Electronic instabilities and structural fluctuations in self-assembled atom wires

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Electronic instabilities and

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Electronic instabilities and

structural ﬂuctuations in self-assembled

atom wires

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magniﬁcus Prof. dr. ir. J. T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op donderdag 8 juni 2006 om 15.00 uur

door

Paul Christiaan SNIJDERS

materiaalkundig ingenieur

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Prof. dr. ir. T. M. Klapwijk Samenstelling promotiecommissie:

Rector Magniﬁcus voorzitter

Prof. dr. H. H. Weitering University of Tennessee, Knoxville, promotor Prof. dr. ir. T. M. Klapwijk Technische Universiteit Delft, promotor Prof. dr. F. Flores Universidad Autonoma de Madrid Prof. dr. F. J. Himpsel University of Wisconsin, Madison Prof. dr. H. W. M. Salemink Technische Universiteit Delft Prof. dr. ir. H. J. W. Zandvliet Universiteit Twente

Dr. S. Rogge Technische Universiteit Delft

Prof. dr. H. W. Zandbergen Technische Universiteit Delft, reservelid

ISBN-10: 90-8593-012-X ISBN-13: 978-90-8593-012-9

Keywords: atom wires, self-assembly, charge density wave Cover design: P.C. Snijders

Front :Empty state STM images of Si(553)-Au at 110 K (left) and Si(112)6× 1-Ga (right) at room temperature

Back : Oostpoort, Delft (2005) Copyright c 2006 by P.C. Snijders

Printed by: Ponsen & Looijen bv, Wageningen, The Netherlands Casimir PhD Series, Delft-Leiden 2006-05

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It is art, a work of joy and beauty. Completely irrational, irresponsible, pointless, with not any justiﬁcation, except for the fact that we just like to do this. J. Christo – Visual artist

Reality, everything we are, everything that envelops us, that sustains, and simultaneously devours and nourishes us, is richer and more changeable, more alive than all the ideas and systems that attempt to encompass it. ... Thus we do not truly know reality, but only the part of it we are able to reduce to language and concepts. What we call knowledge is knowing enough about a thing to be able to dominate and subdue it. O. Paz – Poet

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Chapter 1 Introduction

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1.1 One-dimensional condensed matter systems

One of the first one-dimensional (1D) many-body problems in modern physics, the magnetic chain, was solved already in 1931 by Bethe [1]. This early achievement shows that 1D physics is, at least theoretically, conveniently accessible [2]. It was realized in 1950 by Tomonaga [3] (and later generalized by Luttinger [4, 5]) that electrons confined to 1D behave differently from their counterparts in higher dimensions as a consequence of their increased interaction. In fact, electrons in 1D are so fundamentally different, that the Fermi liquid description of electrons, as characterized by single particle excitations with a distinct charge and a spin [6], is no longer valid. Instead, due to the electron-electron interaction, a 1D system needs to be described by collective excitations with either a spin and no charge or vice versa, called a Luttinger Liquid. Figure 1.1 illustrates the existence of collective excitations in such a correlated electron system in 1D in a very simplified manner. As will be explained in some detail in Section 2.7, 1D systems are not only affected by electron-electron interaction but also by electron-phonon interaction. The latter also leads to fundamental changes of the electronic structure of the system through the formation of a symmetry breaking charge density wave (CDW) with a gapped excitation spectrum.

In condensed matter physics, usually (but not always) experiments that can not be explained within existing theoretical frameworks, instigate theorists to expand our horizon by providing new views on our existing world. The order of the theoretical and experimental advances in 1D physics reﬂects the ease with which 1D systems can be described theoretically and the experimental diﬃculty toward the actual realization of such systems; only after the theory had been developed in the middle of the last century, experiments on 1D systems were initiated. Cross-fertilization with theory has led to a progressively expanding body of knowledge of 1D condensed matter physics.

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1.1 One-dimensional condensed matter systems 3

a

b

Figure 1.1: Simpliﬁed illustration of the collective excitation of electrons conﬁned to

1D. In 2D or higher, repelling electrons can avoid one another (b), whereas in 1D no such possibility exists. In a 1D ”pipe” completely ﬁlled with electrons, an excitation of one electron will aﬀect all electrons in the pipe through these head on collisions, resulting in a collective excitation. Illustration adapted from Ref. [7].

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a

0 10 20 localization metallic Temperature (K) R e si st iv it y

b

absence of long range order

Figure 1.2: (a) Absence of long-range (ferromagnetic) order in a 1D spin chain with

short-range magnetic interactions. Fluctuations induce the chain to break up into ﬂuctuating segments with diﬀerent orientation of the magnetization. (b) Resistivity of a normal metal, and a metal showing Anderson localization.

and as we will see in Chapter 5, phase transitions can be observed. It also turns out that atomic-scale defects play a crucial role in the phase transition.

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1.2 Atom wires at surfaces 5

Experiments on 1D systems are hampered by the fact that we live in a 3D world. A complete detachment of the system under investigation from the 3D world can never be accomplished. Therefore, true 1D systems do not exist and generally one observes a mixing of Fermi liquid and Luttinger liquid properties. Moreover, low temperatures are needed to observe the predicted (low energy) deviations from the Fermi liquid description, and unfortunately, arbitrarily small correlations in higher dimensions produce instabilities toward 2D or 3D ordering at the lowest temperatures [8]. This often precludes the observation of the long sought-after 1D physics. Nevertheless, physicists have shown to be able to pro-duce systems that behave significantly different from the Fermi liquid description. Systems studied experimentally in search of 1D non-Fermi liquid behavior include anisotropic bulk solids such as the inorganic blue bronzes and organic Bechgaard salts, large 1D molecules (carbon nanotubes and polyacetylene), stripe phases in high temperature superconductors, quantum Hall edge states in 2D electron gas systems, lithographically patterned 1D channels in 2D electron gas systems, and short atom wires fabricated using mechanical break junctions. Two basic approaches can be recognized: one being a bottom up method that makes use of extreme anisotropies of the Fermi surfaces of materials provided by nature, and the other being a top down procedure for artificial creation of 1D structures. Still, all 1D systems mentioned here consist of a large number of atoms per 1D channel, ranging from at least ∼ 10 for short the nanotubes to at least ∼ 10000 for the patterned 2D electron gas in for example GaAs/AlGaAs multilayer stacks. The atom wires fabricated using the break junction technique have a diameter of only 1 atom, but this technique is limited to extremely short single wires of only∼ 10 atoms long, thereby severely limiting the parameter space that can be studied. In this Thesis I have used a bottom up method to create macroscopic arrays of the ultimate 1D wires, only one atom wide. In order to do this we exploit the symmetry-breaking properties of surfaces.

1.2 Atom wires at surfaces

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separating two liquids. Surfaces sometimes exhibit the most surprising proper-ties, for example biological surfaces can be inert to highly acidic environments in stomachs; others represent a perfect 2D playground for (renormalized) free elec-trons on e.g. the Si(111)-Ag and Au surfaces. The latter example touches upon the work in this Thesis; it shows that also the electronic structure of a surface can be inherently low dimensional. Nature provides us with perfectly ordered, ther-modynamically stable 2D surface nanostructures. Surface physicists explore this playground of low-dimensional physics in pursuit of novel collective phenomena at the nanoscale. In this Thesis we will show that one can reduce the eﬀective dimensionality even one more by creating arrays of quasi-1D atom wires at sur-faces. Spatial access to these 1D systems with a Scanning Tunneling Microscope (STM) provides a much needed opportunity to investigate the peculiarities of 1D physics.

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1.3 Outline of this Thesis 7

1.3 Outline of this Thesis

In the next Chapter, I will introduce a few concepts that represent the founda-tions of the discussion in the following Chapters. These concepts comprise surface reconstructions, surface states, surface stability, step-edge decoration, atom wires on vicinal Si(111)-Au, and Charge Density Waves. Furthermore, I will introduce the experimental techniques used or often referred to in this Thesis: Scanning Tunneling Microscopy and Spectroscopy (STM and STS), Low Energy Electron Diﬀraction (LEED), Angle-Resolved Photoemission Spectroscopy (ARPES), and Density Functional Theory (DFT). I will also brieﬂy discuss the Ultra-High Vac-uum (UHV) system used for the experiments.

In Chapter 3, I will describe the formation of thermodynamically stable atom wires on the Si(112)n×1-Ga surface, including their intrinsic defects by combining DFT calculations with STM and STS experiments. The Ga atoms passivate all dangling bonds, thereby effectively reducing the total energy of the surface. I will show that vacancies and structural fluctuations fully govern the symmetry and energetics of the surface. In Chapter 4, we explore the possibility to control the energetics of the surface. It will be shown, using a series of LEED experiments, that we can tune the average periodicity continuously via the chemical potential of the Ga atoms on the surface. A detailed statistical STM analysis shows that the distribution of the 6×1 and 5×1 unit cells is random, and provides the repulsive interaction between vacancy line defects and the kink energy of a single vacancy line. Finally, in Chapter 5 I will investigate the influence of structural defects on the electronic structure of metallic atom wires on the Si(553)-Au surface. The defects induce a charge transfer between two atom wires within one unit cell, and as a result we observe a direct competition between two different CDWs orders within a single atom wire. These results can be explained using published band structure data. Lastly, we observe phase slips in the low temperature CDW that carry a fractional charge and spin, thus enabling real space investigations of states with fractional quantum numbers. In short, this Thesis attempts to provide a view of the physical principles playing a role in the formation, stability, and the properties of 1D atom wires realized on vicinal Si surfaces.

References

[1] H. Bethe, Z. f¨ur Physik 71 205 (1931)

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[3] S. Tomonaga, Progr. in Theor. Phys. 5 544 (1950) [4] J. M. Luttinger, J. of Math. Phys. 4 1154 (1963) [5] J. S´olyom, Adv. Phys. 28 201 (1979)

[6] L. Landau, Sov. Phys. JETP 3 920 (1957)

[7] F. J. Himpsel, K. N. Altmann, R. Bennewitz, J. N. Crain, A. Kirakosian, J.-L. Lin, and J. L. McChesney, J. Phys.: Condens. Matter 13 11097 (2001) [8] V.N. Prigodin and Yu.A. Firsov, Sov. Phys. JETP 49 369, 813 (1979) [9] P. Starowicz, O. Gallus, Th. Pillo, and Y. Baer, Phys. Rev. Lett. 89 256402

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Chapter 2 Atomic wire arrays: concepts and

experimental techniques

We review the fundamental concepts playing a key role in atomic wire systems on surfaces, which are needed for a proper understanding of their structural and elec-tronic properties. In addition, we present a short discussion of the experimental methods used in this Thesis.

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2.1 Introduction

In this Thesis, our goal is to investigate aspects of 1D physics in atom wires. The creation of self-supporting and stable atom wires in free space appears not to be possible. Therefore, we exploit the intrinsic symmetry breaking properties of Si surfaces in order to produce atom wires. Because we will work at these surfaces, we will frequently use and comment on various experimental surface science techniques. Surface science is a well developed branch of science that has brought forth an entire collection of measurement techniques. Their applica-tions range from a precise determination of long-range ordered surface structures to electronic structure determination of localized defects. Most of these tech-niques have matured over the years. Thus, we can conveniently employ these well-developed surface sensitive techniques to elucidate the physics of the atom wire arrays realized on surfaces. In this Chapter I will ﬁrst introduce the basic concepts that are needed for a good comprehension of the results presented in later Chapters. Furthermore I will discuss the experimental techniques that are used to study atom wire arrays.

2.2 Reconstructed surfaces

The atom wires studied in this Thesis are formed via self-assembly on vicinal Si surfaces. These surfaces play a very important role. In fact, the fundamental physics governing the physical properties of surfaces fully dictates the formation of atom wires and their resulting characteristics. In this Section we will discuss the most pervasive manifestation of the various mechanisms governing physics at surfaces, namely the surface reconstruction. This discussion is based in part on the excellent review of C.B. Duke [1], and on Refs. [2, 3], and reﬂects an atomistic view; electronic and thermodynamic aspects will be discussed in Sections 2.3 and 2.4.

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2.2 Reconstructed surfaces 11

with a low surface free energy, or by reconﬁguring the atomic arrangement of the exposed surface.

Schlier and Farnsworth [4] were the first to consider that atoms at surfaces can shift from their bulk positions in response to the different chemical and phys-ical environment at the surface. The change in balance of the interatomic forces at the surface leads to a different equilibrium configuration of the surface atoms. A semiconductor surface is considered ”reconstructed” when the (in-plane) sym-metry of the surface structure is lower than that of the bulk-terminated solid. Generally, distortions in the atomic positions increase the size of the surface crystallographic unit cell and the basis vectors of the reconstructed surface struc-ture can be expressed as an (n× m) superstructure of the (often hypothetical) bulk truncated surface [5]. Note that the distortions are not necessarily limited to the topmost atom layer; deeper layers are usually also distorted to some degree. Since Schlier’s work many reconstructions have been identified and their prop-erties have been investigated. These studies revealed several recurring features that, evidently, lower the surface energy efficiently. The most notable ones are dimers, ad-atoms and π-bonded chains. Recently, honeycomb-chains (which look like one-dimensional (1D) graphene) [6] have been identified as a stabilizer of a rather large number of adsorbate induced reconstructions. From this structural information, one can sketch a general picture regarding the driving forces toward surface reconstruction.

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counter-Top view

Side view

dimer

adatom

corner

hole

stacking fault layer

unfaulted half

Figure 2.1: Top view and side view of the dimer-adatom-stacking fault (DAS)

struc-tural model of the Si(111)7×7 surface. Dimers, adatoms, corner holes, and the stacking fault layer in the left half of the unit cell are indicated.

acted by the energy cost associated with the formation of strained bond angles and bond lengths. Apparently, the elements identiﬁed above optimize the energy gain in this delicate interplay of bond energy and bond strain.

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2.3 Surface states 13

reduces the number of partially ﬁlled dangling bonds to only 5 per 7×7 unit cell. The penalty for this considerable reduction of electronic energy is a considerable increase in strain energy (up to the 4th _{layer below the adatom layer) but still a}

large energy gain of 0.403 eV per 1× 1 (!) unit cell is acquired [9]. This clearly demonstrates the eﬃciency of dangling bond removal in decreasing the surface energy.

The phase transition from an array of bulk terminated 1×1 unit cells into the lower energy state of the 7× 7 ordered surface on Si(111) is actually a manifes-tation of self-assembly. Self-assembly is the fundamental behavior where struc-tural organization develops in response to a drive toward a thermodynamic stable state. This diﬀers from a top-down approach of creating nanostructures, in which one artiﬁcially produces ordered (nano-) structures by, for example, lithography. Generally, these man-made structures are thermodynamically unstable, but they can exist because a transition to a lower energy state is kinetically inhibited. Self-assembled (nano-) structures instead can be thermodynamically stable, just because the driving force in their formation is not a search for an ordered system, but it is purely a drive toward the lowest energy state.

2.3 Surface states

As shown in the previous Section, the atomic geometry at the surface generally deviates from that in the bulk.1 Consequently, the potential energy landscape of the surface electrons is also very diﬀerent. New electronic states will exist at the surface. Note that even the bulk terminated (i.e. unreconstructed) surface will exhibit new states due to the broken symmetry at the surface. These states are called surface states. The atomic wave functions of surface atoms have reduced overlap with neighboring atomic wave functions and, consequently, the bond-ing/antibonding splitting will usually be smaller. This often places the bonding and antibonding surface states above the valence band maximum and below the conduction band minimum, respectively. This means that for semiconductors, surface states generally lie inside the band gap, and the surface electronic struc-ture will be decoupled from the 3D bulk electronic strucstruc-ture, thus resulting in a perfect 2D electronic system.

Figure 2.2(a) shows the three types of wave functions present at surfaces; bulk states reaching to the very surface, a surface state decaying away with distance from the surface plane, and a state with mixed characteristics, known as a surface resonance. A bulk state can be characterized by its energy E_b and its wave vector

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bulk surface vacuum bulk state surface resonance surface state ψ a E_F valence band conduction band bulk gap E k b z 0

Figure 2.2: (a) Three types of wave functions present at the surface of a bulk solid.

(b) Location of surface states and surface resonances within a surface projected bulk band structure of a hypothetical semiconductor (gray). One surface state crosses the Fermi level and is, consequently, metallic.

− →

k_b, which has components −→k_b and −→k_b⊥ parallel and perpendicular to the sur-face plane, respectively. Alternatively, a sursur-face state is fully characterized by its energy E and its wave vector parallel to the surface−→k. The lack of periodicity or symmetry in the direction perpendicular to the surface prevents an unambigu-ous definition of −→k_⊥; i.e. −→k_⊥ is not a good quantum number. The solution to the (one electron) Schrödinger equation outside the surface is an exponentially decaying wave function. Generally, wave vectors inside the solid need to be real (otherwise the Bloch waves will exponentially grow to infinity with z and cannot be normalized). But states localized at surfaces are readily normalized, and thus (surface states) wave functions with a complex perpendicular component of the wave vector −→κ (=−→k_⊥) can be matched to the exponential decay in the vacuum perpendicular to the surface (z-direction): ψ(−→r ) = ψ_⊥ψ = e−−→κ ·−→z_u(−→_r

)ei − →

k·−→r

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2.4 Surface stability and energetics 15

band structure onto the surface plane, one typically ﬁnds that the surface state is located within the (projected) bulk band gap, meaning that it is electronically decoupled from the bulk electronic system. Instead, if the surface-state wave function is degenerate with a bulk state, then mixing of the bulk and surface states may occur. The resulting state will have an increased amplitude at the surface but it can penetrate deeply into the bulk, eﬀectively losing its 2D char-acter. Such a state is known as a surface resonance. Figure 2.2(b) illustrates the location of surface states, surface resonances and bulk states in a hypothetical band structure.

Metallic surface states on a semiconductor bulk surface are especially inter-esting because they can be considered as an spatially accessible realization of a 1D or 2D free electron gas. Indeed a wealth of interesting physical proper-ties is displayed by surface reconstructions with isotropic 2D or anisotropic 1D band structures that are nominally metallic and potentially unstable: a Mott insulating state was found in a 2D reconstruction of K on Si(111) [10], the 2D Ge(111)-√3×√3-Sn surface exhibits a charge density wave (CDW) instability [11] and possibly also a Mott insulating state at very low temperatures [12]. The Si(111)-4× 1-In surface displays a complex multi-band Peierls distortion [13] and ﬁnally, as shown in Chapter 5, the 1D Si(553)-Au surface exhibits a complex interplay between competing CDW orders. Clearly, self-assembled periodic sur-face structures exhibit exciting physical phenomena, especially when the single particle band structure and electron count would imply a metallic ground state.

2.4 Surface stability and energetics

In Section 2.2, we discussed the surface energy in terms of a microscopic atomistic picture. Here, we analyze the surface in terms of a macroscopic thermodynamic picture. This discussion is mainly based on Ref. [14].

The stability of matter depends on its (Gibbs) free energy. The higher the free energy, the more likely it is that a competing state with lower energy exists and a phase (state) transition will occur if kinetic limitations do not prevent the transition. The Gibbs free energy F is given by F = E_tot+ P V − T S, with E_tot,2 P, V, T , and S the total energy, pressure, volume, temperature, and entropy of the system. Likewise, the stability of a particular surface reconstruction depends on the surface energy (the total energy of the bulk cancels in a comparison be-tween diﬀerent reconstructions). As long as the surface is in equilibrium with its surroundings, the stability can be evaluated following standard thermodynamics

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from the surface free energy and the chemical potentials of each atom type i present at the surface:3 F = E_tot−µ_iN_i, where µ_i are the chemical potentials of the atomic species i present on the surface, and N_i is the number of these atoms i per unit cell. We assume that the entropy term T S is negligible for relative diﬀerences between free energies of ordered condensed states considered here. Also the P V term is negligible for the pressures in our experiments. The chemical potential µ_i is the partial derivative of the Gibbs free energy F with respect to the number of particles of type i at constant T, P , and N_j=i i.e. it is deﬁned as the change in energy when the number of particles of that species is increased by one. The chemical potential takes into account that the total energy is an extensive thermodynamic quantity. The number of atoms is conserved in reactions at the surface, so that changes in the total free energy of the system can be determined when atoms are added or removed provided that their chemical potentials are known.

The total energy is the sum of kinetic and potential energy of all the atoms, bonds, and conduction electrons in the solid, which can be evaluated from Density Functional Theory calculations (DFT, cf. Section 2.11). However, the value of the chemical potential is generally unknown and needs to be estimated. For the substrate atoms, the bulk chemical potential is used, because the reconstructed surface is assumed to be in equilibrium with the bulk substrate. The bulk chemi-cal potential of the substrate remains constant and cancels in a comparison of the free energies of different reconstructions. If there are atoms of foreign elements present on the surface as adatoms, then their chemical potential cannot be higher than that of the corresponding bulk elemental phase. This sets an upper limit for the value of its chemical potential. If the chemical potential of the adsorbate species is equal to its bulk chemical potential, then the adsorbate must also be present on the surface in some bulk form (e.g. 3D islands or large clusters). In this case equilibrium between the adsorbate-induced reconstruction and the bulk exists. Alternatively, if the reconstruction is in thermodynamic equilibrium with a gaseous phase (for instance vapor evaporating from an effusion cell), then the chemical potential can be determined from the experimental parameters T and P (which do remain constant during the experiment), cf. Chapter 3. It is possible to calculate a thermodynamic stability diagram from DFT total energy calcula-tions. In such a diagram, the total energy of different surface reconstructions is plotted as a function of the chemical potential of the constituent atoms. By esti-3 _{We assume here that the temperature and pressure are constant, which is generally the}

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2.5 Step-edge decoration 17

mating the chemical potential from the experimental conditions, on can compare the experimentally observed phases with the predicted phases from DFT. This has been a key issue in resolving the atomic geometry of the Ga atom wire arrays on vicinal Si(112) in Chapter 3.

2.5 Step-edge decoration

In the previous Sections we have focussed on surfaces that, naturally, are extended in 2D. In order to further decrease the dimensionality of the surface electronic properties, one must try to optimize the overlap between the surface-state wave functions along one direction and minimize the overlap in the perpendicular di-rection (both didi-rections within the surface plane). Then the surface-state band will only disperse along −→k_x, and not along−→k_y. This conﬁnes the electronic mo-tion to an atomic ”track” in the x direcmo-tion; a 1D atom wire has been created. Unfortunately, the 2D translational symmetry at low index surfaces generally prevents the formation of long-range ordered 1D structures. Therefore, in order to study atom wires, one needs to ﬁnd a way to break the 2D symmetry of the surface structure.

The most common 1D structures on single crystal surfaces are line defects consisting of step edges. It was appreciated already more than 15 years ago that such steps might provide a template for atom wire growth. This idea is based on the well known fact that there exists a hierarchy of adsorption sites which is determined by their binding energies for a particular substrate-adsorbate combination. Generally, adatoms ﬁrst move toward the step edges where they can form more bonds with neighboring atoms. When additional adsorbate atoms are deposited, less favorable sites with smaller binding energies (e.g. sites on the planar terrace) will be occupied as well [15, 16]. A regular array of steps can be produced via an intentional miscut α to crystal planes with low Miller indices. As a result, the low index atom planes with an interplanar distance d make an angle α with the macroscopic surface plane. This leads to a surface morphology consisting of ﬂat terraces exposing the low index crystal plane. These terraces are separated by periodic steps spaced d

sin α apart. These ideas have been mainly

developed from experiments on metal surfaces, and only later the idea of step-edge decoration was put to use on semiconductor surfaces [15, 17].

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sites along the kink. Therefore, step edges on Si(111)7× 7 are very straight [18] and provide an ideal template for atom wire growth. In the particular case of vicinal Si(111), arrays of steps along one of the [110], [011], or [101] direc-tions will break the threefold rotational symmetry of the surface. This can be exploited to produce single-domain atomic wire arrays. For instance, quasi-1D surface reconstructions such as the Si(111)-4×1-In and the Si(111)-5×2 surfaces normally grow in three domains, due to the threefold rotational symmetry of the substrate. However, the broken rotational symmetry of the vicinal step arrays forces the atom wires to grow along a single direction, thus producing a single mesoscopic or even macroscopic domain. As long as faceting to more stable high index crystal planes does not occur, one can in principle even control the distance between the step edges and hence the electronic coupling between the atom wires [19].

2.6 Atom wires on vicinal Si(111)-Au

In this Section, we discuss the properties of atom wire arrays fabricated by de-positing Au atoms onto vicinal Si(111) surfaces. These systems created great excitement following the initial report in Nature by Segovia and coworkers that seemed to indicate Luttinger liquid behavior [20]. Photoemission experiments on Si(557)-Au revealed two parallel surface-state bands dispersing toward the Fermi level. These bands were interpreted as the spinon and holon excitations of a 1D metal in the Luttinger Liquid regime [20]. This would have marked the first di-rect observation of Luttinger Liquid physics. Although later experiments refuted the initial interpretation [21], several other gold-induced atom wire systems on vicinal Si(111) surfaces have been explored, revealing electronic properties asso-ciated with these 1D bands that are not yet fully understood. Figure 2.3 shows a cross section of the Si lattice and indicates four of the vicinal surfaces that have been discussed in Ref. [19]. Besides the varying miscut angle and, consequently, the varying interchain distances, the main difference between the four surfaces is the direction of miscut with respect to the (111) plane. Consequently, there is a different number of dangling bonds at the step edge; two for Si(335) and Si(557), and one for Si(553) and Si(775). These dangling bonds are expected to play a role in the reconstruction once the Au is deposited, but their precise influence of the atomic details on the 1D electronic structure is not yet clear because the structural models of these Au-covered vicinal surfaces have not yet converged (cf. Refs. [19, 22, 23, 24]).

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2.6 Atom wires on vicinal Si(111)-Au 19

Figure 2.3: Cut through the silicon lattice in the (110) plane, and perpendicular to

the (111) surface. Dotted lines show cross sections of four diﬀerent vicinal surfaces. Image from Ref. [19].

wire arrays, their electronic structure has been studied thoroughly, especially for the Si(557)-Au and the Si(553)-Au surfaces [25, 19, 26, 20, 21, 23, 24, 27, 28]. Angle-resolved photoemission spectroscopy (ARPES, cf. Section 2.10) experi-ments have provided a full characterization of the electronic band structure E(−→k ) of the atom wire arrays. From the measured dispersion curves one can extract the band ﬁlling, width, dimensionality, and symmetry. In particular the band ﬁlling, band width, and the dimensionality of the bands determine whether these atom wires could display exotic many-body physics (cf. Section 2.7).

Figure 2.4 shows the band structure of the Si(557)-Au and the Si(553)-Au surfaces. Both surfaces show a doublet of surface-state bands, with Si(553)-Au also featuring a third band. These bands appear to cross the Fermi level.4 The Fermi surface shows straight lines, revealing that the electronic structure is indeed quasi-1D. Slight wiggles exist, which are related to a small, but ﬁnite interchain coupling and thus to a mixed-in 2D character. A tight binding ﬁt to these data indicates that the ratio of intrachain coupling to inter chain coupling is∼ 46 and > 60 for the doublet bands in Si(553)-Au and Si(557)-Au, respectively [19]. The single third band in Si(553)-Au exhibits a coupling ratio of 12. A comparison between these coupling ratios and those of bulk chain compounds such as the Bechgaard salts, which have a coupling ratio of ∼ 10 [29], indicates that the atom wire arrays on vicinal Si are much more 1D electronically.

Another interesting parameter that can be extracted from the experiments in 4 _{However, the data published in Ref. [26] seem to indicate that only the outer band of}

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BZ

₁

BZ

₂

BZ

₂

Γ

-0.5

Energy (eV)

Figure 2.4: Fermi surfaces (DOS as a function of −→ky and −→kx, top halves) and band

structure (DOS as a function of−→k_x and energy, bottom halves) of the Si(557)-Au and the Si(553)-Au surfaces, as measured with ARPES, cf. Section 2.10. −→k_x and −→k_y are parallel and perpendicular to the atom wires, respectively. High DOS is shown dark. Adapted from Refs. [25, 19].

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2.6 Atom wires on vicinal Si(111)-Au 21

Si(557)-Au Si(553)-Au

Figure 2.5: STM images of the two atom wire arrays discussed in the text. Si(553)-Au:

-1 V, 0.1 nA, Si(557)-Au: 1.5 V, 0.3 nA. Both images are 35× 35nm2.

Mott states and CDW phases in 1D. For Si(553)-Au, it has been suggested that the unusual fractional band filling could be explained from the ordering of intrin-sic defects which could triple the unit cell [25, 19]. However, low temperature STM observations (cf. Chapter 5) do not reveal regularly spaced defects, thus refuting this suggestion. Instead, STM images of the Si(553)-Au surface indicate continuous 1D chain structures, as shown in Figure 2.5. Indeed the atom wires on both Si(557)-Au and Si(553)-Au appear to be randomly cut by defects which appear either as vacancies or as adatoms. Si(557)-Au has a higher defect den-sity and might show an onset toward defect ordering. The specific role of these structural defects, which seem to be intrinsic to the surface, is not yet clear. In analogy to the planar Si(111)5× 2-Au reconstruction [31], these defects possibly dope the band structure of the atom wires [25]. The observation of slightly differ-ent band fillings for the Si(553)-Au surface in Ref. [28], combined with a variable defect density indeed suggests that these defects can act as dopants. In Chapter 5 we will present convincing experimental evidence that defects in the Si(553)-Au surface indeed modify the band filling of the atom chains, but ordering of defects is not observed. Therefore, the origin of the fractional total band filling in this surface remains as yet unclear.

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STM experiments on Si(557)-Au also reveal the influence of defects on the electronic properties of the atom wires. The surface shows a metal-insulator transition at low temperatures, accompanied by a doubling of the unit cell. It was concluded that the phase transition is due to a Peierls distortion in the quasi-1D bands [26]. As shown in Figure 2.5 this surface also exhibits numerous defects. At room temperature, a faint period doubling already exists near vacancy-like defects [24] which seems to indicate that these defects play a role in the Peierls transition. However at low temperatures, defects also appear to perturb the now fully developed buckling. This may suggest that the defects actually do not affect the Peierls distortion [24], which leaves the issue regarding the influence of defects on phase transitions in 1D electronic systems largely unresolved.

In summary, vicinal Si surfaces with Au induced atom wire arrays show very interesting 1D electronic band structures. Defects present at the surface appear to signiﬁcantly aﬀect the electronic structure of the wires, both locally and globally. However, the precise role of the defects remains rather obscure, largely because the structural models for these surfaces are still subject to debate, which in turn precludes a conclusive interpretation of the available experimental data.

2.7 Charge density waves

In this Section I will focus on the electronic instability that is observed most frequently in 1D atom wire systems on semiconductor surfaces: the Charge Den-sity Wave (CDW). The present discussion is largely based on Refs. [34, 35, 36]. Peierls pointed out already in 1955 that a 1D metal is unstable at low temper-atures against a static perturbation with a wave vector of 2k_F [37]. Using the Lindhard linear response theory this can be explained as follows. When a homo-geneous electron gas is subject to a static perturbation or ”external” potential of wave vector q, then the induced charge density ρind_{(q) will be given by}

ρind(q) = χ(q)V (q) (2.1)

where V (q) is the qth _{Fourier component of the external perturbation V (r), and}

χ(q) the static dielectric susceptibility or ”response function”. χ(q) is given by

χ(q) = e

2

q2

f(E_k)− f(E_k+q)

E_k− E_k+q (2.2)

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2.7 Charge density waves 23

to consider values of k where the state E(k) is occupied and E(k + q) is empty, or vice versa. When q is small (long wavelength perturbation), only a small fraction of the Fermi sphere contributes to the summation. When q becomes larger, more states will contribute to the sum. In particular, when q = 2k_F, all states within the 3D Fermi sphere contribute to the sum. Equation 2.2 also indicates the possibility of a singularity in the expression under the summation: when the wave vector q spans the Fermi surface, the denominator vanishes accordingly. For a spherical Fermi surface, such singularities do not contribute much to the total summation because relatively few k-point pairs contribute to the divergence. However, for non-spherical Fermi surfaces, and in particular for Fermi surfaces with parallel (or nested) segments, there can be many k-point pairs with vanishing denominator. In particular, the Fermi surface of a 1D solid is perfectly nested, and hence all k-point pairs produce a vanishing denominator and the dielectric response function exhibits a singularity. In Figure 2.6(a) we have plotted the dielectric response function χ(q) of a free electron gas as a function of q. The plot shows that for a 1D electron gas, the susceptibility diverges at q = 2k_F. Figure 2.6(b) illustrates the extreme nesting of a 1D system, showing that all states that are located on one sheet of the Fermi surface, are connected to those at the other sheet by a vector q = 2k_F thus causing the divergence in the dielectric response function. This sensitivity to the instability opens a gap at these wave vectors because the 2k_F perturbation exactly couples electrons in states at E(k_F) to holes in states at the opposite branch of the band (E(−k_F)) and doubles the periodicity of the system for a half-filled band, see also Figure 2.7.5 For arbitrary non-commensurate band filling, the coupling between the Fermi wave vector and the lattice periodicity is generally smaller and an incommensurate CDW could develop. Alternatively, the perturbing potential might not be strong enough to induce the transition to a CDW at all, in which case the system remains metallic. Figure 2.6 also shows that CDWs can in principle occur in materials with 1D, 2D, and 3D electronic structures, but the nested portion of the Fermi surface is significantly reduced for higher dimensions and, consequently, the singularity is much stronger for low-dimensional systems. This instability often prevents the observation of enhanced electron-electron interactions in low-dimensional systems at low temperature, because the formation of a correlated Fermi- or Luttinger-liquid state is often preempted by the destruction of metallicity in the CDW state.

As mentioned above, for a half-ﬁlled band the Peierls or CDW instability leads to a doubling of the periodicity. It can also be shown that this doubling 5 _{Phonons can act as such scattering centers for the electrons in the system, described by}

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0 ₁

₂

1

2 χ(

q

)/

χ(0)

q/2k

_F

a

b

Figure 2.6: (a) Lindhard response function for a free electron gas in one, two, and

three dimensions. (b) Fermi surface nesting for a free electron gas in one and two dimensions. Taken from Ref. [36].

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2.7 Charge density waves 25

Γ

k

x

E

_F

k

F

-k

F

-π/a

π/a

a

ρ=constant

2a

ρ(x) = ρ

0

cos(2

k

F

+

φ)

a

b

Figure 2.7: The eﬀect of a Peierls distortion on a half-ﬁlled 1D band structure (a),

the charge density and the atomic structure (b). In (a) the thin continuous band belongs to the undistorted metal and the bold printed gapped band belongs to the Peierls distorted system. (b) shows the periodic modulation of the charge density and the period doubling of the lattice structure.

to u2ln u, where u is the amplitude of the periodic lattice distortion. However, the total energy of the system also increases due to the increase of the lattice strain in the distorted system. In a simple elastic model, this energy cost is proportional to u2, and thus the total energy will decrease upon distortion with u2ln u− u2 and the system will generally be unstable against small symmetry breaking distortions at low temperature.

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In-deed a softening of a phonon mode (i.e. a decrease in phonon frequency ω with decreasing T ) is frequently observed above T_c. For systems with a weak electron-phonon coupling, the thermodynamics of the CDW state resembles that of a superconductor [38] and the CDW energy gap can be directly related to the electron-phonon coupling constant λ_e−ph: 2∆ = 4W exp(−1/λ_e−ph) at T = 0 K where W is the bandwidth of the perturbed metallic band. Within the mean ﬁeld (MF) description, the temperature dependence of the CDW gap 2∆ follows 2∆ = Ck_BTM F

c , analogous to the BCS gap equation for superconductors. C is

a constant which is 3.52 in the limit of weak electron-phonon coupling (and ab-sence of ﬂuctuations). As is the case with superconductors, the spatial variation of the CDW order can be described by a complex order parameter which can be written as ∆(r) exp(i· φ(r)) where ∆(r) and φ(r) are the amplitude and phase of the order parameter (the amplitude is usually taken as the CDW gap or the amplitude of the lattice distortion whereas the phase describes the spatial phase coherence of the condensate).

According to the Mermin-Wagner theorem [39] long-range order is not possible in perfect 1D (and 2D) systems at T > 0 K due to the presence of fluctuations of a two-component order parameter.6 Therefore, a true phase transition is not possible in 1D (and 2D). Thus in principle, the atom wires studied in this Thesis should not exhibit long-range CDW order at finite temperatures. However, the atom wires studied in this Thesis are not perfect 1D systems; they do not have a perfectly 1D Fermi surface and they are coupled to a 3D bulk phonon reservoir. Indeed, phase transitions are possible in quasi -1D systems at T > 0 K and most of the 1D correlations will be preserved in the broken symmetry state below T_c. However, the fluctuations do lower the transition temperature to T_c < TM F

c , but

not to T = 0 K as would be the case for a true 1D system. With the STM, one can also observe local CDW ordering above T_c and observe that the coherence length further decreases for increasing T . This is the precursory phenomenon mentioned above.

Charged impurities can affect both amplitude and phase of the order para-meter near the impurity sites and even prohibit long-range ordering [35]. In fact, theoretical studies have indicated that long-range order can be completely de-stroyed by a ”random external field”, meaning static or quenched disorder, if the dimensionality of the system is 2 or lower. Alternatively, defects can also act as nucleation centers or CDW precursors above T_c. In principle, the STM offers unparalleled capabilities of exploring the spatial variation of both amplitude and 6 _{A loophole in the theorem for 2D systems is that it excludes topological excitations,}

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2.8 Scanning Probe Microscopy 27

phase in reduced dimensionality. Speciﬁcally, one can visualize the important role of defects in establishing CDW order [40].

Finally, in an STM image, the phase transition from the metal to the CDW mainly manifests itself by a periodic modulation of both the empty and ﬁlled DOS with their lattice registry being in anti-phase. In other words, the charge density ρ is periodically modulated; a Charge Density Wave ρ(x) = ρ₀cos(2k_Fx + φ) exists where ρ₀ and φ are the amplitude and phase of the wave, as illustrated in Figure 2.7.

2.8 Scanning Probe Microscopy

2.8.1 Scanning Tunneling Microscopy

Scanning Tunneling Microscopy (STM) is the patriarch of the family of Scanning Probe techniques. STM was invented by Binnig and Röhrer, and the first results were published in 1982 [7]. Other members of the expanding family of Scan-ning Probe techniques include ScanScan-ning Force Microscopies, a whole family in it-self, constituting (non-contact) Atomic Force Microscopy (AFM), Frictional Force Microscopy (FFM), Electrostatic Force Microscopy (EFM) and Magnetic Force Microscopy (MFM), Scanning Near-field Optical Microscopy (SNOM), Scanning Near-field Acoustic Microscopy (SNAM), Scanning Near-field Thermal Microscopy (SNTM), and other more or less related techniques (cf. [41]). It is obvious that the spectrum of scanning probe techniques rapidly expanded following the inven-tion of the STM. Most importantly, it fundamentally changed our understanding of surfaces in widely varying fields as condensed matter physics, chemistry, bi-ology, and nanotechnology because an increasing number of different physical quantities could be measured with atomic spatial resolution. Indeed the inven-tion of the STM was worthy of the Nobel prize, which was awarded in 1986. In this Section we present a short discussion on the basic principles of the STM experiment, based on the books by Wiesendanger [41] and Chen [42].

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X-piezo drive Y-piezo drive Z-piezo drive tip tip-holder sample tunnel bias Feedback loop sample tip path of tip (a) (b)

Figure 2.8: Illustration of the basic operating principle of an STM. Schematic

visual-ization of (a) the piezodrives, tip, and sample including the feedback loop and tunneling circuit, and (b) the tip following the sample corrugation in tunneling contact.

mechanical process, which gives a ﬁnite probability for an electron to be found at the other side of the barrier - i.e. a ﬁnite tunneling current exists. This tunneling current relies on the magnitude of the overlap between the tip and sample wave function which results in an exponential dependence of the tunneling current on the distance between tip and sample:

I ∝ exp(−2d √

2mΦ

) (2.3)

where d is the distance between the electrodes, m is the electron mass and Φ is the height of the tunneling barrier. Applying a bias voltage over the tunneling gap and using a feedback loop, the tunneling current can be held constant while scanning the tip along the surface. The tip then follows the contour of the sur-face, and by recording the voltage on the z-piezo drive as a function of the x and y coordinates in the plane of the surface, we obtain a height proﬁle or topograph-ical image of the surface under investigation. Using a carefully calibrated piezo scanner, the experiment can be performed with sub-˚Angstrom lateral resolution and picometer vertical resolution. In this regime the atomic details of a surface can be observed.

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2.8 Scanning Probe Microscopy 29

imaging atoms, one actually maps the Local Density of States (LDOS) at the surface. This is reﬂected by the model put forward by Tersoﬀ and Hamann [43]:

I ∝ V exp(−2d √ 2mΦ ) _E_F_+eV EF

ρ(E_Ftip+ eV )· ρ(E_Fsample)dE (2.4)

where V is the applied bias, ρ(E_Ftip) and ρ(E_Fsample) are the DOS at the Fermi level of the tip and sample, respectively. This is a simple expression which does not cover all physics, but nonetheless it provides a reasonable understanding of the underlying principles. The current appears to be dependent on the DOS of both sample and tip. Assuming a constant DOS of the tip, the empty LDOS or the filled LDOS is probed depending on the sign of the applied bias. This is nicely illustrated by the voltage dependent imaging of the GaAs(110) and the Si(111)2× 1 surfaces by Feenstra and Stroscio [44]; upon changing the bias sign, the registry of the maxima in the STM corrugation within the unit cell changes as well. For the GaAs(110) case, this was explained by the charge transfer from the Ga dangling bonds to the dangling bonds on the As atoms. Consequently, in the empty state STM image (recorded with a positive bias voltage on the sam-ple) the maxima correspond to the Ga atom sites, and in the filled state image (recorded with a negative bias voltage on the sample) the maxima correspond to the As atom sites. A similar line of reasoning applies to the Si(111)2× 1 sur-face; this surface shows lines of dangling bonds and furthermore reveals a charge transfer between neighboring dangling bonds. The registry of the topographic maxima is different for opposite imaging polarities. Clearly, contrary to the com-mon belief that an STM images atoms, neither image revealed the true atomic structure. Instead, the information in the STM images reflects the LDOS of the surface. Fortunately, this does not prohibit extraction of structural information from STM images, but one has to be extremely careful in doing so. Definitive con-clusions regarding the surface structure can only be obtained by complementing experimental STM/STS data with an in depth theoretical analysis.

2.8.2 Scanning Tunneling Spectroscopy

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in 1973. Quantum mechanical tunneling has since been extended to spectroscopy experiments on superconducting materials, molecules, and surfaces.

From Equation (2.4) it was concluded that the STM probes the LDOS of the surface and not the atomic structure. This opens up the possibility for systematic studies of the LDOS of (low-dimensional) surface structures as a function of en-ergy, i.e. by using the STM as a spectroscopic tool. In such an STS experiment, an I− V -curve is obtained by ramping the voltage bias and recording the result-ing tunnelresult-ing current while maintainresult-ing a constant tip-sample distance. When ramping the voltage, one changes number of states accessible for tunneling, so that spectroscopic information can be extracted from the I− V -data. By record-ing I− V -curves (with the feedback loop switched oﬀ) at known locations during a regular STM scan, one obtains the spatial correlation between the spectroscopy and the topographic image. This technique is also called current image tunneling spectroscopy (CITS) [46].

According to Equation (2.4), the derivative of a measured I− V -curve should reﬂect the surface LDOS (assuming ρ(E_Ftip)=constant). In practice, however, the transmission probability T , described by the term exp(−2d√2mΦ ) in Equa-tion (2.4), is also exponentially dependent on the applied bias. Therefore, a better description a of T would be T = exp(−2d 2m 2 (Φ + eV 2 − E) (2.5)

for electrons with energy E. The tunneling current, is then given by the integra-tion of the tunneling probability times the DOS dependent terms in 2.4 over the appropriate energy window. Figures 2.9(a) and (b) show an I− V and _∂V∂I curve measured with STS from the Si(112)6× 1-Ga surface. The exponential depen-dence on the bias voltage in the ∂I

∂V curve is clearly visible for low bias voltages

near ±0.5 V. Feenstra et al. and Tersoﬀ et al. proposed a normalization pro-cedure that essentially removes the exponential dependence of the transmission probability of the barrier on the applied bias so that the data better reﬂect the true LDOS [47, 43]: ∂I/∂V I/V = ∂(ln I) ∂(ln V ) ∝ ρ sample _(2.6)

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2.9 Low Energy Electron Diﬀraction 31

-1.0

-0.5

0.0

0.5

0

2

4 -1.0

-0.5

0.0

0.5

1.0

0

5

10 15 (c)

(b)

(a)

I (

na)

d

I/

d

V (

n

A/

V)

V (V)

dln(I)

/

dln(

V)

(a

rb

. u

n

its

)

Figure 2.9: (a) I − V -curve measured with STS. (b) Derivative of the I − V -curve

in (a). (c) Normalized derivative of theI − V -curve in (a). Note that only in (c) one observes considerable structure associated with the LDOS.

2.9 Low Energy Electron Diﬀraction

Low Energy Electron Diffraction (LEED) is a standard tool in surface physics used to identify and explore structures present on surfaces.7 Generally, there are two requirements to perform a successful diffraction experiment on a peri-odic surface structure. First, the technique should be surface sensitive, or in other words, the penetration depth of the incident beam (or escape depth of the backscattered particles) should be small so that backscattering from the bulk can be neglected. Second, in order to observe a diffraction pattern the wavelength of the incident particles should be of the order of the unit cell size or less. Free

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electrons easily satisfy both conditions: using the De Broglie relation for a free electron λ =

150.4

E with λ in ˚A and E in eV, one can deduce that the beam

energies needed to resolve periodic surface structures are of the order of 10 eV or higher, generally ranging up to 200 eV. In this energy range the inelastic mean free path of the electrons is on the order of a monolayer, thus satisfying the second condition for a successful surface experiment. Note that X-rays also have wave-lengths comparable to interatomic distances, but they do not satisfy the second condition; x-rays penetrate deeply into the bulk so that x-ray diﬀraction is bulk sensitive. Monochromatic electron beams in energy range of 20-200 eV are easily produced in a laboratory environment.

As depicted in Figure 2.10(a), in a LEED experiment an electron beam is aimed at a sample surface and the backscattered electron beams produce an in-terference pattern on a fluorescent screen. Analysis of this diffraction or the LEED image provides information on the surface structure. As in every diffrac-tion based structural identificadiffrac-tion process, the full determinadiffrac-tion of the atomic structure can be divided into two components. First the periodicity of the system under investigation is determined. In this step the elementary repeat unit and surface symmetry are obtained. The second step involves a detailed analysis of the diffracted beam intensities, which in principle yields the atomic coordinates. We only used LEED to determine the periodicity of the surface, and thus we will only focus on the first part of the full analysis, described by the kinematic diffraction theory.

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2.9 Low Energy Electron Diﬀraction 33 sample grids screen e-gun (a) k‘ k K Ewald _sphere 0 1 2 3 4 5 k (2x π/a) (b) -6 -5 -4 -3 -2 -1

Figure 2.10: (a) Schematic illustration of a LEED setup. The electron gun provides

an electron beam which is diffracted from the sample surface. The diffracted beams pass through grids which filter inelastically scattered electrons, and are subsequently accelerated towards the fluorescent screen producing a visible LEED image. (b) 2D Ewald sphere construction of a LEED experiment.

rod and Ewald sphere intersect. The endpoint of the wave vector of the elastically scattered or diﬀracted beam −→k must also be located on the Ewald sphere. In Figure 2.10(b) only integer order spots (rods) are shown at 2nπ_a just as they would appear on a bulk truncated surface with lattice constant a. However, (adsorbate induced) superstructures on surfaces will often have unit cells that are larger than those of a truncated bulk surface. This would lead to the presence of additional spots or fractional order spots located in between the integer order spots at 2π

na;

for instance a doubling of the periodicity on the surface results in fractional spots lying half-way in between the integer order spots.

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These two effects induce slight differences in the phase of the incoming electron wave for different locations on the surface. If the size of the ordered regions or domains on the surface is larger than the coherence width of the incident beam, then the sharpness of the spots is determined by instrumental limitations, i.e. as if the domain size were no larger than the instrumental coherence width. If, on the other hand, the size of the ordered domains is smaller than the coher-ence width of the LEED beam, the diffracted spots will be broadened due to the limited correlation length of the domains. The coherence width in LEED is estimated to be of the order of 10 nm.

2.10 Angle-resolved photoemission spectroscopy

The development of angle-resolved photoemission spectroscopy (ARPES) is based on photoelectric effect or emission of electrons from a surface caused by incident light.8 Einstein explained this phenomenon in 1905 [50] for which he received the Nobel prize in 1921. Today it is the principal method of gaining informa-tion about the occupied states of solid samples. A simple approach to describe a photoemission experiment is the so-called three-step model within the single electron picture. In the first step (monochromatic) light is used to irradiate a sample. The electron system in the sample can absorb these photons, which will excite electrons from an occupied initial state to an unoccupied final state. The second step is the propagation of the excited electron to the surface. During this step the electrons might lose kinetic energy due to inelastic scattering events with the other electrons or phonons in the solid. Inelastically scattered electrons generally produce a smoothly varying background intensity in the spectrum. The inelastic mean free path of the photoelectrons is typically between 5 and 20 ˚A, which limits this technique to the surface region of solid samples. If the kinetic energy of the electrons in the final state is large enough to overcome the potential barrier imposed by the work function of the material, then the excited electrons will escape from the solid (third step). Subsequently they can be detected by an electron analyzer which measures the kinetic energy of the electrons. Because the energy of the electrons is conserved during the excitation process, the kinetic energy E_kin of the electrons is given by E_kin = hν − E_b − eφ_s, where E_b, hν, and φ_s are the binding energy, the photon energy, and the work function of the sample, respectively.9 The data measured in an (angle integrated)

photoemis-8_{The discussion in this Section is based on Refs. [3, 49].}

9 _{Note that the kinetic energy of the electrons as detected by the analyzer}_Edet

kin is in fact Edet

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2.10 Angle-resolved photoemission spectroscopy 35 k E_F E_vac hν E E DOS E_F electrons (E )_k θ surface normal hν φ

Figure 2.11: Schematic representation of the photoemission process; photons with

energy hν excite electrons from a ﬁlled-state band to above the vacuum level, their kinetic energy is measured and the data are represented as intensity (DOS) as a function of (binding) energy. The inset shows the geometry of an ARPES experiment.

sion spectroscopy experiment are generally presented as intensity versus binding energy (initial state energy) which, in a ﬁrst approximation, can be correlated with the ﬁlled-state DOS. Figure 2.11 shows a schematic representation of the photoemission process including an illustration of a measured DOS.

For single crystal surfaces, one can also determine also the parallel wave vector of the photoemitted electrons. This enables a full characterization of the elec-tronic states (dispersion relation or band structure) of the surface. The parallel wave vector can be obtained by detecting photoelectrons in an angle-resolved pho-toemission experiment. The 2D translation symmetry at the surface conserves the parallel momentum of the emitted electron (i.e. the component of the wave vector parallel to the surface). The parallel wave vector is obtained from the emission angle θ and kinetic energy: k =√2mE_kinsin θ/. The azimuthal angle φ is often chosen such that electron states along the high symmetry directions of the crystal are probed. The inset in Figure 2.11 shows the geometry of an ARPES experiment. Note that the perpendicular component of the electron wave vector

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0 -0.5

Ener

gy (eV

)

Γ= 0

E =

_F

1.23

-1

k (Å )

_x

Figure 2.12: ARPES measurement on Si(557)-Au. Intensity (DOS) is shown as a

function of (binding) energy and wave vector along the chains. High intensity is shown dark. Image from Ref. [19].

of the 2D surface state is not conserved upon transmission through the surface, due to the lack of periodicity in this direction. The perpendicular component of the electron wave vector outside of the crystal therefore does not provide infor-mation on the electronic structure inside the crystal and is determined only by energy conservation: −→k2_⊥= 2m₂E_kin−−→k2.

Figure 2.12 shows ARPES data from the atomic chain reconstruction of Au on Si(557), revealing two parabolic bands that are nearly degenerate in k-space [19]. The emitted electron intensity DOS is shown in dark as a function of both (binding) energy and the wave vector −→k_x along the chains. Data like those presented in Figure 2.12 contain a wealth of information: within the precision of the measurement, both bands cross the Fermi level indicating that the sample is metallic. The difference in dispersion along different symmetry directions reflects the anisotropy in the band structure. From the first and second derivative, the band specific DOS and the effective mass can be determined. Furthermore, the line width of the bands provides information on the ordering of the sample and on the lifetime of excited states.

ARPES and STS experiments provide complementary information on the elec-tronic structure. STS probes the filled and empty state DOS in real space with sub-nm resolution whereas ARPES probes only the filled states averaged over a large sample area, but is able to resolve k-space information, i.e. the band struc-ture. But some overlap exists: both angle integrated PES and filled state STS experiments provide similar information of the filled DOS, and should produce similar data.10

10 _{Note that in an STS experiment one is more sensitive to electrons at the center of the}

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2.11 Density functional theory 37

2.11 Density functional theory

Some of the work described in this Thesis involved Density Functional Theory (DFT) calculations by our theory collaborators,11and a close comparison between theory and experiment was made. Therefore, a short conceptual discussion of DFT is appropriate. The discussion in this Section is based on a few review papers on DFT: the Nobel lecture of Kohn [51], lecture notes of Nogueira et al. [52] and Perdew and Kurth, both from a summer school DFT2001 [53], and a review by Argaman and Makov [54].

The calculation of (electronic) properties of solids is of fundamental impor-tance in order to better understand the origins of measured characteristics. It is also one of the greatest challenges in theoretical solid state physics. The challenge lies in the fact that the Schrödinger equation generally cannot be solved for an interacting many-body system. This limits an approach based on directly solv-ing the Schrödinger equation mainly to small molecules and light atoms having a small number of electrons. DFT offers a possible way out of this, providing the possibility of handling systems with up to 102 − 103 atoms with computer force available at this time.12 The success of the theory, manifested also by the Nobel prize awarded to Kohn as one of its founders in 1998, is that instead of dealing with a wave function of an N-body system depending on 3N variables (coordinates), it deals with only 3 spatial variables describing the electron density. The fundamental theorem used in DFT, the Hohenberg-Kohn theorem, states that the ground-state energy of a system of interacting electrons in an external potential (in our case the lattice potential of the atom cores experienced by the valence electrons) is a unique functional of the electron density; moreover this functional has its minimum value when the electron density is the ground-state electron density. This also means that once the electron density of a ground state is known, the external potential is also known, and thus in principle the ground-state wave function can be calculated. The ground-ground-state expectation value of any observable is also a unique functional of the electron density, and therefore physical properties can also be calculated.

Unfortunately, the precise form of this universal density functional is not known, and the minimization with respect to this density functional is therefore problematic. Kohn and Sham solved this minimization problem by transforming this functional to a ﬁctitious density functional of a non-interacting single body

11_{C. Gonz´}_{alez, J. Ortega, R. P´}_{erez, and F. Flores, Universidad Autonoma de Madrid.} 12 _{Note that by using the Bloch theorem for periodic potentials, DFT allows for a full}

calculation of the band structure of a perfect (i.e. without any defects) macroscopic solid with

Electronic instabilities and structural fluctuations in self-assembled atom wires

Electronic instabilities and

Electronic instabilities and

structural ﬂuctuations in self-assembled

atom wires

Proefschrift

Paul Christiaan SNIJDERS

Contents

Chapter 1

Introduction

1.1

One-dimensional condensed matter systems

a

b

a

b

1.2

Atom wires at surfaces

1.3

Outline of this Thesis

References

Chapter 2

Atomic wire arrays: concepts and

experimental techniques

2.1

Introduction

2.2

Reconstructed surfaces

counter-Top view

Side view

dimer

adatom

corner

hole

stacking fault layer

unfaulted half

2.3

Surface states

2.4

Surface stability and energetics

2.5

Step-edge decoration

2.6

Atom wires on vicinal Si(111)-Au

BZ

BZ

BZ

Γ

-0.5

Energy (eV)

E = 0

-0.2

0.2

0

k (Å )

-1

Energy (eV)

E = 0

-0.2

0.2

0

-2

Γ

BZ

BZ

BZ

BZ

k (Å )

k (Å )

k (Å )

2.7

Charge density waves

0

1

2

1

2

χ(

q

)/

₁

₂