Corso di Laurea Magistrale in Fisica Gaetano

Corso di Laurea Magistrale in Fisica

Gaetano Calogero A VAN DER WAALS DENSITY FUNCTIONAL STUDY OF N-DOPED GRAPHENE/IR(111) FOR CHEMICAL IDENTIFICATION OF INDIVIDUAL ATOMS WITH STM AND AFM

elaborato finale

Relatori: Prof. G.G.N. Angilella Dr. M.A. van Huis Dr. I. Swart

anno accademico 2014/2015

Alla zia

Problems are just opportunities that haven’t presented themselves yet. - Wilson Fisk

Contents Abstract

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Introduction

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1 Graphene interaction with metal substrates and the effect of dopants in graphene 1.1 Basic electronic properties of graphene . . . . . . 1.2 Graphene interaction with metal substrates . . . . 1.2.1 Electronic structure of graphene on metals 1.3 The effect of dopants in graphene . . . . . . . . . 1.3.1 Interplay between doped graphene and a metal substrate . . . . . . . . . . .

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2 Computational methods 2.1 Density functional theory . . . . . . . . . . . . . . . . . 2.1.1 DFT in a nutshell . . . . . . . . . . . . . . . . . 2.1.2 Why DFT? . . . . . . . . . . . . . . . . . . . . 2.1.3 Computational details and accuracy . . . . . . . 2.2 Other methods . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Bader model for atomic charge partitioning 2.2.2 The Tersoff-Hamann approach to STM . . . . . 3 Study of pristine graphene on Ir(111) 3.1 A supercell approach . . . . . . . . . . . . . 3.2 The Ir(111) substrate . . . . . . . . . . . . . 3.2.1 Test of DFT approaches to dispersion 3.2.2 Equilibrium geometry of Ir(111) . . . 3.3 Graphene on Ir(111) . . . . . . . . . . . . . 3

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3.3.1 3.3.2 3.3.3 3.3.4 3.3.5

The moiré pattern . . . . . . . . . . . . . Recap of computational details . . . . . . Equilibrium geometry . . . . . . . . . . . Physisorption with chemical modulation . Electronic structure and imaging contrast

4 Study of N-doped graphene on Ir(111) 4.1 Equilibrium geometry . . . . . . . . . . 4.2 Energetics . . . . . . . . . . . . . . . . 4.3 Electronic charge distribution . . . . . 4.4 Bader charges . . . . . . . . . . . . . . 4.5 Density of states . . . . . . . . . . . . 4.6 The N signature in LDOS and STM . .

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5 Chemical identification of N-dopants with 5.1 Theoretical modeling of non-contact AFM 5.2 Results and discussion . . . . . . . . . . . 5.2.1 Influence of the substrate . . . . . . 5.2.2 Towards the experiments . . . . . .

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AFM . . . . . . . . . . . . . . . .

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63 63 64 66 68 70 71

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76 76 80 80 86

Conclusions and perspectives

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A Long-range interactions in DFT A.1 A failure of standard DFT: the dispersion problem . . . . A.2 Classification of dispersion corrections . . . . . . . . . . . A.3 The Casimir-Polder relationship . . . . . . . . . . . . . .

93 94 96 104

B Additional remarks on the computational details

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C Protocol of geometry optimization for graphene/Ir(111)

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Acknowledgements

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Bibliography

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Abstract As emerges from the Science and technology roadmap for graphene, recently developed within the framework of the European Graphene Flagship, the stunning properties of this 2D carbon allotrope promise to impact and benefit several areas of society in the near future. Many of its brightest prospects depend on the ability to synthesize large highquality samples with controllable electronic properties. At the state-ofthe-art one of the most viable approaches is to grow doped graphene on a close-packed metal surface, such as Cu(111) or Ir(111). The resulting graphene structure is often characterized using scanning probe microscopes such as STM and AFM. However, dopants such as nitrogen atoms in the honeycomb network are not visible in a standard AFM measurement. In this work, by performing an extensive DFT investigation of N-doped graphene on Ir(111), we demonstrate the possibility to fill this gap and enable the chemical identification of individual atoms using AFM. In particular, we use the recent DFT-D3(BJ) approach to accurately simulate the short- and long-range forces that govern the geometric and electronic structure of graphene in this system, the interplay between N dopants and the metal substrate, the characteristic nitrogen signature in STM images and the interactions between this system and a model of an AFM tip. The results of our AFM simulations are in excellent qualitative agreement with the experiments on N-doped graphene/Cu(111) recently carried out by the Condensed Matter and Interfaces group at the Debye Institute in Utrecht, and the measurements on N-doped graphene/Ir(111) are in progress within the same group. We believe that, if the chemical identification capabilities demonstrated in this work will be actually reflected in that context, then a substantial contribution would be given on behalf of the emergence of this identification technique in the AFM field of re1

search. Important repercussions are expected not only in the graphene domain, but also in all those scientific and technological areas in which important functional properties are dictated by a short-range ordering and the chemical nature of defects, dopants, adsorbates or individual atoms. Keywords: chemical identification, atomic force microscopy, density functional theory, graphene, nitrogen, iridium, scanning tunneling microscopy, van der Waals force

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Introduction Since 2004, when a physics professor and his PhD student first isolated it in a laboratory in Manchester just using just a piece of graphite and some scotch tape, graphene has been one of the most appealing materials appeared on the scientific and technological horizon [1–5]. Being just a monolayer of carbon atoms arranged in a two-dimensional hexagonal lattice, it has a high specific surface area (∼ 2630 m2 g−1 ) [6] and can be regarded as the thinnest known material in the universe [3]. It is a flexible, robust and impermeable film, which behaves as an ideal membrane withstanding up to a 6 atm pressure difference [7]. Graphene is more transparent than plastic [8] and exhibits high optical (saturable) absorptivity (∼ 2.3 %) [9, 10]. Its electrons and phonons can travel almost ballistically for micrometers in the structure, even at room temperature, conducting heat and electricity better than any metal (k ≈ 5·103 W m−1 K−1 ; µ ≈ 2·105 cm2 V−1 s−1 ) [11–13]. Graphene exhibits a plethora of astonishing quantum phenomena that are characteristic of 2D Dirac fermions [4], from specific integer and fractional quantum Hall effects [14, 15] to Shubnikov–de Haas oscillations with a π-phase shift due to Berry’s phase [2], as well as a non-vanishing quantum conductivity even for a carrier concentration close to zero [2]. For these and more reasons graphene has been selected as a core element for next generation flexible, low-powered and sustainable technology. Its potential electronic applications include high-frequency devices, touch screens, photonic devices, ultrasensitive sensors and super-dense data storage, among many more [5]. In the energy framework, applications include photovoltaic cells as well as batteries and supercapacitors to store and transport electrical power [5]. Radically new technologies could be also enabled by graphene, from non-charge-based devices (e.g., spintronic [16] or valleytronic [17] devices) to high-TC chiral supercon3

ductors [18]. As emerges from the Science and technology roadmap recently reported by the European Graphene Flagship [5], we are not too far1 from the day when graphene, along with its related2 2D materials, will impact and benefit several areas of society, becoming part of our everyday life. A great deal of the graphene success stems from the ability to control and tune its electronic properties. A simple and effective way to gain such a control is to dope the structure with selected heteroatoms, such as boron (B) or nitrogen (N). The effects on the graphene properties have been shown to mainly depend on the difference in the the number of valence electrons between the dopant element and carbon, as well as the interactions between the dopant and a possible substrate which supports graphene [21]. However recent experimental and theoretical works have also reported that by N-doping one of the two graphene sublattices preferentially it is possible to open a band-gap, which can furthermore be tuned through control of the dopant concentration. In the process, it may also lead to a quasi-ballistic transport of electrons in the undoped sublattice, which is another important quality for any competitive graphene device (see Ref. [22] for a comprehensive review). On the other hand, different bonding configurations of a nitrogen dopant have been shown to cause opposite effects on the graphene electronic structure, hampering its performance in several applications [23, 24]. It is therefore of paramount importance to precisely identify and control the local environment of impurities embedded in graphene. The presence of a dopant atom in a graphene sheet may be assessed by means of various experimental techniques, such as photoelectron (XPS, ARPES) [24], core-level X-ray [23] or Raman spectroscopies [25], which directly probe the electronic structure of the dopant. However, the most widely used approach to characterize the impurity is to compare atomically resolved scanning tunneling microscopy (STM) images with electronic structure calculations [24–28]. By exploiting the electron tunneling occurring between a sharp tip and the surface at 1

In the same document [5] the authors estimate that most products based on graphene or related materials will be massively commercialized around 2025-2030.

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Graphene is the paradigm for a new class of 2D materials, such as the transition metal dichalcogenides (TMDCs) or the hexagonal boron nitride (hBN). These are likely to grow following the rise of graphene technology [19, 20].

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Figure 1 (a) Experimental and (b) simulated STM image of a single graphitic N dopant in graphene adsorbed on a copper foil. The inset in (a) shows a line profile across the dopant where atomic corrugation and apparent height of the dopant are well visible (Ubias = 0.8 V, I = 0.8 nA). A ball-and-stick model of the graphene lattice with a single N impurity is overlapped to the simulated image in (b), which has been obtained setting Ubias = 0.5 V within the Tersoff-Hamann approach [29]. The figure is reproduced from Ref. [25].

cryogenic temperatures, this method has led to the remarkable identification of an unambiguous STM signature for single graphitic N dopants in graphene: a small triangular spot, with lateral dimensions of a few atomic spacings, characterized by a pronounced apparent corrugation which make it distinguishable from the surrounding honeycomb lattice (see Fig. 1) [25]. DFT-based simulations have furthermore associated this triangular shape to the local redistribution of electron density over the N dopant and its three nearest neighbors [25, 26]. However, especially when dealing with more complex bonding configurations, the resulting STM contrast seems to provide ambiguous information about the actual atomic positions in the defective geometry, hampering its exact identification [28]. In addition to this, a strong dependence of the imaging contrast on experimental parameters such as the sample bias or the chemical nature of the tip apex has been reported [27, 28], further complicating the comparison with theoretical models. In general, some complementary or alternative approaches to image a dopant atom in graphene with atomic resolution and, at the same time, unambiguously identify its chemical nature are needed. A suitable candidate is the atomic force microscopy (AFM) [30, 5

Figure 2 (a) Constant-current STM and (b) constant-height frequencymodulated non-contact AFM images of the same area of CVD grown N-doped graphene on Cu(111). The typical triangular signature of a N dopant in graphene is marked in the STM image with red circles, which also indicate the expected positions for the N atoms in the AFM image. Both images were obtained with an Omicron low-temperature AFM-STM hybrid microscope operating at 4.6 K in ultra high vacuum (< 5−10 mbar), using a metallic tip on a qPlus sensor (Q-factor ≈ 30000, f0 = 25.634 Hz, A < 1 Å). The AFM image was obtained at −100 pm with respect to the STM set-point (Ubias ≈ 1.0 V, Is ≈ 50 pA). Courtesy of D. Smith.

31] In fact, by probing the interaction forces generated when a sharp tip is moved close to the surface, this method can actually image any kind of systems providing true atomic resolution [32–34]. However, a fundamental intrinsic shortcoming is that, when the atoms in the surface have very similar chemical properties (like C and N in our case) and identical site preferences, neither a constant-height or a constantfrequency AFM scan of the surface would provide the desired chemical contrast. As a consequence, even the simple identification of graphitic N dopants in graphene, which is an easy task in STM, would not be possible in this framework (see Fig. 2). A possible procedure to get around this limitation and unveil the chemical identity of the imaged atoms still using an AFM has been suggested by Sugimoto et al. in 2007. It is based on detecting and precisely measuring the short-range forces that arise with the onset of chemical bonding between the outermost atom of the tip apex and the probed atoms, while imaging the surface with a non-contact AFM in frequency 6

Figure 3 Curves resulting from analytical expressions for the van der Waals force, the short-range chemical interaction force, and the total force a function of the absolute tip–surface distance. Figure reproduced from Ref. [36].

modulation3 (fm-AFM) [35]. The procedure is relatively simple: the sharp tip, oscillating at a certain resonance frequency f0 , is moved on top of the atom of interest at a certain lateral position (x, y) above the surface, then the frequency shift ∆f induced by the interactions between the tip and the probed atoms is measured as a function of their vertical separation z. The resulting spectrum will resemble a LennardJones- or Morse- like potential, with a single minimum at specific values zmin and ∆fmin [36]. The key point is that, while zmin reflects the distance of maximal attraction between the tip and the substrate, the depth of the well ∆fmin is proportional to the sum of the short-range chemical interactions and the long-range van der Waals or electrostatic interactions between the tip and the probed atom (see Fig. 3). As a consequence, the obtained value for ∆fmin can be uniquely regarded as a specific chemical signature of the probed atom in the surface. In their paper, Sugimoto et al. have demonstrated that this method actually allows to unambiguously discriminate between the three species (topo3

In non-contact (nc-) AFM the sharp probe is moved close (∼ Å) to the surface, and the image is constructed from the forces exerted on the tip during a raster scan across the surface. The tip is connected to a sensor, usually a silicon-cantilever or a quartz tuning fork (qPlus), which is driven to oscillate during the measurement. The force interactions are measured either by measuring the change in amplitude of the oscillation at a constant frequency (amplitude modulation) or by measuring the change in resonant frequency, by using a feedback circuit which always excites the sensor on resonance (frequency modulation).

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Figure 4 (a) Topographic AFM image of a surface alloy composed by Si, Sn and Pb atoms mixed in equal proportions on a Si(111) substrate. (b) Height distribution of the atoms in (a), showing that Pb and Sn atoms with few nearest-neighboring Si atoms appear indistinguishable in topography. (c) Distribution of maximum attractive total forces measured over the atoms in (a). By using the relative interaction ratio (i.e., the ratio of the maximum attractive short-range forces of two curves) determined for Sn/Si and Pb/Si, each of the three groups of forces can be attributed to interactions measured over Sn, Pb and Si atoms, which are therefore marked in (d) by blue, green, and red colors, respectively. The images in (a) and (d) represent an area of (4.3 × 4.3) nm2 . Adapted from Ref. [36]

graphically not clearly distinguishable) of a surface alloy comprised of Si, Sn and Pb atoms (see Fig. 4). According to the authors, a similar analysis may be extended to other systems if the total force between tip and sample is not dominated by long-range interactions and there are no pronounced in-plane local spatial variations of the long-range forces. Inspired by these considerations, van der Heijden et al. have recently investigated the possibility to identify C, N and O atoms in organic 8

TOAT4 molecules, suggesting that the main factor contributing to the observed chemical contrast might be the local electrostatic environment of the probed atom [37]. In particular, in order to get a more complete and accurate chemical map of the surface, they measure a stack of constant-height fm-AFM images at different height z, resulting in a 3D ∆f data cube where every column corresponds to the Lennard-Joneslike spectrum at that particular (x, y) lateral position. Eventually the depth of the well ∆fmin plotted at every pixel yields an actual AFMbased image of the surface with the desired chemical contrast. However, at the state-of-the-art, the real physical mechanisms and range of applicability of this single-element identification technique are still far from understood. In this work we perform advanced electronic structure calculations to gain more insights about the generality of this AFM technique, by evaluating in particular its potential for chemically identifying nitrogen dopants in graphene. Since the epitaxial growth on a metal close-packed surface is proving to be an extremely promising method to produce large and high-quality graphene, we choose to deal with a supported system such as graphene/Ir(111). After a preliminary study of the impact of N-doping on the properties of this system, which has still not been addressed in literature, we undertake a number of DFT-based simulations aiming to unveil (i) whether one can actually discriminate between nitrogen and carbon in the graphene using AFM and (ii) how the metal substrate which supports graphene affects the outcome of the identification procedure. The content of the work is organized as follows. In Chapter 1 we recall the fundamental electronic properties of graphene, reviewing the main modifications induced by dopants and/or a metal substrate. In Chapter 2, we briefly describe the computational methods we used to simulate the system, highlighting the physical and numerical accuracy that characterize our approach. In Chapter 3 we present an accurate reproduction of the state-of-the-art model for graphene/Ir(111), describing in detail all the stages of the development, from the original graphene and Ir unit cells to the corrugated composite structure observed in the experiments. In Chapter 4, after doping the model with 4

“TOAT” indicates 1,5,9-trioxo-13-azatriangulene.

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a substitutional N dopant, the overall system is thoroughly characterized in terms of geometry, energetics, charge density distribution, absolute charge transfers, site- and -orbital projected density of states (DOS) and STM imaging contrast. In Chapter 5 we present our theoretical modeling of non-contact AFM, along with the simulated ∆f (z) curves associated to carbon or nitrogen at various sites in the corrugated graphene and our answers to the aforementioned questions about the possibility of carrying out chemical mapping with AFM. The simulations in this thesis will be compared, wherever possible, to experimental constant-current STM and non-contact frequencymodulated AFM images, courtesy of the Condensed Matter and Interfaces (CMI) group at the Debye Institute in Utrecht. All these images (including those in Fig. 2) have been obtained using an Omicron lowtemperature AFM-STM hybrid microscope operating at 4.6 K in ultra high vacuum (< 5−10 mbar), using a metallic tip on a qPlus sensor with the following characteristics: Q-factor ≈ 30000, resonance frequency (f0 ) = 25.634 Hz, amplitude (A) < 1 Å.

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Chapter 1 Graphene interaction with metal substrates and the effect of dopants in graphene 1.1

Basic electronic properties of graphene

Most of the outstanding physical and chemical properties of graphene are a natural consequence of the way the electrons are distributed in its 2D network. As in any other allotrope of carbon, each atom supplies the system with four valence electrons. In an isolated carbon atom these are arranged in a [He] 2s2 2p2 configuration. In graphene, an sp2 hybridization between the s orbital and two p orbitals (say px and py ) leads to a trigonal distribution of three electrons, which are shared among nearest-neighbor atoms creating in-plane σ bonds. These are responsible for the mechanical strength and flexibility of graphene, as well as its impermeability1 to even the smallest molecules. For each atom, the fourth electron occupies the remaining pz orbital, which is 1

Assuming a carbon vdW radius of 0.11 nm and a C-C bond length of 0.142 nm, only molecules with a diameter smaller than 0.064 nm are able to pass through graphene. Since the diameter of small molecules such as helium and hydrogen are 0.28 nm and 0.314 nm, respectively [7], this means total impermeability of graphene to any gases.

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Figure 1.1 Graphene hexagonal structure in the real (a) and reciprocal space (b). The unit cell of the graphene lattice in (a) is marked in red and includes two C atoms. The lattice vectors module is ∼ 2.46 Å and the nearest-neighbor distance is ∼ 1.42 Å [38]. The correspondent unit cell in the reciprocal space and its Wigner-Seitz cell (or first Brillouin zone) are illustrated in (b), along with the high-symmetry points Γ, K (K′ ) and M.

normal to the planar structure. The superposition of all pz atomic orbitals in the system leads to π ’molecular’ orbitals delocalized over the whole graphene layer, hence allowing the conduction of an electrical current when a bias voltage is applied. Due to the Pauli principle, the σ bands correspond to a filled shell and, therefore, appear as deep valence states in the electronic band structure, whereas the π bands corresponding to the pz orbitals are found at higher energies (see Fig. 1.2b). The peculiarity of graphene is that, unlike common insulators, there is no band gap, but also no partially filled band, unlike metals. The band structure rather exhibits a linear behavior, defining a filled valence band and an empty conduction band connected at one single point (see Fig. 1.2b). When mapped over the 2D k-space, this linear dispersion can be seen as a section of a circular cone, whose vertex is located at one of the highly symmetric K points in the graphene first Brillouin zone (see Fig. 1.2a). A remarkable consequence of such linearity is that the charge carriers in graphene behave in a very peculiar way: they all move at the same velocity and with zero inertia, mimicking the massless particles traveling at the speed of light predicted by the relativistic Dirac equation [38]. For this reason the labels “Dirac cone” and “Dirac point” are commonly 12

Figure 1.2 Electronic band structure of graphene. In (a) the π and π∗ bands are illustrated as E (kx , ky ). The Dirac cones at the K and K ′ high-symmetry points of the graphene first Brillouin zone are clearly visible. In pure graphene the Dirac points correspond to the Fermi energy, making π and π∗ the highest valence and the lowest conduction band, respectively. In (b) the π and σ bands are reported along a high-symmetry path in the first Brillouin zone, and the DOS for each energy is shown, decomposed into single orbital contributions.

adopted. In fact, any possible modification of the graphene electronic properties can be described in terms of a Dirac cone shift or distortion. In a perfect free-standing graphene the states at the Dirac point are degenerate as the two carbon atoms that make out the graphene unit cell are fully identical. But, if there is a potential variation along the graphene unit cell, then the symmetry of states which belong to different atoms in the same cell get different, the sublattice local symmetry in the graphene is broken and a band gap is usually opened at the Dirac point.

1.2

Graphene interaction with metal substrates

Before its first exfoliation in 2004, graphene was actually well known in the solid-state and surface-science community as an unsought layer 13

Figure 1.3 Schemes of the most used methods to synthesize graphene on a metal surface. (a) segregation of dissolved carbon from the bulk metal, (b) chemical vapor deposition (CVD) from hydrocarbons and (c) molecular beam epitaxy (MBE) from evaporation of solid carbon sources. Adapted from Ref. [49].

of soil, which degrades the catalytic activity of metal surfaces [39–42]. Nowadays, however, growing graphene on a metal surface turns out to be one of the most efficient and perspective methods to synthesize large graphene samples with extraordinary quality, controllable properties and easy transferability onto an insulating or polymer support [8, 43, 44]. For instance, Bae et al [8] have recently reported a rollto-roll production, followed by the incorporation into a fully functional touch-screen panel device, of monolayer 30-inch flexible graphene/Cu films whose conductivity and transparency are much superior to commercial transparent electrodes such as indium tin oxides (ITO). Besides, more and more fundamental research concerning graphene/metal interactions has been motivated by the need to fully understand the nature of metal/graphene junctions, which strongly affect the doping level (and, hence, the transport properties) of graphene in any electronic device [45–48]. In the following we will briefly recall the physics governing the interactions between graphene and a metallic substrate. Graphene is usually grown on metals using one of the following methods2 (Fig. 1.3): (i) segregation of dissolved carbon from the bulk metal, (ii) chemical vapor deposition (CVD) from hydrocarbons and (iii) molecular beam epitaxy (MBE) from evaporation of solid carbon sources. In particular, graphene growth on Ir(111) has been widely performed using CVD: contrary to many other transition metals, the high melting temperature of this material indeed enables to keep the surface 2

The interested reader can find a detailed description of each technique in Ref. [49].

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Figure 1.4 A moiré is a superposition of two regular lattices generating a third one. (a) Overlapping two misaligned stripe patterns with slightly different stripe distances creates a diagonally striped pattern with a much larger periodicity. The reciprocal lattice vector k⃗i associated to each pattern is normal to the stripes and its module is inversely proportional to the stripes separation. The reciprocal lattice vector of the moiré is the difference vector between them. (b) Top-view of graphene/Ir(11), showing that the superposition of a graphene and a metal closepacked surface forms a moiré with hexagonal symmetry.

at temperatures up to 1470 K during the exposure [50], thus boosting the hydrocarbons decomposition and achieving large high-quality samples. As for any pure 2D object, these systems can then be studied using various surface science techniques, such as low-energy electron microscopy (LEEM) and scanning probe microscopies and spectroscopies (STM/STS, AFM/AFS), which give information about the morphology and the electronic structure from µm to the atomic scale, or even photoelectron spectroscopies (NEXAFS, ARPES, etc.), which explore the electronic structure on the macroscopic scale (from several hundreds nm to mm scale). Intense investigations carried out in the last ten years [51–54] enabled the description of graphene grown on either single-crystalline or polycrystalline close-packed surfaces of 3d, 4d or 5d metals as a moiré superstructure, whose orientation depends on the relative orientation of the lattice vectors of graphene and metal surface (Fig. 1.4). This is often accompanied by a corrugation of the carbon network, due to the mismatch between the two lattices (Fig. 1.5) . As a consequence, the interaction strength between graphene and the metallic substrate at the interface is spatially modulated, yielding a spatially periodic electron density, which is known to be responsible for a number of fascinating effects. For instance, this modulation offers a periodic template for selective adsorption of organic molecules [55] 15

Figure 1.5 Representative STM images of (a) (1×1)graphene/(1×1)Ni(111), (b) (12×12)graphene/(11×11)Rh(111), (c) (13×13)graphene/(12×12)Ru(0001) and (d) (10×10)graphene/(9×9)Ir(111). The different structural periodicity between graphene and metal surface in the latter three cases is dictated by the lattice mismatch between them. The imaging contrast is represented both by the color scale and the vertical modulation. Further details can be found in Ref. [49].

or metal clusters [56]. In particular, the adsorption of different metals such as Ir, Au, Pt or Ru on graphene/Ir(111) has been intensively studied, showing a preferential nucleation around the graphene sites that lie closer to the substrate, driven by a local sp2 to sp3 rehybridization of C atoms which “nails” the graphene sheet to the substrate [57–59].

1.2.1

Electronic structure of graphene on metals

Adsorption of a graphene layer on a metallic surface always leads to the modification of the electronic valence bands of graphene [60–70]. One of the most exciting and challenging questions in the last ten years focuses on unraveling the link between these modifications and the bonding between graphene and the metal. A huge number of theoretical calculations predict a bonding energy of graphene to metals in the range of 50 − 200 meV/C-atom, which is far below the lower limit 16

of ≈ 500 meV/atom usually assumed for a strong chemical adsorption [49]. Such low values suggest that the interaction between graphene and metal can be accounted as a physisorption in all studied cases, where the only interactions at the interface are the weak van der Waals (vdW) forces. At the same time, however, the same calculations and various ARPES investigations have pointed out a significant modification of the graphene electronic band structure (linear dispersion and the Dirac cone) in some cases. As a result, a graphene on metal has been classified either as “strongly” or “weakly” bound to the substrate. In the former case, graphene is usually n-doped and the strong overlap between the valence orbitals of graphene and metal (e.g., Ru, Ni) completely destroys the linear dispersion of the graphene π states around the Fermi level EF [60–65]. In the latter case (e.g., Al, Ir, Pt), the Dirac cone around EF is almost preserved, and the graphene results either nor p-doped [66–70]. This graphene/metal puzzle has been recently solved by Voloshina et al [71], who proposed a universal qualitative model to describe the whole family of graphene/metal systems (Fig. 1.6). The interaction can be thought as a two-step process: first, a n- or p-doping of graphene, due to mobile sp-electrons and depending on the work functions of graphene and metal as well as the interface geometry, increases or decreases, respectively, the strength of the vdW forces at the interface; afterwards, the effective space-, energy-, and wave-vector-overlapping of the Cpz orbitals of graphene and metallic d orbitals yields a local hybridization. Depending on the relative energy-overlap of the π and d bands, the Dirac cone of graphene is fully destroyed (in the case of open d-shell metals) or a symmetry energy gap is opened directly at the Dirac point (in the case of closed d-shell metals). In both cases a central role is played by the overlap of the graphene pz orbitals of two carbon atoms in the graphene unit cell with metallic d orbitals with different symmetries (e.g., dz2 , dxz or dyz ). It is worth to point out that, by definition, the van der Waals forces acting between graphene and metal cannot directly affect the electronic structure of these systems. However, in all the studied cases these long-range interactions have been shown to play a crucial role in defining the geometry (and hence the orbital space-overlap) at the interface between the graphene and the metal substrate. As a consequence, an indirect contribution of the dispersion forces should always be considered when discussing the 17

Figure 1.6 The universal model proposed by Voloshina et al [71]. In (a), (b) and (c) we report the energy schemes for three representative graphene-metal systems. (a) Graphene/sp-metal: only doping of graphene is observed (usually n-type) without any modification of the Dirac cone. Valid for alkali metals [72] or Al [67]. (b) Graphene/open-d-shell-metal: after the initial doping, a massive π-d hybridization leads to the violation of the sublattice symmetry in the graphene unit cell, destroying the Dirac cone. (c) Graphene/closed-d-shell-metal: as for (b), but the hybrid states form far below the Dirac point, causing the formation of mini-bands and opening a band-gap directly at the Dirac point, while preserving the linear dispersion around it. (d) The hybridization can involve metal d orbitals with different symmetries, such as dz2 , dxz or dyz .

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Figure 1.7 Effects of doping graphene with heteroatoms. In (a) the DOS of freestanding graphene (G, gray area), boron-doped graphene (BG, blue dotted line), nitrogen-doped graphene (NG, red dashed line) and graphene codoped with N and B (BNG, green dash-dotted line) are reported from Ref. [21]. The different effects on the graphene electronic structure are clearly associable to the Dirac cone shift or the formation of new defective states. In (b) the different doping effects of a nitrilic (N1, one σ bond), pyridinic (N2, two σ bonds) and graphitic (N3, three σ bonds) nitrogen bonding configuration in graphene are illustrated in terms of DOS and Dirac cone shift with respect to the Fermi energy (EF ). Adapted from Ref. [23].

electronic properties of a graphene-metal system. A striking example is the comparison between graphene/Ru(0001) and graphene/Ir(111): although both substrates are open d-shell metals, the former is a typical “strongly” bound system (∼ 206 meV/C-atom) [73], whereas the latter (which will be widely discussed in chapter 3) is representative of a very “weakly” bound one, with a binding energy of only ∼ 50 meV/C-atom [53].

1.3

The effect of dopants in graphene

A great deal of the graphene success stems from the ability to control and tune its electronic properties. A simple and effective way to gain such a control, without the use of external voltages, is to dope the structure , similar to what has been done in the past decades for silicon or other semiconductors. In general, doping graphene with selected heteroatoms, such as boron, nitrogen, fluorine, hydrogen, sulfur, or a combination of them, has been shown to extend its potential for energy storage, energy conversion, sensing, and gas storage [74–77]. In 19

particular, as we mentioned in the introduction, their effects on the graphene properties mainly depend on the difference in the the number of valence electrons with respect to carbon, but also on the different bonding environment of the dopant [23]. For instance, as illustrated in Fig. 1.7a, replacing a carbon with a boron (which has three electrons in the external shell) in a graphitic configuration will lead to a p-type doping3 of graphene, whereas replacing it with a nitrogen (five electrons in the external shell) will lead to an n-type doping. At the same time, a nitrogen incorporated in a pyridinic configuration (i.e., forming two σ-bonds and one π-bond with the neighboring C atoms) would rather lead to a p-type doping, as reported by Schiros et al [23] or Usachov et al [24], who actually observed significant variations in the graphene work function, carrier concentration and local electronic structure (Fig. 1.7b). Their core-level spectroscopies and DFT investigations enabled the description of such diverse outcomes in terms of a difference in charge distribution around the dopant site, drawing a simple chemical bond picture. In their model for a N dopant incorporated in a graphitic configuration (which is the only one considered in this work) four of the five electrons form the σ- and π-bonds as for C. Half of the fifth electron is localized on the dopant, producing a single N non-bonding π-state in the local DOS (LDOS) [25]. The remaining half-electron is delocalized over the carbon π-network, enhancing the carrier concentration and, hence, n-doping the system. In contrast, a N dopant in pyridinic configuration has the opposite effect. Two of the five electrons form σ-bonds with C neighbors, two electrons form a lone in-plane pair localized at the vacant C site, and the remaining one goes to the extended Cπ-network. As a result, the N atom has the equivalent occupation of a nominal C atom in graphene, but a π-electron is missing due to the C vacancy, actually reducing the carrier concentration and resulting in overall p-type doping.

3

A p-type doping of graphene corresponds to a shift of the Dirac cone towards higher energies, or a shift of the Fermi level towards lower energies. The opposite effect occurs in case of a n-type doping.

20

1.3.1

Interplay between doped graphene and a metal substrate

We want to conclude this chapter with a focus on the differences in doping effects when the graphene is supported by a metal substrate. At the state-of-the-art, while several works have been published for both free-standing doped graphene or metal-supported defect-free graphene, only few authors have studied the effects of simultaneously doping and metal-supporting graphene [21, 78, 79]. A remarkable example is the study recently reported by Ferrighi et al [21], who performed a thorough vdW-DFT study of boron-doped, nitrogen-doped and codoped graphene grown on Cu(111), unraveling the underlying mechanisms which can affect the electronic properties of this system. It turns out that the interactions in a doped graphene-metal interface involve several factors, such as the lattice mismatch, the dispersion forces and the charge transfers between the graphene and the substrate, the hybridization between the graphene pz orbitals and the metal d orbitals, as well as the relative position between the dopant and the substrate. Similar investigations with respect to different dopants or metal substrates (e.g., Ir) might significantly contribute to figure out how to synergistically use these two functionalization routes during the synthesis and fabrication of graphene-based devices.

21

Chapter 2 Computational methods 2.1

Density functional theory

Since its formal inception in 1964–1965 by P. Hohenberg, W. Kohn and L.J. Sham [80, 81], density functional theory (DFT) has represented the most elegant and ambitious approach to describe the quantum behavior of atoms and molecules. Over the past 35 years1 it has rapidly evolved from a highly specialized branch at the cutting edge of quantum mechanics, practiced only by a few physicists and chemists, to the most popular electronic structure method in computational science, exploited by a large cadre of active researchers in physics, chemistry, materials science, biology, geology, or even astrophysics [83]. In the present chapter we aim to provide a quick primer of the DFT fundamentals, advantages and limitations, so as to become more familiar with the theoretical background of this work. Of course DFT is a much larger subject: we therefore invite the interested reader to refer to the several books2 available on this field, as well as the Perspective 1

An analysis of Web of Science citation data undertaken at Tulane University [82] reveals that DFT has been the clear citation leader of physics during the 30-year period 1980-2010.

2

Some excellent introductions to DFT are F. Jensen’s Introduction to computational chemistry [84], R.M. Martin’s Electronic structure: basic theory and practical methods [85], or D.S. Sholl’s J.A. Steckel’s Density Functional Theory: a practical introduction [83]. For a more formal approach, one can refer to R.M. Dreizler’s and E.K.U. Gross’s Density Functional Theory: an approach to the quantum many-body problem [86], or S. Lundqvist’s and N.H. March’s Theory of

22

recently published by Axel D. Becke [88], where the state of the art of DFT is outlined in a clear and comprehensible way.

2.1.1

DFT in a nutshell

The origins of DFT go back to 1926, several decades before the publication of the seminal papers by Hohenberg, Kohn and Sham [80, 81]. The first hunches are indeed already contained in the theory developed by L.H. Thomas [89], E. Fermi [90] and P.A.M. Dirac [91], where they propose that the exchange and kinetic energies of a many-electron system could be locally approximated by their one-electron ground-state density, ρ(r). Despite its simplicity, this method was unable to reproduce atomic shell structure, and later it turned out to be too rough to bind molecules [92]. The Hartree-Fock (HF) method proposed in 1928 [93], and further improved in the mid 1900s by Slater [94] in the HFS theory, enables the first complete modellization of the electronic structure, even for complex condensed-matter systems, foreshadowing the success of DFT. The main downside of this theory is that electron correlations are completely neglected, and thus some fundamental properties in solid state physics are not reproducible. The solution comes in 1964 with the publication of the first paper on DFT by Hohenberg and Kohn [80]. For a system of N interacting electrons in an external nuclear potential Ve−n , hence described by the Hamiltonian3 H=−

1! 2 ! 1 !! 1 ∇i + Ve−n (ri ) + , 2 i 2 j̸=i i |rj − ri | i

(2.1)

they demonstrate that the ground-state energy from Schrödinger equation is a unique functional of the electron density ρ. (1st Hohenberg-Kohn theorem), and that

the inhomogeneous electron gas [87]. 3

In atomic units, i.e., ! = me = e = 4πϵ0 = 1.

23

the electron density that minimizes the energy of the overall functional is the true electron density corresponding to the full solution of the Schrödinger equation (2nd Hohenberg-Kohn theorem). This means that all the (ground-state) properties of a many-electron system are completely and exactly determined by ρ, which indeed is considered as the basic variable in DFT. In principle, once specified and minimized the energy functional, one can find the relevant electron density of the system. In the most popular version of DFT in use today, based on the results published by Kohn and Sham in 1965 [81], the energy functional can be written as E(ρ) = Eknown (ρ) + Exc (ρ) (2.2) where Exc , called exchange-correlation (xc) term, is defined to include all the quantum mechanical effects that are not included in the “known” term4 . Kohn and Sham also show that the N-electron Schrödinger equation can be decomposed into a set of single-electron equations 1 − ∇2 ΨKS + Veff ΨKS = ε i Ψi i i 2 "

(2.3)

#

where ΨKS is a set of single-particle orbitals whose density ρ is dei fined to be exactly that of the real system, and Veff is a new potential defined to include the effects of exchange and correlation5 incorporated in Exc . Electrons in atoms, molecules, and solids can be thus viewed as independent particles moving in this effective potential. As a result, from an operational viewpoint, Kohn-Sham DFT can be actually considered an independent-particle theory, even simpler than HartreeFock, which delivers, in principle, the exact density and exact total energy of any interacting, correlated electronic system. The key point 4

The term Eknown in eq. (2.2) includes the electron kinetic energies, the Coulomb interactions between the electrons and the nuclei, those between pairs of electrons and those between pairs of nuclei. Under the Born-Oppenheimer approximation, the kinetic contribution by the nuclei is assumed negligible.

5

Note that since this VKS is a rather good representation of both exchange and correlation, the results are usually considerably more reliable than those of the full HF equations, in which correlation effects are not considered.

24

Figure 2.1 Flow chart of the Kohn-Sham iteration procedure. An initial guess for the electron density is assumed, which is required to evaluate the effective potential Veff . This is in turn used to solve the Kohn-Sham equations with respect to the single-particle wave functions, Ψi . The electron density ρ and the total energy related to these wave functions are subsequently calculated and compared to the initial guess values. This procedure is iterated until these values are converged within a certain accuracy criterion. When self-consistency is achieved for this loop, the electronic part of the system is solved, and various physical quantities can be estimated, such as eigenvalues or forces on the ions that make up the system. Image adapted from [95].

is the functional Exc : once its analytical form is specified, then the main DFT computational problem comes down to a self-consistent numerical solution of the KS equations, carried out as outlined in Fig. 2.1. Once the self-consistent loop illustrated in Fig. 2.1 is converged for a certain fixed configuration of nuclei, the forces acting on each of them can be evaluated by invoking the Hellmann-Feynman theorem [84, 85]. The atomic positions, as well as the shape or volume of the unit cell, can then be updated accordingly and a new static calculation 25

can begin, based on the resulting configuration. This procedure can be iterated until a (local) energy minimum is achieved, corresponding to a stable configuration of the system. Such an optimization problem, usually referred to as a geometry relaxation, can be solved by employing proper numerical methods [84], such as quasi-Newton methods or the conjugate gradient method, and the main parameter that needs to be set is a reasonable threshold value for the forces acting on each atom, referred to as a stopping criterion.

2.1.2

Why DFT?

As stated by W. Kohn on the occasion of his Nobel lecture in 1998 [96], all the DFT popularity stems from two kinds of benefits. The first is conceptual. Since 1926, when Erwin Schrödinger published his historic equation marking the formal beginning of quantum mechanics, theoretical chemists and physicists have become accustomed to describe quantum problems in terms of single particle wave functions, Ψi (r), which assign three degrees of freedom to every particle, i.e., electrons, in the considered system. The outstanding advances [97, 98] achieved in this direction, from Hartree-Fock (HF) all the way up to the modern post-HF methods, attest the fruitfulness of such a wave function approach. However, the higher the required accuracy, the more excited or “virtual” orbitals are needed to describe quantum interactions among electrons, and eventually so many Slater determinants are involved (sometimes even 109 !) that comprehension becomes difficult. DFT can be thought as a complementary look from a different perspective. Instead of dealing with a 3N-dimensional Hilbert space of single-particle states, it focuses on physical quantities defined in the three-dimensional coordinate space, easily visualizable even for very large systems. Among these the most fundamental one is the electron density ρ(r). The second benefit is practical. Even the most advanced wavefunction methods, when applied to systems of many particles, such as large organic molecules, large solids, DNA, or condensed systems, are inevitably affected by severe development and computational costs. The computation time of HF calculations scales like N4 , with N the number of electrons, while that of post-HF wave function methods scales with N5 −N8 , which makes calculations with a large number of particles 26

unfeasible [88]. In contrast, more and more accurate and efficient linear scaling [O(N)] algorithms for DFT have been developed during the last few decades, encouraged by the early studies of S. Goedecker published in 1999 [99]. Nowadays their implementation within various DFT codes [100] is enabling researchers to perform electronic structure calculations for several tens of thousands of atoms in a reasonable time.

2.1.3

Computational details and accuracy

In the present section we briefly discuss the fundamental computational details we have used to study N-doped graphene on Ir(111) and simulate the AFM spectra for chemical identification. Some of the information provided below will be taken up and expanded throughout the reminder of this work. The main goal is to provide the reader with the information needed to reproduce our results using any DFT package. At the same time we hope to give a general idea of the numerical accuracy of our calculations. Exchange-correlation functional In principle, especially for small systems, the exchange-correlation functional Exc can be found exactly, but it turns out to be more expensive than directly solving the Schrödinger equation. In practical calculations the xc contribution is approximated, and the reliability of the results depends on the level of approximation used. The simplest and oldest one is the local density approximation (LDA) [81], which has already achieved satisfactory results for systems with a slowly varying electron density, like metals [101]. However, molecules in LDA are typically over-bound by about 1 eV/bond, well below the accuracy required in chemical calculations, and only the evolution to generalized gradient approximation (GGA) [102] can lead to satisfactory results. In the early 1990s, also hybrid functionals were introduced by A.D. Becke [103] replacing a portion of GGA exchange [104] with Hartree-Fock (HF) exchange, and leading to the B3LYP [105] approximation, one of the most popular in the chemistry community. Although these hybrid functional and its advanced versions provide a very impressive accuracy, they are not suitable to describe large systems, such as the one considered in this thesis, due to unfeasible timedemanding. In contrast, the GGA proposed by Perdew, Burke, and 27

Ernzerhof (PBE) [106] has proven excellent to treat exchange and correlation effects in extended systems, representing a reasonable trade-off between accuracy and time savings. Therefore, we decide to adopt this GGA-PBE functional throughout all our calculations. Dispersion correction An important characteristic of our system is that the adsorption of graphene on a metal close-packed surface, such as Ir(111), is mainly governed by van der Waals interactions [53] (also known as dispersion interactions or London forces). The inclusion of such long-range contributions in the standard functionals has always represented a shortcoming of DFT [107], where the binding energy curves usually decay exponentially instead of −C6 /R6 , where R is the interatomic distance and C6 is a collection of proper physical constants. The conceptually simplest solution, recently implemented in almost all the DFT codes, is to add a dispersion-like contribution to the computed ground-state energy, whose correlation part is intrinsically influenced by this kind of interactions. Depending on whether this correction is applied during the self-consistent procedure to solve the Kohn Sham problem or immediately after it, one can distinguish two classes of approaches. One of the most recent, systematic and accurate methodologies, currently implemented in VASP, which falls into the latter category is the so-called DFT-D developed by S. Grimme [108]. It basically provides an empirical correction to the DFT result in the form Edisp = −

!

!

AB n=6, 8, 10

sn

CnAB fdamp, n (RAB ) n RAB

(2.4)

where the sum is over all atom pairs in the system, RAB is the internuclear distance between paired atoms, CnAB and sn are empirical coefficients used to optimize the correction to the particular system and the specific density functional used (i.e., PBE in our case), and fdamp is a damping function which determines the range of the dispersion interactions [109]. In this thesis we adopt the most recent version of this correction, known as DFT-D3 [110], along with the damping function proposed by A.D. Becke and E.R. Johnson [111]. We point out that this work represents the first example of application of the DFT-D3 correction for studying graphene on Ir(111). For 28

this reason in Appendix A we report a comprehensive review about dispersion corrections in DFT, where the differences between this method and the others are discussed in detail. Basis set and core electrons All the theoretical calculations carried out in this thesis have been performed using the Vienna ab initio simulation package (VASP) [112, 113], which is based on a planewave implementation of DFT, i.e., the Kohn-Sham wave functions are expanded in terms of plane waves φk (r) =

! G

ck+G exp [i (k + G) · r] ,

(2.5)

where the summation is over all reciprocal lattice vectors G. According to this expression, evaluating the solution at even a single point in k-space would involve a summation over an infinite number of possible values for G. However, since these functions can be seen as solutions of the Schrödinger equation with kinetic energy E = (!2 /2m) |k + G|2 , and since solutions at lower energies are usually more physically important than those at very high energies in an electronic structure calculation, in practical calculations one can significantly save computational time by truncating the infinite sum in eq. (2.5) to include only plane waves with energy below a certain cut-off value Ecut . This parameter should be carefully defined at the beginning of any DFT study, according to the chemical elements that make up the system, and it is recommended to keep it constant throughout all the calculations to avoid systematic errors when comparing results from different calculations. In our calculations, involving iridium, carbon and nitrogen atoms, we always 6 set a cutoff energy Ecut = 400 eV. The discussion above emphasizes that a large Ecut should be used to include plane waves that oscillate on a short length scale in the real space, such as those associated to the tightly bound core electrons in the atomic shell structure. However, core electrons are not vital in 6

We should mention that, when performing DFT calculations on a new system, it is always advisable to check that the chosen value for the cut-off energy actually provides a sufficient number of plane waves to self-consistently achieve a numerical solution that converges to the true mathematical solution. See Appendix B for a discussion on the energy convergence with respect to the cut-off energy in our study.

29

defining chemical bonding and other relevant physical properties of materials, which are rather dominated by the external valence electrons. Therefore, in DFT, a frozen core approach is often used. In most cases an empirical ultra-soft pseudo-potential (US-PP) can be exploited to replace the electron density from a certain collection of core electrons with a smoothed density, properly chosen to match the most important physical and mathematical properties of the real ion core [112]. An alternative frozen core approach, that avoids some of the disadvantages7 of US-PPs, is the projector-augmented-wave (PAW) method, which combines a good transferability to different atomistic configurations and chemical compositions with an excellent accuracy [114, 115]. In our calculations, we adopt the PAW approach. k-point sampling in the Brillouin zone A great deal of a plane-wave DFT calculation reduces to evaluate integrals in the form ˆ υcell I= g(k) dk, (2.6) (2π)3 BZ where g(k) is a generic function of vectors k in the reciprocal space, υcell is the unit cell volume and BZ stands for Brillouin zone [116]. In order to efficiently solve this formally infinite-dimensional problem, one needs to numerically evaluate the function to be integrated at a discrete set of properly weighted k-points. Most DFT packages offer the option of choosing these points based on the method developed by Monkhorst and Pack in 1976, where all that is needed is to specify how many k-points should be used in each direction of the reciprocal space. The bigger the cell in the real space, the smaller its Brillouin zone in the reciprocal space, and hence the lower the density of k-points needed to span it and achieve accurate results. The possible symmetries of the system can also be exploited to further decrease this number, reducing 7

One disadvantage of using USPPs is that the construction of the pseudo-potential for each atom requires a number of empirical parameters to be specified. Another one is that USPPs has been shown to give unreliable results for materials with strong magnetic moments or with atoms that have a large difference in electronegativity [83]. PAW potentials are generally more accurate for two reasons: first, the radial cutoffs (core radii) are smaller than the radii used for the US-PPs, and second the PAW potentials reconstruct the exact valence wave function with all nodes in the core region.

30

the computational time for the calculations. As will be clear in the next chapter, the size and the symmetries of our cell enable us to use8 a relatively small 3×3×1 k-point mesh, automatically generated with its origin at the Γ point of the BZ, for studying the structural and electronic properties of N-doped graphene on Ir(111), further reduced to a single k-point (Γ) for simulating the AFM spectra for chemical identification.

2.2

Other methods

Of course the outputs of a plane-wave DFT calculation are not only limited to the ground state energy or the optimized geometry of the system. For instance, the fundamental physical quantities involved in the solution of the Kohn-Sham equations, such as the electron wave functions or the charge density, are always reported after being evaluated at a number of points in a real space grid, whose density can be set according to the desired accuracy. The k-point grid sampling the whole Brillouin zone is also accessible, along with the eigenvalues and the DOS evaluated therein, enabling one to study the electronic band structure of the system. A site-, orbital- or spin-decomposition of the DOS can even be returned. As a result, many chemical and physical properties of the system can be derived, after a more or less straightforward post-processing. In particular, the charge density grid in the real space obtained from DFT calculations can be used as an input for a Bader charge analysis (section 2.2.1). Even STM images can be simulated when having access to the LDOS and the wave functions evaluated over the same real space grid (section 2.2.2).

2.2.1

The Bader model for atomic charge partitioning

A DFT calculations produces an estimate of the electronic charge distribution in the system. In principle, this kind of information can predict charge transfers between atoms and the presence of ionic charges or 8

We should mention that the calculation of certain physical quantities, such as the DOS, usually requires a higher density of k-points in order to provide a satisfactory accuracy. Nevertheless, in line with literature, we assume that a 3 × 3 × 1 is dense enough to provide reliable information also in this context.

31

Figure 2.2 (Left) One-dimensional illustration of the Bader partition scheme. Two adjacent fragments are separated by a zero-flux surface where charge density is minimum. (Right) Application of this scheme to a three-dimensional system, namely a water molecule.

electric multipoles on atoms or molecules. However it is not clear how to extract it, because the output of the calculation is a continuous electronic charge density and it is not clear how one should partition electrons amongst fragments of the system (atoms or molecules). The most efficient and elegant approach, entirely focused on the charge density and hence applicable to the results of either a plane-wave or localized orbital calculation, is based on the so called quantum theory of atoms in molecules (QTAIM) proposed by R. Bader in 1990 [117]. In his theory Bader divides the 3D space into volumes separated by 2D surfaces (zero-flux surfaces) that run through minima in the charge density (Fig. 2.2). More precisely, at any point on a zero-flux surface the gradient of the electron density has no component normal to the surface. Typically in molecular systems, the charge density reaches a minimum between atoms, therefore this is a natural place to separate atoms from each other. By integrating the electronic density within a volume, enclosed between zero-flux surfaces, where a nucleus is located, and possibly adding the electronic charge in nearby volumes that do not include a nucleus, the total electronic charge on that atom9 can be estimated. As a consequence, the DFT-based charge distribution 9

In principle, if an atom is isolated or in a homogeneous symmetric structure (where electronic charge is delocalized over the whole system), then its Bader charge would be zero, because the nucleus positive charge and the negative electronic charge around it would cancel. Any deviation from zero indicates that a charge transfer has occurred involving that atom. For instance, a negative value of Bader charge would mean that electrons have been transferred to it.

32

Figure 2.3 (a) Schematic illustration of the steepest ascent paths on a charge density grid to partition the charge using the on-grid method (see text). The paths are constrained to the grid points, moving at each step to the neighboring grid point towards which the charge density gradient is maximized. Every path terminates at a grid point which has already been assigned to a maximum or, otherwise, at a new charge density maximum, mi . (b) When all grid points are assigned, the set of points which terminates on the same maximum (green to m1 and blue to m2 ) constitutes a Bader volume, which is separated by a zero-flux surface (red line) from the adjacent ones.

can be used to quantitatively estimate charge transfers between atoms or molecules, as well as multipole moments or the hardness of atoms, i.e., the cost of removing charge from them. From a conceptual point of view, this approach also provides a new operative definition for chemical bonding, based on the topology of the electron density. In a dynamic framework, an atom can thus be considered as a proper open system, i.e. a system that can share energy and electron density with the threedimensional environment, with the nucleus acting as a local attractor for the electron density. An efficient and robust computational method for partitioning a charge density grid into Bader volumes has been proposed by G. Henkelman et al [118–120]. Their algorithm, which scales linearly with the number of grid points, follows steepest ascent paths along the charge density gradient from grid point to grid point until a charge density maximum is reached. Subsequent paths are terminated when they reach a grid point already assigned to a charge density maximum (Fig. 2.3). In its more recent version, on-grid ascent paths in the algorithm are substituted by off-lattice ascent paths, so as to avoid a tendency for the zero-flux surfaces to be aligned along the grid directions [120]. Thanks 33

to this grid-based approach the analysis of the large grids generated from plane-wave DFT calculations can be efficiently addressed. It is important to mention that, typically, this algorithm finds one charge density maximum at each atomic centre and one Bader volume for each atom, but this is not always the case: sometimes no nucleus is found within a Bader basin [121]. In most cases, especially when covalent bonds are involved, this situation yields a certain degree of uncertainty while integrating the charge density over the volumes, because it is not clear to which atom the charge contained in the aforementioned basin should be assigned.

2.2.2

The Tersoff-Hamann approach to STM

A scanning tunneling microscope (STM) [30] is an instrument for imaging surfaces at the atomic level. In the past decades, its applications have extended far beyond the original function of mapping surface topography, encompassing a wide range of engineering and analytical areas [122–125]. Nowadays, due to the increasing complexity of the studied systems, the interpretation of experimental results is not always straightforward, and theoretical calculations are often needed. An accurate STM simulation can enable one to understand the origins of the observed image contrast, as well as unraveling the fundamental mechanisms governing the performance of advanced devices. From a computational point of view, all that is needed is a fine 3D grid in the real space at which the electron wave functions and the DOS are evaluated. Any theoretical model of STM has to describe the electron tunneling occurring as a result of an applied bias voltage between a sample and a probe tip, which are separated by a thin layer of vacuum (Fig. 2.4). The simplest approach, developed by J. Tersoff and D. R. Hamann in the 1980s [29] and based on the earlier model [127] of J. Bardeen (1961), is currently incorporated in nearly every state-of-the-art DFT code. The tunneling in this framework is approximated with a single electron transport process: the electron propagates from the bulk region of the sample material to the sample-vacuum interface, it tunnels over the vacuum region and then it further propagates from the tip-vacuum interface into the bulk of the tip. If one regards at the electron transport in the bulk regions as due to free electron modes, then 34

Figure 2.4 Principle of a STM. A bias voltage yields electron tunneling between sample and tip. By keeping the tunneling current constant while scanning the tip over the surface, the tip height follows a contour of constant LDOS. Adapted from [126].

the tunneling current is only proportional to the probability of finding the surface electron in the tip apex region. By assuming a constant potential barrier U in the vacuum, and ignoring any electronic correlation effects, the electron can be thus described by a single particle one-dimensional Schrödinger equation −

!2 d 2 ψ(z) + U ψ(z) = Eψ(z) 2m dz 2

(2.7)

where z is the tip-sample distance. It is known that the surface electronic states decay as φ(z) = φ(0) exp (−κz) in$the classically forbidden region E < U , with a constant exponent κ = 2m (U − E)/! which is related to the square root of the material work function for states at the Fermi level. In the Tersoff-Hamann model, the unknown tip electronic structure is approximated by a single atomic s orbital 10 having the same 10

According to Tersoff and Hamann, since the tunneling current depends on the overlap of the wave functions of the tip and the sample, and since the wave function decays exponentially into the vacuum, only the orbitals localized at the outermost tip atom will be of importance for the tunneling process.

35

work function as %the sample. As a result, there is a non-vanishing prob% % %2 2 % % % S % S ability %ψ(z, En )% = %ψ(0, En ) exp (−κz)% for a surface electron from the nth state ψnS with energy EnS to tunnel into the corresponding state in the tip. If the STM bias voltage is small enough to keep the tip and surface electronic structures unchanged, then the tunneling current can be evaluated using the Tersoff-Hamann (TH) formula: I(z) ∝

EF !

EnS =EF −eVbias

% %2 % % %ψ(z, EnS )%

(2.8)

where EF is the surface Fermi level. In practice, since the probability of finding an electron at EnS is proportional to the number of states available within a small neighborhood of that energy, the obtained I(z) map from the TH model is simply the LDOS of the sample at the various positions of the STM tip. It is important to mention that, in reality, the plethora of approximations made in this simple model yield several limitations [124, 128]. The most relevant ones arise from the extreme simplifications made on the STM tip. For instance, the restriction to s-like tip orbitals only, along with the assumption that the work functions of tip and sample are identical, turn out to predict an incorrect corrugation amplitude for various surfaces, that does not explain the observed atomic resolution. Moreover, it has been shown that the presence of molecules, accidentally or deliberately attached at the tip apex, can significantly affect the image contrast [129]. Another problem lies in the restriction to small bias voltages Vbias . The use of a high bias would lead to an important difference between the tip and sample band structures, leading to a spatial variation of the decay parameter κ. Fortunately, this is not too relevant for the application presented in this thesis, since STM experiments on graphene grown on metal surfaces can achieve atomic resolution even operating at very low bias voltages |Vbias | < 1 eV. Despite the several refinements made in the past decades [130], often providing a solution to some shortcomings mentioned above, the TH approach has been shown to provide a fairly good qualitative prediction of the experimental observations [122] and for this reason it represents the most widely used approach for simulating STM. The calculations of the images reported in Chapter 4 have been carried out 36

using the P4VASP software [131]. By selecting a specific isosurface of the electron probability density distribution (namely the LDOS) calculated with VASP, this software enables us to simulate constant current STM images.

37

Chapter 3 Study of pristine graphene on Ir(111) The choice of a model to reproduce graphene grown on a metal surface, as for any other system, is naturally closely connected to the physical properties under investigation. For the purposes of this work the crystallographic model of graphene/Ir(111) presented in Fig. 3.1 was accurately designed. This model has been recently adopted by several experimental and theoretical groups [51, 53, 132–135]. In these contexts accurate data from X-ray photoelectron spectroscopy, low energy electron diffraction and various scanning probe microscopies and spectroscopies have been collected, corroborated by either standard or dispersion-corrected DFT calculations, leading to an almost full understanding of the mechanisms that govern the geometry and the electronic structure of this defect-free (pristine) graphene-metal system. Since our final task is to undertake an original study concerning a system with very similar physical properties - namely, a doped one - then it is clear that all these data provide an excellent source of information for a solid benchmarking in the preliminary stage of our computational investigation. In the following the main stages of the development of the aforementioned model are outlined and the results of a thorough geometrical and electronic characterization are presented in comparison with the stateof-the-art. We will try to rationalize our findings within the general framework concerning graphene on metals introduced in Chapter 1. The results of this analysis will eventually serve as a solid and competi38

Figure 3.1 A 3D view of the crystallographic model of graphene/Ir(111) designed for the DFT calculations carried out in this work. The carbon atoms and the first, second, third and fourth layers of iridium atoms are marked with black, white, red, blue and yellow spheres, respectively. The scale of the figure has been artificially manipulated in order to ensure sufficient visibility to the typical moiré pattern of a graphene layer grown on a metal close-packed surface (see Table 3.2 for a quantitative description of the graphene corrugation).

tive basis for developing a reliable model of N-doped graphene/Ir(111).

3.1

A supercell approach

Our goal is to study a mono-layer graphene grown on a solid surface. Therefore the ideal model is a slice of material that is infinite in two dimensions, but finite along the normal to the surface. In order to achieve this, it would seem natural to take advantage of periodic boundary conditions in the former two dimensions, but not the third. Although such an approach is currently implemented in some DFT codes (e.g., NWChem [136] or GPAW [137]), it is more common to model a surface using a method where periodic boundary conditions are actually applied in all three dimensions (this is what the VASP code does, for instance). The basic idea is to introduce a so-called supercell, where atoms fill the entire available space in two directions but occupy only a fraction of its vertical size. In such a way, when the supercell is replicated in all 39

three dimensions, it defines a periodic sequence of stacked slabs of solid material separated by empty spaces, usually referred as vacuum. For this reason such model is commonly known as a slab model. The first step in the development is to properly shape the supercell in order to enable the study of the effects of a single nitrogen dopant sitting at different positions in a moiré patterned graphene. To this end, both the lateral and vertical extents of the supercell assume great importance and should be carefully determined so as to meet certain requirements. Above all, an almost general practical issue is that the lateral size of a doped supercell should always be large enough to avoid interactions between dopant images in adjacent cells. Besides, since a surface is by definition the exterior boundary of a bulk crystal, the number of iridium layers beneath graphene needs to be adequately chosen in order to model the bulk environment in a realistic fashion. On the other hand, it is extremely useful to gather all the interesting positions in a moiré pattern together in the same supercell, so as to reproduce a realistic spatial modulation of the physical and chemical interactions at the interface between graphene and metal. A good candidate is the supercell shown in Fig. 3.2a, which has a (9 × 9) lateral periodicity and contains one layer of (10 × 10) graphene on a four-layer slab of Ir atoms. Such supercell turns out to have the minimum lateral extent needed to bring up a real-scale moiré corrugated carbon network. In addition, a quick look at the literature on this systems suggests that the energetic isolation of neighboring dopant images in graphene should be always safely achieved using a (10 × 10) lateral periodicity (e.g., see Ref. [21] for a study of N-doped graphene/Cu(111) system using a (4 × 4) supercell). It should also be noted that when using a slab model it is of crucial importance to consider a sufficiently large vacuum space. This arrangement indeed guarantees that the electron density of the slab tails off to zero in the vacuum region, preventing the interaction between the planar dipoles of its neighboring images. For the graphene/Ir(111) system under discussion the vertical size of the supercell has always been set large enough1 to include at least 20 Å of vacuum space separating 1

It should be pointed out that usually an energy convergence test with respect to the vacuum extent in the supercell should be carried out when first modeling a system. This has not been performed in the present case, mainly for practical

40

(a)

(b) Figure 3.2 A 3D view of (a) the supercell and (b) the correspondent slab model designed for the DFT calculations carried out in this work. The supercell has a (9 × 9) lateral periodicity and contains one layer of (10 × 10) graphene (black) on a four-layer slab of Ir atoms (colored). In (b) the supercell, whose boundaries are marked by red solid lines, is spatially replicated 3 × 3 × 3 times along the directions defined by the supercell lattice vectors (red arrows). A vacuum layer of at least 20 Å separates each replicas along the surface normal direction, so as to avoid any possible interaction between the planar dipoles of neighboring slab images. The scale of the figure has been artificially manipulated in order to enhance the visibility of the moiré pattern (refer to Table 3.2 for a quantitative description of the graphene corrugation).

41

Figure 3.3 Top and side views of the Ir fcc and hcp unit cells. The colors correspond to those in Fig. 3.2a.

two slab replicas, as illustrated in Fig. 3.2b. Such large vacuum has been maintained even when including the model of AFM tip into the supercell (see Chapter 5.1), simply by increasing its vertical extent.

3.2

The Ir(111) substrate

The second step is building the iridium substrate. The atoms in a natural iridium crystal are most commonly arranged in a face-centered cubic (fcc) Bravais lattice, like in most of the metals [116]. The primitive lattice vectors are well known, along with the experimental lattice constant, afcc 0, exp = 3.86 Å. The atomic coordinates have been directly downloaded from the Crystallography Open Database[138] in the first place. A proper unitary rotation matrix has then been applied to the fcc primitive vectors in order to obtain a new hexagonal close-packed reasons such as the extremely high computational effort. Nevertheless, as reported both in literature and in the next sections, a vacuum size of ∼ 20 Å does not seem to negatively affect the reliability of the results at neither the structural nor the electronic level.

42

(hcp) unit cell, where the vertical lattice vector lies along the [111] direction of the fcc structure, as illustrated in Fig. 3.3. The number of atoms per unit cell after this transformation is 3. The experimental values for the equilibrium lattice parameter along the (111) plane and the interlayer distance of the rotated cell (derived from the fcc unit (111) cell through proper geometric considerations) are a0, exp = 2.73 Å and d0, exp = 2.22 Å, respectively. As a rule, such values should be regarded as the main references when assessing the reliability of DFT or any other electronic structure calculations.

3.2.1

Test of DFT approaches to dispersion

Among the tunable parameters in a DFT calculation, of course the most important is the functional used to link the ground state exchangecorrelation energy of a system to its electron density (see Chapter 2). In this work we the GGA functional developed by Perdew, Burke and Ernzerhof (PBE), that is known for its general applicability to solidstate materials and has been widely used in the literature related to graphene/Ir(111) [51, 53, 132–135]. Since a graphene/metal interface in fact represents a van der Waals bound system, a proper dispersion correction scheme should be applied to provide a sound feedback to experiments. Considering that in a generic DFT study it is necessary to keep the adopted approach unchanged throughout all the stages of the development, it would be advisable to set up various simulations using different schemes and eventually give more credit to the one which provides the most realistic results. However it will soon become clear that such a comparative analysis would require a huge computational effort. We therefore opt for testing the most important dispersion corrections only in this preliminary substrate-development stage. Inspired by the literature, we set a cutoff energy of 300 eV and a Γ-centered 24 × 24 × 16 k-point mesh2 to test four different correction approaches among those currently implemented in VASP: vdWDF, Grimme’s DFT-D2, DFT-D3 with zero-damping and DFT-D3 with Becke-Jonson damping (See Appendix A for further details). The calcu2

The chosen cutoff energy and the number of k-points are assumed to be high enough to provide a good energy convergence.

43

Table 3.1 Estimate of the bulk iridium equilibrium lattice constant and interlayer distance when using PBE only and the vdW-DF or PBE + DFT-D approaches to dispersion corrections. Approach

(111)

a0

(Å)

(111)

a0

(111)

− a0,exp (Å)

d0 (Å)

d0 − d0,exp (Å)

PBE

2.74

0.01

2.24

0.02

vdW-DF

2.75

0.02

2.24

0.02

PBE + DFT-D2

2.66

-0.07

2.17

-0.05

PBE + DFT-D3 (Zero-damping)

2.71

-0.02

2.22

0

PBE + DFT-D3 (BJ damping)

2.72

-0.01

2.22

0

lations that we have carried out consist in self-consistently minimizing the energy while allowing the ionic positions and the cell shape and volume to relax, until the forces on each ion are below the threshold of 10−3 eV/Å. In this way all the internal degrees of freedom in the system (e.g., lattice constant, interlayer distance) can be optimized. Our main goal was to check whether one of these methods stands out more than the others when comparing the computed equilibrium lattice constant and interlayer distance in bulk iridium with those observed in the experiments. Table 3.1 suggests that the choice of the correction scheme does not affect significantly the reliability of the results. Although not reported here, it is worth to mention that we have observed a similar situation when simulating a simple graphite unit cell. The reason for this could be that van der Waals forces are not the main source of interaction between the ions. However, we know that these forces will play a role when graphene will be included in the cell (see Chapter 1). Therefore, based on the up-to-date literature, we choose to adopt the DFT-D3(BJ damping) throughout the rest of the development. The main developer of the DFT-D methods, S. Grimme, has indeed shown that the recent D3 approach, provided with the damping function developed by Becke and Jonson, is more accurate and stable than the earlier D2 version [109]. Furthermore, although the vdW-DF is also expected to give accurate results, it has been shown to predict irrationally weak binding of graphene on metals surfaces in most cases [139]. 44

3.2.2

Equilibrium geometry of Ir(111)

As already mentioned, the corrugation occurring when graphene is grown on a metal surface is governed by the lattice mismatch at the interface. Therefore at this stage an accurate estimation of the Ir(111) lattice constant is highly desirable, since this parameter will have a significant influence on the way the graphene atoms will be arranged upon it. For this reason, in addition to the value achieved in the previous section by letting VASP optimize the forces on the atoms in a single dynamic calculation, we give a second independent estimate of (111) the Ir(111) equilibrium lattice constant a0 . In particular, we perform a series of single self-consistent calculations at different values for the lattice constant, manually selected within a range reasonably close to the expected experimental value. These calculations provide a curve of the iridium ground state energy as a function of a(111) , also known as Rose curve (see Fig. 3.4a). This is a particular example of a more general binding law in material science known as the “universal binding energy curve” [140]. The shape of the curve is quite simple, showing a (111) single minimum at a certain lattice constant a0 . If the system exists (111) with any value larger or smaller than a0 , then its total energy may (111) be reduced by changing the lattice parameter to a0 , which can thus be thought as the equilibrium lattice parameter for Ir(111). In our case the exact value, obtained by interpolating the data points with a cubic spline, is 2.720 Å. This is only 0.4% smaller than the experimental value of 2.730 Å and in excellent agreement with the previous estimate achieved by dynamically relaxing the structure. The previous series of calculations can also be exploited to predict some other important equilibrium properties of the material. Specifically, by fitting the calculated E(V ) or its derivative with respect to volume (i.e., the pressure) to a properly chosen form of equation of state (EOS), one obtains the theoretical equilibrium unit cell volume V0 and other mechanical properties, such as the isothermal bulk modu& ' ∂P lus B0 = −V0 ∂V , which measures the resistance of the material to P =0 a uniform compression. Iridium, which is a high melting, high density element, has a bulk modulus3 B0, exp = 366 GPa at 4.2 K [141]. Such a 3

It should be noted that the thermodynamic environment in our DFT calculations is by definition at 0°K in vacuum.

45

Energy /atom (eV)

-2 -3

-9.576

-4

-9.577

-5

-9.578

-6

-9.579 2.71

-7

2.72

2.73

-8 -9 -10 2.2

2.4

2.6

2.8

3.0

3.2

Lattice constant a (Å)

(a) 0 -9.5778

Energy /atom (eV)

-1

Energy /atom (eV)

-2 -3 -4 -5 -6

-9.5780 -9.5782 -9.5784 -9.5786 -9.5788 14.15 14.20 14.25 14.30 14.35

-7

3

Volume /atom (Å )

-8 -9 -10 8

10

12

14

16

18

20

22

3

Volume /atom (Å )

Universal EOS (Vinet; 1989) Murnaghan EOS (Murnaghan; 1937) Birch-Murnaghan 3rd order EOS (1947) Birch-Murnaghan 4th order EOS Natural strain 3rd order EOS (Poirier; 1998) Natural strain 4th order EOS Cubic polynomial in (V − V0 )

(b) Figure 3.4 (a) Total energy (per atom in the unit cell) as a function of the lattice parameter a(111) in bulk Ir. Each data point is from a DFT calculation. The blue line is a cubic spline interpolation of the data points. The inset shows a zoom of the energy curve around the equilibrium lattice parameter. The minimum of the curve yields the equilibrium lattice parameter, a(111) = 2.720 Å, and the energy per atom in the unit cell, E = −9.58 eV. (b) Energy vs. volume data computed from those presented in (a) using the formula reported in the text. Seven different forms of equation of state (EOS) are reported as fit curves, also zoomed in a range close to the equilibrium volume. References for each EOS are reported in the text.

46

high value is due to the high cohesive energies arising from the unfilled 5d-shell of this element. Therefore, if the model developed hitherto is realistic enough, we expect to find a similar value from our DFT calculations. In Fig. 3.4b we report some very well known equations of state (EOS) fitting our calculated E(V ) data points. Here√the volumes √ 3 2 have been computed using the formula V = 2 a c = 23 a3 ac which refers to the Ir hexagonal unit cell, where the vertical to planar lattice parameters ratio c/a has been extracted from the optimized geometry. It is clear that all the curves fit quite well the DFT data, except the most trivial cubic polynomial in (V − V0 ), which begins to deviate quite strongly as the lattice parameter increases. A closer look at the range around the minimum suggests that the Birch-Murnaghan EOS expanded to the 4th -order [142], which provides a value of B0 = 372 GPa, represents the best fit curve, although at least two among the others (universal EOS [143] and natural strain 4th -order EOS [144]) seem to reasonably fit the data points in a more general volumes range, still providing a value for B0 within 2% from the experimental one. On the one hand, a detailed study of the theoretical background concerning the EOS used in this context and a thorough investigation of their applicability to Ir or other elements go far beyond4 the topic of this work. On the other hand, we point out that the previous analysis represents the first calculation of the Ir bulk modulus using the DFT-D3(BJ) approach and, according to the obtained results, we can conclude that the chosen functional and the computational settings used to simulate the Ir substrate definitely provide a reliable model of what is observed experimentally. By increasing the vertical extent of the unit cell and properly adjusting the coordinates of the atoms within it, the vacuum layer mentioned in Section 3.1 can be included in the model, thereby shaping an actual four-layer Ir(111) surface ready to support graphene. An optimization procedure performed on this slab unit cell confirms that there is no significant difference in the geometry of the Ir atoms with respect to the bulk. Specifically, no relevant inward, outward or in-plane displacement of the atoms in the outmost layer have been observed. This is taken 4

We suggest the interested reader to refer to the original papers by Vinet et al [143], Birch and Murnaghan [142, 145], Poirier and Tarantola [144], as well as the computational studies carried out by Hebbache et al [146] or Ziambaras et al [147].

47

as an evidence that four layers of Ir atoms are sufficient to simulate a real Ir substrate (which in reality is made out of more than only four atomic layers).

3.3

Graphene on Ir(111)

The next step is to “grow” graphene on our Ir(111) model. This shall be done by first modeling a hexagonal carbon layer that, after being properly placed upon the surface, will be relaxed along with the underlying Ir atoms, so as to reproduce the corrugated moiré superstructure observed in nature and depicted in Fig. 3.1. As mentioned in Section 3.1, the graphene/Ir(111) system should be modeled using a (9 × 9) supercell for the Ir slab and a (10 × 10) for graphene.5 The former supercell can be easily achieved by generalizing the optimized structure obtained in Section 3.2. Specifically, one needs to include a vacuum layer into the supercell, so as to achieve a total number of 9 · 9 · 4 = 324 Ir atoms in the substrate (i.e., 81 surface atoms in the top-most Ir layer). On the other hand, we know that the structure of graphene can be seen as a triangular lattice with a basis of two atoms per &unit cell (see Fig. 1.1). The unit cell vectors can be & √ ' √ ' δ δ written as a1 = 2 3, 3 and a2 = 2 3, − 3 , where a δ ≈ 1.42 Å [38] is the nearest-neighbor distance and |a1 | = |a2 | = aC ≈ 2.46 Å is the lattice constant [148]. As for the Ir substrate, a simple repetition of this unit cell in the directions of a1 and a2 leads to the desired supercell, which is made out of 10 · 10 · 2 = 200 carbon atoms.

3.3.1

The moiré pattern

The moiré pattern that forms when graphene is epitaxially grown on Ir(111) is a consequence of the energy minimization driven by the lattice mismatch between graphene (2.46 Å) and Ir (2.73 Å). From a theoretical point of view, when adsorbing carbon layers on metals, it is possible either to adapt the metal to the graphene lattice parameter or, vice versa, adapt the graphene to the metal lattice parameter. We have

5

Note that both lattices have hexagonal symmetry

48

chosen to match6 the graphene supercell to the Ir(111) supercell. In this way we do not need to perform any preliminary accurate optimization of the isolated graphene geometry, because its lattice constant will be dictated by the underlying substrate, which we have simulated with high accuracy. The effect of this matching on the graphene network is a 0.6% compression of the lattice constant (from 2.460 Å to 2.446 Å), which may be considered as a measure of the strain (and therefore the vertical displacement) in the system. From an experimental point of view, a number of slightly different moiré superstructures have been observed for graphene/Ir(111), especially as a function of the carbon coverage and the growth temperature. During CVD, for instance, various graphene patches could nucleate on the metal surface, resulting, at sufficiently high coverages, in the formation of domains with different crystal orientations [150] or vertical modulations [132]. In this regard, we note that the choice of a particular patch among the others is actually not relevant for the purposes of this work: although the pattern periodicity would be different, the vertical modulation of the carbon layer, which is ultimately the most relevant physical quantity, is not expected to change significantly. Therefore, in line with other DFT studies of graphene/Ir(111) [51, 53, 133, 134] and adopting the notation used by Meng et al [150], we have decided to reproduce the moiré pattern of R0 graphene7 , shown in Fig. 3.5, which appears to provide remarkable diffraction spots and the largest corrugation of all the possible rotated configurations reported in literature.

3.3.2

Recap of computational details

In the following we want to summarize the main computational details we use to simulate graphene on Ir(111). Most choices have been justified in Chapter 2. The same settings shall be considered in Chapter 4, where 6

Matching the graphene supercell to the Ir(111) supercell results in a commensurate ratio of the surface unit cell lengths of aIr(111) /aC = 10/9 = 1.111. It should be noted, however, that the experimental values for aIr(111) and aC make an Ir(111)-(9 × 9)-graphene-(10 × 10) model slightly incommensurate: for instance, aIr(111) /aC ≈ 1.107 for flakes with an average size of 1000 Å [53]. Nevertheless the error made by forcing a commensurate structure is well below 1%. ( ) 7 The label “R0” indicates that the direction 11¯0 of Ir is parallel to the direction ( ) ¯ 1120 of the moiré superstructure and graphene.

49

Figure 3.5 (a) Atomically-resolved STM image of R0 graphene on Ir(111) and (b) corresponding fast Fourier transform (FFT) pattern. The black rhombus in (a) indicates the 10 × 10 graphene ( ) unit cell of the moiré superstructure. The blue arrow shows the direction of Ir 1¯10 , identified by a combination of low energy electron diffraction patterns; the solid and dashed black arrows indicate the close( (LEED) ) packed 11¯ 20 direction of the moiré superstructure and graphene, respectively. The observed FFT spots are marked in (b) by small dashed circles and in (c) by solid circles, where they are depicted around one K point of the graphene first Brillouin e zone. In (c) the reciprocal vectors aiIr , bR0 and cmoir´ of Ir(111), graphene unit i i cell and moiré superstructure are marked in blue, red and black, respectively.( It is) clearly visible(that )in the R0 rotational variant of graphene the directions of Ir 1¯10 and graphene 11¯ 20 are parallel and the respective Brillouin zones are aligned. The figures are adapted from [66, 149, 150].

50

a nitrogen dopant will be included in the system. Our theoretical calculations are based on density functional theory as implemented in the Vienna ab initio simulation package (VASP) [112, 113]. Wave functions in the Kohn-Sham equations are expanded in terms of a plane-wave basis set. The projector augmented wave (PAW) potentials [114, 115] are used to simulate the ions cores and the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE) is adopted to treat exchange and correlation effects [106]. We use Grimme’s DFT-D3 empirical correction scheme with the BeckeJonson damping profile, as implemented in VASP, to take into account the van der Waals (vdW) interactions [108, 109]. The periodic slab model includes four layers of iridium with (9 × 9) lateral periodicity, one layer of (10 × 10) pristine or N-doped graphene and a vacuum layer of ca. 20 Å. Plane waves with a kinetic energy up to 400 eV have been included in the basis set, and the first Brillouin zone was sampled by a 3×3×1 k-point mesh, automatically generated with the origin at the Γ point. Further details about energy convergence with respect to these parameters are reported in Appendix B. For the DOS calculations, we use the Gaussian smearing method to set the partial occupancies for each orbital (see Ref. [151] for further details).

3.3.3

Equilibrium geometry

Optimizing a structure of 524 atoms per unit cell is not trivial. Due to the extremely high number of degrees of freedom, a simultaneous optimization of the forces on all the atoms often turns out to be a longlasting and cumbersome solution, where only a local energy minimum in the configurational spectrum is actually achievable. A smarter approach is to perform gradual and selected simulations, whereby, while making the most of the available computational resources, one can progressively relax the atoms until a global energy minimum is eventually reached. In line with this strategy, we have first tested statically (i.e. without optimizing the atomic positions) various distances dG/Ir(111) of the graphene sheet from the out-most Ir layer, seeking the value that minimizes the ground-state energy of the system. A cubic spline interpolation of the data, reported in Fig. 3.6, suggests that the minimum energy per atom is found when dG/Ir(111) = 3.440 Å. Of course this value 51

-9.101 -9.102 E0 /atom (eV)

-9.103 -9.104 -9.105 -9.106 -9.107 -9.108 -9.109

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

dG/Ir(111) (Å)

Figure 3.6 Energy per atom vs. distance dG/Ir(111) between the graphene and the outmost Ir layer. Each point is the result of an independent self-consistent calculation. The blue dashed line is a cubic spline interpolation of the data. The minimum energy per atom is found at dG/Ir(111) = 3.440 Å. The colors of the atoms on the right correspond to those in Fig. 3.2a.

has no direct correspondence with experiments, since graphene always corrugates when adsorbed on a metal surface: the only values available in literature is an average of the carbon vertical coordinates. Nevertheless, the previous estimate makes the whole optimization procedure significantly faster, because the graphene atoms are now located closer to their equilibrium configuration. In order to include our model in the current established scientific framework, we decide to optimize the position of all the 524 atoms in the system until the net force on every atom is less than 10−2 eV/Å (while keeping the cell shape and volume unchanged throughout the calculations). By performing a series of selected simulations (see Appendix C) we create a model which reproduces very well the main features of graphene/Ir(111) observed in the experiments, both from the structural and electronic point of view. The geometry resulting from this optimization is illustrated in Fig. 3.7. It can be characterized by focusing on four high-symmetry stacking positions8 for C atoms in the layer: • ATOP, where carbon atoms surround the metal atom of the top 8

We note that different notations are used in literature to mark the four high-symmetry positions. Here we use the notation adopted in Ref. [51].

52

Figure 3.7 (a) Top view of the graphene/Ir(111) system in Fig. 3.1, where the moiré pattern is visible. The colors correspond to those in Fig. 3.1. The white solid rhomb indicates the Ir(111)-(9 × 9)-graphene-(10 × 10) supercell illustrated in Fig. 3.2a. (b) Side view of the same system along the dashed white line drawn in (a), after the geometry optimization. The magnitude of the corrugation has been artificially manipulated in order to enhance the visibility of the vertical modulation. (c) Top view of the local high-symmetry positions of the graphene/metal interface, indicated in (a) and (b). A detailed description can be found in the text.

Ir layer and are placed in the hcp and fcc hollow sites of the (111) stack, above the 2nd and 3rd Ir layers, respectively; • FCC, where carbon atoms surround the fcc hollow site of the Ir(111) surface and are placed in the top and hcp hollow positions of the (111) stack, above the 1st and 2nd Ir layers, respectively; • HCP, where carbon atoms surround the hcp hollow site of the Ir(111) surface and are placed in the top and fcc hollow positions of the (111) stack, above the 1st and 3rd Ir layers, respectively; • BRIDGE, where carbon atoms are bridged by the Ir atom of the 1st layer. 53

From the experimental point of view, various spectroscopic or imaging techniques have been exploited to analyze this structure. For instance, Hämäläinen et al. have recently studied the local surface topography with pm accuracy by means of LEED I(V) measurements and noncontact AFM observations with a CO-terminated tip [132]. Moreover, Busse et al. have performed x-ray standing wave (XSW) experiments ¯ at the interface and the to determine both the average bond distance h buckling amplitude ∆h [53]. In Table 3.2 we report their main results along with other experimental and DFT data collected on this system, in comparison with the optimized geometry we have achieved. Similar to other graphene/metal systems [51], and in excellent agreement with the literature, the largest C-Ir distance is found at the ATOP site and the smallest at the HCP site, which turns out to be slightly lower (∼ 1 pm) than the distance at the FCC site, whereas the carbon atoms at ¯ of the BRIDGE site lie at an intermediate height. The average height h the graphene is well within the error margins of the published values, as well as the amplitude of the corrugation, quantifiable either through the difference ∆h between the furthest and the closest graphene atom from the Ir substrate, or the standard deviation σh of the height distribution. It is worth to note that no significant stretching of the C-C bonds was observed after the structure relaxation, as expected from the extremely high rigidity of the graphene sheet.

3.3.4

Physisorption with chemical modulation

As pointed out in Chapter 1, most DFT calculations predict binding energies between graphene and a metal surface in the range of 50 − 200 meV/C-atom, depending on the substrate [52]. In particular, from our calculations we estimate9 a value of ∼ 88 meV/C-atom for the graphene/Ir(111) system, obtained while taking into account both the 9

We calculate the adsorption energy per C atom as suggested in [21], Eads =

( )+ 1 * EG/Ir(111) − EIr(111) + EG , NC

(3.1)

where EG/Ir(111) , EIr(111) and EG are the energies of the interacting graphene/Ir(111) system, clean Ir slab and pristine graphene, respectively, and NC = 200 is the number of C atoms included in the supercell.

54

55

3.62 n/a

DFT XSW DFT DFTe

Busse et alb

Voloshina et alc

This workd

from PBE + DFT-D2 calculations [134]

from PBE + DFT-D3(BJ) calculations

d Data

the atoms that constitute the hexagon in the ATOP position have the same height.

3.403h

n/a

3.41 3.38 ± 0.04

3.39 ± 0.03 n/a

¯ (Å) h

0.343i

n/a

0.35 n/a

0.43 ± 0.09 0.47 ± 0.05

∆h (Å)

i This

to the Ir maximum height.

value was extracted by comparing the z cartesian coordinate of the topmost and the lowest C atom in the graphene layer.

h Referred

values for hF CC , hHCP and hBRIDGE correspond to the height of the lowest atom among those in a FCC, HCP and BRIDGE position, respectively.

g The

f All

3.323

3.315

n/a n/a

n/a n/a

value of h in the various positions is computed using the height of the nearest Ir atom as a reference.

from XSW measurements and PBE + DFT-D2 calculations [53]

c Data

e The

from LEED I(V ) and AFM (with a CO-terminated tip) measurements [132]

3.311

3.27

3.27 n/a

3.27 n/a

b Data

3.318g

3.28

3.29 n/a

3.29 n/a

hHCP (Å) hBRIDGE (Å)

a Data

3.633f

3.58

3.71 n/a

LEED AFM

hATOP (Å) hFCC (Å)

Hämäläinen et ala

Reference

Table 3.2 Corrugation of the graphene moiré pattern on Ir(111) in comparison with experiments and simulations reported in literature.

0.103

n/a

0.09 n/a

n/a n/a

σh (Å)

local and non-local contributions to the overall binding energy.10 This value is slightly higher than the 50 meV/C-atom predicted in Ref. [53] using DFT-D2, however it is still much below the lower limit usually considered for chemical adsorption on a metallic surface [49]. In addition to this, looking at the data in Table 3.2 we note that the average distance of the graphene from the top-most Ir layer (3.403 Å) is quite similar to the distance between carbon layers in pure graphite (3.36 Å [152]). All this confirms that the binding between graphene and Ir(111) is mainly governed by weak physical interactions. However, as already mentioned in Chapter 1, despite the impression given by these averaged values, here we are not dealing with just a pure physisorption, but rather with a chemically modulated interaction [53]. This becomes clear when analyzing the electronic charge redistribution caused by adsorption of the graphene layer on Ir(111). In Fig. 3.8a we plot the charge density difference (CDD) calculated using the formula ∆ρ = ρG/Ir(111) − ρIr(111) − ρG , where ρIr(111) is the charge density of the isolated Ir substrate, ρG is the charge density of the isolated graphene and ρG/Ir(111) is the charge density of the overall system graphene/Ir(111), plotted in units of e/Å3 . A positive value of ∆ρ (red in Fig. 3.8a) represents an enhancement in electron density caused by adsorption; vice versa, a negative value (blue in Fig. 3.8a) indicates a loss of electron density. In the HCP and FCC regions we observe a small charge transfer from graphene towards the substrate. This can be related to the fact that the C atoms, especially those around the HCP and FCC high-symmetry which sit directly above Ir atoms, hybridize their pz -orbitals with the top-most Ir d-orbitals having out-of-plane components, such as dz2 , dxz or dyz (Fig. 3.8c). A Bader charge analysis, in excellent agreement with the literature [53], indicates that graphene gives away in total ∼ 0.01 e per Ir surface atom, resulting in a slight p-type doping.

10

We emphasize this because in some cases, for instance when using a vdW-DF dispersion-correction approach (see Appendix A), one can distinguish between local and non-local contributions to the binding energy. Such an analysis has enabled to separate the attractive and repulsive contributions to the adsorption of graphene on Ir(111), demonstrating that the only reason why this system is stable is the existence of the van der Waals forces [53].

56

(a)

(b)

(c)

Figure 3.8 (a) Charge density difference upon adsorption, ∆ρ = ρG/Ir(111) − ρIr(111) − ρG , plotted in units of e/Å3 . It represents a cut through the atoms along a line of maximum corrugation in graphene/Ir(111). The color scale ranges from −0.013 e/Å3 (blue) to +0.007 e/Å3 (red). A positive value of ∆ρ indicates an enhancement in electron density, whereas a negative value indicates a loss. Carbon and iridium atoms are represented as dark and light grey spheres, respectively. The black dashed lines indicates the border of the graphene/Ir(111) supercell. (b) Zoom of the red dashed area in (a). (c) C 2pz and Ir 5dz2 , 5dxz and 5dyz orbitals of graphene and top-most Ir atoms.

3.3.5

Electronic structure and imaging contrast

It is interesting to investigate the effects of the aforementioned C-Ir hybridization on the electronic structure of graphene, trying to separate these effects from the possible contributions only due to the corrugation. In Fig. 3.9 we therefore compare the DOS of a freestanding pure and flat graphene with the DOS projected11 on the atoms of a corrugated graphene adsorbed on Ir(111), obtained with and without including the Ir atoms in the supercell during the calculation. We find that new electronic states appear near the Fermi level (EF ) in the projected DOS 11

The total DOS extracted from DFT calculations can be decomposed into contributions from single atoms or even single atomic orbitals in the system. By summing up all the contributions from the pz orbitals of the graphene atoms in our supercell we obtain “projected” DOS such as those illustrated in Fig. 3.9.

57

25 Freestanding - flat Freestanding - corrugated

DOS (states/eV)

20

Supported - corrugated

15 10 5 0

-1

-0.5

0

0.5

1

E-EF (eV) Figure 3.9 DOS (projected on pz orbitals) of freestanding flat graphene (grey filled curve) in comparison with those projected on a corrugated graphene adsorbed on Ir(111), obtained with (black solid line) and without (red dashed line) including the Ir atoms in the supercell.

obtained when including the Ir atoms in the supercell. In addition, we observe a broad similarity between the DOS for a free-standing (flat) graphene and the DOS for the corrugated graphene obtained without including the metal substrate in the supercell, suggesting that the C vertical distortion alone has only a limited impact on the electronic structure. The natural conclusion is that the observed modifications are due to the interaction with the substrate, in particular the local hybridization between C pz - and Ir d-orbitals around the HCP and FCC sites of the moiré. These findings are in qualitative agreement with Refs. [53, 134] and, in a more general framework, with the universal model for graphenemetal systems described in Chapter 1. In principle, the process responsible for the modifications observed here may be assumed analogous to the one hypothesized for graphene on another open d-shell metal such as Ru(0001), with the difference that, in the latter case, the dramatically stronger attractive vdW contribution leads to a much smaller ¯ G/Ru ≈ 2.195 Å [73]) distance between graphene and the substrate (h and, hence, to an enhanced spatial overlap between carbon and metal 58

orbitals, which completely destroys the Dirac cone of graphene. As pointed out in Ref. [49], the existence of hybrid states caused by this small but finite chemical interaction between graphene and Ir(111) is the basis of the mechanism that enables the visualization of the moiré structure in constant-current STM (cc-STM) experiments. During these kind of measurements, when a negative bias voltage is set, the electrons tunnel from occupied states of the sample into unoccupied states of the tip (i.e., the occupied states of the sample are probed); on the contrary, at a positive bias, the tunneling occurs in the opposite direction and the unoccupied states of the sample are thus probed. As a consequence, the electronic structure of the surface has a crucial effect on the feedback current detected by the microscope. More precisely, since the tunneling probability is proportional to the surface LDOS at the position of the tip (see Chapter 2), the local C-Ir hybridization at the HCP and FCC positions leads to a spatially modulated imaging contrast which resolves the actual moiré pattern of graphene/Ir(111) (see Fig. 3.10a). Moreover, it has been shown that the apparent corrugation of the moiré is strongly dependent on the imaging bias voltage and that, at some tunneling conditions, an inversion of the imaging contrast is observed (Fig. 3.10b) [153]. Interestingly, this phenomena can also be associated to the existence of hybrid states in the graphene electronic structure. In Fig 3.11 we report the calculated local DOS (LDOS) projected on the pz orbital of a representative C atom in all the four high-symmetry positions. In line with literature [134], we associate the contrast observed for high (negative) bias voltages12 to the peaks in LDOS located at E − EF ≈ −1.49 eV , E − EF ≈ −0.95 eV and E − EF ≈ −0.43 eV. Indeed in correspondence of these energies the LDOS for the ATOP position is higher than the LDOS for the other positions, explaining the higher tunneling current (i.e., the brighter contrast) around the ATOP regions in experimental cc-STM images (on top in Fig. 3.10b). On the contrary, we find that the the inverted contrast observed at small bias voltages (on bottom in Fig. 3.10b) can be associated to the peak at E − EF ≈ −0.15 eV where the LDOS for the HCP and FCC positions is more pronounced. 12

Setting a high (negative) bias voltage in STM corresponds to probe (occupied) electronic states whose energy is far from EF (see 2).

59

Figure 3.10 STM images of defect-free graphene on Ir(111). In (a) the moiré of a graphene flake attached to an iridium step edge is clearly visible. The image, reported from [153], shows a (100 × 100) Å2 area scanned at a voltage UT = −0.17 V and tunneling current IT = 21 nA. The moiré supercell is marked as a white rhombus. In (b) an STM scan of the same system at two different bias voltages is shown, UT = −1.8 V on top and UT = −0.3 V on bottom, as an evidence of the contrast inversion due to the physisorption with chemical modulation typical of graphene/Ir(111). The ATOP (A), FCC (F) and HCP (H) high-symmetry positions are marked in the image. The details of the measurements are reported in the paper by Dedkov et al [51].

In conclusion to this discussion, it is interesting to compare our DOS with the band structure of the same (10×10)graphene/(9×9)Ir(111) model discussed in Ref. [49] (Fig. 3.12a-b), which has been calculated using similar computational settings. Since the use of such a large supercell leads to the folding of the energy bands into a very small Brillouin zone [116], the authors had to undertake a considerable postprocessing on the original band structure in order to unfold it back to the usual Brillouin zone associated to the graphene (1×1) primitive cell [154, 155]. The resulting band structure shows a rough preservation of the linear band dispersion for the graphene π and σ states, along with a scattered dispersion around EF that might be regarded as an effect of the local hybridization at the graphene-metal interface. In addition to this, a 150 meV shift of the Dirac point above EF (i.e., p-type doping) is observed, along with an energy gap of ∼ 300 meV for the π states at the K point. This is qualitatively confirmed by some recent ARPES measurements (Fig. 3.12c), where a band-gap of ∼ 100 meV is found 60

0.16

C C C C

LDOS (states/eV)

0.14 0.12

@ @ @ @

ATOP FCC HCP BRIDGE

0.10 0.08 0.06 0.04 0.02 0.00

-1.5

-1

-0.5

0

0.5

1

E-EF (eV) Figure 3.11 DOS projected on the pz orbital of a C atom located at the four high-symmetry positions of the graphene moiré.

at the same point [66, 149]. The fact that such an energy gap is not detectable in our calculated DOS in Fig. 3.11 might be related to the presence of a small residual Ir component which is responsible for a nonzero DOS at the Dirac point [21]. A calculation of the effective band structure, once properly unfolded and orbital-decomposed, might actually make it visible.

A quick recap The main goal of this chapter was to reproduce the state-of-the-art model of graphene on Ir(111). The obtained results concerning the geometry, charge density distribution, DOS and imaging contrast demonstrate that this attempt has been proved successful. The widely reported physisorption with chemical modulation of graphene on Ir(111) has been reproduced. At the same time the Grimme’s DFT-D3(BJ) approach to include dispersion in DFT has been shown for the first time to reasonably reproduce the geometric and electronic properties of graphene grown on a metal surface. Interestingly, it predicts a binding which is slightly stronger than the DFT-D2 result, further contributing to put aside the original belief that graphene is nearly free-standing on 61

Figure 3.12 (a) Band structure of (10×10)graphene/(9×9)Ir(111), unfolded into the graphene (1×1) Brillouin zone using the BandUp code [154]. (b) Zoom of the electronic structure where the energy gap ∆E between the graphene π bands (solid lines) is open. (c) ARPES intensity map along the Γ–M–K–Γ path in the Brillouin zone for graphene/Ir(111). Replicas of the Dirac cone, associated to the superperiodicity of the moiré, are clearly visible around the most intense one, related to the (1×1) graphene unit cell. Figures are adapted from [49, 52].

Ir(111) [66]. It should be noted, however, that the present analysis does not give any information about the optimal correction scheme for this system. More insights in this direction might be gained by performing the same simulations using various dispersion correction approaches (e.g. those reported in Table 3.1) and then comparing the results with physical or chemical properties which are experimentally accessible, such as the average distance between graphene and Ir(111). Overall, we consider the achievements in this chapter as a solid and competitive basis for simulating N-doped graphene/Ir(111), whose properties have never been investigated so far.

62

Chapter 4 Study of N-doped graphene on Ir(111) In the previous chapter we have presented how to build a sound and valuable model for defect-free graphene adsorbed on Ir(111). We have reproduced all the structural and electronic features observed and simulated in literature, thus assessing the reliability of the specific computational settings used through the development and the optimization processes. In what follows we generalize this model so as to include a substitutional nitrogen dopant in the graphene. In fact, while several theoretical works have been reported for freestanding N-doped graphene [24, 76, 156–158], only few works are dealing with doped graphene on a metal surface [21, 79], with almost no reference to Ir(111) as a substrate. We will try to understand how the presence of a N dopant affects the geometry and the charge transfer at the interface between graphene and Ir(111), by analyzing charge density and Bader charge. A study of the LDOS at the N site and at the neighboring C sites will be also presented, aiming to identify the electronic states that give rise to the main features observable with a scanning tunneling microscope.

4.1

Equilibrium geometry

We develop four new models by simply substituting a carbon atom with a nitrogen at all the four high-symmetry positions in the moiré 63

of graphene/Ir(111), described in Chapter 3.3.3. As for the defect-free system, a structure optimization is performed by updating the atomic configurations until the Hellmann-Feynman forces on all the atoms are less than 10−2 eV/Å. The lateral stretching in the graphene network induced by the substitutional dopant is only a 1% compression of the nearest-neighbor distance and a 0.6% compression of the lattice parameter a, as expected due to the similar radius for carbon and nitrogen. In Table 4.1 we compare the geometry of the pristine model with that ¯ and the corruof the nitrogen doped ones. While the average height h gation ∆h (σh ) do not seem to be significantly affected by the presence of a dopant, the local vertical modulation of the graphene appears to change in correspondence of the high-symmetry position when nitrogen is located therein, except for the ATOP positions. Specifically, a nitrogen introduced at an ATOP position almost retains the z coordinate of a carbon located at the same place, whereas in all the other places the substitution of a carbon with a nitrogen moves the ion upwards of approximately 5 pm. In addition, we note that the top-most iridium layer, which is intrinsically corrugated in the order of the pm, is also slightly dependent on the position of the nitrogen in the moiré.

4.2

Energetics

From an experimental point of view it is interesting to investigate the stability of the defected system. Various formation energy calculations reported in literature suggest that graphitic bonding configuration of a substitutional nitrogen atom into a graphene sheet is energetically the most favorable among the possible nitrogen bonding configurations [23, 159]. Moreover, since the relative position of the nitrogen dopant with respect to the substrate can indeed significantly affect the energetics [21], it would be useful to provide a reasonable prediction of where the dopant would be most likely observed after the deposition. Therefore we calculate the difference in the adsorption energy Eads of graphene on Ir(111) when comparing the four different locations of N in the moiré. From the expression of Eads given by Eq. 3.1, we calculate the variation NG G in adsorption energy ∆Eads = Eads − Eads due to the substitution of a single C atom with a N dopant as ,

-

∆Eads = ENG/Ir(111) − EG/Ir(111) − [ENG − EG ] , 64

(4.1)

65

The value for hAT OP in the first row refers to the height of the N atom in that position. Idem for the other positions.

0.332

This value was extracted by comparing the z cartesian coordinate of the topmost and the lowest C atom in the graphene layer.

3.404

0.332

f

3.382

3.405

0.341

Referred to the Ir maximum height.

3.334

3.334

3.405

0.345

e

3.331

3.374

3.327

3.401

0.343e

d

3.630

3.330

3.321

3.326

3.403d

The values for hF CC , hHCP and hBRIDGE correspond to the height of the lowest atom among those in a FCC, HCP and BRIDGE position, respectively.

0.042

N@BRIDGE

3.630

3.383

3.311

3.323

∆h (Å)

All the atoms that constitute the hexagon in the ATOP position have the same height.

0.048

N@HCP

3.629

3.323

3.311

¯ (Å) h

c

0.047

N@FCC

3.633f

3.318c

hBRIDGE (Å)

The value of h in the various positions is computed using the height of the nearest Ir atom as a reference.

0.031

N@ATOP

3.633b

hHCP (Å)

b

0.027

Pristinea

hATOP (Å) hFCC (Å)

a

∆hIr (Å)

N position

Table 4.1 Corrugation of the first Ir layer and graphene for different dopant positions in the moiré. Bold numbers indicate that the atom considered in that specific high-symmetry position is a nitrogen. Data are from PBE + DFT-D3(BJ) calculations.

0.101

0.100

0.100

0.103

0.103

σh (Å)

under the assumption that the cell size is large enough to simulate a very AT OP F CC low dopant concentration. We find ∆Eads = −0.373 eV, ∆Eads = HCP BRIDGE −0.241 eV, ∆Eads = −0.240 eV and ∆Eads = −0.261 eV. A comparison between these results suggests that a N-doped graphene is generally more tightly bound to the substrate with respect to a pristine one, and that the relatively most stable configuration for the N dopant in the graphene/Ir(111) system is the ATOP high-symmetry position.

4.3

Electronic charge distribution

The variations in geometry and stability described so far can be related to the presence of an additional localized electronic charge provided by the nitrogen dopant. To justify this statement, in Fig. 4.1 we present the calculated charge density difference upon adsorption, ∆ρ = ρNG/Ir(111) − ρIr(111) − ρNG , where ρIr(111) is the charge density of the isolated Ir substrate, ρNG is the charge density of the isolated doped graphene and ρNG/Ir(111) is the charge density of the overall doped graphene on Ir(111). We note that an additional charge actually accumulates in the region immediately below the nitrogen for all the considered configurations, with a substantial loss of charge density around the nearest carbon neighbors and a slight local enhancement below the closest iridium atoms. Especially around the FCC, HCP and BRIDGE places, which are the closest to the substrate, this enhanced charge density can lead to a further modulation of the graphene network, moving upwards the dopant ion located therein, explaining the data in Table 4.1. In line with this argument, the relatively large distance from the substrate can explain why a dopant lying at an ATOP position appears to retain the coordinates of a carbon located at the same position (see Section 4.1). In order to gain more insights, we have also calculated the charge density difference upon doping, ∆ρ = ρNG/Ir(111) − ρIr(111) − ρ[N#→C]G . Here ρ[N#→C]G is the charge density of a fictitious isolated graphene which retains the optimized geometry obtained for the doped structure, but where the nitrogen atom has been substituted by a carbon atom. In this way we can qualitatively figure out how the extra nitrogen charge is distributed over the doped system. In Fig. 4.2 we present our results for doped graphene with N in ATOP position. We find a significant local 66

Figure 4.1 Charge transfer upon adsorption, ∆ρ = ρNG/Ir(111) − ρIr(111) − ρNG , in units of e/Å3 , calculated for the pristine system (top) and the four doped systems with a substitutional nitrogen located at ATOP, FCC, HCP and BRIDGE (bottom) high-symmetry positions [a side view was taken as a cut through atoms along the line of maximum corrugation in the model]. A positive value of ∆ρ represents an enhancement in electron density, whereas a negative value means a loss. Nitrogen, carbon and iridium atoms are represented as green, dark and light grey spheres, respectively. The black dashed line indicates the border of the graphene/Ir(111) supercell used in this work.

modification in the electron density close to the dopant center, with a noticeable threefold symmetric delocalization, involving substantially the three nearest C atoms. Interestingly, a similar delocalization is also observed in other defects with threefold symmetry, such as impuritydoped and impurity-adsorbed freestanding graphene [23, 159–162] (see for instance Fig. 4.2a). Therefore, despite the loss of perfect planarity due to the moiré corrugation, we conclude that the local sp2 -bonding configuration usually reported for a substitutional N in graphene is substantially retained.

67

Figure 4.2 Gain (red) and loss (blue) of charge in the (a) freestanding and (b,c,d) supported graphene caused by doping with a substitutional N dopant [which lies at an ATOP position in the case of graphene/Ir(111)]. The charge density difference (CDD) in (a) has been computed by Schiros et al [23] for a freestanding model as ∆ρ = ρNG − ρG . It can be qualitatively compared to the CDD calculated in this work for the supported system, ∆ρ = ρNG/Ir(111) − ρIr(111) − ρ[N!→C]G , shown in (b), (c) and (d) in 3D, side and top view, respectively. The isosurface value of the electron density in (d) is set to ±0.017 e/Å (red is for positive values). Nitrogen, carbon and iridium atoms are represented as green, dark and light grey spheres, respectively.

4.4

Bader charges

In order to provide a more quantitative understanding of the dopinginduced charge transfer, we have performed a Bader charge analysis1 on one of the considered doped systems, for instance the one with nitrogen in ATOP position. It turns out that there is a slight charge transfer of ∼ 0.907 e from the graphene to the Ir substrate, corresponding to a tiny 1

The total charge density has been calculated by sampling the supercell in the real space with a dense 756 × 756 × 1134 grid (31 grid points per Angstrom), so as to minimize the errors associated to the definition of the Bader volumes boundaries (see Chapter 2).

68

Bader charge (atoms @ Line of Maximum Corrugation)

0.4 Bader charge (e− )

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4

N ATOP

FCC

HCP

ATOP

Figure 4.3 Bader charge plotted in correspondence of the atoms located along the direction of maximum corrugation in the graphene, reported below the graph. The value associated to the N dopant is clearly marked. The calculation details and a complete discussion can be found in the text.

∼ 0.011 e transfer per Ir surface atom.2 Such a small charge transfer is consistent with the large electronegativity of iridium and indicates that a weak binding between the graphene and the substrate is going on, leading to a slight p-doping of the graphene. Besides, it is also in qualitative agreement with the results reported in literature for the pristine structure (see Chapter 3). In Fig. 4.3 we also report the Bader charge in correspondence of the atoms located along the direction of maximum corrugation in the graphene, which includes the N dopant. Far from the dopant site, this graph shows a charge scattered around zero, with a mean deviation of ∼ 0.1 e. Such deviation might be a consequence of assigning the charge volumes to the wrong atoms. This is mostly, although not exclusively, expected for the C atoms, as all of their valence electrons participate in 2

It should be noted that this small value might rather be due to the inaccuracy in the internal definition of the Bader volumes boundaries.

69

DOS (states/eV)

50 45 40 35 30 25 20 15 10 5 0

Pristine N @ ATOP N @ FCC N @ HCP N @ BRIDGE

-3

-2

-1

0

1

2

3

E-EF (eV) Figure 4.4 DOS projected on the pz orbital of the graphene atoms in our graphene/Ir(111) model, with (grey filled line) and without (colored lines) including a substitutional N dopant in the structure. The DOS calculated for different locations of N in the moiré are reported for comparison.

covalent bonding (see Chapter 2.2.1). A somewhat larger charge redistribution seems to occur around the N dopant and its nearest neighbors in the graphene. We actually find that N withdraws almost 73% of its additional ∼ 1.268 e directly from the three nearest C atoms, which actually are found to transfer in average ∼ 0.3 e each to the dopant. The remaining charge might be provided by the next to nearest neighbors.

4.5

Density of states

In the previous chapter we have shown that the chemical interaction between a pristine graphene and the Ir substrate leads to the formation of new electronic states near the Fermi level, and a slight p-doping of the carbon sheet. We now want to investigate the effects of N-doping on the electronic structure. Fig. 4.4 shows a projection of the calculated DOS on the atoms making up the pristine and doped graphene. We observe that N-doping has various consequences. On the one hand we always observe a shift of the Fermi level towards higher energies 70

with respect to the pristine case, indicating an n-type doping of the graphene sheet, as expected from the presence of the N extra valence electron [21, 24, 159]. Interestingly, this shift turns out to be almost the same, ∼ 56 meV, for all the considered dopant locations in the moiré, except for the ATOP one, which results in a slightly larger amount, ∼ 75 meV. Our interpretation, based on the information gained in the previous sections, is that the difference in EF -shift between ATOP and the other N configurations is driven by the stronger chemical interaction occurring in the latter regions, which lie closer to the substrate: there the extra charge provided by N is massively shared between the C network and the metal, occupying the hybridized orbital that is formed between its pz orbital and the Ir d ones. In contrast, farther from these regions, the nitrogen charge is almost entirely distributed over the local graphene network, resulting in a stronger n-doping when projecting the DOS only on its atoms. On the other hand, we note that the aforementioned discrepancy between the ATOP and the other configurations is actually visible all over the energy spectrum (the blue curve in Fig. 4.4 is slightly shifted to the left with respect to the others), except for the region right above the Fermi level, where we observe a very localized state whose energy is exactly the same for all the considered doping configurations (E − EF ≈ 25 meV). Inspired by a recently reported observation of N localized states in substitutionally doped graphene/SiC(0001) [26], we identify this state as a characteristic defective state: a N signature, independent from the chemical interactions occurring at the interface between graphene and Ir(111). Very similar features have been observed as a result of XES measurements on freestanding N-doped graphene, where a single N non-bonding π-orbital resonance has been observed at ∼ EF [23].

4.6

The N signature in LDOS and STM

We now present our simulations of STM images of the nitrogen dopant in our system, obtained in the framework of the Tersoff-Hamann approximation (see Chapter 2.2.2). In this method, it is assumed that the tunneling current is proportional to the LDOS of the surface at the tip position, integrated over an energy range defined by EF and the 71

0.18

0.35 N

0.16

C1

0.14

0.25

LDOS (states/eV)

LDOS (states/eV)

0.3 Cbulk

0.2 0.15 0.1 0.05 0

DOS@N - DOS@C1

0.12 0.1 0.08 0.06 0.04 0.02

-3

-2

-1

0

1

2

0

3

-3

-2

-1

E-EF (eV)

0

1

2

3

1

2

3

E-EF (eV)

(a) N @ ATOP 0.25

0.4 N

DOS@N - DOS@C1

C1

0.3

LDOS (states/eV)

LDOS (states/eV)

0.35

Cbulk

0.25 0.2 0.15 0.1

0.2 0.15 0.1 0.05

0.05 0

-3

-2

-1

0

1

2

3

0

-3

E-EF (eV)

-2

-1

0 E-EF (eV)

(b) N @ HCP Figure 4.5 (Left) LDOS projected on the pz orbital of a C atom far away from the N substitutional dopant (Cbulk ), on a neighboring C atom (C1 ) and on the N atom at (a) ATOP or (b) HCP sites, marked by green filled, blue solid and orange dashed lines, respectively. (Right) Difference between the LDOS projected at the N site and at the C1 site. The black solid arrow indicates the energy of maximum LDOS difference, E − EF ≈ +0.547 eV, which is found identical for all the considered N configurations in the moiré (FCC and BRIDGE are not reported).

applied bias voltage. By tuning this voltage, one can select the states involved in the tunneling process. As a result, by properly setting this parameter, one can actually adjust the observed contrast in order to emphasize the contribution of a specific electronic state. In our particular case, we would like to yield the maximum contrast between nitrogen and the surrounding C atoms. We therefore project the DOS on the N site and its neighboring C site (Fig. 4.5, left), and then we seek the energy at which the difference between them is the largest (Fig. 4.5, right). Interestingly, we find that this condition is fulfilled at E ≈ EF + 0.547 eV (i.e., an unoccupied state) for all the considered positions of N in the moiré (ATOP and HCP are for instance shown in 4.5). By integrating the LDOS from E = 0 to this value we 72

actually manage to simulate STM images, associable to high-resolution constant-current measurements at applied bias voltage ≈ 0.5 − 0.6 eV, with a contrast that makes the N dopant well distinguishable3 (see Fig. 4.6a). The N signature can be identified as the typical bright triangular spot extended to the three nearest C neighbors of the N dopant. These result is in very good agreement with the reported experimental and theoretical studies of N-doped graphene deposited on SiO2 [25], SiC(0001) [27, 159], or Cu(111) [163] and, moreover, it is confirmed by recent experimental images of N-doped graphene on Ir(111) (see Fig. 4.6b). We further note that this triangular feature extends to a few lattice spacings from the N, also where the honeycomb lattice of graphene is recovered, and that its spatial orientation strongly depends on which one of the two graphene sublattices is occupied by the N dopant, as actually reported by Zabet-Khosousi et al [163] in their recent observations of N-doped graphene on Cu(111) (see Fig. 4.7a). Interestingly, in addition to this, an overall sublattice asymmetry is in general observed all over the honeycomb lattice: more precisely, the areas surrounding the graphene atoms which belong to the sublattice not occupied by the N atom seem to present a slightly brighter triangular shape with respect to the others. Lastly, on a larger scale, we note that the graphene moiré pattern is actually resolved, and that the typical bias-dependence of the imaging contrast found for the defect-free graphene on Ir(111) (see Chapter 3.3.5) is still observable (see Fig. 4.7b).

3

It has been shown that a single graphitic N dopant incorporated in graphene grown on a C-terminated 6H-SiC(000¯1) surface induces resonances at positive energies, ≈ 0.95 eV above EF in the experimental scanning tunneling spectroscopy (STS) spectrum, but ≈ 0.8 eV above EF in the calculated LDOS [28]. A similar difference between theory and experiments is observed for N-doped graphene grown on a copper foil [25]. This suggests that the N resonance for our system, which is found at ≈ 0.5 eV above EF in our LDOS calculations, might be observed at slightly higher energies (≈ 0.6 − 0.7 eV) in a real STS experiment.

73

(a)

(b) Figure 4.6 (a) Simulated constant-current (cc-) STM images of a substitutional N dopant located at four high-symmetry positions in the moiré of graphene/Ir(111). The images are generated at applied bias voltage of 0.547 V, which corresponds to probe an electronic state in the conduction band which is found to provide the maximum contrast between N and the surrounding C atoms (see Fig. 4.5). The apparent out-of-plane height is represented by a grey scale contrast, ranging from 0 (black) to 60 pm (white) when N lies at ATOP position, and from 0 to 40 pm in all the other cases. (b) Experimental atomically-resolved cc-STM image (0.1 V, 500 pA) of a N dopant in graphene deposited on Ir(111) by means of CVD. The image has been generated using an Ir-coated tip obtained by controlled contact with the Ir substrate. Courtesy of D. Smith.

74

[pm]

(a)

(b) Figure 4.7 (a) On the left, two simulated constant-current (cc-) STM images of a substitutional N dopant occupying opposite sublattices of graphene/Ir(111). The simulated applied bias voltage is 0.547 V. Red and blue in the overlapped balland-stick model indicate graphene atoms on different sublattices. On the right, an experimental cc-STM image (7 × 7 nm2 , 1 V, 1 nA) of N-doped graphene on Cu(111), showing three dopants on different sublattices, marked by red and blue triangles (adapted from [163]). (b) On the left, a 5×5 repetition of the simulated ccSTM image for our supercell (black rhombus) along the directions of its two lattice vectors. The simulated bias voltage is −0.1 V on top and −1 V on bottom. On the right, the contrast inversion observed experimentally for pristine graphene/Ir(111) [134].

75

Chapter 5 Chemical identification of N-dopants with AFM As discussed in the introduction, the chemical nature of a specific surface atom can be probed in frequency-modulated non-contact AFM (fm-AFM) by measuring the curve ∆f (z) of the oscillating-tip-frequency shift (∆f ) as a function of the vertical tip-sample distance (z). The depth of the well in the resulting Lennard-Jones- or Morse-like curve depends on the sum of the short-range chemical interactions and the long-range electrostatic and van der Waals interactions acting between the tip and the sample: the larger the attractive contributions, the deeper the well. By comparing the spectra obtained by probing nitrogen and carbon atoms at various sites in the graphene moiré, we can actually address the two questions posed in the introduction: 1. Is it possible to discriminate nitrogen and carbon atoms in our system? 2. How does the interaction with the substrate affect the outcome of the identification procedure?

5.1

Theoretical modeling of non-contact AFM

The first step to simulate an AFM is to include the model of a tip in our supercell. Despite the several complex models adopted in litera76

Figure 5.1 Schematic representation of the four-atom tetrahedral Ir tip model used to simulate the AFM ∆f (z) curves. The vertical distance z between of the tip from the graphene/Ir(111) surface is defined as the distance between the top-most Ir layer in the Ir(111) substrate and the top-most Ir atoms in the tip model. The scale of the tip is intentionally magnified. The graphene atomic coordinates have been artificially manipulated in order to enhance the visibility of the moiré corrugated structure.

ture, the large size of the graphene moiré cell on Ir(111) and the huge computational resources required to perform this kind of simulations oblige us to use a very simple approximation for the AFM tip apex, which consists of four Ir atoms arranged in a tetrahedron oriented as shown in Fig. 5.1. This model is assumed to be a good approximation of an Ir-coated tip1 such as those typically used to image epitaxial graphene on Ir(111) with atomic resolution [33]. From a first geometry optimization of the tip structure we find a face-apex distance of ∼ 2 Å. Considering that in a real ∆f (z) measurement the tip is moved within a range of ∼ 2−10 Å above the surface, and that a ∼ 15 Å vacuum layer 1

In a practice, it can be realized by controlled contact with the Ir surface before the actual scanning of the sample, and its metallic nature can be confirmed by conductance spectroscopy on the Ir surface [33].

77

is still required in the calculations to isolate slab images in the vertical direction, we decide to extend the size of the supercell to ∼ 50 Å. Although our DFT calculations cannot directly simulate an oscillating tip, we can predict the experimental data from a real fm-AFM via calculation of the interaction energy E(z) between the tip and the surface. We define this energy as E(z) = Etip/S (z) − Etip − ES ,

(5.1)

where ES is the total energy of the isolated (doped) graphene/Ir(111) system, Etip is the total energy of the isolated model tip, and Etip/S (z) is the total energy of the overall system computed for a distance z between the tip and the sample. In order to make our data directly comparable to experimental results, we then fit the obtained E(z) points with a Morse potential & ' c1 + c2 1 − e−c3 (z−c4 ) , (5.2)

which has been shown to be the most suitable to describe the tipsample interaction [133, 164]. The resulting curve is further converted into force F (z) acting between the tip and the probed atoms, by simply making the first derivative (F (z) = −∂E(z)/∂z), and eventually used to predict the correspondent frequency shift ∆f (z) using the following relation, derived by F.J. Giessibl [165]: ˆ 1 f0 u ∆f (z) = − F [z + A (1 + u)] √ du, (5.3) πAk −1 1 − u2 where k is the spring constant of the oscillating sensor in the AFM, f0 is its unperturbed resonant frequency and A is the amplitude of oscillation.2 These three parameters are particular to the specific setup in use and, hence, they represent the main link between our calculations and the experiments. In this work we use k = 1800 N/m, f0 = 24 kHz andA = 2 Å. It has been demonstrated that the consideration of a dynamic environment in the framework of microscopy simulations can be crucial for the correct reproduction of the imaging contrast in some cases [166]. 2

This formula is valid under the assumption that the frequency shift ∆f is much smaller than the unperturbed resonant frequency f0 .

78

Therefore, although for the sake of computational efficiency the coordinates of most atoms in the system are kept fixed during the calculations, we allow the atom at the tip apex, the probed atom in the graphene layer and its three nearest-neighbors to relax, until the forces exerted on them are below 10−2 eV/Å. We point out that for this reason, since the tip-sample distance z needs to be unambiguously set out during the calculations, we define it as the distance between the top-most Ir(111) layer and the top-most Ir atoms in the model tip (see Fig. 5.1). When comparing experimental data with our results, one should therefore be careful to take an additional ∼ 5.6 Å z-offset into account, corresponding to the sum of the graphene-Ir(111) maximal separation (∼ 3.6 Å; see Table 4.1) and the face-apex distance in the tip (∼ 2 Å). For the generation of the E(z) curves, we stick to the following protocol: 1. we move the tip at a particular lateral position (x, y) on top of the surface; 2. we place it close to the surface at a small distance (2 Å separating the tip apex and the probed atom, corresponding to 7.3 Å under our definition of z) and we optimize the system as described above, so as to extract the value of Etip/S (z) to be used in Eq. 5.1; 3. the same calculations are performed while moving the tip away from the surface in small steps of 0.1 Å (as long as the tip is located at relatively short distances) or 0.5 Å (when it is already at larger distances), up to z = 15.3 Å. In this way the E(z) curve is reproduced with denser data points around the expected minimum. In order to simulate a realistic tip movement, we always start a new calculation assuming the optimized geometry obtained in the previous step. We point out that all the calculations are performed adopting the same computational details used in the previous chapters, except for the number of k-points used to sample the Brillouin zone: due to the huge computational requirements, we reduce it to the Γ-point only. Considering the large size of our supercell, we assume the use a single k-point does not affect significantly the E(z) data.

5.2

Results and discussion

We begin our investigation by placing the tip above a carbon atom located at an ATOP or HCP site in the pristine structure and comparing the obtained ∆f (z) spectrum with the one resulting from probing a nitrogen atom located at the correspondent site in the doped system.3 The E(z), F (z) and ∆f (z) curves calculated for the two elements in an ATOP or HCP high-symmetry position are reported in Fig. 5.2 and characterized in Table 5.1a-b. We first note that the depth of the well in the E(z) curves is in the order of 1 − 2 eV, already suggesting that the nature of the interaction between the tip and the sample is mainly electrostatic and chemical rather than dispersive. In addition to this, the position of the well in the curves is almost the same for carbon and nitrogen, and only depends on their position in the moiré (see below). However, the most remarkable result is that the well is always less deep for nitrogen than for carbon, regardless of the position of these atoms in the moiré, with a significant difference in frequency shift in the order of ∼ 10 Hz. We therefore conclude that a nitrogen dopant in graphene/Ir(111) is clearly distinguishable from carbons using AFM, exhibiting a site-independent chemical contrast.

5.2.1

Influence of the substrate

The previous observations indicate that the influence of the substrate on the outcome of the identification procedure is basically not relevant for our graphene/ Ir(111) system. However, in order to assess the generality of these results, it is interesting to dig a little deeper into the role played by the substrate in the chemical identification process. We thus focus on the comparison between the calculated curves associated to the same element, either carbon or nitrogen, located at a different position in the moiré. From Fig. 5.3 and Table 5.1c we observe that the differences in the position of the well for both elements roughly reflect the ∼ 34 pm corrugation of graphene adsorbed on Ir(111) (see Table 3.2). As a result of this corrugation, indeed, the tip starts to be repulsed by the surface at a larger distance z (from the top-most 3

In these calculations we refer to the optimized geometry for the pristine and the doped systems reported in Table 3.2 and Table 4.1, respectively.

80

(a)

(d) 0.5

N @ ATOP C @ ATOP

Interaction energy (eV)

Interaction energy (eV)

0.5 0.0 -0.5 -1.0 -1.5 -2.0

N @ HCP C @ HCP

0.0 -0.5 -1.0 -1.5 -2.0

7.0

7.5

8.0

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9.0

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(b)

(e)

0.4

N @ ATOP C @ ATOP

0.2

0.4

9.5

N @ HCP C @ HCP

-0.4 -0.6 -0.8

-0.2 -0.4 -0.6 -0.8

-1.0

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(f) N @ ATOP C @ ATOP

8.5

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5.0 Frequency shift (Hz)

0.0 -5.0 -10.0 -15.0 -20.0 -25.0 -30.0 -35.0

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0.0 -5.0 -10.0 -15.0 -20.0 -25.0 -30.0

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9.5

Distance (Å)

7.0

7.5

8.0

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9.5

Distance (Å)

Figure 5.2 (From top to bottom) Calculated interaction energy E(z), force F (z) and frequency shift ∆f (z) as a function of the distance z between a 4-Ir-atom tetrahedral model of an AFM tip and a (N-doped) graphene/Ir(111). In particular, a comparison between the nitrogen- and carbon-related curves is shown for (a-c) ATOP and (d-f) HCP high symmetry positions in the moiré. Each point in (a) and (d) represents the result of a single DFT-D3(BJ) calculation, normalized according to Eq. 5.1. The points have been fitted with a Morse potential and the resulting curve has been manipulated to extract the corresponding force and frequency shift. See text for further details.

81

Table 5.1 Main features of the AFM spectra calculated by probing carbon or nitrogen atoms located at ATOP or HCP high-symmetry positions in the moiré of graphene/Ir(111). (a) Absolute coordinates of the minima for the E(z) curves and the correspondent ∆f (z) curves. (b) Differences in position and depth of the well when comparing the curves for different elements located at the same site (see Fig. 5.2). (c) Differences in position and depth of the well when comparing the curves for the same element located at different sites (see Fig. 5.3). All the values have been estimated while allowing the tip apex, the probed atom in the graphene layer and its three nearest-neighbors to relax during DFT-D3(BJ) calculations. For further details we refer to the text.

(a)

(E)

z0

(Å)

E0 (eV)

(∆f )

z0

(Å)

∆f0 (Hz)

NATOP

7.907

-1.158

8.212

-19.2

CATOP

7.703

-1.980

8.116

-31.2

NHCP

7.585

-1.265

7.813

-22.9

CHCP

7.386

-2.030

7.772

-31.7

∆z (E) (pm)

∆E (meV)

∆z (∆f ) (pm)

∆(∆f ) (Hz)

20.4

822.1

9.6

12.0

19.9

764.5

4.2

8.8

∆z (E) (pm)

∆E (meV)

∆z (∆f ) (pm)

∆(∆f ) (Hz)

31.8

50.0

34.5

0.5

32.2

107.6

39.9

3.7

(b) NATOP − CATOP NHCP − CHCP (c) CATOP − CHCP

NATOP − NHCP

82

(a)

(d) C @ ATOP C @ HCP

N @ ATOP N @ HCP

0.5 Interaction energy (eV)

Interaction energy (eV)

0.5 0.0 -0.5 -1.0 -1.5

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Force (eV/Å)

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(f)

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-5.0 -10.0 -15.0 -20.0 -25.0

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8.0

8.5

9.0

7.0

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7.5

8.0

8.5

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Figure 5.3 Same curves of Fig. 5.2, but plotted so as to compare (a-c) carbon or (d-f) nitrogen located at two different high-symmetry positions in the moiré. Each point in (a) and (d) represents the result of a single DFT-D3(BJ) calculation, normalized according to Eq. 5.1. The points have been fitted with a Morse potential and the resulting curve has been manipulated to extract the corresponding force and frequency shift. See text for further details.

83

Ir(111) layer!) when it is approaching an ATOP site of the moiré rather than a HCP one. Interestingly, a relevant variation in depth of the well is only observed for nitrogen, despite the difference is not as pronounced as that resulting from the comparison between the N and C spectra at the same site (see Table 5.1b). Such a discrepancy might be triggered by two possible mechanisms. One of these is the difference in the vdW interactions between the tip and the sample induced by the different curvature of the graphene layer in the ATOP and HCP regions. However, the long-range character of these forces, along with the relatively small vdW interactions associated to single Ir-C or Ir-N dimers (∼ 2 − 3 meV), suggest that these contributions should be negligible with respect to the electrostatic and chemical local ones. And besides, a DFT investigation aiming at unambiguously assess the role played by this mechanism in the chemical identification process would be actually workable only by re-running all the calculations (including the cumbersome geometry optimization) from scratch without adopting the DFT-D3(BJ) or any other approach for including long-range interactions in DFT. Due to the huge computational effort which would be required, we choose to put this strategy aside and focus instead on the only possible alternative explanation: an influence of the substrate. A fast and simple way to check the impact of the substrate in our simulated AFM spectra is to literally “remove” it and re-run the calculations described in this chapter, while probing only the fictitious free-standing corrugated graphene sheet. As a result, if the substrate is actually the main cause for the discrepancy observed for N, we expect the difference in depth of the well to vanish. A shortcoming of this strategy is that we need to keep the coordinates of all the atoms in the system fixed during the calculations: the absence of the substrate beneath the graphene would indeed fake the eventual relaxed geometry. But nonetheless, we find that the error made on the AFM spectra when fixing the atomic coordinates is not relevant for our purposes: the curves shown in Fig. 5.4 actually demonstrate that performing a static calculation only results in a stronger repulsion at short tip-sample distances and in a slight upwards shift of the minimum, whose amount (∆E < 100 meV) is still almost the same in all the studied cases. 84

1.5

0

1 Energy (eV)

Energy (eV)

0.5

-0.5 -1 C @ ATOP (rel)

-1.5

0.5 0 -0.5 N @ ATOP (rel)

-1

C @ ATOP (stat) -2

7

8

9

10

11

12

13

14

15

N @ ATOP (stat) -1.5

16

7

8

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1

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-2

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12

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8

9

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16

0 -0.5 N @ HCP (rel)

-1

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13

z (Å)

Energy (eV)

Energy (eV)

z (Å)

13

14

15

N @ HCP (stat) -1.5

16

z (Å)

7

8

9

10

11

12

13

14

15

16

z (Å)

Figure 5.4 Calculated interaction energies E(z) as a function of the vertical distance z between a 4-Ir-atom tetrahedral model of AFM tip and a (N-doped) graphene/Ir(111). The results obtained by fixing the coordinates of all the atoms in the system (blue crosses) are shown in comparison with those obtained by allowing the tip apex, the probed atom in the graphene layer and its three nearest-neighbors to relax (red dots). Nitrogen (right) and carbon (left) lying at ATOP (top) and HCP (bottom) high-symmetry positions in the moiré are considered. In each case the energies have been obtained from DFT-D3(BJ) calculations and are normalized according to Eq. 5.1. The points that appear slightly departed from the Morse behavior correspond to calculations that have not managed to converge to the global energy minimum. See text for further details.

After performing these new series of calculations with a fixed geometry, we compare the results with those obtained for the global graphene/Ir(111) structure (see Fig. 5.5). Remarkably, we find that the substrate removal does not affect the depth of the carbon-related curves, whereas it actually smooths out the difference originally found for nitrogen. A further insight about the substrate impact on the dopant identification comes from the charge density redistribution occurring when the tip approaches the surface. In Fig. 5.6 we plot the charge density difference ∆ρ = ρtip/S − ρtip − ρS calculated for nitrogen at ATOP and HCP sites, where ρS is the charge density of an isolated (N-doped) graphene/Ir(111) system, ρtip is the charge density of the isolated model 85

Figure 5.5 Calculated frequency shift as a function of the distance between a 4-Ir-atom tetrahedral model of AFM tip and a (N-doped) graphene/Ir(111). The curves obtained by probing (a) carbon and (b) nitrogen atoms located at ATOP or HCP sites are shown, either including or not including the Ir(111) substrate during the calculations. See text for further details.

tip, and ρtip/S is the charge density of the overall system, with the tip at its smallest distance from the surface. A careful comparison indicates that the observed discrepancy in the AFM spectra might be related to the strong hybridization which occurs between the Ir(111) substrate and the nitrogen atom at the HCP site. This hybridization, extensively discussed in the previous chapters, here leads to the formation of a chemical bond which extends to the tip atoms lying close to the surface, boosting the attractive contributions in the tip-sample interaction and, hence, yielding a deeper well.

5.2.2

Towards the experiments

In order to qualitatively check the reliability of our simulations, we broadly compare them with the results of a real experiment performed4 on a very similar system, namely N-doped graphene/Cu(111), using a Cu-coated AFM tip obtained by controlled contact with the substrate. By measuring a stack of constant-height fm-AFM images at different height z, and extracting the depth of the well (∆fmin ) in the ∆f (z) 4

The experiment was carried out by N.J. van der Heijden, D. Smith and I. Swart at the Condensed Matter and Interfaces (CMI) group, Debye Institute, Utrecht (NL).

86

Figure 5.6 Charge density difference ∆ρ = ρtip/S − ρtip − ρS , plotted in (e/Å3 ), for nitrogen at (a) ATOP or (b) HCP sites in the moiré of graphene/Ir(111). ρS is the charge density of the isolated (N-doped) graphene/Ir(111) system, ρtip is the charge density of the isolated model tip, and ρtip/S is the charge density of the overall system, with the tip at small distance from the surface. The color scale ranges from −0.013 e/Å3 (black; loss in electron density) to 0.013 e/Å3 (white; gain in electron density), passing through zero (gray). The charge densities have been calculated after allowing the tip apex, the probed atom in the graphene layer and its three nearest-neighbors to relax. Iridium, carbon and nitrogen atoms are marked in green, brown and blue, respectively. The yellow arrow highlights the relatively stronger chemical bond which is formed between nitrogen and Ir(111) at the HCP site.

curves associated to every pixel, it has been actually possible to chemically map the surface (see Fig. 5.7b). Remarkably, in good qualitative agreement with our calculations, we note that the well in the ∆f (z) spectrum measured above a nitrogen site is less deep than the well found above a carbon site, regardless of the spatial separation between the two elements in the imaged area. Moreover, the contrast observed here in the 2D map for the position of the well (Fig. 5.7a) likewise reflects the moiré corrugation of ∼ 35 pm observed in STM for graphene adsorbed on Cu(111) [167]. 87

Figure 5.7 (a) zmin and (b) ∆fmin maps of CVD grown N-doped graphene on Cu(111). The images were obtained from a stack of constant-height fm-AFM images at different height with gap voltage 0 V. From this 3D ∆f data cube a ∆f (z) spectrum could be obtained for each (x, y) lateral position. From these ∆f (z) curves a minimum was fitted and the coordinates zmin and ∆fmin of the minimum are mapped. The value z = 0 corresponds to the STM feedback set-point 50 pA at bias 1.0 V. See the Introduction for more experimental details. Courtesy of N.J. van der Heijden and D. Smith.

In view of a quantitative comparison between our simulations and measured data on N-doped graphene/Ir(111), in the remainder of this chapter we briefly point out the possible sources of discrepancy, both from the computational and experimental viewpoints: • the values we have used for A, f0 and k in the Giessibl formula (Eq. 5.3) were just rough approximations. • by definition our DFT calculations simulate only the ground-state electronic properties of the system at 0 K and in perfect vacuum. • the size of the Ir cluster which models the tip might be too small to mimic all the real short- and long-range interactions between the sample and a real AFM tip. A quick simple way to estimate the error in depth of the well due, for instance, to the addition of a single Ir atom to the cluster is to calculate the variation induced by this additional atom in the energy (Etip ) of the isolated tip only. Such variation, once accounted for in Eq. 5.1, will indeed affect the normalization of the E(z) curves, giving an idea of the 88

overall error without the need for a time-consuming re-calculation of the whole spectrum. • our approach, which amounts to a z-scan for a given (x, y) lateral position above the surface, might be different from the experimental one, where a (x, y)-scan is usually carried out for a given z, in order to obtain 2D maps like those in Fig. 5.7. A discrepancy between our results and the experiments might therefore indicate that dissipation effects, such as friction or hysteresis, play a significant role during the acquisition of the experimental images, which indeed would be dependent on the actual tip trajectory. In this case a possible way to gain more insights is to simulate the AFM spectra both by approaching and then retreating the tip from the surface. If dissipation is actually relevant, then the finite area enclosed between the two branches of the curve would give an estimate of its amount. • it is known that the use of a reactive (e.g., Ir) or a non-reactive (e.g., CO) tip apex can significantly affect the final results [33, 132]. However, usually there is no way to assess the effective structure of the tip apex during the ∆f (z) measurement. • there is always an uncertainty in both the z-coordinate of the piezo scanner and the resonance frequency of the oscillating sensor during the measurement [133], especially during the timeconsuming acquisition of 2D maps such as those in Fig. 5.7. • from a numerical point of view, the use of a single k-point in our calculations might marginally affect the final result. • the intrinsic numerical accuracy of the Giessibl formula might also play a small part [165].

89

Conclusions and perspectives In this work we present a thorough DFT study of N-doped graphene on Ir(111), which mainly aims at exploring the possibility of carrying out chemical mapping with AFM. In the first part of the work we systematically reproduce the state-ofthe-art model for graphene/Ir(111), from the typical moiré corrugated structure to the expected physisorption with chemical modulation, as well as the observed bias-dependence of the STM imaging contrast. In order to compensate the well known shortcoming of standard DFT calculations regarding the contribution of the van der Waals interactions, we use the recent DFT-D3(BJ) correction scheme, showing for the first time that this enables to accurately reproduce the structural and electronic properties of a graphene-metal system. Against the background of this sound basis we undertake the study of the N-doped system, aiming at unraveling the impact of doping on the geometric and electronic structure of graphene, also with respect to different high-symmetry positions occupied by the dopant in the moiré. We find that the electronic properties of N-doped graphene/Ir(111) are substantially similar to those reported for a free-standing graphene: the usual n-type doping and sp2 -rehybridization involving the three neighboring carbon atoms are still observed, even leading to the typical triangular N-signature in the simulated STM. Slight differences are found in the local geometry and DOS when the N dopant is located at valley sites in the moiré, suggesting that larger modifications might occur when growing graphene on a more strongly interacting metal such as Ru(000¯1) or Ni(111). On the practical side, our calculations predict that the most stable site for a N dopant, over the four considered in the moiré of graphene/Ir(111), is the top site, and our LDOS analysis suggests that an optimal N-C imaging contrast in constant-current STM 90

might be achieved in principle by setting a bias voltage of ≈ 0.5−0.6 V, because it provides a resonance with the most distinctive N electronic state. Based on the interaction energy between a model of Ir-coated tip and our system, we have indirectly simulated the frequency shift spectra ∆f (z) measurable with a non-contact frequency-modulated AFM. By probing N and C atoms located at top and valley sites in the graphene moiré we demonstrate that it is possible to chemically identify a N dopant in graphene using AFM. Interestingly, the chemical contrast is shown to be site-independent, with a small influence of the metal substrate only observed when probing nitrogen. We associate this effect to the hybridization occurring at the interface between N pz -orbital and metal d-orbitals. As occasionally emerged from the discussion, our simulations might be reinforced or improved on several accounts. For instance, in order to gain a more complete understanding of the electronic structure of N-doped graphene on Ir(111), it would be interesting to calculate its precise band structure: once properly unfolded, this would provide additional information about the effects of doping in the corrugated graphene, corroborating our DOS analysis and our understanding of the observed STM contrast. Furthermore, in order to improve our theoretical model for AFM, it would be advisable to pinpoint the real impact of the tip size and shape on the calculated AFM spectra. In addition, it would be interesting to adopt a recently developed molecular approach to simulate dynamic models for the tip, which has been shown to realistically reproduce, for instance, the effects of using a functionalized tip such as a CO-terminated one [37, 166]. At the same time, the simulations performed in this work may serve as a valuable input for further computational investigations in several directions. Just as one example, the same analysis carried out here for a single graphitic N dopant might be extended, with a relatively small effort, to more complex doping configurations (e.g., pyridinic, nitrilic), and it would be just as easy to characterize different dopant species such as boron or sulfur. In fact, while the effects of such chemical modifications on the free-standing graphene properties are well known, the investigation of their interplay with a metal supported graphene is still in its infancy. 91

We are fully aware that, ultimately, only experiments will be able to assert the real scientific merit of our simulations. To date, we can only claim an excellent qualitative agreement with the experiments about Ndoped graphene/Cu(111) recently performed by the Condensed Matter and Interfaces group at the Debye Institute in Utrecht. However, similar measurements on N-doped graphene/Ir(111) are in progress within the same group. We believe that, if the chemical identification capabilities demonstrated in this work will be actually reflected in that context, then a substantial contribution would be given on behalf of the emergence of this identification technique in the AFM field of research, with important repercussions not only in the graphene domain, but also in all the scientific and technological areas in which important functional properties are controlled by the short-range ordering and chemical nature of defects, dopants, adsorbates or individual atoms.

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Appendix A Long-range interactions in DFT One of the common features in the plethora of density functionals available for DFT calculations is that while the exchange energy of a collection of electrons can, in principle, be determined exactly, the correlation energy can only be approximated. Therefore it is understandable to be concerned about the reliability of DFT for those physical properties that depend significantly on electron correlation. In particular the long-range interactions between atoms and molecules [168] need to be carefully dealt with.1 The connection between electron correlation and such long-range interatomic forces was originally pointed out by F.W. London in the 1930s [83]. For this reason one usually refers to these forces as London dispersion interactions. He was the first to realize that although the time-average electron density around an atom or non-polar molecule has no dipole moment, electron oscillations lead to deformations of the density resulting in a transient dipole moment. This temporary dipole moment can in turn induce one on other atoms or molecules by distorting their electron density, leading to a net attractive interaction. 1

Dispersion interactions (also known as van der Waals interactions or London forces) play an crucial role in the everyday life. For instance, the global infrastructures that deliver and use fuels such as diesel or gasoline rely on the fact that they are liquids. The fact that non-polar molecules such as hexane are stable in a liquid phase is a proof of the net attractive forces acting upon hexane molecules, arising directly as a result of electron correlation [83].

93

Figure A.1 Potential energy curves for the Kr2 (left) and the benzene dimer (right, D6h symmetry) with two different density functional approximations in comparison with accurate CCSD(T) reference data. Adapted from [108].

London’s main contribution was to provide a general expression for the long-range interaction between two spherically symmetric atoms at a distance R: C6 Vdispersion = − 6 (A.1) R where C6 includes various physical constants.2

A.1

A failure of standard DFT: the dispersion problem

For several years it is known that standard density functionals, such as those mentioned in section 2.1.3, are not able to describe London dispersion interactions properly. Originally it was noticed for raregas dimers [169] but recently it turned out also for base pair stacking [170] or N2 dimers [171]. In Fig. A.1 we report two examples, for Kr2 and benzene dimers, that clearly illustrate the problem. Here B3LYP [103] (Burke, three-parameter, Lee-Yang-Parr) and PBE [106] 2

Some of these physical constants refer to the dispersion of the refractive index of the materials included in the system; hence the labelling of these interactions as dispersion interactions.

94

(Perdew-Burke-Ernzerhof) are widely used versions of hybrid and GGA exchange-correlation functional, respectively, whereas CCSD(T) is a very accurate coupled cluster (non-empirical) numerical technique used to describe many-body systems, whose results are used as a reference for a benchmarking [172, 173]. For both dimers, B3LYP is only repulsive, yielding no binding at all. The PBE functional provides a minimum for Kr2 , although with an unrealistically small interaction energy and at a too large distance. For the benzene dimer, both functionals yield no minimum. Moreover for both functionals the interaction potential is exponentially decaying and almost zero for R > 6 Å when looking at Kr2 , which is very strongly dominated by dispersion interactions, whereas the decay of the reference data is quite slow, so that the reference potential is significantly bound up to about 8 Å. Recently much effort has been devoted to developing approaches that could accurately model the London dispersion interactions within density functional theory. It has now become glaring that the involvement of this interactions in physical and chemical theoretical simulations is crucial for achieving realistic results. One important feature of the current dispersion-corrected density functionals is the incorporation of empirical terms. As illustrated in Figure A.2a the main reason for this is the attempt to merge in a smooth and reasonable fashion the short- and long-range asymptotic regimes that are actually well understood. On the one hand, at short correlation length the standard functionals work quite well due to the high sensitivity to electron density fluctuations. On the other hand, at large distances the asymptotic behavior should approach the semiclassical limit given in Equation (A.1).

95

(a)

(b) Figure A.2 (a) Scheme of the correlation and dispersion problems on different electron correlation length scales. (b) Overview of the most commonly used approaches that include dispersion corrections in density functional theory. EKS and VKS are the pure Kohn-Sham energy and potential, respectively. Adapted from [108].

A.2

Classification of dispersion corrections

In Figure A.2b the three most widely used3 approaches to dispersion corrections in DFT are outlined. They include the non-local vdW-DFs [174–176], “pure” density functionals (DFs), which are highly parame3

It is worth to point out that all the approaches cited in this context work with standard or slightly modified density functionals (DFs). There is a whole different class including virtual orbital-dependent methods [i.e., random phase approximation (RPA)] or fragment-based methods [e.g., DFT–symmetry adapted perturbation theory (SAPT)] that could also be dealt with. However, due to their current lack of a complete generalization, and due to the still provisional development stage, these class has been excluded from the discussion.

96

terized forms of standard meta-hybrid approximations (e.g., the M0XX family [177]); the DFT-D methods, which provide an atom pairwise sum over −C6 R−6 potentials [110, 178, 179]; finally, the atom-centered oneelectron potentials (1ePOT), e.g., DCACP [180] or the local variants LAP [181] and DCP[182]. A comprehensive and comparative discussion on these methods can be found in some recent review papers, e.g., the one by S. Grimme [108]. In the following paragraphs only the vdW-DFs and the DFT-D approaches will be illustrated in detail, pointing out the physical concepts behind them, advantages and disadvantages as well as some details regarding their implementation in VASP (Vienna ab initio Simulation Package [113]), which is the DFT package used in this thesis (see section 2.1.3). The other methods will be only briefly introduced in the end of this section. The vdW-DF method The vdW-DF method is a non-empirical method to include dispersion corrections in a DFT calculation, only based on the electron density of the system. In all the currently used versions it performs a super-molecular calculation of the total energy of the complex and the fragments to obtain the actual interaction energy, employing the following approximation for the total exchangecorrelation energy: Exc = ExLDA/GGA + EcLDA/GGA + EcNL

(A.2)

Here the standard exchange and correlation components of a local (i.e., LDA) or semilocal (i.e., GGA) approximation to the density functional account for the short-range interactions, whereas EcNL represents the additional correlation energy due to nonlocal dispersion interactions. This term takes the form of a double space integral ˆ ˆ 1 NL Ec = ρ(r)φ(r, r′ )ρ(r′ ) dr dr′ (A.3) 2 where ρ is the electron density, while r and r′ are electron coordinates. φ(r, r′ ) is a nonlocal correlation kernel which depends on two electron coordinates simultaneously and it constitutes the discriminant term among the various vdW-DF versions existing in literature. From a physical point of view this kernel is based on local approximations to 97

the (averaged) dipole polarizability at frequency ω [i.e., α(r, ω)], which when integrated yields the total polarizability α ˆ α(ω) = α(r, ω) dr. (A.4) By means of the Casimir-Polder relationship (see section A.3) a link between the polarizability (known at all imaginary frequencies) and the long-range term of the dispersion energy can be established, yielding the following expression for the dispersion coefficient C6AB associated to two fragments A and B: ˆ 3 ∞ AB C6 = αA (iω)αB (iω) dω. (A.5) π 0 This integral also provides the physical basis of DFT-D approaches to the dispersion problem (see below). It is worth to point out that the polarizability is strongly connected to the electron density through the dielectric function ϵ(ω) = 1+χ(ω) = 1 + 4πnα(ω), where χ is the electric susceptibility and n is the electron numerical density. A proper choice of a model to describe the dielectric function is thus fundamental and it constitutes another difference between various vdW-DF reported in literature. At the state of the art the biggest advantage of vdW-DF methods over all other approaches mentioned in A.2 is that they apply an actual “a priori” correction to the charge density computed within the KohnSham self-consistent problem. This also means that charge-transfer (hence the atomic oxidation state) dependence of dispersion is automatically included in a physically reasonable way. These factors are not accounted for in the class of DFT-D methods. However, it is still not completely clear whether double-counting of correlations at short range are present in the vdW-DFs algorithms. The method is currently implemented in VASP thanks to the efforts of J. Klimeš [176], who exploited an algorithm by G. Roman-Perez and J. M. Soler [183] which transforms the double integral in the real space to a single integral in the reciprocal space, hence reducing the computational effort. A set of optimized parameters for different exchangecorrelation functionals is available (see the VASP manual [151]).

98

The DFT-D method The DFT-D (also known as DFT-disp) refers to a class of semi-empirical methods that enable one to correctly describe the van der Waals interactions by pragmatically adding a correction to the conventional Kohn-Sham DFT energy, such that EDFT−D = EDFT + Edisp . The basic idea is to treat the dispersion interactions semi-classically and to combine the resulting potential with a quantum chemical approach. This idea dates back to the standard Hartree-Fock theory developed in the 1970s [184, 185]. It has been discarded for 30 years and only recently it has been reconsidered in the context of density functional theory. Since the first developed version [186] many modifications have been proposed. In chronological order, the most widely used versions have been the DFT-D1 [178] (2004) and its updated versions DFT-D2 [187] (2006) and DFT-D3 [110] (2010), originally proposed by S. Grimme. A general overview of these approaches4 is given below. The basis of DFT-D methods is an atom pairwise additive treatment of the dispersion energy, whose most general form is Edisp = −

!

!

AB n=6, 8, 10

sn

CnAB fdamp, n (RAB ) n RAB

(A.6)

where the sum is over all atom pairs in the system,56 RAB is the internuclear distance between paired atoms, CnAB denotes the averaged (isotropic) nth-order term to the approximated dispersion coefficient (n = 6, 8, 10, ...) for atom pair AB, and sn are global (DF-dependent) scaling factors, usually used to optimize the correction to the repulsive 4

It is worth to mention also the method proposed by A. Tkatchenko and M. Scheffler [188] (known as DFT-TS), which represents one of the most efficient DFT-D related approaches, where the expression for the dispersion energy is formally identical to that of DFT-D2, but the C6 coefficients are computed specifically for each atom pair in a system, so as to improve the system transferability. In addition, both the C6 coefficients and the damping function are charge density-dependent.

5

For DFT simulations relying on periodic boundary conditions (e.g., VASP . .Nat .Nat .′ simulations) the sum AB = 12 i=1 j=1 L is over all Nat atoms within the unit cell and over all translations L = (l1 , l2 , l3 ) of the unit cell (the prime indicates that i ̸= j for L = 0). In such cases, RAB = Ri,j,L represents the distance between atom i located in the reference cell L = 0 and atom j in the cell L.

6

For extensions to include three-body non-additive dispersion effects, see [110, 189].

99

behavior of the chosen DF [178]. The higher-ranked multipole terms Cn>6 contribute significantly in the short- and mid-range regions and they interfere rather strongly with the (short-ranged) DF description of the electron correlations. For this reason they can be used to adapt the potential specifically to the chosen DF in this mid-range regions. The question is how many high-order terms are required to achieve the best results. Indeed, recent insights suggest that C6 alone may not be sufficient to describe medium/short-range dispersion [110, 190], and at least C8 contributions should be taken into account (as in DFT-D3 [110], described below). A very important role in Eq. (A.6) is played by the damping function fdamp . It avoids divergent behavior for small R and double-counting effects of correlation at intermediate distances, setting the range of the dispersion correction. Various expressions for this damping function have been proposed within the framework of DFT [191]. The most used are: • the original Fermi-like damping function introduced by S. Grimme in his DFT-D2 approach [187], fdamp, n (RAB ) = 1+e

−γ

/

1

0,

RAB −1 sr, n RAB 0

(A.7)

where sr, n is a global DF-dependent scaling factor, γ is a global constant that determines the steepness of the function for small RAB and R0AB is a pair-specific cutoff radius, often represented by (averaged) empirical atomic van der Waals radii or determined by means of sophisticated ab initio procedures [110]; • the function proposed by J.D. Chai and M. Head-Gordon [192] and implemented in Grimme’s DFT-D3 method with zero damping [110], i.e., DFT-D3(zero): fdamp, n (RAB ) =

1+6

/

1 RAB sr, n R0AB

0−γ ;

(A.8)

• the rational damping proposed by A.D. Becke and E.R. Johnson [111], also provided in the recent releases of Grimme’s DFT-D3 100

Figure A.3 Dispersion correction for two carbon atoms (dispersion coefficients from [110]) obtained applying zero- and finite-damping (Becke–Johnson) approaches (black and blue lines, respectively) to the Tao-Perdew-Staroverov-Scuseria (TPSS) exchange-correlation functional [193], in comparison with the undamped van der Waals R−6 behavior (red line). Adapted from [108].

[108], i.e., DFT-D3(BJ): fdamp, n (RAB ) =

n RAB , n RAB + (a1 R0AB + a2 )n

where a1 and a2 are adjustable parameters. A fundamental difference between the above functions is their behavior for small interatomic distances R, as illustrated in Fig. A.3. A known disadvantage of the DFT-D3(zero) approach is that at small R the atoms experience a repulsive force which may lead in some (rare and special) cases to overestimated interatomic distances when considering dispersion correction. On the other hand, although the Becke-Jonson approach seems theoretically well-founded [194], it basically requires to properly adapt the values of the parameters associated to the particular exchange-correlation DF considered. This is a downside of the BJ damping with respect to the “zero-damping” method, which works 101

fairly well with the standard functionals without any special adjustment. Regardless the damping function used, the DFT-D3 presents various improvements compared with its previous version DFT-D2. Although the physical basis has remained unchanged, the algorithm has been refined in terms of higher accuracy, less empiricism and a broader range of applicability. This has been achieved thanks to a whole new set of atom pairwise-specific dispersion coefficients and cutoff radii for all elements up to Z = 94, both computed from first principles. In addition, an eighth-order dispersion term has been introduced, computed recursively using currently established relations [195], and employing the new concept of fractional coordination numbers, that enables one to account for the different hybridization states of atoms in molecules and to achieve geometry-dependent information. Furthermore, in comparison with other approaches, DFT-D3 presents various advantages. It is particularly intended for efficient geometry optimization, allowing the calculation of energy gradients during relaxation. Contributions to the global energy from individual atom pairs, or for a particular distance range, can be readily isolated, enabling a quantitative analysis of the dispersion effects. Moreover, more than 45 different functionals have been accurately parametrized and coupled to the DFT-D3 method. A careful comparison between DFT-D3-computed molecular dispersion coefficients and the experimental values reveals a mean average deviation of about 5% [108], which can be considered the asymptotic limiting accuracy of the method. At the state of the art, DFT-D3 (in particular the version with Becke-Johnson damping) seems to be the best “simple” way to account for dispersion effects in DFT simulations, providing reasonable results at a rather low computational effort. It has been implemented in VASP by J. Moellmann and it is updated in every released version of the package, according to the information provided by S. Grimme and coworkers in their official up-to-date open-source database [196]. Highly parametrized conventional functionals (DFs) If no special corrections for dispersion effects are included, all current Kohn– Sham density functional approximations that are based solely on the electron density and occupied orbitals do not accurately account for 102

the long-range interactions in the weakly overlapping regime. However, as long as only equilibrium structures of small systems are considered, even these non dispersion-corrected DFs, if properly optimized, can “emulate” nonlocal dispersion effects on the medium-range, providing in some cases fairly accurate results. They are purely based on the electron density and thus share some advantages with vdW-DF approaches. On the other hand the dispersion energy in this framework asymptotically vanishes, therefore they cannot be recommended for extended systems (e.g., solids or biomolecules) in which long-range (asymptotic) contributions are important. For this reason it is still difficult to isolate and quantify the dispersion contributions from such calculations. See Grimme et al [108] for further details and a non-exhaustive list of the currently most used highly parametrized functionals. One-electron corrections (1ePOT) Dispersion is inherently interconnected to electron correlations, thus it origins from a many-body problem. As for vdW-DFs, it can be modeled by a nonlocal, twoparticle kernel that acts directly on electron densities. Whilst on the one hand it may sound odd to describe dispersion by means of an effective single-electron potential, on the other hand DFT-D approaches show that dispersion coefficients (hence polarizabilities) of molecules can be reproduced rather well by adding local, atom-like quantities. Thus, if properly optimized, atom-centered nonlocal potentials (e.g., DCACP[180]) can actually provide reasonable results at a reasonable computational cost. Similar to DFT-D, an additional local contribution is included to account for dispersion; in contrast, however, this correction is defined in terms of a potential and so affects the electronic charge density. Similar to highly parameterized DFs, it seems difficult to extract reasonable insight about the dispersion effects from such calculations. Moreover, as the vdW-DFs, the currently used potentials do not show the correct asymptotic R−6 behavior, decaying too fast (exponentially) with increasing interatomic distance. These potentials still need to be generalized in order to cover a larger fraction of the periodic table, reduce the number of free parameters and account for hybridization or oxidation state dependence of the potentials parameters.

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A.3

The Casimir-Polder relationship

When two neutral atoms are located at a distance R large with respect to their intrinsic size some attractive forces act on them. In this case, due to the large distance, the effect of delay on these forces can become relevant. Such delay can be taken into account by means of quantum electrodynamics, yielding to the following expression for the potential energy between the two neutral atoms ˆ ∞ i U (R) = ω 4 αA (ω)αB (ω) exp(2iωR) × 2 πR 0 1 2 2i 5 6i 3 × 1+ − − + dω. (A.9) ωR (ωR)2 (ωR)3 (ωR)4 Here αA/B (ω) are the atomic statistical polarizabilities, which are strongly connected to the electron density through the dielectric function ϵ(ω) = 1 + χ(ω) = 1 + 4πnα(ω), where χ is the electric susceptibility and n is the electron numerical density. The general formula stated in Eq. (A.9) can be simplified when looking at the important limits of short (a ≪ R ≪ λ0 ) and large (R ≫ λ0 ) distances, where a is the atomic radius and λ0 are the characteristic spectral wavelengths of the isolated atomic species. The mathematical details will not be reported in favor of a discussion more qualitative, but the interested reader can find them in the original paper by Casimir and Polder [197] or, in the context of fluctuation theory, in Chapter VIII of “Statistical physics - Part 2” in Vol. 9 of the Course of Theoretical Physics Series by Landau, Lifšitz and Pitaevskij [198]. For R ≪ λ0 the two atoms are very close to each other. A charge density fluctuation occurring in one of them (say A) acts somehow on the surrounding environment, where also the atom B lies. In a perturbative framework, the modifications induced on the atom B can be thought as due to a signal radiated by A and characterized by the frequency ω0 = 2πc/λ0 which is directly related to the spectral signature of the (unperturbed) atomic species A and B. In the case at hand this radiation does not have to “travel” too long to be felt by B. The interaction occurs almost immediately, i.e., not suffering any significant delay with respect to the instant t0 when the radiation has been emitted. In terms of frequencies this leads to the fact that only the frequencies 104

ω ∼ ω0 ∼ λc0 will give a relevant contribution to the integral, hence to the potential energy itself. In this case one can consider ω0 R ∼ ωR ≪ 1 in Eq. (A.9), which thus simplifies to ˆ ∞ 3i U (r) = αA (ω)αB (ω) dω (A.10) 2πR6 −∞ where the parity symmetry α(−ω) = α(ω) has also been exploited. Rather, for R ≫ λ0 the radiation can be thought as if received at some instant which is sufficiently delayed with respect to the emission, due to the larger distance to bridge. This is reflected in the fact that now only the frequencies ω ! c/R ≪ ω0 will contribute to the integral.7 Furthermore, the integral vanishes for ω " ω0 due to the exponential factor.8 The polarizabilities αA (ω) and αB (ω) can now be substituted by their static values αA (0) and αB (0), thus by solving the remaining integral one obtains the definitive Casimir-Polder relationship[197] which determines the interaction energy of two neutral atoms at large distances: 23!c αA (0)αB (0) U (r) = − (A.11) 4π R7 The interaction potential in the form −C6AB /R6 is then achievable by integrating Eq. (A.11) in the space domain, leading (up to constants) to Eq. (A.5) for the coefficient C6AB .

7

Because a delay in the time domain corresponds to a reduction in the frequency domain.

8

This could also be thought as due to the observation of causality. Indeed, assume atom A emits at t0 . Atom B would receive the signal at t > t0 , so that the frequency would be ω < ω0 . If it was possible to receive it with ω > ω0 , then the instants t and t′ would be reverted, i.e., cause and effect would be reversed.

105

Appendix B Additional remarks on the computational details In order to provide a whole coherent framework where both pristine and doped systems can be included, it is crucial to discuss and pinpoint some computational details already at an early stage of the graphene/Ir(111) development. When comparing the results from different DFT calculations, indeed, it is advisable to set some input settings identical, especially when it comes to the k-point mesh used to sample the Brillouin zone in the reciprocal space or the cut-off kinetic energy for the planewave basis set. The k-point mesh is mainly dependent on the supercell size: the bigger the supercell, the smaller the first Brillouin zone, hence the lower the number of k-points needed to span it. Therefore, in line with the literature where the same type of supercell has been developed [51, 53, 133, 134], we use a 3 × 3 × 1 grid of k-points, automatically generated with their origin at the Γ point of the first Brillouin zone (recommended mesh for hexagonal lattices [151]). On the other hand, the optimal cut-off energy for the plane-wave basis set is usually dictated by the method (PAW in our case) used to describe the core electrons in the system, and it strongly depends on the atomic species therein involved. Since there is no literature concerning a (10 × 10) N-doped graphene on Ir(111), we carry out an energy convergence test on an isolated (10 × 10) supercell of N-doped graphene, in order to investigate how the presence of nitrogen alters the optimal cut-off energy. After a geometry optimization at a high cut106

(E0 − E0, @1000 )/atom (eV)

-9.300 -9.305 E0 /atom (eV)

-9.310 -9.315 -9.320 -9.325 -9.330 -9.335 -9.340

0.010 0.008 0.006 0.004 0.002 0.000 -0.002

-9.345 300 400 500 600 700 800 900 1000

300 400 500 600 700 800 900 1000

Cut-off energy (eV)

Cut-off energy (eV)

Figure B.1 Calculated energy per atom at different cut-off energies for the planewave basis set. Each point represents an independent self-consistent calculation of an isolated (10 × 10) supercell of N-doped graphene. The blue dashed line is reported to guide the eye. On the right we show the difference between the data and the value at highest cut-off energy (1000 eV). The greyed region highlights the confidence interval of ±1 meV per atom within which the energy can be considered converged.

off energy (950 eV), we perform a series of self-consistent calculations as a function of this parameter, so as to identify the minimum value which provides a good energy convergence. We consider that a safe convergence is achieved when the energy becomes constant within ±1 meV per atom. The results, illustrated in Fig. B.1, seem to suggest that a cutoff energy of at least 830 eV would better be used when nitrogen is included in the structure. However, as stated in Section 3.3.2, we rather choose to set a cutoff energy of 400 eV. This choice is primarily driven by limitations in terms of computational time and resources, due to the huge number of atoms involved. In addition we want to compare our study of a pristine graphene with the existent literature, where a cutoff energy of 400 eV has always been used, and at a later stage we also aim at comparing the pristine system with the doped one. Therefore the use of a cutoff energy of 400 eV appears to be the only way to enable a thorough and comprehensive study of the structural and electronic properties of N-doped graphene on Ir(111).

107

Appendix C Protocol of geometry optimization for graphene/Ir(111) Below we list all the stages of the complex geometry optimization we perform to reproduce the moiré corrugated structure of the pristine graphene/Ir(111). The minimum number Ni of ionic steps necessary to achieve the force threshold of 10−2 eV/Å is indicated at each step. 1. With all Ir atoms fixed, a first global relaxation of the graphene atoms is performed (Ni > 50 ); 2. The z coordinate of the C atoms, now slightly displaced in the vertical direction, is artificially modulated to resemble the corrugated structure reported by Busse et al. [53]; 3. With the lowest two Ir layers fixed, the graphene atoms are further relaxed (Ni > 50 ); 4. Only the z coordinates of first Ir layer and the z coordinates of graphene are allowed to relax (Ni > 50 ); 5. While keeping the previous settings, also x and y coordinates of graphene are included in the relaxation process (Ni ∼ 30);

6. Also x and y coordinates of the first Ir layer are included ( Ni ∼ 25); 108

7. Also the z coordinate of the second Ir layer is included ( Ni ∼ 25);

8. Also x and y coordinates of the second Ir layer are included (Ni ∼ 20); 9. The last two steps are repeated for the other two Ir layers; 10. A global relaxation of the whole system is performed until the convergence criterion is fulfilled by all the 524 atoms (Ni ∼ 40).

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Acknowledgements All the results presented in this thesis have been collected from March 14th to August 14th 2015, while working in the Soft Condensed Matter (SCM) and the Condensed Matter and Interfaces (CMI) groups of the Debye Institute for Nanomaterials Science at Utrecht University (NL). These five months have been co-funded by the Erasmus+ programme of the European Union and the University of Catania. Like many others in my past, I owe this experience to Giuseppe Angilella. By showing complicity, care and loyalty he has been able to feed my passion for this wonderful subject from year to year. I believe there is no better advisor and mentor than a good friend. With Peppe, from the first “cozzata” around the hallways of the department till the pdfa-day in his room 233 just a few days ago, I could not have asked for a better fate. The ultimate implementation of this project would never have been possible without the goodwill shown by Dr. Marijn van Huis, assistant professor in the SCM group, who committed himself to supervising me during my stay in Utrecht. Besides the computational time he granted me within his cluster, I am extremely grateful to him for the continuous trust and for educating me on what it means to design simulations at a scientific level. I will always remember the warm welcome and the warmer goodbye I got from him and his team of “simulators” in the SCM group. Among these, I would like to express my sincere gratitude to Rik Koster, my daily supervisor, for guiding with immense patience my first steps in the world of DFT simulations. He showed me the tools and tricks of the trade, from the simplest Bash scripting to the most sophisticated calculation, always willing to solve any technical problems or suggest a smart way for saving computational time. He was always 110

there to support me with expertise and experience, even putting up with the dumbest among my questions. A very special thank you goes to Dr. Ingmar Swart, Nadine van der Heijden and Daniel Smith, assistant professor, PhD student and MSc student in the CMI group, respectively. It is because of them that I have managed to fulfill my strong desire to combine theory and experiments within this thesis project. They have kindly provided me with all the experimental images presented here, and they have significantly helped me in making this whole story credible. In particular, I thank Ingmar for being a powerful motivator, for his insightful comments and for his confidence. On the other hand, I thank Nadine and Daniel for their friendly cooperation and for the enthusiasm with which they have taken on board my contributions to their projects on chemical identification with AFM. I would also like to take this opportunity to thank all those who have provided help, ideas and useful discussions during the preparation of this work, with special mention to Wu Fan (Murphy) Li, Somil Gupta, (Lagr)Angelo Pidatella, Peter Bøggild and Daniele Stradi. Without the love, trust and understanding that only my family and my friends may offer, even when I am miles away from home, this thesis (and the person who wrote it) would not exist: thank you all from the bottom of my heart. In particular I want to thank my mom, my dad, my brothers and Giacomo, for their unwavering support. Last but not least, I would like to thank the fantastic international team of new friends with whom I have had the good fortune to toast in Utrecht.

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