Atomistic simulations of electronic structure and coherent transport in 1D & 2D carbon-basednanoarchitectures

Universidad del Pa´ıs Vasco / Euskal Herriko Unibertsitatea Doctoral Thesis Atomistic simulations of electronic structure and coherent transport in 1D & 2D carbon-based nanoarchitectures Author: Xabier Diaz de Cerio Palacio Supervisor: Aran Garcia-Lekue 2025 (cc) 2025 Xabier Diaz de Cerio Palacio (cc by 4.0) This thesis has been carried out at Donostia International Physics Center (DIPC). Abstract The miniaturization of silicon-based field effect transistors (FETs) over the last 60 years has led to remarkable improvements in the performance of electronic devices. However, these basic electronic components have already reached sizes of a few nanometers, where their characteristics start to be affected by quantum mechanical phenomena. One-atomthick materials, and specially those based in graphene, emerge as promising alternatives to silicon. In particular, it has been predicted that atomically precise nanometer-size pores can introduce important modifications in the electronic properties of graphene, for example opening a gap in the band structure or introducing quantum interference effects in the electron transport. In this thesis, we theoretically study the electronic properties of porous 1D and 2D carbon-based nanoarchitectures fabricated with atomic precision using the on-surface synthesis technique. We employ density functional theory and tight-binding models in combination with non-equilibrium Green’s functions to simulate the electronic structure and coherent quantum electron transport in different size scales, ranging from the atomic to realistic device levels. We shed light on a new synthetic route to fabricate porous graphene nanoribbons (GNRs) and we investigate how those nanopores affect the electronic structure of the system. Besides, we explore new strategies to tune the electronic anisotropy and quantum electron transport in nanoporous graphene (NPG), a lateral array of coupled GNRs. We show that engineering the molecular bridges connecting adjacent GNRs can be used to enhance the electronic anistropy and control the electronic Talbot interference effect, a phase coherent phenomena governing large-scale electron transport in NPG. Apart from intrinsic modifications of the atomic structure, we explore the stacking of NPG on graphene. We analyze the dependence of the electronic properties and the Talbot effect on the twist angle between the layers and provide fingerprints for the experimental verification of our predictions. Finally, we address the origin of the deceptive electronic density confinement in spectroscopic measurements performed with scanning tunneling microscopy on GNRs and NPG. All in all, the simulations reported in this thesis provide support for the understanding and analysis of experimental measurements. Moreover, our predictions open new pathways towards the design of electronic devices based on the nanoscale control of electronic currents and phase-coherent transport phenomena in carbon-based nanoarchitectures. Resumen (summary in Spanish) Durante los ´ultimos 60 a˜nos, la miniaturizaci´on de transistores de efecto de campo (FET) basados en silicio ha producido extraordinarios avances en el rendimiento de dispositivos electr´onicos. Sin embargo, estos fundamentales componentes electr´onicos ya han alcanzado tama˜nos de unos pocos nan´ometros, donde sus caracter´ısticas comienzan a estar influenciadas por fen´omenos mec´anico-cu´anticos. En este sentido, los materiales de un solo ´atomo de grosor, especialmente aquellos basados en grafeno, surgen como alternativas prometedoras al silicio. En particular, se ha predicho que la creaci´on con precisi´on at´omica de poros de tama˜no nanom´etrico puede introducir modificaciones importantes en las propiedades electr´onicas del grafeno, como la apertura de un “gap” de energ´ıa en su estructura de bandas o la introducci´on de efectos de interferencia cu´antica en el transporte electr´onico. En esta tesis, realizamos un estudio te´orico sobre las propiedades electr´onicas de nanoarquitecturas porosas 1D y 2D basadas en carbono, las cuales son fabricadas con precisi´on at´omica mediante la t´ecnica de s´ıntesis en superficie. En particular, empleamos la teor´ıa del funcional de densidad y modelos de enlace fuerte en combinaci´on con funciones de Green fuera de equilibrio para simular la estructura electr´onica y el transporte cu´antico de electrones en escalas de tama˜no que var´ıan entre la escala at´omica y la de dispositivos realistas. Arrojamos luz sobre una nueva ruta sint´etica para fabricar nanocintas de grafeno (GNRs) porosas y analizamos c´omo los nanoporos afectan a la estructura electr´onica del sistema. Adem´as, exploramos nuevas estrategias para optimizar la anisotrop´ıa y transporte electr´onicos en grafeno nanoporoso (NPG), una matriz lateral de GNRs acopladas. Demostramos que la ingenier´ıa de los puentes moleculares que conectan GNRs adyacentes puede ser utilizada para aumentar la anisotrop´ıa electr´onica y controlar el efecto de interferencia Talbot electr´onico, un fen´omeno de coherencia de fase que gobierna el transporte de electrones a gran escala en NPG. Aparte de las modificaciones intr´ınsecas de la estructura at´omica, exploramos el apilamiento de NPG sobre grafeno. Analizamos el efecto que el ´angulo de rotaci´on relativo entre NPG y grafeno tiene sobre las propiedades electr´onicas y el efecto Talbot y proporcionamos caracter´ısticas distintivas del sistema para la verificaci´on experimental de nuestras predicciones. Finalmente, abordamos el origen de la confinaci´on enga˜nosa de densidad electr´onica en mediciones espectrosc´opicas realizadas con microscop´ıa de efecto t´unel en GNRs y NPG. En conjunto, las simulaciones reportadas en esta tesis proporcionan soporte para el entendimiento y an´alisis de mediciones experimentales. Adem´as, nuestras predicciones abren nuevas v´ıas hacia el dise˜no de dispositivos electr´onicos basados en el control de la corriente electr´onica y fen´omenos de transporte coherente en nanoarquitecturas de carbono. List of publications This thesis is based on the following published contributions: I. C. Moreno, X. Diaz de Cerio, M. Tenorio, F. Gao, M. Vilas-Varela, A. Sarasola, D. Pe˜na, A. Garcia-Lekue, and A. Mugarza. “On-surface synthesis of porous graphene nanoribbons mediated by phenyl migration”. Communications Chemistry 7, 219 (2024). II. C. Moreno, X. Diaz de Cerio, M. Vilas-Varela, M. Tenorio, A. Sarasola, M. Brandbyge, D. Pe˜na, A. Garcia-Lekue, and A. Mugarza. “Molecular bridge engineering for tuning quantum electronic transport and anisotropy in nanoporous graphene”. Journal of the American Chemical Society 145, 8988-8995 (2023). III. X. Diaz de Cerio, A. Bach Lorentzen, M. Brandbyge, A. Garcia-Lekue. “Twisting nanoporous graphene on graphene: electronic decoupling and chiral currents”. Nano Letters (2025) (accepted). IV. I. Piquero-Zulaica*, E. Corral-Rasc´on*, X. Diaz de Cerio*, A. Riss, B. Yang, A. GarciaLekue, M. A. Kher-Elden, Z. M. Abd El-Fattah, S. Nobusue, T. Kojima, K. Seufert, H. Sakaguchi, W. Auw¨arter, and J. V. Barth. “Deceptive orbital confinement at edges and pores of carbon-based 1D and 2D nanoarchitectures”. Nature Communications 15, 1062 (2024). (*These authors contributed equally to this work). Other publications not covered in this thesis: •D.M.A. Mackenzie, M. Galbiati, X. D. de Cerio, I.Y. Sahalianov, T.M. Radchenko, J. Sun, D. Pe˜na, L. Gammelgaard, B.S. Jessen, J.D. Thomsen, P. Bøggild, A. Garcia-Lekue, L. Camilli and J.M. Caridad. “Unraveling the electronic properties of graphene with substitutional oxygen”. 2D Materials 8, 045035 (2021). 6Chapter 1. Introduction (b)(a) Figure 1.3: (a) On-surface synthesis route (top) and STM images of NPG (bottom). The bottom right STM image was taken using a CO-tip. Figure taken from Ref. [111]. Reprinted with permission from AAAS. (b) Multi-scale DFT+TB electron transport simulation showing the Talbot interference effect in NPG. Figure reproduced from Ref. [112] with permission from the Royal Society of Chemistry. Adapted with permission from Ref. [113]. Copyright 2019 American Chemical Society. achieve atomically precise graphene nanostructures [81]. OSS procedures start from small building-block molecules that are assembled as “lego”-pieces into complex target geometries. These molecular precursors are assembled via thermally activated reactions catalyzed by chemically active metallic surfaces (e.g. Au(111)) (see Fig. 1.2a). The elegance of this bottom-up fabrication method relies on the fact that it allows to systematically reproduce atomic level modifications of the product geometry through chemically controlled adjustmets of the precursor monomer. As a consecuence, since the fabrication of the first atomically precise GNR 15 years ago [82], a myriad of novel carbon-based nanoarchitectures exhibiting all kinds of sizes and geometries has been designed and fabricated using this method. For example, GNRs with armchair (AGNR) [82], zigzag (ZGNR) [83] and chiral (ch-GNR) [84] edge orientations have been realized, whose width, edge structure and chemical composition can be further modified (see Fig. 1.2b) [85–89]. Equally, a wide range of 0D nanographenes [90–93] and 2D molecular [94,95] assemblies have been obtained. Most importantly, this has led to a remarkable variety of tailored electronic properties, including tunable semiconducting bandgaps [85,86,96], localized and entangled spins [97–107], and topological boundary states [99,100,102,107–110], highly attractive features for nanoelectronics, sensing and quantum information technologies. Particularly attractive is the ability to engineer pores at the nanoscale in graphene [70, 94,114]. In 2018, a fully covalent atomically precise NPG was synthesized for the first time by laterally fusing AGNRs of alternating 7-atom and 13-atoms width (7-13-AGNR), 7 setting a significant milestone in the OSS community (see Fig. 1.3a) [111]. This 2D structure can be viewed as a lateral perodic array of semiconducting GNRs that are connected by interribbon molecular bridges. Interestingly, because the interibbon electronic coupling is weak compared to the kinetic energy along the ribbon growth-direction, the resulting NPG exhibits remarkable electronic anisotropy. Moreover, theoretical studies have revealed that such a weak interribbon coupling is responsible for electronic currents in NPG exhibiting the electronic analog of the Talbot interference effect [113], an optical phenomena arising from the interference between weakly coupled optical waveguides (see Fig. 1.3b). This extends the observation of electron analogs of optical phenomena to semiconducting nanostructured graphene. Furthermore, designing new strategies to optimize these effects and handle nanoscale currents in NPG arises as a promising avenue of research. An additional striking feature of NPG is the emergence of so-called pore states, electronic states that are strongly confined to the nanopores [111]. These electronic states have been described as 1D analogs of the image potential states (IPSs) of graphene [115], emerging on the side of GNR edges rather than above the graphitic surface. Due to their potential sesitivity to foreign molecular agents, pore states have boosted the perspective of using NPG as a molecular nanosensor. However, these states lie at very high energies above the Fermi surface, which may hinder their applicability [115,116]. Thus, understanding their properties and controlling their energetics are important issues to be addressed. However, before graphene nanostructures can be used in practical applications, an atomic-level characterization of their electronic properties is required. From the experimental side, scanning probe methods (SPMs), such as the atomic force microscope (AFM) [117] and the scanning tunneling microscope (STM) [118], have demonstrated exceptional performance in tackling this task [119]. By measuring the current tunneling between a metallic tip and the sample, the STM can provide both local atomic and spectroscopic characterization of the sample with remarkable sensitivity to atomic-scale features, making the electronic structure and its real-space properties accessible [119,120]. By functionalizing the tip with a CO molecule, STM and AFM can provide high resolution images of the atomic structure, as shown in Fig. 1.3a for NPG, in which even individual C-C bonds are identified [121,122]. Along with experimental advances, theory and computation has developed as an essential partner in the characterization of graphene-based nanoarchitectures. As such, simulations and theoretical studies play a key role in understanding and predicting the chemical-physical processes of the novel functional units, e.g. GNRs or NPG, at the nanoscale. In particular, Density Functional Theory (DFT) has been one of the most widely employed method to study the atomic and electronic structures of materials, as it offers accurate ab initio predictions in exchange of acceptable computational cost. However, using DFT may become unfeasible when studying the properties of large-scale systems consisting of hundreds of thousands or even millions of atoms. The challenge then relies on building simpler models whose accuracy is comparable to that provided by ab initio methods. Interestingly, tight-binding (TB) models can be employed as a first approximation of the problem. Despite their accuracy being often significantly lower than DFT, the TB approximation allows for systematic investigations of large systems that 8Chapter 1. Introduction would take ages to carry out using DFT [123,124]. Besides, the fully quantum mechanical non-equilibrium Green’s function (NEGF) approach can be used to calculate electron transport properties. The underlying electronic structure can be calculated using DFT or simplified models such as TB. Remarkably, a multi-scale approach combining DFT, TB and NEGFs has recently allowed electron transport simulations with ab initio accuracy in graphene-based devices of realistic size, for example made from NPG (see Fig. 1.3b) [112, 113,125]. All in all, the above-described atomistic simulation tools have been successful for understanding experiments and predicting novel electronic properties in a wide variety of carbon-based systems. Outline of the thesis In this thesis, we combine DFT, TB and NEGFs simulations in order to rationalize experimental measurements and predict novel phase-coherent electronic and transport phenomena in 1D and 2D carbon-based nanoarchitectures, from the atomic level to a realistic device-scale. In particular, the thesis is organized as follows: In Chapter 2, we describe the theoretical methods applied in this thesis. In particular, we explain the basic theorems and concepts of DFT, as well as relevant implementation details. We also introduce the main features of different kinds of TB models. Additionally, the non-equilibrium Green’s function (NEGF) method and its application to quantum electron transport calculations is presented. Chapter 3 is focused on the study of two synthesized porous carbon-based nanoarchitectures: a porous GNR and a NPG structure with tailored interribbon molecular bridges. We scrutinize the phenyl migration reaction enabling the formation of nanopores in GNRs and examine their effect on the electronic properties. Besides, using a combined DFT+TB multi-scale approach, we address the role of engineered molecular bridges in the electronic anisotropy of NPG. In particular, we explore how the chemical and structural configuration of the bridges can be used to fine-tune the electronic Talbot intereference pattern in NPG. In Chapter 4, we propose an alternative approach to modify the electronic properties of NPG by considering bilayers composed of NPG stacked on graphene with different interlayer twist angles. Based on a simple TB model, we show that the interlayer coupling is determined by the twist angle. We further demonstrate that the electronic transport and the Talbot interference pattern in NPG are highly sensitive to the interlayer arrangement. Next, in Chapter 5 we disentangle the origin of electronic density confinement at the edges and nanopores of carbon-based nanoarchitectures, a common effect that has led to dubious interpretations in the field of OSS. Combining DFT calculations with a Fourier analysis of the electronic wave functions, we rationalize scanning tunneling spectroscopy (STS) measurements in gulf-type GNRs and NPG, showing that electronic density confinement is likely to arise due to experimental limitations of the STM rather than from intrinsic wave function properties. Finally, in Chapter 6, the main conclusions of the thesis are summarized and the future perspectives are discussed. CHAPTER 2 Methods In the present chapter we introduce the theoretical methods applied in this thesis. These include density functional theory (DFT), the tight-binding (TB) method and the nonequilibrium Green’s function (NEGF) formalism for quantum electron transport simulations. 2.1 Density Functional Theory DFT is an ab initio variational theory that provides an efficient approach to study manybody systems. Its elegance and power relies on the fact that the total energy of a system composed of many electrons and ions can be expressed only in terms of the electronic density. Minimizing the total energy with respect to the electron density provides the ground state energy and density, which determine all the properties of the system [126]. Although the basic concepts of the theory were introduced in the 1960s, it was thanks to the developments implemented in the following decades that DFT became a practical method. Since the 1990s, boosted by an exceptional improvement of the computational capabilities, it has become one of the most widely employed methods for studying the electronic structure of atoms, molecules and materials [127]. Likewise, DFT is one the main methods applied to calculate the electronic structure of carbon-based nanoarchitectures, and it plays a central role in this thesis. Therefore, after introducing the many-body problem of electronic structure, this section is devoted to explaining the basic theorems and concepts of DFT, as well as the approximations and methods adopted for its practical application. 2.1.1 The many-body problem Studying the electronic structure of molecules or crystals, implies solving the Schr¨odinger equation for systems composed of a large number of ions and electrons, what is commonly known as the many-body problem. For such a system composed of M ions and N electrons, 10 Chapter 2. Methods the many-body Hamiltonian operator is defined as H=−ℏ2 2me N X i=1 ∇2 i−1 2MI M X I=1 ∇2 I+ N X i=1 M X I=1 ZIe2 4πϵ0|ri−RI| + N X i=1 N X j>i e2 4πϵ0|ri−rj|+ M X I=1 M X I>J ZIZJe2 4πϵ0|RI−RJ|, (2.1) where ions, with mass MI, charge ZIeand position RI, are denoted by upper case subscripts, while electrons, with mass me, charge eand position ri, are denoted by lower case subscripts. Thus, the system consists of a total of 3(M+N) spatial degrees of freedom: 3M corresponding to the ions and 3N to the electrons. For many purposes where the interactions between electrons and ionic vibrations can be neglected, this extremely complex Hamiltonian is commonly simplified by applying the Born-Oppenheimer approximation [128]. The latter states that, as ions are much heavier than electrons, ionic positions can be considered to be fixed and behave as parameters. The ionic kinetic energy (second term in Eq. (2.1)) can be accordingly neglected and the ion-ion interaction term (fifth term in Eq. (2.1)) can be taken as a reference energy. By adopting Hartree atomic units ℏ=me=e= 4πϵ0= 1, the fundamental Hamiltonian for the electrons is then written as follows: ˆ Helec =ˆ T+ˆ Vext +ˆ Vint,(2.2) where the electronic kinetic energy operator is ˆ T=−1 2 N X i=1 ∇2 i,(2.3) the external potential ˆ Vext is the ionic potential acting on electrons, ˆ Vext =− N X i=1 M X I=1 ZI |ri−RI|,(2.4) and ˆ Vint is the electron-electron Coulomb interaction, ˆ Vint = N X i=1 N X j>i 1 |ri−rj|.(2.5) The Born-Oppenheimer approximation provides a significant reduction in the number of degrees of freedom of the many-body system. Yet, it still yields a complex many-electron Schr¨odinger equation with no straightforward solution. The early approaches by Hartree (1928) [129] and Fock (1930) [130] achieved a remarkable simplification of this problem by treating electrons as independent particles. Applying the variational method they obtained self-consistent single-particle equations describing the motion of electrons in an effective potential due to the ions and the rest of electrons. Although their approaches laid 2.1. Density Functional Theory 11 the foundations of effective single-particle methods, they failed to address the electronic correlations that arise from electron-electron Coulomb interactions (Eq. (2.5)). Accurately describing such electronic correlations is the main challenge of electronic structure methods, which has motivated the search of efficient alternative methods to solve the many-electron problem [126]. 2.1.2 Hohenberg-Kohn theorems Instead of attempting to calculate the many-body wave function of the system, the problem can be remarkably simplified by expressing it only in terms of the electronic density. Thomas [131], Fermi [132] and Dirac [133] made the first attempts to approximate the total energy of a many-body system as an explicit functional of the density. However, it was not until 1964 when Hohenberg and Kohn introduced the theorems that demonstrated that the total energy of the system could indeed be defined as a functional of the electronic density [134]: •Theorem I: The external potential, Vext(r), (the ion-electron interaction potential, in our case) is uniquely determined, except for a constant, by the ground state electronic density, n0(r). •Theorem II: A universal functional for the energy E[n(r)] can be defined in terms of the electronic density n(r), for any external potential. The global minimum of E[n(r)] will be the exact ground state energy of the system, and the density n(r) that minimizes that energy will be the exact ground state electronic density, n0(r). It follows that the full Hamiltonian is fully determined, except for a constant shift, by the ground state electronic density n0(r). This implies that the many-body wave functions for all states and all the properties of the system are fully determined by n0(r). Moreover, the ground state density n0(r) can be exactly achieved from the energy functional E[n(r)], which can be written as: E[n] = T[n] + Eext[n] + Eint[n],(2.6) where T[n] is the kinetic energy and Eext[n] = ZVext(r)n(r)d3r, (2.7) is the external energy. Eint[n] is the Coulomb energy of interacting, correlated electrons, and is composed of the classical Hartree term and a non-classical term describing exchange and correlation: Eint[n] = EHartree[n] + Encl[n],(2.8) where EHartree[n] = 1 2Zn(r′)n(r) |r−r′|d3rd3r′.(2.9) As shown by Eq. (2.6), the key idea of expressing the total energy as a functional of the electronic density provides a remarkable simplification of the many-body problem 12 Chapter 2. Methods (Eq. (2.2)), as the complexity of the latter is reduced to the three degrees of freedom of n(r) [126]. The Hohemberg-Kohn theorems are the basis of DFT. Importantly, unlike previous approaches to the many-body problem, DFT is an exact theory of correlated many-body systems, as no approximations have been made apart from the Born-Oppenheimer approximation. Nevertheless, despite the beautiful simplicity of Hohenberg and Kohn’s proposal, T[n] and Encl[n] are still unknown many-body functionals. For instance, there is no known expression of the kinetic energy in terms of the density, and the only way of computing the exact kinetic energy Tis using the many-body wave function corresponding to the interacting system. The challenge then relies in finding a density functional form of the total energy that allows for a practical application of the theory. 2.1.3 The Kohn-Sham ansatz In 1965, Khon and Sham presented an approach for obtaining an explicit expression of the total energy functional [135]. This consists in replacing the interacting many-body system by an auxiliary independent-electron system, assuming that the interacting ground state density of the former is equal to the independent-electron density of the latter. This assumption is called “non-interacting-V-representability”, and leads to a description in terms of independent-electron equations. In particular, the electronic density n(r) of an auxiliary independent-electron system is given by n(r) = N X i=1 |ψKS i(r)|2,(2.10) where ψKS i(r) are the independent-electron wave functions, or Kohn-Sham wave functions. For practical reasons, here we neglect the electronic spin degree of freedom. The KohnSham energy is then written as follows: EKS =Ts[n] + Eext[n] + EHartree[n] + Exc[n],(2.11) where the kinetic energy, Ts[n] = −1 2PN i=1 Rd3rψKS* i(r)∇2ψKS i(r) =1 2PN i=1 Rd3r|∇ψKS i(r)|2, (2.12) is simplified to the sum of the kinetic energy of each electron, and Exc[n] is the exchange and correlation energy. Considering that the Kohn-Sham energy represents the total energy of the system and equating both Eqs. (2.11) and (2.6), Exc[n] is expressed as Exc =T[n]−Ts[n] + Eint[n]−EHartree[n].(2.13) This expression describes the physical meaning of Exc[n], which accounts both for the non-classical effects of the interacting Coulomb energy (see Eq. (2.8)) and the kinetic terms beyond the independent-electron description of the kinetic energy. 2.1. Density Functional Theory 13 Interacting many-body system Auxiliary independent-electron system Electrons Interactions/correlations External potential Effective Kohn-Sham potential: Independent Kohn-Sham electrons Equivalent Same electronic density Figure 2.1: Illustration of the Kohn-Sham ansatz, where the many-body interacting system is replaced by an auxiliary independent-electron system with the same ground state density. One can then apply a variational minimization of Eq. (2.11) with respect to ψKS* i(r) employing the Lagrange multiplier method, which leads to the so-called Kohn-Sham equations: h−1 2∇2+Vext(r) + VHartree(r) + Vxc(r)iψKS i(r) = εKS iψKS i(r),(2.14) where VHartree(r) = Zn(r′) |r−r′|d3r′,(2.15) Vxc(r) = δExc[n] δn(r),(2.16) and εKS iare the Kohn-Sham eigenvalues. Eq. (2.14) is a Schr¨odinger-like equation for independent electrons moving in an effective potential VKS(r) = Vext(r) + VHartree(r) + Vxc(r),(2.17) which depends self-consistently on the electronic density (see Fig. 2.1). The Kohn-Sham equations must be solved iteratively to find the ground state solution of the single-electron wave functions and the electronic density, as described in Fig. 2.2. In addition, the equations are general for any expression of Exc[n], and if Vxc was known, solving the equations would provide the exact solution of the interacting system. The central problem of DFT is that there is not an exact known expression for Exc[n], which has motivated decades of research focused on finding suitable approximations to exchange-correlation effects [126,136]. 2.1.4 Exchange and correlation functionals A very convenient point of the Kohn-Sham ansatz is that independent-particle kinetic energies and the long-range Hartree potential enter as separate terms in Eq. 2.14. All 14 Chapter 2. Methods Initial Guess n(r) Calculate the effective potential VKS(r)=Vext(r)+VHartree(r)+Vxc(r) Solve the Kohn-Sham equations [−1 2∇2+VKS]ψKS i=εKS iψKS i Compute the electronic density and the total energy n(r)=∑ i |ψKS i(r)|2→E[n(r)] Converged? Compute output quantities n(r),E[n(r)]→ YES NO Figure 2.2: Self-consistent iterative cycle for solving Kohn-Sham equations. Upon making an initial guess of the electronic density the effective Kohn-Sham potential is obtained. By solving the Kohn-Sham equation the single-particle wave functions are obtained, and the corresponding new electronic density and the total energy are computed. If these are the same as those obtained in the previous iterative step, convergence is achieved and output quantities are calculated. Otherwise, the new electronic density is employed to build a new effective potential and perform another step. the rest of electronic effects included in Exc[n] are then typically expected to be shortranged and can be approximated via local or nearly local functionals. Exc[n] can then be expressed as: Exc[n] = Zn(r)ϵxc([n],r),(2.18) where ϵxc([n],r) is the exchange-correlation energy per particle at position rand depends on the density in some neighborhood of such position. Very good results have been obtained employing remarkably simple approximations to ϵxc([n],r). Next, we review some of the most commonly used ones. Local density approximation The simplest way to describe Exc is the local density approximation (LDA), which takes the exchange and correlation energy per particle in a uniform electron gas, ϵunif xc [n], and 2.1. Density Functional Theory 15 substitutes in that expression the varying electronic density of our system n(r). In particular, the exchange energy per particle in the LDA is given by ϵunif x[n] = −3 43 πn(r)1 3.(2.19) Although the correlation term ϵunif c[n] cannot be determined analytically, approximate parametrized expressions have been proposed which are fitted to numerical results obtained for example using quantum Monte Carlo methods [137]. The LDA exchangecorrelation functional ELDA xc [n] is then calculated by adding the exchange and correlation terms [126]: ELDA xc [n] = Zd3rn(r)hεunif x[n(r)] + εunif c[n(r)]i =Zd3rn(r)εunif xc [n(r)]. (2.20) Generalized gradient approximation As a next step to LDA, the generalized gradient approximation (GGA) suggests εxc[n] to be not only a function of n(r), but also of its gradient. Consequently, EGGA xc [n] can be expressed as EGGA xc [n] = Zd3rn(r)εxc[n(r),|∇n(r)|].(2.21) Numerous forms for εxc have been proposed. Probably the simplest GGA functional is the one by Perdew, Burke and Ernzerhof (GGA-PBE) [138], which we employ in most of the calculations reported in this thesis. Although GGA provides significant improvements over LDA [126], both approximations have been successfully tested in systems where n(r) varies slowly. However, they can be inadequate in systems with rapidly varying electronic densities. This is illustrated by the fact that different GGA functionals can diverge notably in those situations, while they yield almost identical results otherwise [126]. Besides, LDA and GGA approximations fail to describe long-range van der Waals forces, which can be important for instance in calculations of molecules adsorbed on substrates. Accordingly, additional extensions to the above-described approximations have been proposed. Approximations including dispersion forces Electron dispersion forces, also known as van der Waals forces, are long range attracting interactions produced by instantaneous variations of the electronic density in one region of the system. These fluctuations induce a corresponding redistribution of the density in another region, giving rise to an attractive force decaying with a leading term −1 r6. In contrast, the locality of the LDA and GGA provides an exponential decay of the binding curves, consequently failing to address a wide variety of systems where non-local dispersion forces play a crucial role [139]. In order to accurately describe these systems, several corrections to LDA and GGA have been developed. These can be mainly classified in two groups. The first one consists 22 Chapter 2. Methods where g0 α(E) = [ESα−Hα]−1is the Green’s function of the isolated electrode α. In practice, SE matrices truncate our problem to finite dimensions, while they account for the unperturbed, semi-infinite part of the electrode. Physically, they can be viewed as effective Hamiltonians arising from the coupling of the device region with the electrodes. They induce a renormalization and broadening of the device region eigenvalues, which reflects the fact that device region eigenstates are perturbed and their lifetime is reduced. Such a finite lifetime indicates the rate at which electrons are injected into or drained from the device by the electrodes. The probability of an incoming electron propagating into the scattering region and being drained by an electrode again determines the transport characteristics of the device. Importantly, the Landauer-Buttiker formalism and the NEGFs method can be extended to an arbitrary number of electrodes [125,156,158]. 2.3.2 Important quantities from the Green’s function Some important quantities that can be written in terms of the device Green’s function are the transmission function, spectral function, density of states and the bond-transmissions [125,156,157]. Tranmission function The transmission function provides the probability of an electron being transmitted between electrode αand β. It can be calculated from the Green’s function as follows: Tαβ(E) = Tr[GDΓαG† DΓβ],(2.38) where Γα=i[Σα−Σ† α].(2.39) Γαis the broadening of the isolated device region eigenvalues due to coupling to electrode α, and represents the rate at which electrons are injected into or drained from the device. According to the Landauer-B¨uttiker formalism, the current between the two electrodes is given in terms of the corresponding transmission function as: Iαβ =2e hZdETαβ(E)[fα(E;µα, τα)−fβ(E;µβ, τβ)],(2.40) where fα/β are the Fermi distributions of each electrode, with chemical potential µα/β and temperature τα/β. Spectral function and density of states The spectral function corresponding to electrode αreads: Aα(E) = GDΓαG† D.(2.41) From the spectral function, the spectral contribution to the density of states coming from each electrode can then be written as: ADOSα(E) = 1 2πRe Tr[Aα(E)S].(2.42) 2.3. Quantum electron transport with non-equilibrium Green’s function 23 Besides, the device density of states can be expressed as: DOS(E) = −1 πIm Tr[GD(E)S].(2.43) Often the device density of states can be expressed as the sum of spectral density of states of each electrode such that DOS(ϵ) = ΣαADOSα(ϵ). However, if a bound state exists in the device region that does not couple to the electrodes, this will not be captured by the sum over the spectral density of states of all electrodes, and the previous equality will not hold. Bond transmissions Bond transmissions provide a useful way of visualizing real-space current densities coming from a particular electrode [113,159,160]. They represent the local current flowing between two atoms. In particular, the orbital transmission, i.e. the transmission between two orbitals iand joriginating from an electrode α, is given by: Tij(E) = i[(HDji −SDji)Aαij(E)−(HDij −ESDij)Aαji(E)].(2.44) The bond transmission between two atoms µand νis then given as a sum over the transmission between all orbitals in each atom: Tµν(E) = X i∈µX j∈νTij(E).(2.45) It follows from Eq. (2.40) that bond transmissions can be extrapolated to non-zero bias conditions by integrating orbital transmissions in the pertinent bias window. The corresponding quantities are called orbital currents Jij(E) = 2e hZdETij(E)[fα(E;µα, Tα)−fβ(E;µβ, Tβ)],(2.46) and bond currents Jµν(E) = X i∈µX j∈ν Jij(E).(2.47) There are different ways of plotting bond transmissions [112,123,159,160]. Following the method employed in previous works studying NPG structures [112,113,161,162], in this thesis we choose to represent them as 2D colormaps in which bond transmissions originating from each site are summed and interpolated onto a 2D grid. Although information about the directionality of currents is lost using this method, it provides a detailed visualization of electron flow in large devices of realistic size [125]. 2.3.3 Details of TB+NEGF simulations In this thesis, we perform quantum coherent electron transport simulations by combining TB Hamiltonians and NEGFs as implemented in the TRANSIESTA utility TBTRANS [157]. System geometries and Hamiltonians are set-up using the SISL python software 24 Chapter 2. Methods CAP Self-energy ri rf ∆r CAP Self-energy rf ri ∆r Transport direction Figure 2.4: Example of a NPG device equipped with top and bottom electrode SEs (yellow area), and left and right CAPs (red area). CAPs avoid backscattering off the left and right device edges. riand rfindicate the positions where CAPs start and end, respectively. [163]. In our calculations, the calculated device Green’s function is expressed in the following general form: GD(E, k) = "SD(k)(E+iη)−HD(E, k)−δH−X α Σα(E, k)#−1 .(2.48) where kis the Bloch wave-vector in transverse periodic directions perpendicular to the transport direction. There are various ways to describe the TB device Hamiltonian HD. On the one hand, pruned Hamiltonians can be obtained, only retaining a reduced set of the full DFT matrix elements. This is the strategy adopted in Chapter 3. On the other hand, one can define a TB Hamiltonian whose matrix elements depend on parameters that have to be fitted to ab initio calculations or experiments, as we do in Chapter 4. The term Σα(E, k) can be used to introduce the effect of electrodes (see Fig. 2.4) and other perturbing components in the system. Following a recently developed multi-scale method, such perturbations can be treated with DFT and then be embedded via SE objects into large-scale devices described by TB [112,113]. For example, this strategy allows to include the effect of a STM tip in contact with some atoms in the device from DFT calculations, as illustrated in Fig. 2.5. Alternatively, such DFT-based perturbations 2.3. Quantum electron transport with non-equilibrium Green’s function 25 Figure 2.5: Illustrative representation of the multi-scale method employed for embedding DFT-calculated components into a large-scale device described by TB. Figure reproduced from Ref. [112] with permission from the Royal Society of Chemistry. can be replaced by parametrized SE objects defined by hand, so that they approximately reproduce the effect of an ab initio SE at a lower computational cost [125,161]. The latter is the approach adopted in this thesis to simulate the presence of a STM tip in the transport simulations. The δHobject can be used optionally to account for the effect of external perturbations to the Hamiltonian, such as charge-carrier doping or magnetic fields [123,125]. Here, we use it to introduce complex adsorbing potentials (CAPs) in regions around the finite device edges, as shown in Fig. 2.4. The position dependent CAP terms are given by: W(r) = ℏ2 2m2π ∆rf(r),(2.49) where f(r) = 4 c2∆r rf−2ri+r2+∆r rf−r2−2.(2.50) Here, c= 2.62, riand rfare the positions where the CAP region starts and ends, respectively, and ∆r=rf−riis the CAP region width [164]. In the selected region, the CAP introduces on-site imaginary terms −iW(r) whose magnitude grows progressively as rapproaches rf, i.e. as the distance to the edge decreases, eventually diverging on the device edge (r=rf). This introduces an on-site decay rate that adsorbs electrons in the atoms close to the edge. The efficiency of CAPs improves with their width ∆r, and it has been shown that few nanometers-wide regions are able to completely drain electrons and avoid backscattering off the device edges [123,125]. CAPs can therefore be employed as a computationally less costly alternative to SEs in order to mimic open boundary conditions in transverse directions, and they indeed play a crucial role in many of the simulations reported in this thesis. CHAPTER 3 Porous 1D and 2D nanoarchitectures: disentangling phenyl migration and electronic anisotropy The NPG obtained by fusing AGNRs of alternating 7-atom and 13-atom width (7-13AGNRs) exhibits a highly anisotropic electronic structure. On the one hand, VBs and CBs have a marked longitudinal (L) character, their dispersion being strong along the growth direction of the ribbons and weak along the perpendicular direction. Moreover, due to the weak interribbon coupling, the otherwise degenerate frontier bands coming from the two fusing ribbons (see unit cell of NPG in Fig. 3.1a) hybridize and exhibit a small energy splitting [111]. This gives rise to the electronic Talbot interference effect, which can be understood as a standing wave pattern that arises from the superposition of equal-energy wave functions coming from two longitudinal bands with a small momentum difference (see Fig. 3.1b) [113]. On the other hand, although not explored in this work, NPG also hosts electronic bands of transversal (T) character at higher energies, whose dispersion is small along the growth direction of the ribbons and stronger along the perpendicular direction. In addition, the porous structure leads to the emergence of so-called pore states (P) at high energies. These resemble the superatom molecular orbitals (SAMOs) found in fullerenes [165] and have generated significant interest for their potential application in molecular sensing (see Fig. 3.1a) [111]. Interestingly, the remarkable flexibility provided by OSS has allowed to optimize the promising electronic properties of NPG through atomic-scale variations of its geometry [166–168]. For example, lateral superlattices of electrostatically gated GNRs have been fabricated upon substitution of individual N atoms in their interribbon molecular bridges [168]. From a theoretical perspective, similar strategies of molecular bridge engineering have also been considered in order to tune the Talbot effect. In particular, chemical coupling modifications or chemical functionalization can cause destructive quantum interference in the bridges, abruptly quenching interribbon electronic transmission and enhancing the anisotropy of electronic currents [161,162]. Thus, developing a synthetic strategy towards experimental molecular bridge engineering as well as devising alternative control knobs for fine-tuning quantum interference effects are key for harnessing coherent transport phenomena in NPG devices. Optimizing the energetics of pore states by modifying the size and shape of nanopores is another important requirement in order to fully exploit the multifunctional capabilities of NPG and other porous nanoarchitectures. In this regard, the design of new synthetic mechanisms within the OSS toolbox may contribute to keep creating innovative 28 Chapter 3. Porous 1D and 2D nanoarchitectures (a) (b) Lateral fusion X Z Figure 3.1: (a) Electronic band structure of single a 7-13-AGNR (left) and NPG (right). Longitudinal (L), transversal (T) and pore (P) states are highlighted in yellow, purple and green, respectively. The atomic structures of the 7-13-AGNR and NPG are schematically depicted below the band structures. Figure taken from Ref. [111]. Reprinted with permission from AAAS. (b) Interference pattern emerging from the superposition of wave functions coming from equal-energy states in the two longitudinal CBs. Adapted with permission from Ref. [113]. Copyright 2019 American Chemical Society. geometries. In particular, surface-assisted thermally-activated phenyl migration has been recently reported to induce internal transformations in molecules [169,170]. Achieving selective control on this type of reactions is a necessary condition to reach the target products systematically and reliably. In this chapter, we introduce two original, experimentally synthesized carbon-based nanoarchitectures fabricated from the same parent GNR via different synthetic routes: 1) a porous GNR obtained through a phenyl migration reaction at the ribbon edge; and 2) a NPG structure with engineered molecular bridges synthesized by the lateral crosscoupling of parent ribbons. After introducing their synthesis procedure and supported by experimental measurements, we theoretically unveil the particular reaction path leading to phenyl migration, as well as the effect of pore formation on the electronic structure of the GNRs. In the case of NPG, we explore the influence that its different structural parameters have on the electronic anisotropy and quantum coherent transport phenomena. 3.1 On-surface synthesis of porous GNRs and NPG The synthetic route followed to obtain porous GNRs and NPG is summarized in Fig. 3.2, along with STM images of the products after each reaction step. Both resulting struc- 3.1. On-surface synthesis of porous GNRs and NPG 29 tures share the same precursor monomer, called 2’-di([1,1’-biphenyl]-4-yl)-10,10’-dibromo9,9’bianthracene (DBP-DBBA), which is very similar to the DP-DBBA monomer employed to synthesize the NPG structure in Ref.[111]. The only difference between them is the addition of single phenylene units at the left and right sides of the DBP-DBBA, which are key elements for the synthesis of the porous GNRs and NPG introduced in this chapter. The OSS of both structures share the same initial synthesis steps described below, all the way until the formation of so-called Ph-7-13-AGNRs. The procedure is performed in ultra-high vacuum conditions, and starts by depositing precursor DBP-DBBA molecules on a Au(111) surface held at room temperature (Fig. 3.2a). Subsequent substrate annealing at T1= 200◦C triggers the so-called Ullmann coupling reaction between monomers, leading to the formation of 1D polymeric chains (Fig. 3.2b). By increasing the annealing temperature to T2= 400◦C the cyclodehydrogenation reaction in the polymers is activated, which induces their planarization and the formation of Ph-7-13-AGNRs. The latter consist of 7-13-AGNRs formed of alternating 7-C-atom and 13-C-atom-wide sections [111], with additional phenyl rings that appear single-bonded to the edges of 13-C-atom-wide sections (Fig. 3.2c). Importantly, Ph-7-13-AGNRs retain the prochiral configuration of their parent monomer (i.e. they are chiral when confined in 2D), and can appear in Sand Rorientations. At this point, experiments reveal that further annealing the substrate can trigger two different scenarios depending on the surrounding environment of Ph-7-13-AGNRs. 1) If Ph-7-13-AGNRs are aligned and free to diffuse laterally, dehydrogenative cross coupling is thermally induced at T3= 450◦C and adjacent ribbons fuse to form covalent 2D NPG structures (Fig. 3.2e). In contrast to NPGs obtained from 713-AGNRs [111], three different bonding configurations are obtained depending on the chirality of the ribbons and the site in the phenyl side-group participating in the covalent bond (ortho,para or meta, see Fig. 3.2c): homochiral ribbons create para-para (pp) or meta-meta (mm) configurations, while heterochiral ribbons bind via para-meta (pm) coupling. Bonds between ortho positions are not found, as they are sterically hindered by the hydrogens at the adjacent phenyl unit and the fact that it would involve the formation of more than one bond. The observation of large defect-free NPG flakes demonstrates the successful formation of NPG with distinct interribbon bridges. 2) When the lateral coupling between GNRs is inhibited, a new reaction path opens at T3= 450◦C. In this scenario, the outermost phenylene ring of the ribbon migrates from the C2(C3) site to the neighboring equivalent position C3(C2) at the edge of the 13-C-atom-wide section (see Fig. 3.3a and Fig. 3.4a). In some cases, this migration is followed by a dehydrogenative cross coupling with an adjacent phenyl side group, leading to the formation of GNRs functionalized with [18]-annulene pores ([18]-annulene GNRs) (Fig. 3.2d). In the next two sections we present combined experimental and theoretical studies of each of these nanostructures. An special emphasis is put on the role of our DFT-based simulations in corroborating the experimental STM characterization and predicting the electronic properties of these materials. 30 Chapter 3. Porous 1D and 2D nanoarchitectures DBP-DBBA monomer Polymer Ph-7-13-AGNR T1 = 200 ºC Ullmann Coupling T2 = 400 ºC Cyclodehydrogenation NPG T3 = 450 ºC T3 = 450 ºCDehydrogenative Cross CouplingPhenyl migration [18]-annulene GNR (a) (b) (c) (d) (e) Figure 3.2: Step-by-step synthetic route followed to obtain [18]-annulene GNRs and NPG, along with STM images of the products at each step. (a) DBP-DBBA precursor monomer. Sand Rindicate the two possible enantiomers emerging when confining the monomer to 2D. (b) Polymer chain formed via Ullmann coupling of DBP-DBBA molecules. (c) Ph-7-13-AGNR. Illustration (right): Ortho,meta and para labels in the phenyl side-group indicate the 3 different positions at which the ribbon can form covalent bonds. STM images (left): right STM image is a bond-resolved STM (BR-STM) image taken with a CO-tip. (d) [18]-annulene GNR. Illustration (top): shaded phenyl side-groups indicate the original position of phenyl rings before migration takes place. STM images (bottom): right STM image is BR-STM. (e) NPG. Illustration (top): the three possible bonding configurations, namely para-para,meta-meta and para-meta, are depicted, and the chirality S/R is indicated on top of each ribbon. Reaction names and activation temperatures and indicated on each step. Figure taken from Refs. [171,172]. 3.2. Porous GNRs: phenyl migration and electronic structure 31 3.2 Porous GNRs: phenyl migration and electronic structure 3.2.1 Atomic structure characterization using STM Fig. 3.3a shows a STM image obtained after the phenyl migration reaction has taken place at T3= 450◦C on a flat Au(111) surface. Besides the formation of [18]-annulene pores at the ribbon edges (configuration 1), two other scenarios are also found. When a migrated phenyl does not find another side-group in the adjacent unit-cell to which it can couple, the phenyl ring remains free at the edge (configuration 1’). On the other hand, phenyl migration from C2(C3) sites to C1(C4) sites in the bay areas of the ribbon can also occur. In that case, the migrated phenyl ring fuses with the ribbon backbone via Grown on flat Au(111) Grown on stepped Au(111) (a) (b) (c) Figure 3.3: Experimental results of [18]-annulene-GNR formation. (a) STM image of GNRs grown on a flat Au(111) surface after phenyl migration reaction has been thermally induced at T3= 450◦C. The different product configurations found in the experiment are indicated with numbers in the STM image and schematically represented on the right side of the panel. (b) STM image of GNRs grown on a curved Au(111) surfaces after phenyl migration reaction has been thermally induced at T3= 480◦C. (c) Statistical analysis of the selectivity (top) and yield (bottom) obtained on the flat Au(111) annealead at T3= 450◦C (green), and the curved Au(111) annealead at T3= 480◦C (blue). Selectivity is measured as 1/(1 + 1′+ 2) ×100. The yield accounts for the relative ratio of phenyls forming [18]-annulene pores. Figure taken from Ref. [172]. 38 Chapter 3. Porous 1D and 2D nanoarchitectures Ph-7-13-AGNR [18]-annulene GNR Bonding Bonding Antibonding Antibonding Figure 3.9: Band structure and lowest-energy vacuum-state wave-functions of Ph-7-13AGNRs (left) and [18]-annulene GNRs (right). The unit-cell of the former was artificially doubled to facilitate comparison. Vacuum state bands are highlighted with thicker lines. shows the band structure and lowest-energy bay and pore state wave-functions of Ph7-13-AGNRs and [18]-annulene GNRs. The unit-cell of the former has been artificially doubled to facilitate comparison. In each of the ribbons, vacuum-states (highlighted with thicker lines) consist of 4 bands, corresponding to two pairs of states with bonding and antibonding character in the periodic direction, labeled as “bonding” and “antibonding”. Note that in the case of Ph-7-13-AGNRs this is only due to artificial band-folding, and bonding and antibonding states belong to the same band but have different Bloch phases. Consequently, bonding and antibonding bands are degenerate at the X-point. On the contrary, the introduction of pores in [18]-annulene GNRs doubles the unit-cell, and their vacuum-state bands are not degenerate at X. Besides, the two states within each of these pairs labeled as “bonding” and “antibonding” also exhibit bonding and antibonding character across the ribbon backbone. This is associated with a tiny band splitting which actually reflects some weak interaction across the ribbon backbone between vacuum states of different edges. However, this splitting is imperceptible in the energy scale selected for representing the band structure. 3.3 NPG: molecular bridge engineering After thoroughly studying the formation and electronic structure of [18]-annulene GNRs, we now turn to the case of lateral fusion of Ph-7-13-AGNRs giving rise to NPG structures (see Fig. 3.2e). In order to study the electronic properties of pp-pmand mm-NPG, it is essential to first provide a detailed description of their atomic structure. 3.3. NPG: molecular bridge engineering 39 3.3.1 Atomic structure of molecular bridges Bond-resolved STM (BR-STM) images of the three different molecular bridges (see Fig. 3.10a; see also Sec. 3.1 and Fig. 3.2e) obtained in the synthesis of NPG are shown in Fig. 3.10b. Interestingly, the image contrast depends on the type of bond formed by the phenylene units in the bridges. p-phenylenes appear as sharp contrast features, which is a common characteristic of nonplanar molecular groups measured with a CO-tip in STM [183]. On the other hand, m-phenylenes exhibit the same contrast as the C hexagons composing the honeycomb structure of the 7-13-backbone. These experimental results are corroborated by comparison to DFT calculations for pp-pmand mm-NPG struc- (a) (b) (c) (d) ExperimentTheory Figure 3.10: (a) Schematic representations of pp-, pmand mm-NPG structures. The arrows and the θand αangles represent the rotation of freedom of phenyl rings. (b) Experimental BR-STM images of pp-, pmand mm molecular bridges. The arrows indicate twisted phenyl rings. (c) Phenyl-ribbon (θ, green-pand red-m) and inter-phenyl (α, black) angles in the free-standing (solid circles) and Au(111)-supported (open circles) NPG structures. (d) Side view of free-standing (top) and Au(111)-supported (bottom) NPG structures, where the colors indicate out-of-plane displacements of each atom with respect to the average vertical position. The Au(111) substrate is omitted in the figure. Figure taken from Ref. [171]. 40 Chapter 3. Porous 1D and 2D nanoarchitectures tures in a free-standing configuration and adsorbed on Au(111)1, as shown in Figs. 3.10c,d. Specifically, the bridge conformation is characterized by two different twist angles in the relaxed structures: one between the plane of the outermost benzene ring of the GNR backbone and the adjacent phenyl ring (θ), and the second between the two phenyl rings at the bridge (α). In contrast to experiments, calculations on free standing NPGs reveal a nonplanar conformation driven by steric hindrance for the three types of bridges [184], with θand αangles exceeding 15◦and 30◦, respectively. This result suggests that the substrate plays a crucial role in stabilizing the specific bridge conformations observed in BR-STM images. In fact, calculations of NPG adsorbed on Au(111) show an overall planarization of the three NPG structures (see Fig. 3.10c,d). This is specially prominent for m-phenylenes, while p-phenylenes retain a notably tilted conformation. This effect is most visible in pm-bridges, but it can also be noticed from the more pronounced planarity of mm-bridges compared to pp-bridges. In particular, it is worth noting that the interphenyl twist angle αis as high as ∼30◦in pp-NPG. The selective planarization of m-phenylenes via interaction with the substrate and the concomitant twist of p-phenylenes constitutes an important structural feature with crucial consequences in the electronic properties of NPG, as we will explain in the following. 3.3.2 Band structure and interribbon coupling Aimed at understanding the influence of the chemical bonding configuration and the conformation of molecular bridges on the electronic structure of the synthesized NPGs, we perform DFT simulations of free-standing pp-, pmand mm-NPGs. Periodic NPG structures are arranged in the XY plane, where Y and X axes correspond to the growth direction of the ribbons and the transverse direction, respectively. The rectangular unitcells are 48.14 ˚ A×8.69 ˚ A, 48.05 ˚ A×8.69 and 48.01 ˚ A×8.69 for the pp-, pmand mm-NPG, respectively, and a vacuum region of 20 ˚ A is used in the vertical Z-direction in all of them. The unit-cells consist of 104 C and 36 H atoms. In order to disentangle the role played by the bonding configuration from that of its specific conformation (i.e. the twist angle between the phenyl rings), we start by considering planar geometries with fixed vertical atomic coordinates. Atomic positions and lattice vectors are then relaxed in the XY plane with force and pressure tolerances of 0.01 eV/˚ A and 0.25 GPa, respectively. The atomic structure of relaxed unit-cells are provided in Fig. 3.11a. Exchange-correlation is treated in the GGA-PBE approximation [138]. Core electrons are simulated using Troullier-Martins pseudopotentials [177], while valence electrons are represented using a linear combination of localized NAOs. We employ DZP orbitals, with an energy shift parameter of 0.01 Ry [148]. The real-space integration grid is defined by a cutoff energy of 400 Ry, while we use a Monkhorst-Pack grid of 4 ×15 k-points for BZ sampling [181]. Effect of chemical bond The calculated band structures of the three planar NPGs are shown in Fig. 3.11a. The case of pp-NPG qualitatively resembles that of NPG in the absence of phenyl side-groups 1DFT calculations shown in Fig. 3.10 were performed by Dr. Ane Sarasola I˜niguez. The rest of the simulations in this thesis were performed by Xabier Diaz de Cerio Palacio, unless otherwise stated. 3.3. NPG: molecular bridge engineering 41 [111]. In particular, it exhibits a finite splitting of degenerate valence and conduction single-ribbon bands. This behaviour is due to the interribbon electronic coupling, and indicates a finite transmission probability in the transversal direction. This band splitting is also responsible for a slight reduction of the bandgap with respect to the parent Ph-7-13-AGNR. On the contrary, the energy splitting of the frontier bands is drastically reduced upon the inclusion of m-bonds in the bridges, demonstrating an effective quenching of interribbon crosstalk. Accordingly, the band structure of both pmand mm-NPG closely mimic that of single-Ph-7-13-AGNR around the frontier band onsets, with negligible dispersion in the transversal X-direction. The fact that a single m-phenylene is enough to almost completely prevent interribbon coupling is in good agreement with a previous work on polyphenylene chains, where electron propagation is greatly hampered across single m-connections [185]. Similar conclusions were also extracted from theoretical studies on NPG structures with 7-13-AGNRs bonded by single m-phenylene units [161]. In particular, the electronic decoupling between adjacent ribbons is attributed to destructive quantum interference ocurring in m-bonds, as extensively explored in experimental [186,187] and theoretical [184,188–191] investigations of electron transport across molecular junctions. Moreover, quantitative information on the interribbon coupling is extracted from the energy resolved momentum splitting of valence and conduction frontier bands ∆k. This (a) (b) Figure 3.11: (a) Band structure of planar pp-(green), pm-(black), and mm-NPG (red), represented in directions parallel (Γ →Y) and perpendicular (Γ →X) to the ribbon growth direction. Atomistic models of the relaxed structures are provided below each band structure. (b) Energy-resolved interchannel coupling coefficient κc= ∆k/4 extracted from the momentum splitting of the longitudinal bands for each NPG structure. The black horizontal line at E=EVBM −0.2 eV indicates the energy position at which κcis analyzed as a function of the phenylene twist-angle in Fig. 3.12. Figure taken from Ref. [171]. 42 Chapter 3. Porous 1D and 2D nanoarchitectures quantity is directly related to the interchannel coupling coefficient, κc= ∆k/4, which characterizes the Talbot interference effect [113]. Fig. 3.11b shows κcas a function of the energy for the three different NPG structures. The contrast between pp-NPG and pmand mm-NPG is evident, showing that the bridge bonding configuration acts as chemical knob for the interribbon crosstalk. More specifically, pp-bridges lead to qualitatively higher momentum splitting over the whole energy range spanned in the figure. For example, −0.2 eV below the VBM, κcis reduced by more than a factor of 5 by introducing a single meta-connection in the molecular bridge. Effect of atomic conformation in pp-NPG The combined experimental and DFT characterization of the atomic structure in Fig. 3.10 reveals the tendency of p-phenylenes to be twisted with respect to the planar ribbon backbone. That indicates that the electronic properties of NPG might be further tuned by manipulating this conformational degree of freedom [192,193]. To analyze the relation between bridge conformation and interribbon coupling, we perform additional DFT calculations on free-standing pp-NPG. Starting from the planar geometry shown in Fig. 3.11a, we systematically increase the twist angle of the two phenyl rings in the bridge by the same amount and in opposite direction (i.e. α= 2θ). No further relaxation of the geometry is performed in order to isolate the effect of the twist angle on the electronic structure. The band structures corresponding to θ= 0◦,20◦and 45◦(α= 0◦,40◦and 90◦, respectively) rotations are shown in Fig. 3.12. The results show that the phenylene twist angle acts as a conformational knob for the interribbon crosstalk. In particular, the band splitting exhibits a gradual reduction of the interribbon coupling with increasing θ. Note that the θ= 45◦case almost exactly behaves as an array of isolated Ph-7-13-AGNRs with negligible transversal group velocity. Because the twist angle can be continuously varied, it is interesting to track the interribbon crosstalk in the whole range of twist angles. Fig. 3.12b (bottom panel, green dots) shows the momentum splitting of pp-NPG longitudinal bands at an energy of −0.2 eV below the VBM for 25 different twist angles θ(α) uniformly distributed between 0◦(0◦) and 90◦(180◦). In contrast to the abrupt on/off switching offered by the chemical knob, the conformational knob provides continuous control on the momentum splitting. In particular, κcis maximized in the planar configuration (θ= 0◦) and initiates a smooth decrease as the phenyl rings are twisted. It reaches a minimum at θ≈41.75◦, where it becomes negligible. Nevertheless, the evolution of κcis non-monotonic, and it experiences a slight growth over the next ∼15◦. This trend is again reversed around θ= 60◦, where κcbegins a new decrease that culminates at θ≈82.5◦. This specific behaviour can be understood by mapping the molecular bridge to a simple π-orbital TB model in the two-center approximation, where the four phenyl rings of the bridge are substituted by a linear chain of four π-orbitals. The hopping tbetween two orbitals in the chain varies with the twist angle (θ) between them as ∼t×cos θ. Thus, the evolution of the total hopping across the pp-bridge, and therefore the momentum splitting, can be approximated as κc(θ)∼κc(0◦)×cos2(θ)×cos(2θ), where we only considered hoppings between NNs and the fact that α= 2θ. The result obtained from this model by fitting κc(0◦) to the DFT result is represented by the blue curve in Fig. 3.12b, showing 3.3. NPG: molecular bridge engineering 43 (b) EVBM - 0.2eV (a) Figure 3.12: (a) Band structure of pp-NPG structures with planar conformation (θ= 0◦) and with phenyl rings twisted by θ= 20◦and θ= 45◦. (b) Corresponding total energy (top) and interchannel coupling coefficient κc= ∆k/4 evaluated at E=EVBM − 0.2 eV (as indicated in panel (a)) (bottom) as a function of the phenyl-ribbon (θ) and interphenyl (α) twist angles. Thick gray and yellow lines indicate the angles calculated for the free-standing and the Au(111)-supported structures. The inset in top panel provides an schematic representation of phenylene rotation in the bridges and the corresponding θ and αtwist angles. Figure taken from Ref. [171]. good qualitative agreement with DFT simulations. In particular, the first minimum at θ= 45◦corresponds to an interphenyl angle of α= 90◦, for which the hopping between the two central phenyl rings is fully quenched due to a complete loss of local π-conjugation. Note that the hopping between the two central phenyl rings is symmetric with respect to α→180◦−α, because they gradually recover coplanarity as the interphenyl angle α grows beyond 90◦. For θ > 45◦, this leads to a situation where the progressive breaking of the π-conjugation in opposite ends of the bridge competes against the gradual recovery of local π-conjugation at the central phenylene units. The outcome of that competition determines the non-monotonic behaviour of κc(θ), which eventually drops to zero for θ= 90◦, as the π-conjugation breaks completely at the two ends of the bridge. Despite the successful explanation provided by this model, DFT results exhibit slight deviations from this oversimplified picture (see for instance the small downshift of the angles at which the splitting is minimized). This quantitative disagreement may be assigned to the fact that we are neglecting hopping processes beyond NNs, as well as the effect of the atomic environment and the σ-orbitals [192]. Based on the behaviour of the interribbon crosstalk as a function of θ/α, the range of angles α∈[0◦,90◦] represents the region of interest for tuning interribbon coupling. The top panel in Fig. 3.12b shows the DFT total energy of free-standing pp-NPG as a function of the twist angle (green solid circles). The total energy landscape exhibits a ∼200 meV well around the minimum at α= 47◦, which is within the twist angle range of 44 Chapter 3. Porous 1D and 2D nanoarchitectures interest. The shape of the energy landscape around the minimum can be used to estimate thermally induced oscillations of α. For example, room temperature rotational energy (1 2kBT= 12.7 meV) could induce oscillations of ±8◦around the minimum. Beyond the vicinity of the its minimum, the total energy increases with α, and does not display any other local minima that are thermally accessible. Besides, we find from Fig. 3.10c that the ground state interphenyl angle in the Au(111)-supported pp geometry also lies within the range α∈[0◦,90◦], as schematically represented by the dashed curve in Fig. 3.12b (top panel). Remarkably, the substrate induced planarization, which reduces αfrom 47◦ to 29◦, causes a 44% increase in κc(thick gray and yellow lines in Fig. 3.12b). This result suggests the possibility of tuning interribbon coupling by depositing NPG on different substrates. 3.3.3 Quantum electron transport simulations Based on the band structure analysis reported above, it is expected that the chemical and conformational knobs will significantly influence the current propagation in NPG, particularly the electronic Talbot interference pattern emerging at large length scales [113,161]. We address this question by performing electron transport simulations of NPG devices of realistic size. Current-point injection and propagation into the device is simulated by combining NEGFs with TB Hamiltonians through the TRANSIESTA utility TBTRANS. The latter are obtained by projecting unit-cell DFT Hamiltonians on a reduced subset of atomic orbitals that is relevant to reproduce the electronic structure in the energy range of interest. This multi-scale approach allow us to treat devices in the ∼100 nm scale while preserving DFT-level of accuracy [113,161,162]. We construct this model using the SISL python software[163] to retain s,px,py,pzorbitals in the C atoms in the bridges and pzand polarization Pdxz and Pdyz orbitals in the rest of C atoms. Including s,px,pyorbitals enables an accurate description of the electronic structure of pp molecular bridges with non-planar conformations. The reliability of the model is tested by comparing the corresponding band structures to those obtained with DFT, as shown in Fig. 3.13. The selected reduced basis set provides excellent agreement with full DFT simulations for longitudinal bands within |E−EF|<1 eV. Importantly, the evolution of longitudinal band splitting with phenyl rotation is very well captured by the inclusion of s,pxand pyorbitals. We then build devices consisting of 42.759 nm ×64.950 nm NPG flakes (9 ×75 NPG unit-cells, ∼70,200 atoms). Current is injected from a metallic tip in point-contact to a single atom in the middle of a NPG ribbon. We use the wide-band limit to approximate the metallic tip via an on-site imaginary self-energy (SE) iΓ in the contact atom, where Γ represents the decay rate into the tip. This approach has demonstrated to be a reliable workaround to explicitly modelling the tip [125,161]. The device is further equipped with top and bottom drain electrodes described by NPG SEs and left and right 7.5 nm CAPs that mimic open boundary conditions, avoiding undesired backscattering off the finite device edges [113,123,161]. A full schematic representation of the transport set-up showing all the different components is provided in Fig. 3.14. The real-space current-flow of electrons injected from the tip into the device can be visualized computing bond-transmissions (see Methods in Chapter 2) [113,123,161,162, 3.3. NPG: molecular bridge engineering 45 para-para para-para para-meta meta-meta s + px + py + pzpz + Pdxz + Pdyz s + px + py + pzpz + Pdxz + Pdyz (a) (b) Figure 3.13: DFT vs reduced TB Hamiltonians. (a) Band structure of planar pp- ,pm-, and mm-NPG as calculated with DFT (black solid lines) and the reduced TB Hamiltonian (red dashed lines). A schematic top-view representation of the unit-cell structure and the basis orbitals employed for constructing reduced TB Hamiltonians is given below their corresponding band structure. (b) Band structure of pp-NPG with bridge phenylenes twisted by θ= 0◦(planar configuration, same as panel (a)), θ= 20◦ and θ= 40◦, as calculated with DFT (black solid lines) and the reduced TB Hamiltonian (red dashed lines). A schematic side-view representation of the unit-cell structure and the basis orbitals employed for constructing reduced TB Hamiltonians are given below their corresponding band structure. 194,195]. Fig. 3.15 shows bond-transmission maps evaluated at −0.2 eV below the VBM for different devices built from pp-NPGs with different twist angles (a), as well as planar pm- (b) and mm-NPG (c). The fact that frontier bands in planar pp-NPG disperse more strongly in the longitudinal (Y) than in the transversal (X) direction (Fig. 3.11) leads to a predominantly longitudinal anisotropic propagation. Nevertheless, similar to other NPG structures [113,161,162], finite interribbon coupling is responsible for the emergence of a Talbot interference pattern in the large scale propagation (Fig. 3.15a). This behaviour is abruptly modified by the introduction of m-phenylene units in the bridges (Fig. 3.15b,c). 46 Chapter 3. Porous 1D and 2D nanoarchitectures iΓ (a) (b) Tip CAP Self-energy Figure 3.14: Transport set-up. (a) Illustration representing the use of an on-site imaginary SE to simulate current injection from a metallic tip into NPG. (b) An NPG device equipped with top and bottom SEs (yellow area), and left and right CAPs (red area). The position of the tip is indicated with a yellow dot. The drastic suppression of interribbon coupling (Fig. 3.11) prevents electrons from leaking to adjacent ribbons even in the ∼100 nm scale, and current flow is almost completely confined to a single GNR. In contrast, the pp-bridge conformation reveals a more gradual modification of electron propagation, in line with the momentum splitting analysis (Fig. 3.12b). As shown in Fig. 3.15a, when the interphenyl twist angle increases the propagation becomes more anisotropic and the wavelength of the Talbot standing wavepattern increases (see pp(40◦)), slowly approaching the limit in which current is confined to a single GNR (see pp(80◦)). 3.4. Summary 47 (a) (b) (c) Bond transmission conformational chemical Figure 3.15: Large-scale bond-transmission maps evaluated at E=EVBM −0.2 eV for (a) pp-NPG with interphenyl twist angles α= 0◦(θ= 0◦, coplanar), α= 40◦(θ= 20◦) and α= 80◦(θ= 40◦), (b) planar pm-NPG, and (c) planar mm-NPG. Figure taken from Ref. [171]. 3.4 Summary In summary, we have theoretically studied experimentally synthesized [18]-annulene GNRs and NPG structures, which are fabricated from the same parent Ph-7-13-AGNR via two different on-surface chemistry reactions. By performing a DFT-based energetics analysis, we have proposed a reaction path leading to the migration of phenylene side-groups in Ph-7-13-AGNRs, which is the key distinctive step enabling the formation of [18]-annuleneGNRs. In particular, we highlight the crucial role of surface Au adatoms in facilitating the initial steps of the reaction. Additionally, our calculations reveal that closing the bays of Ph-7-13-AGNRs into [18]-annulene pores significantly affects the energetics of SAMO-like vacuum states. On the other hand, the lateral coupling of Ph-7-13-AGNRs leads to the fabrication of NPG structures exhibiting interribbon phenylene bridges with para-para,para-meta and meta-meta chemical bonding configurations. DFT-simulated band structures reveal two control knobs to vary the interribbon electronic coupling and the resulting electronic anisotropy. While the bridges containing m-bonds act as chemical knobs that abruptly suppress the coupling between adjacent nanoribbons, the conformational knob in pp-NPG 54 Chapter 4. Twisting nanoporous graphene on graphene (a) (b) Figure 4.2: (a) TB vs DFT band structure of the aligned NPG/graphene bilayer in the AB stacking configuration. (b) TB vs DFT DOS of a NPG/graphene bilayer twisted by θ= 21.78◦. Black curves are the total DOS, while red and blue curves are the projected DOS (pDOS) on NPG and graphene, respectively. Solid lines correspond to DFT and dashed lines to the TB model. size. The TB model and electronic structure calculations have been set up using the SISL python package [163]. DFT calculations DFT simulations are performed by treating exchange-correlation energies within the GGA-PBE framework [138]. Core electrons are addressed by norm-conserving TrouillerMartins pseudopotentials [177], while a linear combination of atomic orbitals is used for valence electrons. We employ a DZP basis set, with the basis orbitals range defined by a 0.01 Ry energy shift [148]. The BZ is sampled using Monkhorst-Pack k-grids with 5×19 k-points in aligned bilayers and 11 ×7k-points in the θ= 21.78◦case [181]. The real-space grid is defined by a 400 Ry mesh-cutoff. The employed TB model does not consider effects derived from atomic structure relaxations, and optimizing the geometry in the DFT simulations would most likely introduce quantitative effects that will hamper a proper fitting of the model parameters. Geometry and lattice relaxations are therefore not consider, and the C-C distance and interlayer separation are fixed to a= 1.42 ˚ A and d= 3.35 ˚ A, respectively. The aligned NPG/graphene bilayer forms a rectangular 31.97 ˚ A×8.52˚ A unit-cell, while the lattice vectors of the bilayer twisted by θ= 21.78◦are a1= (−22.14,55.38) ˚ A and a2= (−20.91,138.45) ˚ A. The first consists of 184 C and 20 H atoms, while the latter has 1288 C and 140 H atoms. We include a vacuum region of 30 ˚ A in the direction perpendicular to the bilayers in order to avoid spurious effects between periodic images. 4.2. Electron transport simulations 55 4.2 Electron transport simulations 4.2.1 Device set-up and simulation details Based on the above-described TB model, we simulate quantum transport as a function of the interlayer twist angle on NPG/graphene devices of realistic size (∼100 nm) in point contact with a metallic tip. The calculations are performed within the NEGF formalism as implemented in the TRANSIESTA utility TBTRANS [157]. Contrary to the method applied in Chapter 3, we do not employ self-energies (SEs) of the bilayers as electrodes to drain the current injected from the metallic tip. On the one hand, the non-rectangular unit-cells of twisted NPG/graphene bilayers can take very different shapes for different twist angles, which would prevent us from building device geometries systematically. Besides, generating SE objects for the huge supercell sizes (∼1000 atoms) considered here would have a high computational cost, making the approach practically unfeasible. For these reasons, we consider a single-electrode set-up composed of a finite circular bilayer disk in point-contact to a metallic tip in the wide-band limit. The tip is in contact with a single site in the NPG layer and is simulated via an on-site imaginary SE term, as described in Chapter 3and Ref. [161]. In particular, we consider contact to a site in the center of a ribbon and closest to the rotation center of the bilayer. This choice of contact position ensures that, for all twist angles, the current is injected in a region with local AB stacking, providing consistency among devices. The circular bilayer disk has a diameter of 70 nm and is surrounded by an isotropic 10 nm wide complex adsorbing potential (CAP) ring (see Chapter 2) which prevents backscattering off the finite device edges and mimics open-boundary conditions by adsorbing propagating electrons. 10 nm 50 nm CAP (a) (b) Tip Y (longitudinal) X (transversal) Z X Y Figure 4.3: Transport set-up. (a) Illustration representing the current injection from a metallic tip in contact to a single NPG atom. (b) Schematic representation of the whole device set-up, consisting of a circular NPG/graphene flake equipped with a surrounding CAP (red area). The yellow dot indicates the position of the tip. 56 Chapter 4. Twisting nanoporous graphene on graphene NPG (a) (b) Figure 4.4: (a) Bands structure of single-layer NPG, as obtained from the single-layer TB Hamiltonian used in this chapter. The red horizontal line indicates the energy at which bond-transmissions are evaluated in panel (b). (b) Bond-tranmissions in a singlelayer NPG device, where we use a circular ≈130 nm NPG flake surrounded by a 10 nm-wide surrounding CAP, as explained in the text and illustrated in Fig. 4.3. Bond transmissions are evaluated at E= 0.5 eV, as indicated in panel (a). A schematic representation of the transport set-up is provided in Fig. 4.3. Importantly, the use of a CAP as the only drain for electrons is in contrast to the accurate SE method used to simulate open-boundary conditions along the transport direction in Chapter 3. In order to test the reliability of our approach, we first simulate the electron propagation in a single-layer NPG using the transport set-up shown in Fig. 4.4. The simulated bond-transmissions evaluated at an energy within the longitudinal bands of NPG (0.5 eV) exhibit the expected large-scale Talbot interference pattern, in excellent agreement with Ref. [113] which combines SEs and CAPs as illustrated in Fig 3.14 of Chapter 3. We extend this test to aligned NPG/graphene bilayers, by comparing bond transmissions obtained using a surrounding CAP (Fig. 4.3) to those obtained combining SEs and CAPs (Fig 3.14). These two approaches are in very good qualitative agreement, as demonstrated by their bond transmissions in both NPG and graphene layers in Fig. 4.5. Although we restrict our problem to commensurate interlayer rotations, the use of surrounding CAPs allows to simulate large-scale electron transport properties for any arbitrary commensurate or incommensurate twist-angle, as opposed to electrode SEs that require the scattering region to be periodic. 4.2.2 Large-scale current propagation Fig. 4.6 shows the real-space in-plane bond-transmission of electrons injected at the position of the red dot in NPG and propagating into the bilayer. We represent the bond transmissions evaluated at an energy E=−0.4 eV, lying in the energy range of longitudinal bands of single-layer NPG at which Talbot interference is known to emerge 4.2. Electron transport simulations 57 Graphene Graphene NPG NPG (a) (b) Surrounding CAP Top and Bottom SEs + Left and Right CAPs Figure 4.5: Bond transmissions in NPG (top) and graphene (bottom) evaluated at E=−0.4 eV, as obtained using (a) a surrounding CAP (Fig. 4.3), and (b) a combination of top and bottom SEs and left and right CAPs (Fig 3.14). The scale of the bondtransmission magnitude is the same in both panels. [113]. In the aligned case (θ= 0◦in AB stacking) such a characteristic interference effect is observed in NPG, although significantly smeared out by the presence of the underlying graphene substrate. Remarkably, such smeared Talbot interference pattern is also imprinted into graphene, inducing anisotropic electron propagation in an otherwise quasi-isotropic crystal. When NPG is rotated by a small twist angle (θ= 1.92◦), the interference pattern is heavily perturbed and further smeared out. At the same time, the interlayer transmission remains strong, indicating an enhanced interlayer coupling. Interestingly, for larger twist angles the transmission pathways become strongly asymmetric with respect to the contacted GNR (see θ= 5.78◦,10.53◦), which can be explained by the twist-induced breaking of the in-plane mirror symmetry around it. Besides, this introduces chirality in the system, as the predominant direction of propagation with respect to the contacted GNR would be reversed for θ→ −θ. Simultaneously, the interlayer transmission decreases upon increasing the twist angle and, consequently, the electronic propagation is gradually 58 Chapter 4. Twisting nanoporous graphene on graphene θ=0∘ θ=1.92∘ θ=5.78∘ θ=10.53∘ θ=21.78∘ Graphene Bond transmission Bond transmission NPG X Y Figure 4.6: Real-space bond-transmission maps at an energy E=–0.4 eV in NPG/graphene for different twist angles. Top and bottom rows shows bond-transmissions through NPG and graphene, respectively. The red dot indicates the position in the XY plane at which electrons are injected in NPG. Scale bar in the bottom-right panel (θ= 21.78◦) is 10 nm. restricted to NPG. Eventually, the asymmetries disappear and the bond-transmissions exhibit a perfect Talbot interference pattern along NPG for θ= 21.78◦, mimicking singlelayer-like transport behaviour and providing evidence of the electronic decoupling from graphene. In order to elucidate whether the predicted transport behaviour depends on specific structural details of the device, such as the initial stacking, rotation center or the current injection position, we perform additional quantum electron transport simulations for devices with a modified set-up. In particular, starting from NPG/graphene bilayers in AA stacking (θ= 0◦) we rotate the NPG layer with respect to an axis crossing atomic sites in both layers, as shown in Fig. 4.7(a). The metallic tip is then placed in contact to the NPG atom at the rotation center, which ensures injection in an AA site regardless of the twist angle. Fig. 4.7(b) shows, for different twist angles, bond-transmissions of electrons injected with an energy E=−0.4 eV. The previously observed trends are qualitatively reproduced by this set-up. Namely, interlayer transmission is progressively quenched as the twist angle increases and the electrons propagate asymmetrically for small finite rotations (θ≲10◦). We therefore conclude that the observed transport behaviour is robust against details of the device set-up. 4.3. Electronic structure: origin of decoupling and chiral currents 59 (AA stacking) θ=0∘ θ=1.92∘ θ=5.78∘ θ=10.53∘ θ=21.78∘ Graphene NPG Bond transmission Bond transmission (a) (b) X Y Figure 4.7: (a) Atomic structure of a twisted NPG/graphene bilayer initially in an AA stacking configuration, where the black dot indicates the selected rotation center. (b) Real-space bond transmission maps at an energy E=–0.4 eV for different twist angles of the NPG/graphene bilayer shown in (a). The red dot indicates the injection point in the NPG layer, which coincides with the rotation center (black dot in panel (a)). Scale bar is 10 nm. 4.3 Electronic structure: origin of decoupling and chiral currents 4.3.1 Momentum-space analysis We have focused our analysis of large-scale quantum electron transport simulations on the evolution of the Talbot interference pattern and its interlayer transmission as a function of the twist angle. In order to achieve deeper understanding of the observed transport behaviour, it is key to track the evolution of the longitudinal frontier bands of NPG and their coupling to graphene Dirac cones in the band structure. However, twisted commensurate NPG/graphene bilayers give rise to skewed supercells that break the mirror-symmetry of the aligned geometry, and it is not straightforward to systematically define a convenient BZ path to investigate the NPG bands in the bilayer. Besides, the high number of single-layer unit-cell repetitions required to build the new periodic supercell of the bilayer leads to a large increase of the number of states upon band-folding into a smaller BZ. Consequently, systematically disentangling the band structure of twisted NPG/graphene bilayers becomes a cumbersome task. Here we project the band structure of NPG/graphene bilayers on NPG, and we unfold them to the BZ of the latter, more particularly to the X→Γ→Ypath commonly employed to study its characteristic electronic anisotropy. A detailed description of the method is given in Fig. 4.8. This simple reconstruction of the NPG band structure allows us to access the evolution of its frontier states under the influence of the underlying graphene. Fig. 4.9 shows the band structure of twisted NPG/graphene bilayers, projected on NPG and unfolded to its BZ, for the twist angles considered in transport simulations (see Fig. 4.6). The band structure of single-layer NPG is provided for comparison in the panel corresponding to θ= 0◦(gray dashed lines). The aligned case (θ= 0◦, AB stacking) exhibits a strong hybridization between the longitudinal bands of NPG and the 60 Chapter 4. Twisting nanoporous graphene on graphene Γ Y X bz (NPG) BZ (bilayer) qi  ki(qi) Sample convenient path in bz: qi Diagonalize monolayer NPG Hamiltonian for each : HNPG qi HNPG(qi)→({ϕnqi},{ϵnqi}) Repeat monolayer NPG cell to match NPG layer geometry in the NPG/graphene bilayer. Apply Bloch theorem to accordingly. {ϕnqi} Fold to BZ: qi qi→ ki Diagonalize bilayer Hamiltonian for each : H  ki H( ki)→({Ψm ki},{ϵm ki}) Obtain the NPG contribution to each bilayer eigenstate by projecting on NPG eigenstates: PNPG(m,qi)=∑ n |⟨ϕn,qi|Ψm, ki(qi)⟩|2 Plot the bilayer energy-momentum dispersion in the selected bz path by weighting the eigenvalues according to their NPG contribution . ϵm, ki(qi) PNPG(m,qi) Method for unfolding of projected band structure folding Figure 4.8: Schematics of the method to unfold the projected band structure of NPG/graphene bilayers. (a) Representation of the Brillouin zone of a twisted NPG/graphene bilayer (BZ, black) and single-layer NPG (bz, red). (b) Flowchart of the method. Dirac cones of graphene, upon folding of the graphene K(K′) points to ±2/3(π/Lx) into the ΓXline, where Lx= 13√3ais the lattice constant of NPG in the direction across the ribbons. The interlayer coupling leads to a finite contribution of NPG to graphenelike states even near the Fermi level, and to the opening of a tiny bandgap in the Dirac cone. The latter effect is enabled by the small band misalignment between the layers (see Sec. 4.1.2) which, similarly to AB bilayer graphene [211,212], can be tuned using external electric fields perpendicular to the surface [210]. Besides, longitudinal NPG bands exhibit significant energy renormalization, effectively reducing the NPG bandgap at the Γ point. Remarkably, inter-GNR band splitting in the ΓYdirection remains weak, i.e. longitudinal states retain their anisotropic character. The band structure gets notably modified by twisting to θ= 1.92◦. On the one hand, graphene-like features are no longer visible. On the other hand, longitudinal NPG bands are strongly perturbed, exhibiting numerous avoided crossings and weakly dispersing branches predominantly around the onset of the frontier states. Remarkably, some of these branches appear at energies as low as E≈ −78 meV in the occupied region, well 4.3. Electronic structure: origin of decoupling and chiral currents 61 Projection on NPG Figure 4.9: Band structure of NPG/graphene at different θ, projected on NPG and unfolded to its Brillouin zone. Unfolded bands are represented in directions across (X→ Γ) and along (Γ →Y) the NPG ribbon axis. The band structure of single-layer NPG is plotted as a reference in the panel corresponding to θ= 0◦(gray dashed lines). inside the single-layer bandgap. As the twist angle keeps increasing, avoided crossings become weaker and are restricted to higher energies, eventually leading to NPG singlelayer-like dispersion at θ= 21.78◦. This evolution of the band structure demonstrates an effective electronic decoupling between NPG and graphene for increasing values of θ, and satisfactorily explains the gradual suppression of interlayer transmission and simultaneous recovery of single-layer-like Talbot interference obtained from our quantum electron transport simulations. However, the results shown in Fig. 4.9 do not explain the origin of asymmetric in-plane propagation (see Fig. 4.6). The reason is that our unfolding method provides the projected band structure in the 1st BZ (1BZ) of single-layer NPG, which is not a true physical BZ of the system, but a conveniently selected auxiliary reciprocal-space region that allows a meaningful representation of NPG bands. As such, projected band structures unfolded to that auxiliary BZ are not expected to respect the symmetries of single-layer NPG, unless the interaction with graphene becomes negligible. Consequently, the information provided in Fig. 4.9 is incomplete regarding the specific band dispersion in different directions. We thus extend our method to the whole 2D momentum-space and plot Fermi surfaces corresponding to the band structure of NPG/graphene bilayers projected on NPG and unfolded to its 1BZ. Fig 4.10 shows these Fermi surfaces evaluated at an energy of E= −0.4 eV, which allows to establish a direct connection to the electron transport simulations shown in Fig. 4.6. At AB stacking (θ= 0◦), the Fermi surface projected on NPG contains graphene-like branches emerging from the folding of Dirac cones. The latter appear as two circles centered on k= (±2π 3Lx,0) and are coupled to each other mediated by the hybridization with NPG bands. This is accompanied by an additional pair of NPGlike branches at larger ky. These correspond to anisotropic longitudinal bands, whose splitting in the ky-axis decreases as kxapproaches the BZ boundary at π/Lx, where they are degenerate. Importantly, the Fermi surface formed by this set of branches is characterized by the mirror symmetry of the aligned geometry, thus exhibiting symmetric band dispersion with respect to the Y-axis (i.e. the axis parallel to the GNR growth direction). As a combination of interlayer coupling and twist-induced mirror-symmetry-breaking, the Fermi surface appears strongly and asymmetrically (with respect to the Y-axis) mod- 62 Chapter 4. Twisting nanoporous graphene on graphene Projection on NPG Figure 4.10: Fermi surfaces of NPG/graphene at E=–0.4 eV and different θ, projected onto the electronic states of NPG and unfolded to its Brillouin zone. X(Y)-axis corresponds to the direction across (along) the NPG ribbon axis. ified at small θ(see θ= 1.92◦and 5.78◦). As θkeeps increasing, the Fermi surface of single-layer NPG is gradually restored, with weaker avoided crossings asymetrically distributed with respect to the Y-axis (see θ= 10.53◦). This effect breaks the symmetry of the dispersion relation in the direction across GNRs, and is the key feature leading to the asymmetric propagation shown in Fig. 4.6. For θ= 21.78◦, the interlayer coupling is so weak that the mirror-symmetry-breaking becomes irrelevant and a single-layer transport behaviour is restored. Despite demonstrating an effective electronic decoupling upon twisting, Figs. 4.9 and 4.10 do not answer the question on why the interaction between NPG and graphene is quenched. In order to answer this question, it is instructive to provide a continuum description of the tunneling processes between Bloch states in different layers. In particular, we follow an approach that has been widely used to derive continuum models for TBLG [51,151,225–228]. We start by considering a system composed of any two parallel layers separated by a constant interlayer distance. The two layers differ from each other, their lattice constants and atomic indices being different. In a single-orbital TB approximation, a Bloch state of the top layer with band index nand crystal momentum kis written as |ψb nk⟩=1 √NX i a(nk) iX Rb eik·(Rb+τb i)|Rb+τb i⟩,(4.7) where Rbare lattice vectors, τb iis the sublattice vector indicating the position of orbital iinside the unit cell, |Rb+τb i⟩is an orbital centered on position Rb+τb i, and Nis the number of unit cells in the layer. Similarly, a Bloch state on the top layer with band index mand crystal momentum pis given by |ψt mp⟩=1 √MX i c(mp) iX Rt eip·(Rt+τt i)|Rt+τt i⟩.(4.8) The rotation of the top layer with respect to the bottom layer is described by a rotation matrix Mθ.|Rt+τt i⟩is then replaced by |R′t+τ′t i⟩, and real-space lattice and position vectors and crystal momentum are replaced by r′t=Mθrtand p′=Mθp, respectively. By using primes to indicate rotated top layer variables, the hopping term describing the coupling between a Bloch state with band index mand crystal momentum p′in the 4.3. Electronic structure: origin of decoupling and chiral currents 63 top layer, and a Bloch state with band index nand crystal momentum kin the bottom layer is given by T(bt) nm (k,p′) = ⟨ψb nk|ˆ Hinter|ψt mp′⟩ =1 √NM X i,i′ (a(nk) i)∗c(mp′) i′X Rb,R′t e−ik·(Rb+τb i)eip′·(R′t+τ′t i′)×⟨Rb+τb i|ˆ Hinter|R′t+τ′t i′⟩. (4.9) According to the two-center approximation employed in our TB model ⟨Rb+τb i|ˆ Hinter|R′t+τ′t i′⟩=tinter(Rb+τb i−R′t−τ′t i′),(4.10) and T(bt) nm (k,p′) = 1 √NM X i,i′ (a(nk) i)∗c(mp′) i′X Rb,R′t e−ik·(Rb+τb i)eip′·(R′t+τ′t i′) ×tinter(Rb+τb i−R′t−τ′t i′), (4.11) where tinter(r) is given by Eq. 4.6. Because tinter(r) is a smooth function of the planar projection of the interatomic distance, we can Fourier expand it in the continuum of momentum: tinter(Rb+τb i−R′t−τ′t i′) = 1 NΩbX q tqeiq·(Rb+τb i−R′t−τ′t i′) =1 NΩbX q∈BZ X G tq+Gei(q+G)·(Rb+τb i−R′t−τ′t i′), (4.12) where tqis the Fourier transform of tinter(r), momentum qin the bottom row runs over the 1BZ of the bottom layer, Gis a bottom layer reciprocal lattice vector, and Ωbis the area of the bottom layer unit-cell. Inserting Eq. 4.12 into Eq. 4.11 yields T(bt) nm (k,p′) = 1 √NM 1 NΩbX i,i′ (a(nk) i)∗c(mp′) i′X qX G tq+Gei(q+G−k)·τb iei(p′−q−G)·τ′t i′ ×X Rb ei(q+G−k)·RbX R′t ei(p′−q−G)·R′t. (4.13) Then, using the relations X Rb ei(q+G−k)·Rb=NX Gb δk+Gb,q+G X R′t ei(p′−q−G)·R′t=MX G′t δp′+G′t,q+G,(4.14) 70 Chapter 4. Twisting nanoporous graphene on graphene get gradually confined to a single layer and the IPR decreases. 4.4 Spectroscopic signatures of interlayer coupling Different effects of the interlayer coupling, such as twist-dependent van Hove singularities and moir´e states, are commonly measured in TBLG [231–233] and other van der Waals heterostructures [218] using dI/dV point-spectroscopy and mapping in STM. Similarly, we expect that the results presented in this chapter could be explored using dI/dV characterization. In particular, in a STM experiment, we expect the layer placed further from the tip to represent a negligible contribution to the differential conductance, due to the fast decay of the electronic density over the relatively large interlayer distance. Assuming that the STM tip approaches the sample from above, the measurements will be dominated by the pDOS of NPG [234,235]. Consequently, we look for spectroscopic signatures of the interlayer coupling by computing the pDOS on NPG, as shown in Fig. 4.16. At θ= 0◦, the hybridization significantly modifies the pDOS of NPG with respect to the singleTwist angle Figure 4.16: Density of states of NPG/Graphene projected on NPG at different θ. The DOS of single-layer NPG is shown as a reference in all cases (gray shaded curve). The pDOS is normalized with respect to the number of atoms in each unit-cell. Red and blue arrows in the spectrum corresponding to θ= 1.92◦indicate the energies at which local DOS is represented in Fig. 4.17,E=−78 meV and E=−400 meV, respectively. 4.4. Spectroscopic signatures of interlayer coupling 71 layer behaviour (gray shaded curve). In particular, new peaks arise inside the bandgap of single-layer NPG and around its frontier band onsets, with electronic density slowly decaying towards the Fermi level. This result is in good agreement with the remarkable energy renormalization of frontier NPG-like states and the concomitant NPG contribution to Dirac-like states observed in the projected band structure (Fig. 4.9). When a small twist angle (see θ= 1.92◦) is introduced between the layers, the peaks emerging in the aligned case split in a bunch of new additional resonances, originating from the multiple avoided crossings in the band structure (Fig. 4.9). These resonances gradually move out from the single-layer NPG bandgap as θis further increased, and the spectrum eventually resembles that of single-layer NPG. Additional information about the spatial distribution of hybrid bilayer states can be extracted by representing the local DOS (LDOS) evaluated at relevant energy positions. Fig. 4.17(b) shows the LDOS on NPG for NPG/graphene at θ= 1.92◦, evaluated at the energies indicated by blue (E=−400 meV) and red (E=−78 meV) arrows in Fig. 4.16. These correspond to the energies selected in our electron transport simulations (Fig. 4.6) and to the lowest energy shoulder observed in the pDOS (Fig. 4.16), respectively. At E=−400 meV, the LDOS on the NPG layer very closely resembles the VB of single-layer NPG, represented in Fig. 4.17(a), and does not display any appreciable modulation despite the strong interlayer hybridization at such energy (Fig. 4.15). On the contrary, at E=−78 meV, the electronic density exhibits a marked long-range modulation in the scale of the moir´e pattern of the bilayer, while conserving an NPG-like character. In particular, the density at this low-energy resonance locally vanishes around the domain walls separating regions of local AB and BA stacking that are perpendicular to the nanoribbon axis. E = -400 meV E = -78 meV E = -195 meV Monolayer NPG Twisted NPG/graphene bilayer Y X (a) (b) Figure 4.17: (a) LDOS map of single-layer NPG evaluated at the VB onset energy (E=−195 meV). The LDOS of single-layer NPG does not change appreciably over the energy range E∈[−500,−195] meV. (b) LDOS on NPG for NPG/graphene at θ= 1.92◦ evaluated at E=−400 meV and E=−78 meV. These energies are indicated in Fig 4.16 by blue and red arrows, respectively. The scale bar in the right panel is 1 nm and the NPG geometry is overlaid in the bottom right corner of all panels. 72 Chapter 4. Twisting nanoporous graphene on graphene 4.5 Effect of out-of-plane corrugation in the interlayer coupling As discussed in Section 4.1.1, lattice and out-of-plane atomic relaxations play a significant role in the electronic properties of TBLG at small twist angles [220–222]. While in-plane moir´e pattern reconstructions may be neglected, out-of-plane atomic corrugation might be significant in NPG/graphene bilayers twisted by θ= 1.92◦[222]. To estimate the influence of out-of-plane distorsions on the interlayer electronic coupling, we build a simplified model based on the phenomenology observed in TBLG. In particular, the interlayer distance in TBLG is enhanced in regions of AA stacking, while it remains unchanged in regions of AB stacking [220,221]. Thus, we propose to correlate the interlayer distance with the local stacking order at each atom of the NPG layer. For a NPG/graphene bilayer twisted by θ= 1.92◦, we quantify the local stacking order siof each atom iin the NPG layer by si=1−max(di, dN) a×1−min(di, dN) a×1−min(di, dN) a.(4.20) Here, diis the planar projection of the distance from atom iin NPG to the closest atom in the graphene layer, while dNis the average of the planar projection of the distance from the NNs of atom iin NPG to their closest atoms in the graphene layer. ais the in-plane C-C distance, and siis bounded between 0 and 1. If the local stacking resembles a perfect AB configuration, si≈0, while a perfect local AA stacking corresponds to si≈1. We then correlate the local stacking order with the local corrugation by applying an outof-plane displacement of ∆zi=si×r×d0on each atom i, where ris the proportion by which the reference interlayer distance d0= 3.35 ˚ A is increased in the position of maximum corrugation. This procedure leads to a periodic corrugation of the bilayer, as shown in Fig. 4.18a,b for NPG/graphene with θ= 1.92◦twist angle, which in turn results in a periodic modulation of the interlayer hopping integrals. Using this model, we compute the IPR for different degrees of corrugation (up to 8% of the interlayer distance for the planar bilayer) for NPG/graphene with θ= 1.92◦. As a reference, changes in the interlayer distance of ∼7−8 % have been predicted for graphene bilayers twisted by θ < 3◦[220,221]. The results shown in Fig 4.18c indicate that the IPR and, thus, the interlayer hybridization is reduced upon increasing the corrugation. However, even in the case of 8% corrugation, the hybridization at the energies of interest is only reduced by ∼10%. Furthermore, the IPR at E = -0.4 eV remains above the value for the aligned NPG/graphene bilayer at all degrees of corrugation. Hence, we can assume that the out-of-plane distorsions do not qualitatively modify the main conclusions of this chapter. In fact, the fact that corrugation induces a reduction of the interlayer coupling reinforces the ability of decoupling NPG and graphene by twisting. 4.6. Summary 73 AA AA AA AB AB AB AB AB (a) (b) (c) Figure 4.18: Effect of corrugation in NPG/graphene bilayers twisted by θ= 1.92◦. (a) Atomic structure showing the moir´e pattern, where AA and AB regions are indicated. (b) Colormap showing the interlayer distance in each atom of the NPG layer. (c) IPR for different values of the corrugation. The horizontal blue dashed line indicates the IPR of aligned NPG/graphene (AB stacking) at an energy E = -0.4 eV. 4.6 Summary In this chapter, we have combined a simple TB model and NEGFs to demonstrate that the electronic and transport properties in NPG/graphene bilayers can be efficiently tuned via an interlayer twist angle. At small angles (θ≲11◦), electronic currents injected in NPG exhibit strong asymmetric perturbations in the Talbot interference pattern, which is transmited to the underlying graphene layer. This behaviour reveals a significant interlayer coupling, as confirmed by the strong hybridization of single-layer electronic bands. On the contrary, large twist angles (θ≳11◦) effectively decouple the layers and lead to a single-layer-like propagation in NPG, which is characterized by a perfect Talbot interfer- 74 Chapter 4. Twisting nanoporous graphene on graphene ence pattern. The progressive suppression of the interlayer hybridization with increasing θ is explained by means of twist-induced momentum separation between single-layer eigenstates of NPG and graphene. We also present the IPR as a useful method to quantify the hybridization state of the bilayers. From a computational perspective, we have demonstrated the effectiveness of using surrounding CAPs, instead of computationally heavier and more constraining SEs, in order to mimic open boundary conditions in electron transport simulations. Besides, we have made the otherwise cumbersome dispersion relation of twisted NPG/graphene accessible by implementing a method for unfolding their layer-projected band structures. Finally, we show that the pDOS and LDOS provide spectroscopic signatures of the twist-dependent electronic coupling between NPG and graphene, which could be probed in future STM experiments. Remarkably, recent experimental advances enable accurate measurements of momentum-resolved electron bands via the Quantum Twisting Microscope (QTM), which may be used to confirm the coupling mechanism exposed in this chapter [236]. CHAPTER 5 Disentangling deceptive orbital confinement at the edges and pores of carbon-based nanoarchitectures As we have shown in Chapter 3, the electronic structure of carbon-based nanoarchitectures can be experimentally investigated with atomic-level precision via STS. For example, dI/dV point-spectroscopy and mapping have been extensively used to determine the electronic bandgap of a wide variety of systems and characterize the energy and spatial distribution of their frontier electronic orbitals [83,85,86,96]. However, STS measurements can also yield ambiguous results that lead to erroneous interpretations of the electronic properties. In 7-AGNRs, for instance, S¨ode et al. revealed that the VB and the CB+1 wave functions appeared confined to the vacuum region along the edges in dI/dV maps (see Fig. 5.1a) [237]. Paradoxically, DFT predicted these states to be delocalized over the Exp. dI/dV (b) Theory (a) Exp. dI/dV Theory VB CB VB CB Figure 5.1: Electronic orbital confinement in AGNRs and ZGNRs. (a) STS measurements in a 7-AGNR reveals valence states that are localized in the ribbon edges, while the corresponding DFT-calculated wave function is delocalized over the entire ribbon backbone. Top figure taken from Ref. [82]. Reproduced with permission from Springer Nature. Bottom figures reprinted with permission from Ref. [237]. Copyright 2015 by the American Physical Society. (b) Both experimental dI/dV maps and DFT calculations reveal the existence of low-energy states inherently localized in the edges of ZGNRs. Figure taken from Ref. [83]. Reproduced with permission from Springer Nature. 76 Chapter 5. Deceptive orbital confinement in carbon-based nanoarchitectures entire 7-AGNR backbone. The apparent confinement of VB and CB+1 wave functions was attributed to their particular oscillation pattern across and along the backbone. Regions of positive and negative wave function weight would lead to a local cancellation of the STS signal, leading to the observation of a deceptive distribution of the electronic density [237]. dI/dV mapping has also revealed a strong localization of the VB and CB at the edges of ZGNRs (see Fig. 5.1b) [83]. In contrast to 7-AGNRs, electronic localization in ZGNRs is unambiguously attributed to their so-called edge states. These are frontier electronic states that are inherently localized at the zigzag edges [57,238], and their experimental observation has been corroborated by DFT simulations [83]. Besides, a similar electronic confinement is also characteristic of bay and pore states (b) 2.2 V 3.2 eV Exp. dI/dV Theory Exp. dI/dV (d) STM Exp. dI/dV Theory Exp. BR-STM (a) (c) STM Exp. dI/dV Figure 5.2: Electronic orbital confinement in carbon-based nanoarchitectures exhibiting notched edges and nanopores. (a) In a porous nanographene, the experimental dI/dV signal obtained at 2.4 V is strongly confined to the nanopore. This measurement was assigned to a vacuum pore state predicted by DFT at 3.25 eV. Reprinted with permission from Ref. [239]. Copyright 2018 American Chemical Society. (b) In NPG, the strong confinement of dI/dV features to the pore regions at 2.2V was attributed to the existence of intrinsically localized pore states, which were predicted at 3.2 eV by DFT. Figure taken from Ref. [111]. Reprinted with permission from AAAS. (c) dI/dV maps reveal electronic states confined to the bays of a GNR with notched edges. Such a localized signal was assigned to electronic states delocalized over the entire carbon-network. Figure taken from Ref. [240]. Reproduced with permission from Springer Nature. (d) The electronic density appears confined to the coves of a chevron GNR. This measurement was attributed to the formation of a standing wave pattern by the underlying Au(111) surface state. Reproduced from Ref. [182] with permission from the Royal Society of Chemistry. 5.1. Synthesis of gulf-GNRs and NPG 77 originating from IPSs (see Fig. 5.2a,b). These are inherently localized at the bays and pores of nanostructures [111,239], as we have described for the Ph-7-13-AGNRs and [18]-annulene GNRs studied in Chapter 3. Nevertheless, compared to the localized dI/dV signals observed at energies around the bandgap in 7-AGNRs and ZGNRs, combined STS and DFT characterization has determined that bay and pore states appear at much higher energies. More generally, localization effects like those described above have been routinely observed in many other on-surface synthesized carbon-based nanoarchitectures exhibiting notched edges and nanopores (see Fig. 5.2c,d). Contrary to bay and pore states, these localized STS signals have been usually observed at low energies around the frontier state onsets. Regarding their origin, it has been often assigned to states whose wave functions are actually delocalized over the entire carbon-network according to theoretical predictions (Fig. 5.2c)[175,240–246]. Additionally, electronic localization has also been related to the underlying Au(111) substrate, alleging that the surface state forms a standing wave pattern at the coves of chevron-GNRs [182] (Fig. 5.2d). The existence of multiple, apparently plausible interpretations suggests that a better understanding of edge and nanopore-localized STS signals is still lacking. In this chapter, we provide a thorough DFT-based investigation of the recurrent experimental observation of electronic localization, using as reference systems on-surface synthesized gulf-type GNR (g-GNR) and NPG structures. We find that the orbital confinement observed in dI/dV maps is caused by the decay of delocalized sample wave functions into the vacuum, and is enhanced with increasing tip-sample separation. Supported by additional simulations of related systems, we show that this effect, determined by the specific edge and nanopore geometry, is present in a wide range of carbon-based nanoarchitectures. Furthermore, we confirm the validity of DFT-LDOS map simulations in the Tersoff-Hamman approximation to address deceptive orbital confinement. 5.1 Synthesis of gulf-GNRs and NPG We start by briefly describing the fabrication and atomic strucure of the systems that serve as reference for our study. The on-surface synthesis of g-GNRs has been performed starting from the precursor 4’,5”-dibromo1,1’:2’,1”:2”,1”’-quaterphenyl (DBQP) monomer and following thermally-activated polymerization (T= 250◦C) and cyclodehydrogenation (T= 500◦C) reactions on Au(111), as shown in Fig. 5.3a. Fig. 5.3b exhibits the noncontact AFM (nc-AFM) image of a g-GNR, which consists of a 3-ZGNR backbone decorated with periodic single-phenyl-ring protrusions in a staggered arrangement at both sides of the ribbon. Further annealing the substrate to T= 600◦C triggers the lateral dehydrogenative cross-coupling of g-GNRs into NPG structures. The resulting 2D structures consist of periodic out-of-phase arrays of small nanopores where a single phenyl ring is missing (≈4.7˚ A in diameter) (Fig. 5.3c). Notched gulf edges and nanopores make these platforms ideal for the study of electronic confinement in well-defined vacuum regions. Accordingly, next sections focus on the study of the electronic structure of g-GNRs and NPG using STS characterization and DFT simulations. 78 Chapter 5. Deceptive orbital confinement in carbon-based nanoarchitectures (b) (c) 250 ºC Au(111) 500 ºC 600 ºC NPG g-GNR (a) Figure 5.3: (a) Schematic representation of the synthetic path followed to fabricate gGNRs and NPG. (b,c) Nc-AFM images of g-GNR and NPG. Figure taken from Ref.[180]. 5.2 Electronic structure of gulf-GNRs 5.2.1 Experimental STS characterization Fig. 5.4a shows dI/dV point-spectroscopy measurements performed with a CO-tip on different positions of a g-GNR. The spectra exhibits a semiconducting bandgap, with VB and CB onsets at −1V and 1.7V, respectively. Interestingly, the CB onset is remarkably more intense when measured in the gulfs (red curves), while the VB onset is enhanced in the gulfs and protruding phenyls (blue curves). This distinct behaviour is clearly reflected by the electronic density distribution in dI/dV maps taken around the VB and CB onset energies (Fig. 5.4b). While occupied states (-1.3 V and -1.1 V) are more intensely located along the edges, unoccupied states (1.9 V and 2.1 V) appear as intense conductance features confined to the gulfs, notably extending into the lateral vacuum region and with negligible weight in the carbon-network. As described above, this observation resembles experimental dI/dV maps of other carbon-based nanoarchitectures found in the literature, where electronic density confinement has been attributed to a standing wave pattern formed by the Au(111) surface state [182], bay states originated from IPS [111], delocalized molecular orbitals (MOs) [240,242,244], or a superposition of non-interacting molecular states [247]. 5.2.2 DFT simulations In order to disentangle the electronic structure of g-GNRs and clarify the origin of the observed electronic localization, we perform DFT calculations of free-standing periodic g-GNRs. The unit-cell consists of 24 C and 8 H atoms with a 7.54 ˚ A lattice parameter 5.2. Electronic structure of gulf-GNRs 79 (b) (a) III III IV Figure 5.4: Experimental STS characterization of g-GNRs. (a) dI/dV pointspectroscopy of a g-GNR measured with a CO-tip. The inset shows a BR-STM image of a g-GNR. Spectroscopy is measured at the different positions indicated by solid and open dots in the inset BR-STM image. (b) dI/dV maps of a g-GNR measured with a CO-tip at the selected bias voltages indicated in each map. Figure taken from Ref.[180]. along the Y-direction (Fig. 5.5a). Vacuum spacings of 40 ˚ A are introduced in transversal (X) and vertical (Z) directions in order to avoid spurious effects from periodic images. All atomic coordinates are optimized until forces below 0.01 eV/˚ A are achieved, and the 1D lattice is relaxed below a pressure threshold of 0.25 GPa. Core electrons are treated using norm-conserving Troullier-Martins pseudopotentials [177], while valence electrons are described using a linear combination of NAOs. We employ a DZP basis set with a 0.01 Ry energy shift [148], which is further expanded with slowly decaying 3sand 3p orbitals in order to account for the possible emergence of SAMOs or IPSs in gulfs and nanopores, as explained in Chapter 3. Exchange-correlation energies are approximated by the GGA-PBE functional [138]. A 400 Ry cutoff is employed to define the real-space grid, while the BZ is sampled using a Monkhorst-Pack grid with 51 k-points [181]. The DFT calculated band structure of the g-GNR, shown in Fig. 5.5b, reveals a wide 1.84 eV bandgap that confirms the semiconducting character of the ribbon. The bandgap is bounded by pairs of dispersive frontier bands. In stricking contrast to experimental observations, Γ-point valence and conduction frontier states appear delocalized over the entire carbon-network and do not extend into the vacuum region along the edges, as depicted in Fig. 5.5c. Perhaps the only agreement between experiment and theory is found for the VB. The dI/dV map at -1.1 V shows pairs of lobes located in the protruding phenyl rings, in pretty good agreement with similar features of the calculated wave function. Besides, as in 7-13-AGNRs [111] and Ph-7-13-AGNRs (Chapter 3), the band structure 86 Chapter 5. Deceptive orbital confinement in carbon-based nanoarchitectures (b) (c) GNR tip z = 2 Å z = 5 Å tip (a) X Y Figure 5.10: Wave function decay in a finite 24-unit-cell-long 7-AGNRs. (a) Atomic structure of the finite 7-AGNR considered in the simulations. (b,c) DFT Kohn-Sham Γ-point wave functions corresponding to VB-1, VB, CB and CB+1 states evaluated at distances of (a) z= 2 ˚ A and (b) z= 5 ˚ A above the ribbon surface. Figure taken from Ref.[180]. corresponding to Gdecreases with z. Consequently, for large enough tip-sample distances, only the lowest Fourier components Gwill survive in the expansion, and the wave function will present a slowly oscillating appearance. Likewise, because the intra-unit-cell nodal structure of the wave function is described by the expansion over G, those wave functions characterized by complex and rapidly oscillating nodal structures, dominated by fast Fourier components, will be measured less intensely than smoother wave functions. This well-known effect leads to the fact that STM is more sensitive to states closer to the Γ point (low k∥) [254], but it also has a significant influence on the resolution of dI/dV point-spectroscopy and mapping [237,255]. With the aim of illustrating the significance of Eq. 5.1, we first explore the extensively studied 7-AGNR, where the detection of the CB onset remained elusive for years due to its specific orbital decay [237]. In order to compare our results to those reported in Ref. [237], we consider a finite 24-unit-cell-long 7-AGNR in our simulations, as shown in Fig. 5.10a. We first perform DFT calculations of the molecular orbitals corresponding to the VB-1, VB, CB and CB+1. Considering that the cutoff radius of the 2pzorbital 5.4. Wave function decay: origin of deceptive orbital confinement 87 employed in the calculation is ∼2.9˚ A, we use the wave functions evaluated at z= 1 ˚ A as references, where the orbital is physically accurate. That reference wave function is then extrapolated to larger distances from the ribbon. The extrapolation is performed, as implemented in the SIESTA STM tool, which Fourier expands the wave function evaluated at the reference plane (z= 1 ˚ A) and applies the z-dependent Fourier filtering defined in Eq. 5.1. Fig. 5.10b,c shows the corresponding extrapolated wave functions at distances of z= 2 ˚ A and z= 5 ˚ A from the plane. The results are in good agreement with those reported by S¨ode et al. [237]. In particular, all states are delocalized over the entire ribbon backbone and exhibit similar intensities at z= 2 ˚ A. However, because all states oscillate strongly in the direction across the ribbon backbone, their wave functions tend to decay towards the vacuum region next to the ribbon edges and vanish in the backbone. In addition, VB-1 and CB oscillate rapidly also along the growth direction of the ribbon + - 0 VB-1 VB CB CB+1 z≈ 2.0 Åz≈ 3.0 Åz≈ 4.0 Åz≈ 5.0 Åz≈ 6.0 Å Increasing tip-sample distance Figure 5.11: Wave function decay in g-GNRs. DFT Kohn-Sham Γ-point wave functions corresponding to VB-1, VB, CB and CB+1 states evaluated at distances of z= 2,3,4,5 and 6 ˚ A above the ribbon surface. Figure taken from Ref.[180]. 88 Chapter 5. Deceptive orbital confinement in carbon-based nanoarchitectures and, consequently, their intensity at z= 5 ˚ A is strongly suppressed in comparison to VB and CB+1, which do not oscillate along the ribbon backbone. Consequently, the VB-1 and CB can be hardly detected using STS at typical tip-sample distances, while the VB and CB+1 are typically measured confined to the ribbon edges [237]. In the case of g-GNRs, due to their notched edge structure, this decay effect exhibits its own particularities. Note that, in order to avoid finite size effects, we simulated periodic g-GNRs as described in Sec. 5.2.2. Fig. 5.11 shows the decay of DFT-calculated Γ-point wave functions for VB-1, VB, CB and CB+1 evaluated at various distances from the surface. As the distance increases, all these delocalized wave functions progressively vanish in the ribbon backbone and decay towards the vacuum region along the edges. + - 0 VB-1 VB CB CB+1 z≈ 2.0 Åz≈ 3.0 Åz≈ 4.0 Åz≈ 5.0 Åz≈ 6.0 Å Increasing tip-sample distance Figure 5.12: Wave function decay in NPG. DFT Kohn-Sham Γ-point wave functions corresponding to VB-1, VB, CB and CB+1 states evaluated at distances of z= 2,3,4,5 and 6 ˚ A above the ribbon surface. Figure taken from Ref.[180]. 5.5. Generalization to other porous carbon-based nanoarchitectures 89 At large tip-sample distances, VB and CB, which oscillate along the ribbon backbone, exhibit a pair of opposite sign lobes in each protruding phenyl ring. In contrast, VB-1 and CB+1 exhibit a continuous feature that perfectly follows the shape of the edge at both sides of the g-GNR. In the case of NPG, the porous structure significantly influences the decay effect. The decay of DFT-calculated Γ-point wave functions for VB-1, VB, CB and CB+1 evaluated at various distances from NPG is shown in Fig. 5.12. Similar to g-GNRs, wave function intensities gradually vanish in the 2D carbon-network as zincreases. In the absence of a 1D edge, however, wave functions decay towards the nanopores, eventually exhibiting maximun intensity within them. It is worth noting that the intensity of the VB wave function is significantly reduced compared to the rest of states. Such wave function oscillates along the ribbon backbone, while it exhibits a change of sign from one g-GNR to the next, i.e. it is antisymmetric between adjacent ribbons. This results in a strongly oscillating wave function in both directions of the NPG, which decays faster than the rest of states in the vertical direction. The rest of states are either symmetric between adjacent ribbons or do not oscillate along the ribbon backbone, and are therefore notably more intense at large distances from the surface. In particular, CB+1 does not change sign within the nanopore, closely resembling theoretical LDOS and experimental dI/dV maps. The decay features presented in Fig. 5.11 and Fig. 5.12 explain the localization of electronic density obtained from the LDOS maps, and are the key factor leading to deceptive orbital confinement in experimental STS measurements. Importantly, such strong localization arises from the combination of two factors: (1) the decay of the g-GNR and NPG orbitals in the vertical direction, and (2) the confinement of the orbitals into 1D and 0D vacuum regions (e.g., edges, gulfs and nanopores), as determined by the exact edge structure of the system. While the first is a general and well-known effect reported in many surfaces and molecular assemblies, the second is expected to be particularly relevant in graphene-based 1D and 2D nanostructures, for many of their edge and nanopore morphologies. Next section will serve us as a support of the latter statement. 5.5 Generalization to other porous carbon-based nanoarchitectures In order to demonstrate the general validity of the results presented in the previous section, we next show that the decay of the wave function towards the vacuum can also explain similar localization effects emerging in other types of carbon-based nanostructures. In particular, we study GNRs functionalized with double-void pores and divacancies. 5.5.1 Double-void GNRs We perform DFT simulations of an 18-AGNR with double-void nanopores periodically embedded along the carbon-backbone (see inset in Fig. 5.13a). The ribbon is periodic in the Y-direction, with a lattice constant of 17.39 ˚ A, and the unit-cell consists of 132 C and 26 H atoms. We employ a 51 k-point Monkhorst-Pack grid for the BZ sampling [181]. The rest of simulation parameters are those used in previous sections of this chapter. 90 Chapter 5. Deceptive orbital confinement in carbon-based nanoarchitectures a b VB-1 VB CB CB+1 z≈ 6.0 Åz≈ 6.0 Å 1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 E-EF (eV) ΓY 3.71 pA -2.56 pA LDOS max. min. 3.71 pA -2.56 pA LDOS max. min. (a) (b) VB CB CB+1 VB-1 Figure 5.13: (a) DFT band structure of double-void 18-AGNRs. The energy ranges used for LDOS map simulations in (b) are indicated by pink shaded regions. Inset schematically represents the system geometry, with red dashed lines indicating the unit-cell. (b) DFTLDOS maps at a distance of z= 6 ˚ A above the ribbon surface and evaluated in the energy regions indicated in (a), which include VB-1, VB, CB and CB+1 onsets. Figure taken from Ref.[180]. + - 0 VB-1 VB CB CB+1 z≈ 2.0 Å z≈ 6.0 Å Figure 5.14: Wave function decay in double-void 18-AGNRs. DFT Kohn-Sham Γ-point wave functions corresponding to VB-1, VB, CB and CB+1 states evaluated at distances of z= 2 and z= 6 ˚ A above the ribbon surface. Figure taken from Ref.[180]. Fig. 5.13 shows the corresponding (a) band structure and (b) LDOS map simulations of VB-1, VB, CB, and CB+1. The band structure exhibits a ∼0.95 eV bandgap. The 5.5. Generalization to other porous carbon-based nanoarchitectures 91 bandgap is bounded by dispersing frontier states, which counterintuitively lead to strongly localized DFT-LDOS features in nanopores and edges for tip-sample distances of z= 6˚ A (see Fig. 5.13b). Fig. 5.14 shows the VB-1, VB, CB and CB+1 wave functions at the Γ-point at distances close (z= 2˚ A) and far (z= 6˚ A) from the surface. All states exhibit an intrinsic delocalization over the entire carbon-network at z= 2˚ A, while their confinement into the nanopores is evident at z= 6˚ A. This result correlates very well with the DFT-LDOS maps shown in Fig. 5.13, and corroborates the validity of the wave function decay mechanism reported above in the case of double-void pores. 5.5.2 Divacancy GNRs We have also studied the wave function decay in a 8-ZGNR decorated with periodically embedded divacancy pores. These are arranged according to a 14.98 ˚ A lattice parameter in the Y direction, with 94 C and 16 H atoms in each unit-cell. A 51 k-point MonkhorstPack grid is employed for the BZ sampling [181]. The rest of simulation parameters are those used in previous sections of this chapter. The atomic structure is shown in Fig. 5.15 and Fig. 5.16a. Due to the small size of the divacancy, steric effects drive the H atoms in the pores as well as the nearby C atoms into an out-of-plane configuration, as shown by the corrugation of the GNR in Fig. 5.15. [175]. The DFT-calculated band structure and LDOS maps of the divacancy-8-ZGNR are depicted in Fig. 5.16. When electron spins are not considered, ZGNRs host doubly degenerate zero-energy flat bands, whose wave functions are intrinsically localized at the ribbon edges [57,238]. It is well-known that upon inclusion of an on-site Coulomb repulsion in TB models or taking into account spins in DFT calculations, a bandgap opens in ZGNRs [60,238]. However, because we are just interested in the wave function decay of states that are intrinsically delocalized over the entire carbon-backbone, we do not consider the spin X Y Z Y Y X Y Z Y Y Z Y (a) (b) Figure 5.15: Side (a) and top (b) views of a divacancy-8-ZGNR unit-cell. The color code represents the displacement (corrugation) in the Z-direction of each atom with respect to the average Z-coordinate. Figure taken from Ref.[180]. 92 Chapter 5. Deceptive orbital confinement in carbon-based nanoarchitectures a b VB-1 VB CB CB+1 z≈ 6.0 Åz≈ 6.0 Å 1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 E-EF (eV) ΓY cc 3.71 pA -2.56 pA LDOS max. min. 3.71 pA -2.56 pA LDOS max. min. a b VB-1 VB CB CB+1 z≈ 6.0 Åz≈ 6.0 Å 1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 E-EF (eV) ΓY cc LDOS max. min. LDOS max. min. (a) (b) FBs VB CB CB+1 VB-1 Figure 5.16: (a) DFT band structure of divacancy-8-ZGNRs. The energy ranges used for LDOS map simulations in (b) are indicated by pink shaded regions. Side image is an schematic representation of the system geometry, with red dashed lines indicating the unit-cell. (b) DFT-LDOS maps at a distance of z= 6 ˚ A above the ribbon surface and evaluated in the energy regions indicated in (a), which include VB-1, VB, CB and CB+1 onsets. The intrinsically localized zero-energy flat bands are not consider in the LDOS analysis. Figure taken from Ref.[180]. + - 0 VB-1 VB CB CB+1 z≈ 2.0 Å z≈ 6.0 Å Figure 5.17: Wave function decay in divacancy-8-ZGNRs. DFT Kohn-Sham Γ-point wave functions corresponding to VB-1, VB, CB and CB+1 states evaluated at distances of z= 2 and z= 6 ˚ A above the ribbon surface. Wave functions of intrinsically localized zero-energy flat bands are not consider. Figure taken from Ref.[180]. 5.6. Summary 93 degree of freedom in our calculations. Accordingly, we obtain a metallic band structure with a pair of approximately flat zero-energy bands (FBs). These are not consider in our analysis, as they are intrinsically localized at the ribbon edges. In contrast, we compute LDOS maps at energy integration windows corresponding to dispersive VB-1, VB, CB and CB+1 at a distance of z= 6 ˚ A above the ribbon surface. As in previous cases, the electronic density vanishes in the inner part of the carbon-network and gets confined to edges and divacancies. The fact that the LDOS is asymmetric in the H-passivated divacancies is due to the aforementioned corrugation around them, which enhances the intensity around the atoms that are closer to the tip. The LDOS localization can be understood by looking at the decay of the wave functions from z≈2˚ A to z≈6˚ A, as shown in Fig. 5.17. 5.6 Summary We have addressed the origin of the electronic confinement at the edges and nanopores of various carbon-based nanoarchitectures, as frequently observed in STS experiments. Based on the Tersoff-Hamann theory of STM, we have reproduced dI/dV maps of g-GNR and NPG structures, linking localized LDOS features to intrinsically delocalized electronic states. This deceptive confinement effect is attributed to the filtering of high momentum components in the Fourier expansion of GNR and NPG wave functions as these are evaluated at increasing distances from the sample. The effect becomes critical at large tipsample distances, and it is strongly influenced by the specific edge and nanopore structure of the system under consideration. Importantly, DFT-LDOS maps allow us to correctly interpret the experimental dI/dV measurements taken in g-GNRs and NPG. Additional calculations also show that this effect holds for other edge and nanopore geometries. We thus expect deceptive orbital confinement to be present in the vast majority of STS experiments that are performed in carbon-based nanoarchitectures employing typical tipsample distances. This effect provides an alternative, plausible explanation to many localization effects reported in the literature. CHAPTER 6 Conclusions & outlook This thesis provides an extensive theoretical study of electronic and transport phenomena in atomically precise, porous carbon-based nanoarchitectures. In order to rationalize experimental results and predict electronic properties from atomic to device scale, we have appropriately combined DFT, TB and Green’s functions methods, which have been described in Chapter 2. In Chapter 3, we have focused on the study of on-surface synthesized porous GNR and NPG structures. Performing a DFT-based energetics analysis we have unveiled the mechanism leading to the migration of phenyl side-groups in Ph-7-13-AGNRs, which eventually leads to the formation of [18]-annulene pores in the ribbons. We have shown that the phenyl migration cannot be initiated without the presence of Au(111) surface adatoms, which is in line with the crucial role of surface adatoms as catalyzers of other OSS reactions highlighted in the literature. Combined with experimental STM imaging, our theoretical results demonstrate that phenyl migration can be incorporated into the toolbox of OSS reactions in order to produce controllable internal transformations in carbon-based nanostructures. This may ultimately enrich the catalogue of atomically precise carbon-based nanoarchitectures by promoting the synthesis of more sophisticated geometries. Besides, we have shown that the formation of [18]-annulene pores leaves the bandgap of the ribbon unchanged, while the vacuum states localized at the bays and pores undergo a significant energy shift. Instead, if Ph-7-13-AGNRs are laterally fused, NPG structures containing molecular bridges with either para-para,para-meta, and meta-meta bonding configurations are synthesized. Our DFT simulations demonstrate that the meta bonding configurations act as chemical knobs inducing an abrupt enhancement of the electronic anisotropy in NPG. In addition, the twisted conformation revealed in para-para bridges is shown to behave as a gradual conformational knob of the interribbon coupling and the electronic anisotropy. Based on pruned TB Hamiltonians extracted from DFT calculations, we have performed NEGF simulations of the electronic transport in realistic device scales, predicting rich quantum electron transport phenomena in NPG. While the chemical knob abruptly quenches the Talbot interference effect by confining large-scale currents to a single GNR, the conformational knob can be used to fine tune the wave length of the Talbot standing wave pattern. 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