using the bias voltage 1.6 V (sample negative) at the constant turmeling
current of 30 pA, the STM image (0) was obtained using 1.3 V (sample negative)
bias and a 25 pA current. Domains are labeled by numbers. The size of individual
molecules is indicated by a black rectangle in each panel. The OH and COOH terminal
groups are emphasized in red in (b) and (d), and the domain boundaries are indicated
by the heavy dashed lines.

87

the middle of the domains, in accordance with predicted STM images for this molec—
ular arrangement, shown in Fig. 6.10(a). Combining all this information is sufficient
to present a well-defined structural counterpart to the STM image of Fig. 6.11(a) in
Fig. 6.11(b).

A similar interpretation can be applied to the STM image of docosanoic acid
monolayers in Fig. 6.11(c). The predicted equilibrium separation between P-oriented
docosanoic acid chains in the P-orientation is 5.50 A, incommensurate with the
graphene lattice, which agrees well with the inter-chain separation in the domains
of Fig. 6.11(c). Also in this case, the domains are stripe-shaped and contain two
sub-domains, each spanned by an array of docosanoic acid molecules. As in the case
of the alcohol terminal group, the acid groups favor a head-to-head configuration
at the interface of the sub-domains, which provides the domain with a stabilizing
backbone. Unlike in alcohols, where the chains may either be parallel or form a 120°
angle, the alkyl chains in carboxylic acid SAMs have a strong energetic preference
for being parallel. Also, the interaction between head-to-head acid terminal groups
of alkanes is more than three times larger than in the alcohols, resulting in a very
rigid, straight domain backbone that is normal to the chain direction. The postulated
head-to-head alignment of the acid groups in the middle of the domain can not be
verified easily, since the predicted apparent height of the acid terminal group in STM
images of monolayers is only slightly smaller than that of the alkane, showing very

little contrast.

6.5 Discussion

The favored structure of bulk polymer melts at high temperatures is known to
depend largely on entropy. I believe that the relevant degrees of freedom are largely

frozen out when polymers self-assemble to ordered monolayers on a substrate, thus

88

significantly reducing the configurational part of entropy. I furthermore assume that
other contributions, including configurational entropy, may be nearly equal in systems
I compare, thus justifying a posteriom‘ my discussion of stability differences based on
total energ calculations performed at T = 0.

The formation of ordered overlayers of long chain alkanes and their derivatives
on graphite out of their solution in phenyloctane is a slow process. After an initial
fast adsorption stage, it takes several minutes at room temperature to complete the
ordering in the self-assembled monolayers. Once a domain structure is formed, do-
main walls are observed to move in time [49]. Assuming that the presence of the
solvent does not affect the adsorption energy hierarchy in Table 6.1, the most stable
structure at T = 0 would consist of infinite stripe domains. The favored lamellar
structure of long chain alcohols would be a 120° chevron pattern, locally represented
in Fig. 6.9(g). In the corresponding structure of long chain acids, the lamellae should
be all parallel and aligned normal to the domain walls, as seen in Fig. 6.9(d). At
nonzero temperatures, configurational entropy should introduce roughening of the
domain walls. Moreover, the assumption that the solvent does not affect the ad-
sorption geometry hierarchy is not very well founded, since the adsorption energy of
phenyloctane molecules on graphite is only slightly smaller than that of alkanes and
their derivatives of a comparable size. Thus, the real SAM patterns may deviate on
energy and entropy grounds from the T = 0 equilibrium structure.

In reality, the origin of the complex patterns in Figs. 6.11(a) and 6.11(b) is only
partly related to stability issues, since the relatively slow self-assembly process has
an important kinetic component. Alkane chains are likely to adsorb simultaneously
at different substrate locations and assemble to domains that grow in size, until they
hit the next domain. At this moment, the two domains will attempt to minimize
the interface energy by local rearrangements. Since the assembly process is relatively

slow, I expect these rearrangements not to yield the globally optimized structure.

89

In general, I expect an energy-related compromise structure of the adsorbate layer
subdivided into large domains with straight edges. Energetic compromises, resulting
in a complete monolayer coverage, may include unfavorable adsorbate orientation, in-
cluding the S-oriented domains (1) and (2) in Figs. 6.11(a) and 6.11(b) that may form
under lateral pressure. Rather than forming void islands, the system may accept the
formation of energetically unfavorable domain wall boundaries, such as that between
domains (2) and (3) in Figs. 6.11(a) and 6.11(b).

As mentioned above, the strong interaction between head-to-head terminal groups
of long chain acids favors only one domain structure with all chains parallel and the
domain boundaries normal to the chain direction. Due to the high stability and
rigidity of this assembly, I expect straight stripe domains to cover the entire sample,
with diminished effects of entropy, presence of the solvent, or kinetics. Indeed, the
pattern observed in Fig. 6.11(c) is nearly free of defects and identical to the optimum

structure expected at T = 0.

6.6 Summary and conclusions

In summary, this study combined scanning tunneling microscopy observations
with ab initio calculations to study the self-assembly of long chain alkanes and their
derivatives on graphite. I identified the optimum adsorption geometry of polyethylene,
long chain alkanes, alcohols and carboxylic acids on graphite. Calculated adsorption
energies in different arrangements were used to predict the optimum structure of
self-assembled monolayers. Theoretical predictions for STM images of optimized ad-
sorbate overlayers were compared to the experimental data for docosane, docosanol
and docosanoic acid on graphite. My total energy results provided a sound base
for interpreting the observed domain wall structure in the self-assembled monolay-

ers. In particular, I found that the observed domain wall structure in docosanol

90

SAM superstructures should vary from sample to sample, since kinetic processes are
as important as structural stability. In docosanoic acid SAM superstructures, the
interaction between COOH terminal groups at the domain wall boundary is much
stronger, resulting in patterns reflecting the globally optimized structure, with no
sample-to-sample domain variation. These results can be directly used when design-
ing templates to be used in template directed nanoassembly and transfer processes.
Finally, since nearly any nanoparticle can be attached to long chain molecules, in—
cluding those described here, self-assembly of such molecules or polymers provides an

efficient means to prepare ordered arrays of uniformly spaced nanoparticles.

91

Chapter 7

Molecular self-assembly of
functionalized fullerenes on a metal

surface

The following discussion is closely related to a publication by Bogdan Diaconescu,
Teng Yang, Savas Berber, Mikael Jazdzyk, Glen P. Miller, David Tomanek, and
Karsten Pohl, Phys. Rev. Lett. 102, 056102 (2009) [130]. This study is a collabora-
tive project with an experimental group at the University of New Hampshire. Even
though my contribution is limited to calculations in the theoretical part, results of the
entire study are reproduced in the following as an example, how a Theory-Experiment

collaboration can help to understand nanoscale phenomena.

7. 1 Introduction

Ordered arrays of molecules or nanoparticles such as C60 fullerenes have a great
number of practical applications, ranging from sensors and biological interfaces [50]
to organic electronics [51], photovoltaics, and novel computational methods such as

quantum dot cellular automata and, with endohedral fullerenes, a spin-based quantum

92

computer [52, 53]. Generally speaking, self-assembled monolayers of molecules (SAM)
form as a result of a delicate balance between competing molecule—substrate and
intermolecular interactions [54]. Therefore, to control such self-assembly processes
and grow molecular arrays of designed geometries, it is mandatory to understand
how this balance reflects onto the SAM’s final structure [55].

Here I describe a novel way of controlling the self-assembly process and design-
ing the structure of self-assembled monolayers via chemical functionalization of the
fullerene cages. Self-organization of fullerene cages in ordered arrays at desired non
close-packed distances can be achieved by controlling the hierarchy of interactions
within the SAM, in accordance with ab initio total energy calculations. The scanning
tunneling microscopy (STM) images, obtained by my experimental colleagues, show
that C60 fullerene cages functionalized with two long alkane tails (Fig. 7.1(a)), once
grown onto the close-packed surface of Ag, self-assemble into parallel zigzag rows
(Fig. 7.1(b)), the separation of which is determined by the length of the tails.

My colleagues synthesized [131] this fullerene derivative with multiple supramolec-
ular synthons in order to exploit several intermolecular interactions during assembly
on Ag(111). These interactions include fullerene—fullerene 7r-7r stacking interactions
[132, 133], alkyl—alkyl inter-chain interactions and substrate—molecule interactions.
The peculiar dimerization pattern in the zigzag rows can be traced back to a confor-
mational change of the functionalized C60 molecule (F-C60) upon adsorption. The
conformational change and the self-assembled molecular pattern are a consequence
of a hierarchy of interactions within the SAM, which can be explained by ab initio
density functional theory calculations (DFT).

In this combined experimental and theoretical investigation, we studied self-
assembled monolayers on an atomically flat thin film of silver used as substrate.
The Ag film, with a thickness of two atomic Ag layers or more, was grown on the

(0001) surface of a single crystal ruthenium sample by physical vapor deposition under

93

 

Fig. 7.1: Self-assembled monolayer of F—C60 on Ag(111). (a) Equilibrium structure
of an isolated F-C60 molecule; carbon atoms are shown in green, oxygens in red, and
hydrogens in grey. (b) STM constant current image of F—C60 SAM (295 K, —1 V
sample bias, 0.4 nA tunneling current) also showing the geometry of the unit cell of
the molecular structure. (c) Island of pristine 060 on Ag(111) imaged in identical
conditions (295 K, —1 V, and 0.4 nA).

94

     
  

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top view A—\

side view

(d) pm
680]
20
15
10 nm
15
10
0 5
0

Fig. 7.2: F—C60 STM signature. Wavefunctions of (a) the HOMO and (b) the LUMO
of the free F—C60 molecule, superposed with the atomic structure. (0) Calculated con—
tours of constant local density of states for an F-C60 molecule on Ag(111) representing
constant current STM images under experimental conditions. ((1) 3D representation
of the STM data showing the corrugation of the F-C60 SAM (295 K, —1 V sample
bias, 0.4 nA tunneling current).

(b). .

 

ultrahigh vacuum (UHV) conditions and has a (111) orientation. Functionalized C60
[131] molecules (Fig. 7.1(d)) have then been deposited in situ on freshly prepared Ag
films by low temperature sublimation at about 520 K. Following the F‘CGO deposition,
annealing of the sample at 420 K to 470 K was performed for a few minutes in order to
increase the size of the ordered SAM domains. The sample preparation and analysis
took place in a home-built variable temperature STM operating in UHV at a base
pressure of IX 10‘10 torr [134]. The STM data presented here were acquired at room
temperature (295 K).

Figure 7.1(b) shows a constant current STM image of an ordered domain of F-

060 molecules. The molecular 2D superstructure forms a 4 mm x 2.5 nm oblique unit

95

cell with a sharp angle of about 80°, in stark contrast to the triangular close-packed
arrangement of pristine C60 grown in UHV conditions on an identically prepared
Ag(111) surface (Fig. 7.1(c)). The bright spherical symmetrical features in the F-
C60 SAM image, shown in Fig. 7.1(b), are the fingerprint of the fullerene cages, as
tested against pure fullerenes imaged in identical tunneling conditions (Fig. 7.1(c)).
The darker areas between those features correspond to the regions where the alkane
chains lie down on the Ag(111) surface. The measured distances between the first
and second nearest neighbor fullerene cages in the STM images are about 1.6 nm and
2.0 nm. These distances are larger than the C60‘C60 close-packed interaction dis—
tance of 1.0 nm (Fig. 7.1(c)) [135]. It is somewhat surprising that the functionalized
fullerenes do not assemble in such a manner as to maximize fullerene-fullerene 7r — 7r
stacking interactions given that fullerenes are known to assemble on Ag into hexag-
onal close packed 2D island domains [135] as seen in Fig. 7.1(c). The implication is
that fullerene-fullerene 7r — 7r stacking interactions are weaker than other prevailing
interactions that drive the patterning during assembly.

Since experimental data had low contrast associated with individual F-C60 alkane
tails (presumably because of the high corrugation of the SAM), a detailed analysis
of the F-C60 STM signature is needed. Interpretation of the STM images is often
counter-intuitive since the tunneling current is correlated with wavefunctions near the
Fermi level rather than atomic positions. This is best illustrated by plotting the charge
distribution associated with the highest occupied molecular orbital (HOMO) and the
lowest unoccupied molecular orbital (LUMO) as shown in Figs. 7.2(a)-7.2(b). When
adsorbed on the Ag(111) surface, the nature of states imaged by STM changes dra-
matically due to the hybridization between the electronic states of the substrate and
the molecular orbitals of F-C6O. Calculated contours of constant tunneling current
for F-C60 shown in Fig. 7.2(c), match the experimentally observed high corrugation
of 0.65 nm of the SAM, shown in Fig. 7.2(d).

96

 

Ag(111)
compact
direction

Fig. 7.3: Large scale STM image of F—C60 SAM on 2 ML Ag/Ru(0001). The terrace
edges of the Ag(111) substrate, corresponding to one of its compact directions, are
marked by black dotted lines. The F-C60 self-assemble in domains with six different
orientations, shown schematically in the bottom diagram. We distinguish two domain
groups, characterized by the solid bright green and dashed dark blue lines. Each group
contains three types of domains with distinct orientations, separated by 60°. Any
domain of one group is related to a domain of the other group by reflection symmetry
on a compact direction of Ag(111). The sharpest angle between any domain direction
and a Ag(111) compact direction is 14°.

Significantly more crystallographic information about the F‘CGO SAM structure
can be obtained from larger scale STM images. Fig. 7.3 shows all the observed
orientations of the F-C6O ordered domains. The image also contains two Ag steps
oriented along a compact direction of the (111) surface. I see that the F‘CGO SAM
domains are oriented in six distinct directions, identified by lines coinciding with the
direction of the short side of the unit cell in Fig. 7.1(b). These domains can be sub-
divided into two groups containing three distinct directions, separated by 60° (since
lines have no direction, there is a trivial reduction from 6 to 3 angles in each group).
These two groups of sixfold symmetries suggest a preferential orientation of the F-C60
molecules with the alkane tails aligned along a high symmetry direction on Ag(111),
such as the most and the least compact direction of the substrate. Any domain of
one group is related to a domain of the other group by a reflection operation on a
compact direction of Ag(111), under which the Ag(111) surface is invariant. This
finding suggests that the F-C60 molecule possesses two mirror image configurations

once adsorbed onto the metal surface.

7 .2 Computational details

To obtain fundamental insight into the origin of the SAM pattern of F-C60
molecules on Ag(111), I determined the equilibrium geometry, binding energy and
electronic structure of these systems via ab initio DFT calculations [125]. I use
the local density approximation and the Perdew—Zunger [136] parametrization of the
exchange-correlation functional. Most total energy calculations were performed using
the Quantum-ESPRESSO code [81] with ultrasoft pseudopotentials [137] and em—
ploying a plane wave basis with a 25 Ry kinetic energy cutoff. I calculated the charge
density in real space on a mesh equivalent to a 200 Ry cutoff energy, and used Ag

pseudopotentials containing 4d semicore orbitals. I sampled the Ag(111) Brillouin

98

zone by a 32x32x1 k-point mesh and adjusted the mesh density for F-CGO/Ag(111)
systems with larger unit cells. The vacuum region between 2D slabs was taken larger
than 1 nm to minimize the inter-layer interaction.

Selected structure optimizations were performed using the SIESTA code [87]
with a double-C polarized basis localized at the atomic sites. The valence electrons
were described by norm-conserving Troullier-Martins pseudopotentials [89] in the
Kleinman-Bylander factorized form [92]. I used the counterpoise method [94] to avoid
basis-set superposition errors (BSSE) introduced by the localized basis and found the
BSSE corrected local basis results in good agreement with my plane-wave results. I
used 1— 4-layer slab representations of Ag(111) and periodic boundary conditions. In
the superlattice geometry, the slabs were separated by 2 nm in the normal direction
and represented by orthorhombic unit cells containing 2 — 8 silver atoms. The super-
cells were sampled by a 32x32x 1 k-point mesh. The self-consistent charge density
was obtained using a real-space grid with a mesh cutoff energy of 250 Ry, sufficient

to achieve a total energy convergence of better than 0.05 meV/ atom.

7 .3 The hierarchy of interactions

We next discuss separately the mutual interactions between the substrate, alkane
chains, and fullerenes. The final structure of the F-CGO/Ag(111) monolayer will be
determined by the hierarchy of interactions between these constituents.

Since the functional tail of F—C60 contains alkane chain segments, I performed
total energy calculations of infinite polyethylene chains adsorbed on Ag(111) along
various directions. I found the adsorption energy of 0.22 eV along the most compact
direction, with the carbon backbone parallel to the surface, and with the C atoms
in the trough, to be the most stable. All adsorption energies are normalized per

C2H4 segment of polyethylene. The energy cost to displace the polymer across the

99

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Fig. 7.4: Equilibrium configurations of free and adsorbed F‘CGO molecule: (a) top
view in the I-shape configuration and (b) top view in the V-shape configuration. (c)
Detail of the F'C60 assembling in the optimum geometry. ((1) Adsorbtion geometries
of the F-C60 SAM resulting from the proposed model. Besides the backbone of the
F—C60 molecules, I also depict the envelope associated with the Van der Waals radii of
the atoms. The primitive unit cell depicted by the black lines, containing two F—C60
molecules, agrees with the unit cell shown in Fig. 7 .1(b).

100

surface, while maintaining its direction and orientation, is less than 0.02 eV per C2H4.
Keeping the carbon backbone parallel to the surface, but changing the orientation
from the most compact to the least compact direction, occurs at an energy cost of
0.01 eV. Displacing the chain across the surface in this orientation occurs at an even
lower energy cost of less than 0.001 eV. Comparing polymers aligned along the same
direction on the Ag(111) surface, I found that polymers with the backbone normal
to the surface are less bound by 0.05 eV than polymers with the backbone parallel to
the surface.

Although the alkyl chains of the free F-C60 molecules have many conforma-
tional degrees of freedom, the optimized geometry representing the global minimum
(Fig. 7.1(a)) places the C backbones of the adjacent alkyl chains normal to each other.
My calculations indicate that the alkane tail will gain 0.05 eV per C2H4 segment, thus
favoring both alkane tails to adsorb with the C backbone plane parallel to the surface.
In the free molecule, rotating one of the chains around the virtual hinge of a C—O
bond to make the backbones co—planar, is associated with a moderate energy gain of
about 0.2 eV. However, during this rotation, the chains assume the V-configuration
(Fig. 7.4(b)), therefore eliminating the attractive interaction between the neighbor-
ing chains and as such necessitating a net energy investment of 0.42 eV for the entire
F—C60 molecule. I find the equilibrium opening angle between the alkane chains in
the optimized structure to be 30° — 40°, which would be expected for rigid bonds and
bond angles. The improved interaction between the chains and the substrate results
in a net energy gain of 0.48 eV for the entire molecule, suggesting that adsorbed
F-C60 molecules undergo a conformational change from the I-shape (Fig. 7.4(a)) to
the V-shape (Fig. 7.4(b)) upon adsorption on Ag(111).

The fullerene adsorption energy on Ag(111) is sensitive to substrate relaxation
and increases from 0.84 eV on a monolayer to 1.22 eV on a 4-layer slab, with energy

differences below 0.03 eV depending on the orientation of the C60, in qualitative

101

agreement with earlier results [138]. In equilibrium, depending on orientation, the
center of the fullerene cage lies about 0.60 — 0.65 nm above the Ag(111) surface.

Addressing separately the intermolecular interactions between various constituents,
I found that adjacent C60 cages bind to each other with 0.15 eV at the equilibrium
distance of 1 nm. Also, long chain alkanes bind with 0.27 eV to a fullerene cage, but
interact very weakly with the phenyl rings connecting the C60 to the alkane tails.
Finally, two infinite polyethylene chains, both with their C backbones parallel to the
Ag(111) surface, experience an attractive interaction of 0.06 eV per C2H4 segment
at their equilibrium separation of 0.43 nm [125] and a separation from the substrate
of 0.32 nm.

The equilibrium structure of the F-C6O self-assembled monolayer on Ag(111) is il-
lustrated in Figs. 7.4(c)—7.4(d). As discussed above, the chemisorbed F-C60 undergoes
a conformational change from the I-shape (Fig. 7.4(a)) to the V-shape (Fig. 7.4(b))
configuration. The structure in Fig. 7.4(c) furthermore maximizes the interaction
between the alkane chains and the C60 ends of the F-C60 molecules. The optimum
angle between the chains of the F-C60 molecules, which is about 30° in Fig. 7.4(c),
may be modified to some degree in order to maximize the inter-chain interaction.
When optimizing the SAM geometry, I furthermore must consider the fact that the
alkane tails lie much closer to the substrate than the fullerene cages, suggesting an
apparent overlap between the two when viewed from top as probed by the STM tip.
The orientation of the SAM on Ag(111) is likely given by the preference of at least one
of the alkane tails in each F-C60 molecule to align along the most compact direction,
as shown in Fig. 7.4(c). Moreover, the model generates all the possible orientations
of the SAM with respect to the Ag(111) surface (Fig. 7.4(d)), observed as simultane-
ously occurring domains in Fig. 7.3. Thus, the self-assembly process selects molecules
of given mirror configuration once adsorbed on the surface in individual domains.

The overall calculated adsorption pattern therefore explains the observed unit cell

102

 

 

size and shape, including the distances between neighboring C60 ends of the F—C60
molecules. The alkane chain conformations are the reason for the formation of the

intriguing asymmetric zigzag pattern observed here.

7 .4 Summary and conclusions

Given the fundamental nature of my calculations and the good agreement with
the experimental data, I can use the obtained insight to predict the behavior of related
systems on other substrates. The pattern depicted in Figs. 7.4(c)-7.4(d) should not
only occur on Ag(111), but on any substrate where the energetic preference for the
alkane chains to adsorb with their backbones parallel rather than normal to the
substrate is sufficiently large to initiate the V-shape configuration of Fig. 7.4(b). I
also expect patterns with oblique, but larger unit cells, if the chain length in F-C6O
increases. Other monolayer patterns could be obtained by changing the relative length

of the alkyl chains attached to the C60 molecule.

103

- 4 '

Chapter 8

Optically induced transient

structures in graphite

The following discussion is closely related to a publication by Ramani K. Raman,
Yoshie Murooka, Chong—Yu Ruan, Teng Yang, Savas Berber and David Tomanek,
Phys. Rev. Lett. 101, 077401 (2008) [139]. This study is a collaborative project
with Prof. Ruan’s experimental group at Michigan State University. Even though
my contribution is limited to calculations in the theoretical part, results of the entire
study are reproduced in the following as an example, how a Theory-Experiment

collaboration can help to understand nanoscale phenomena.

8. 1 Introduction

There is growing interest in displacing atoms in materials by photo—excitations
[56]. Observations of transient structures thus formed offer a glimpse into the trans-
formation pathways between different structures. Carbon, with its propensity to form
a wide range of bonding networks (sp, spz, sp3), is ideal to study the dynamics of
bond formation and rupture. Of particular interest is the conversion of graphite to

diamond [58, 59], which is believed to involve the rhombohedral phase of graphite as

104

intermediate state [60, 61]. Whereas pioneering ultrafast optical studies of graphite
have provided evidence for photo-induced melting [62, 63] and generation of coherent
phonons [64, 65] by observing changes in the electronic properties, direct determi-
nation of lattice structural dynamics from optical data proves difficult, especially in
the far-from—equilibrium regime. X—ray diffraction has been successful in observing
nonthermal structural changes in semiconductors [66], relying primarily on the inte-
grated Bragg intensities. However, direct observation of atomic motion in nanostruc-
tures with low atomic number, such as carbon, has not yet been achieved. Electron
diffraction, with its five orders of magnitude enhanced scattering cross-section and
advanced schemes achieving femtosecond temporal resolution [67, 68, 69, 70], offers
a new window into the realm of photo-excited structural dynamics, with resolution
down to 51 nm [67, 70, 71].

In this combined experimental and theoretical study, we report the first direct
determination of structural changes induced in graphite by a femtosecond laser pulse.
At moderate fluences of S21 mJ/cmz, after a short thermalization period of #8 ps,
we can attribute a temperature value to the graphitic layers and find the interlayer
vibration amplitudes to match those reported in X-ray and neutron scattering stud-
ies [140]. At higher fluence values approaching the damage threshold, we observe
lattice vibration amplitudes to saturate. Following a marked initial contraction of
the interlayer spacing by 36%, graphite is driven nonthermally into a transient state
with sp3-like character, forming 1.9 A long interlayer bonds. Using ab initio density
functional theory (DF T) calculations, we trace the structural changes back to a non-
thermal heating of the electron gas, followed by a photo-induced charge separation
causing a compressive Coulomb stress.

The experimental setup, shown in Fig. 8.1(a), has been described in detail else-
where [70]. Freshly cleaved samples of highly oriented pyrolytic graphite (HOPG)

were placed inside an ultrahigh vacuum chamber through a load lock system at room

105

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 8.1: Ultrafast Electron Crystallography (UEC) of highly oriented pyrolytic
graphite. (a) The UEC pump—probe setup. (b) The layered structure of graphite.
(0) Ground state diffraction pattern of graphite. (d) Ground state layer density
distribution function (LDF), obtained via a Fourier transform of the central streak
pattern in (c).

temperature. A mode-locked Ti-Sapphire laser (p—polarized, 45 fs, 800 nm, 1 kHz)
generated the pump pulse, which was marginally focused (@600 um diameter) onto
the sample at z45° relative to the HOPG c-axis. The range of the laser fluence
0.5 < F < 90 mJ/cm2 was below the in situ determined optical damage threshold of
~120 mJ/cmz.

A photo-generated 30 keV electron beam (wavelength A, = 0.069 A), with di-
ameter demagnified to 35 um, was directed onto the sample at grazing incidence
of 24° to serve as the probe. A translational optical delay stage was used to vary
the relative delay between the arrival of pump and probe pulses at the sample, to
observe the changes induced by laser irradiation. The electron pulse sampled the top
3 — 5 layers of the graphite structure, depicted in Fig. 8.1(b), and produced oriented
molecular diffraction patterns in the central streak region, shown in Fig. 81(0). The
layer density distribution function (LDF), shown in Fig. 8.1(d), was obtained from

the Fourier analysis of the interference pattern. The primary peak at 3.35 A and

106

less pronounced peaks at 6.7 A and ~10 A were found to be in good agreement with
the layered structure of bulk graphite [140]. The decay of higher order LDF peaks
suggest a probing depth of ml nm.

To study the photo-induced structure dynamics, my experimental colleagues first
examined the near-equilibrium regime at low fluences of 0.5 mJ/cm2SFS21 mJ/cm2.
Random atomic displacements diminish the intensity of the diffraction spectra by a
Debye-Waller factor e'2M , where M = —32212/4, 212 is the mean-square atomic dis-
placement perpendicular to the reflecting planes, and s = (47r / Ac) sin(6 / 2) is the mo-
mentum transfer associated with the maxima located at the scattering angle 6. Thus,
the change in the mean-square atomic displacement A212, measured relative to the
unperturbed state at negative times, determines the rise in lattice temperature and
can be calculated from the diffraction intensities as ln(I,(t)/I,(t < 0)) = —32Afi2/4.
In Ultrafast Electron Crystallography (UEC), we monitor the integrated intensity of
the (0,0,6), (0,0,8) and the (0,0,10) maxima arising from the interference between the
graphitic layers, shown in Fig. 8.2(a). The intensity of all 3 maxima drops within a
timescale of 8:1:1 ps, indicative of increased thermal motion of the graphitic planes.
As we probe reflections from the basal planes, the observed intensity drops mainly
due to the increased out-of—plane displacement of the atoms. Recent optical studies
[64, 65] have associated near-infrared optical excitations in graphite with a femtosec-
ond generation of coherent phonons with E29 symmetry (interlayer shearing mode).
Hence, this 8 ps timescale of interlayer thermal excitation is a direct measure of the
phonon-phonon interactions in HOPG, and is in good agreement with reports of hot
phonon relaxation times in graphite [141].

Quantitative analysis of the intensity drops in Fig. 8.2(a) indicates that the
mean square atomic displacement A312 increases linearly with the applied fluence in
the near-equilibrium regime, as seen in Fig. 8.2(b). Comparing our data to a theo-

retical model of temperature dependent A212, benchmarked with X—Ray and neutron

107

(a) (b) Temperature (K) (C)
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F=77mJ/cm2
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I'WIITIF‘Irr

0 50 100 150 200
t ' tro (P3): tr°=13 Time (ps)

Figure 8.2: (a) Decay of the diffraction intensity. (b) Change in mean-squared atomic
displacements perpendicular to the graphite planes, estimated from the Debye-Waller
analysis of the intensity drops in (a). (c) Effective surface potential V3 at the HOPG
surface as a function of the applied fluence F. (d) Decay of the effective surface po-
tential with fit to the drift-diffusion recombination model (see text). (e) Contraction
of the interlayer spacing Ad (left axis) and the corresponding increase in the effective
surface potential (right axis) in the far-from equilibrium regime.
scattering data [140], we estimate a temperature rise of 950 K at F = 21 mJ/cm2 in
Fig. 8.2(b). This is also in good agreement with temperature extracted from optical
studies using heat capacity and absorbance [63].

As we enter the far-from-equilibrium regime by increasing the excitation fluence
beyond 21 mJ/cmz, the amplitude of atomic vibrations saturates and no longer in-
creases with increasing fluence, as seen in Fig. 8.2(b). This is rather surprising, for

one would expect a monotonic increase in the thermal motion as graphite approaches

its melting threshold at 130 mJ/cm2 [63]. Furthermore, the (0,0,6) maximum is found

108

to shift by almost 0.34 231-1 at F = 40 mJ/cm2, which would correspond to an inter-
planar expansion of 6%. Such a large peak shift cannot arise from structural changes
alone.

We believe that the key effect of the optical excitation is the repopulation of
electronic states, causing re—bonding and redistribution of charge at the surface[142].
This charge redistribution gives rise to a Coulomb field near the graphite surface,
which induces a collective shift of the electron diffraction pattern. We measure this
Coulomb field here directly by modeling the ‘refraction’ effect commonly associated
with the existence of a surface potential at the vacuum-material interface [142]. We
deduce the effective surface potential V, as a function of the applied fluence and find
it to increase linearly up to Fz80 mJ/cm2, as shown in Fig. 8.2(c). This linear trend
indicates that the surface charging is mainly driven by the non-equilibrium diffusion
of photogenerated carriers rather than multiphoton ionization. The decay of this
transient surface potential resulting from the recombination of space charge follows a
power-law decay with the exponent —1, as shown in Fig. 2(d) and predicted by the
drift-diffusion recombination model [142].

To examine the effect of optical excitations on the atomic structure at thase ele-
vated fiuences, we monitored the interlayer separation following the photo-excitation
by a time—resolved LDF analysis, after having accounted for the surface potential ef-
fects. In contrast to the thermal expansion at low fiuences, we observed the interlayer
separation to contract rapidly, as seen in Fig. 8.2(a). The change in the interlayer spac-
ing Ad is found to correlate well with the rise and fall of V, at F = 77 mJ/cmz, with
a maximum value V, w 12 V corresponding to an interlayer contraction of z 6%. The
potential rise V, sampled within the electron probe depth of 1 nm yields an internal
field of E z 1.2 V/ A, which causes Coulomb stress. Using this value of E and 6,. = 10
for graphite, we estimate a maximum energy density of Ue = ereoE2 / 2 = 0.2 eV / atom,

close to the no.3 eV/atom value based on DFT for the activation barrier in the

109

 

 

 

 

 

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4 6 8 1o 12 o 4 . 8 12 o 20 40 60 80
s (K‘) a (A) Time (ps)

Figure 8.3: Signature of sp3-like bonding in HOPG. (a) Molecular interference pat-
tern M (s) and the corresponding LDF curvas at selected time stances following strong
photo-excitation. The transient peak at 1.9 A in the LDF indicates formation of an
interlayer bond. (b) Ratio of peak intensities obtained from the diffraction patterns.
The enhancement of the (0,0, 12) with respect to the (0,0, 10) peak cannot be ex-
plained by thermal vibrations and indicates a transient structure consistent with sp3
bonding. (c) Ratio of peak intensities at R = 1.9 A and 3.35 A in the LDF.

graphite-to-diamond transformation [60, 59].

To further elucidate the structural evolution, we examined closely the diffraction
spectra. Figure 8.3(a) shows the time evolution of the molecular interference curves
M (s) and the corresponding LDF curvas at F = 77 mJ/cm2, just below the damage
threshold. Starting at m6 ps, as V, rises beyond 4 V, we observe the onset of peak
shouldering and broadening in the M (3) curve and appearance of a new peak at
Rzl.9 A in the LDF curve. Concurrent with this, we also observe a drastic change of
relative intensities between the even and odd peaks in the M (3) curves. In graphite
with the interlayer spacing of do = co/2 = 3.35 A, the even Bragg peaks n = 4,6
are labeled (0, 0, 8) and (0,0,12), and the odd peaks n = 3,5 are labeled (0, 0, 6)
and (0, O, 10), respectively. The observed higher intensity of the (0, 0, 12) peak with

110

 

respect to the (0, 0, 10) peak in Fig. 8.3(b) cannot be explained by thermal vibrations
and suggests a significant structural change. Consistent with the fact that only the
even diffraction peaks should be present in the structure of diamond, and also with
the observation of a well-defined interlayer bond at Rx1.9 A, we conclude that the
transient structure is consistent with sp3 bonding. We find the transient sp3-like
structure, associated with the relative intensity of the Rz1.9 A peak in the LDF curve,
to reach a maximum fraction of 47% at 14 ps, as seen in Fig. 8.3 (c). Similar 3p2 — sp3
hybrid structures have also been identified in pressure-driven pathways of graphite-
to—diamond transition[143]. By 45 ps, the structure recovers its 3102 character, but
remains hot even after 3 us with an average value A112 = 0.033 A2, corresponding to

a lattice temperature Tz1000 K.

8.2 Theory

To clarify the origin of the structural changes, I studied the effect of laser pulses
on the electronic structure and bonding in hexagonal graphite using ab initio DFT cal-
culations in the local density approximation (LDA). While much of the basic physics
underlying photo-induced structural changes can be understood in the bulk structure,
I used graphite slabs representing the surface for quantitative predictions. The total
energy was determined using the ABINIT plane-wave code [144] with a 64 Ry en-
ergy cutoff, Ttoullier—Martins pseudopotentials, and the Ceperley—Alder form of the
exchange-correlation functional. The Brillouin zone of the 4—atom bulk unit cell was
sampled using a fine mesh of 24x24x 12 k—points that included the K — H line at the
Brillouin zone edge, close to the Fermi surface. This approach correctly reproduced
the observed [140] in-layer bond length doc = 1.42 A and the interlayer spacing
do = 3.34 A in the bulk system at T = 0, shown schematically in Fig. 8.4(a). The

electronic density of states of graphite near the Fermi level is shown in Fig. 8.4(b),

111

and the total charge density is depicted in Fig. 8.4(c) in a plane normal to the lay-
ers. In the following, I will study the effect of two types of photo-induced electronic
excitations on the structure.

First, I will consider the thermalization of the initial nonequilibrium population
of electronic states in the laser-irradiated target over a sub-picosecond time scale
[145] to that of a very hot electron gas [146]. The electronic temperature (T8) is
probably lower than the limiting value kBTeShV = 1.55 eV and depends on the
laser fiuence. For the sake of illustration, I compare the Fermi-Dirac distribution in
graphite at kBTe = 0 and kBTe = 1.0 eV in Fig. 8.4(b), indicating the electronic
excitations in the 7r — 7r“ manifold of the hot electron gas. Populating initially empty
conduction statas by valence electrons leads to a change in the total electron density
Ap(r) = p(r; kBTe) — p(r; 0), which is depicted in Fig. 8.4(d) for kBTe = 1.0 eV.

As stipulated by DFT, changes in the charge density modify the force field in
the system. Our results in Fig. 8.4(d) suggest an increased population of C21), or-
bitals that may hybridize to ppa bonding states connecting neighboring layers, in-
creasing their attraction. The depopulation of in-layer bonding states, on the other
hand, should cause an in-layer expansion. To obtain a quantitative estimate of
photo-induced structural changes, I performed a set of global structure optimiza-
tion calculations of graphite with electrons subject to effective temperatures in the
range 0 eV< kBT < 1.55 eV. I used a stringent convergence criterion, requiring
that all strass components lie below 5.0x 10‘7 Ha/ag and that no force exceeds
5.0)(10'5 Ha/ao. In bulk graphite, I indeed found that nonzero electronic tempera-
ture leads to a maximum interlayer contraction Ad/do = -—1% and in-layer expansion
Argo/r00 = +1% for kBTezlfl eV. In the slab geometry, I obtained a slightly larger
interlayer contraction of up to Ad/do = —1.5%.

To further take into account the effect of Coulomb stress induced by the laser

pulse, I represent the charge separation in the electronic ground state, correspond-

112

 

 

(a) g I . (b)5= DOS(aro.units)

 

 

 

 

 

 

A90")
(e/ao3)

+0.01

 

Figure 8.4: (Color online) Electronic structure changes during photo—excitations in
graphite. (a) Schematic view of the structure. (b) Electronic density of states (solid
line) and the Fermi-Dirac distribution of graphite at T = 0 (dashed line) and kBT =
1.0 eV (dotted line). (c) Total pseudo-charge density p(r) and ((1) change in the
pseudo-charge density Ap(r) due to an electronic temperature increase to kBT =
1.0 eV. The plane used in (c) and (d) intersects the graphene layers along the dashed
lines in (a).

113

ing to the observed internal field of zl.2 V/A. To model this system, I immersed
three-layer graphite slabs, separated by 30 A, in a uniform electric field and adjusted
its strength to reproduced the observed field value. With the polarized charge dis-
tribution frozen in, the external electric field was switched off. Then, the internal
field was only due to the polarized charge distribution. The slab calculations were
performed using the SIESTA code [87] with double-polarized triple—C local basis and
a fine 24x24 k-point mesh. The charge density was obtained on a real-space grid
with a mesh cutoff energy of 250 Ry. From total energy calculations at different
interlayer separations, I found that charge density redistribution associated with the
internal field may further reduce the interlayer separation by 2 — 3%. Combined with
the contraction induced by the initial non-equilibrium electron heating, I thus can

explain the observed contraction of the topmost interlayer separation by up to z5%.

8.3 Summary and conclusions

In conclusion, a nonthermal pathway of photo-induced structural changes in
graphite has been observed through multidimensional crystallographic determination
in space, time and energy. Beyond a threshold fluence, a transient sp3-like structure
with well-defined interlayer bonds emerges. The modified force field in the excited

state and the Coulomb stress are the main forces driving this structural change.

114

 

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