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Università degli Studi Roma TRE e Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia Dottorato di Ricerca in Scienze Fisiche della Materia XXIII ciclo Challenges for first principles

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Università degli Studi Roma TRE e Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia Dottorato di Ricerca in Scienze Fisiche della Materia XXIII ciclo Challenges for first principles methods in theoretical and computational physics: multiple excitations in many electrons systems and the Aharonov Bohm effect in carbon nanotubes Tesi di dottorato del dott. Davide Sangalli Relatore: Prof. Giovanni Onida Correlatore: Dott. Andrea Marini Coordinatore Dottorato: Prof. Settimio Mobilio a.a / 20111 Contents Introduction v I Theoretical background 1 1 Many Body Systems Looking for the ground state Perturbing the ground state The macroscopic and the microscopic world Green s Function approach The time evolution operator and the role of the interaction Equilibrium properties The quasiparticle concept Neutral excitations: the Bethe Salpeter equation Density Functional Theory The Hohenberg Kohn theorem The Kohn Sham scheme The local density approximation Time dependent density functional theory The electron electron interaction ii CONTENTS II Double Excitations 49 4 Introduction to the problem Double excitations in quantum chemistry Double excitations in the many body approach The dynamical Bethe Salpeter equation (step I) A new approach The second random phase approximation The dynamical Bethe Salpeter equation (step II) A number conserving kernel for correlated systems Numerical results on model molecular systems Conclusions III Carbon nanotubes in magnetic fields 95 6 The Aharonov Bohm effect in carbon nanotubes What is the Aharanov Bohm effect? An introduction to carbon nanotubes Theoretical predictions and experimental results Numerical results Details of the implementation Gap oscillations The band structure Persistent currents Conclusions Conclusions 141 Appendices 147 A Connection to the experiments: extended systems 147 A.1 The Dielectric constant CONTENTS iii A.2 Electron energy loss spectroscopy and absorption B On the quasiparticle concept 153 C Gauge transformations 157 D DFT and magnetic fields 159 Acknowledgments 171 iv CONTENTS Introduction Many body physics is a branch of physics whose scope is to understand physical phenomena where a number of interacting bodies is present. The presence of the interaction is what makes the description of such systems challenging but at the same time exciting. Interacting particles can give birth to new physical process which cannot be simply described as the sum of the behaviour of each single element. The superconducting phase at low temperatures, plasmon peaks in the absorption spectrum, Mott transitions are only a few examples. New physics emerge as a result of the coherent behaviour of the many body system. The description of interacting particles requires sophisticated many body techniques and the exact mathematical solution to the problem is almost never available: approximations are needed. To construct a practical approximation one need to have some clue as to which are the most relevant phenomena, which are the physical aspects that can be discarded and which can be treated in an approximate way as perturbations. Often even the approximate equation cannot be solved analytically, a computational approach is needed. Computational Physics can be seen as an approach which stands in the middle between Theoretical Physics and Experimental Physics. Some of the results presented in the present work have been obtained through a numerical approach. In the present work we will describe some of these techniques with a focus on a specific phenomenon: the description of double excitations in the absorption spectrum. Double excitations are a peculiar effect of interacting systems which does not have a counterpart in non interacting ones. The optical absorption spectrum of a system is obtained by shining light on it. At the microscopic level photons hit the electrons which sit in the ground state and change their configuration. If the light vi Introduction source is not too intense this can be described in linear response; that is only one photon processes are involved, only one electron per time can be influenced. Here is where the interaction comes in. The hit electron is linked to the others and so other process take place, one of these is the appearance of multiple excitations. These are, strictly speaking, virtual processes as the real time evolution of the system is different from the one described. Nevertheless the physical effect is there and can be measured as extra peaks in the absorption spectrum. Double excitations is not the only subject of the present work and understanding is not the only scope of many body physics. The same techniques can be used to make accurate quantitative predictions of the behaviour of a material. These allow us to control physical phenomena and possibly to use them in technological applications. In the second part of the Thesis we focus on the application of more standard techniques to the description of carbon nanotubes (CNTs). In particular we focus on the effects of magnetic fields on CNTs. CNTs are quasi 1D-systems composed by carbon atoms which have been discovered in They have the shape of a hollow cylinder with a nanometric diameter (10 9 m), a micrometric length (10 6 m) and the thickness of a single atomic layer 2. What makes such objects so interesting is that they are mechanically very strong and stable. These properties makes them ideal system both for many possible technological approaches and for testing the physical behavior of electrons in 1D system as well as in cylindrical topologies. In this work we are in particular interested in the effect of magnetic fields related to topology. Under the effect of a magnetic field electrons delocalized on a cylindrical surface display a peculiar behaviour, known as Aharonov Bohm effect. The Aharonov Bohm is a pure quantum mechanical effect which does not have any counterpart in classical physics. In CNTs the Aharonov Bohm modify the electronic gap and so can be used to tune the electronic properties. Though a model able to account for 1 A large percentage of academic and popular literature attributes their discovery to Sumio Iijima of NEC in 1991 [1, 2], however already in 1952 L. V. Radushkevich and V. M. Lukyanovich published clear images of 50 nanometer diameter tubes made of carbon in the Soviet Journal of Physical Chemistry, the publication however was in Russian. For a detailed review on the discovery of CNTs we address the reader to Ref. [3]. 2 Here we refer to single walled CNTs. Multi walled CNTs, which are composed by concentric single walled CNTs exist too. vii such process is available in the literature, in the present work we will describe the effect of magnetic fields ab initio. Ab initio is any approach which describes the physics starting from first principles and without the use of any external parameter. As pointed out in the first part of the introduction the exact solution to the many body problem is in practice never available and approximations are needed. In the description of CNTs we will use standard approximations which are by far much more accurate and general than any approximation introduced in phenomenological descriptions based on model systems. In part I the general many body problem is introduced. In particular Density Functional Theory (DFT) and Many Body Perturbation Theory (MBPT) are described according to our needs for the forthcoming parts. In part II the problem of double excitations is presented together with experimental evidence and the state of the art. In this part we will propose a new approximation which could be used in the standard approach for the description of absorption spectra in both the MBPT and DFT framework. This approximation is able to describe double excitations. Finally in part III the general problem of CNTs in magnetic fields will be considered. After a brief overview on the main experimental evidence of the Aharonov Bohm effect, the Zone Folding Approach (and the Tight Binding model) will be introduced. Then we will describe how magnetic field effect are included in our ab initio approach and finally we will compare our results with the predictions of the models. viii Introduction Notation and conventions This is a brief overview of the conventions used to express operators and their related physical quantities. The same conventions are introduced in Ch. 1. Atomic units are used in Part I of the thesis, while in Part II the international system (SI) of units is used. The one body operators are written in second quantization according to the following expression: Â = d 3 x 1 d 3 x 2 dt 1 dt 2 A σ1 σ 2 (x 1 t 1,x 2 t 2 ) ˆψ σ 1 (x 1,t 1 ) ˆψ σ2 (x 2,t 2 ). σ 1,σ 2 Here σ is a spin variable, while x and t are space and time variables respectively and A σ1 σ 2 (x 1 t 1,x 2 t 2 ) is the kernel of the operator. A compact notation is also often used Â = d1d2 A(1, 2) ˆψ (1) ˆψ(2), Â = d1d2 A(1t 1,2t 2 ) ˆψ (1t 1 ) ˆψ(2t 2 ). The compact notation will be preferred wherever it will not be source of confusion. In this notation repeated primed variables are supposed to be integrated, i.e. Σ (1, 2) = ig(2, 1)W(1, 2 )Γ (2, 2 ; 1 ), means Σ (1, 2) = d1 d2 ig(2, 1)W(1, 2 )Γ (2, 2 ; 1 ). The symbol d1 stands for σ dx dt. However, when only partial integration x Introduction will be performed (i.e. only on space, time or spin variables), the sums / integrals will be written explicitly. The notation G(1, 2 + ) will be used for lim G(x 1,t;x 2,t + ǫ), ǫ 0 + and the notation Â for the expectation value on the ground state of an operator: Ψ 0 Â Ψ 0. (1) For the coulomb interaction we will use w(1, 2) = for the one particle part of the Hamiltonian 1 x 1 x 2 δ(t 2 t 1 )δ σ1 σ 2 ; Ĥ 0 = V I (x). A functional will be expressed with the following notation E[ρ] which means that the energy E is a functional of the density ρ(x,t). Finally we list here some (but not all) of the symbols used in the thesis: B is the magnetic field, A is the vector potential, Φ is the magnetic flux H is the magnetic induction field, E is the electric field, D is the electric displacement field, P is the total polarization, M is the total magnetization, xi j is the current density, with j (p) the paramagnetic and j (A) the diamagnetic component ρ is the density ǫ is the dielectric constant, α is the polarizability, χ is the response function, χ 0 the independent particle one, and χ KS the Kohn Sham one, L is a four point response function, while L is four point in space and two point in time, T is the time ordering operator while, ˆT and T[ρ] are the kinetic energy operator and the kinetic energy of the system respectively, Σ is the self energy, Σ H = Σ + v H, that is the sum of the self energy and the Hartree potential (v H ), Σ is the reduced self energy, Σ s is a static self energy while Σ d is a dynamical self energy, Π is the polarizability, which correspond to the density density response function 3, Π is the reduced polarizability, Γ is the vertex function and Γ is the reduced vertex function; W is the screened coulomb interaction, 3 The two quantities differ only because Π is T ordered, while χ ρρ is a retarded quantity xii Introduction g is the independent particle Green s function, g H the Hartree Green s function, and G the many body Green s function, Z is the renormalization factor, E xc is the exchange correlation energy, v xc the exchange correlation potential and f xc the exchange correlation kernel, Ξ is the kernel of the Bethe Salpeter equation (BSE) while K the kernel of a generalized equation (including the dynamical BSE) for the four point response function, µ is the magnetic susceptibility or the chemical potential, will be clear from the context Ψ the many body wave function and Ψ 0 the many body ground state, Ψ s the Kohn Sham ground state. Φ 0 will be used for the non interacting many body ground state or for the magnetic flux quantum Φ 0 = h, with h the Planck s constant and e the electron e charge, when explicitly specified Ψ S will be the wave function in the Schrödinger s picture, Ψ H in the Heisenberg s picture, and Ψ I in the interaction picture, δa stands for a variation of the quantity A, while δ(1, 2) is the Dirac s delta function, Part I Theoretical background Chapter 1 Many Body Systems Any known many body system is constituted of interacting particles and, at least in principle, any physical aspect can be understood describing their dynamics. Elementary particles are in general ruled by the equations of quantum mechanics and special relativity (or general relativity) and four possible kind of interactions are known to exist: the Electromagnetic interaction, the Nuclear Weak interaction, the Nuclear Strong interaction and the Gravitational interaction. For this reason even the description of a single atom, where in principle all forces have to be taken into account, appear an almost impossible problem. Moreover any macroscopic body is constituted of an enormous number of interacting particles, for a reference the Avogadro s number; N A = [particles/moles]. So any attempt to solve the many body problems seems doomed to fail. Despite this discouraging scenario there are two factors which in fact make possible to tackle the many body problem from a microscopic point of view. First the majority of the processes which happen in everyday life involve a thin energy window such that only the electromagnetic interaction plays a role and, quite often, only the dynamics of the electrons needs to be described. Secondly the majority of the macroscopic objects are constituted by fundamental building blocks which almost completely determine their properties: these are the molecules which constitutes the gases and the liquids and the unitary cells which are repeated an infinite number of times in many solid systems 1. 1 For gases and liquids only the properties related to the electronic dynamics can be studied from a microscopic point of view. Other properties require a statistical description of the system. 4 Many Body Systems A crucial role is, then, played by the equation which describes the dynamics of few interacting electrons immersed in the Coulomb potential of the nuclei that, in a first approximation can be considered frozen in their instantaneous positions: the Scrödinger Equation (SE) in the Born-Oppenheimer (BO) approximation. Possibly the electrons can interact with external fields which can be used either to explore or to tune the properties of the materials. The first part of this thesis will be dedicated to the description of light absorption experiments where an external light source is used to investigate the optical properties of a many electrons system. The second part will be focused on the study of magnetic field effect on Carbon Nano-Tubes (CNTs) and how the external field can be used to tune the electronic properties of the CNTs. The SE can be obtained applying an Hamiltonian operator to the electronic wave function. The operator can be divided into two terms Ĥint + Ĥext. The first term describes the electrons nuclei interaction, while the second term is due to the presence if an external perturbation that, in this work, is the electromagnetic field. The SE in atomic units reads: (i t ) ˆV ext Ψ = 1 [ ( ) ] 2 i 2 Âext + ˆBext σ + ŵ + ˆV I Ψ. (1.1) ˆV ext and Âext accounts for the external potentials and the term B ext σ accounts for the interaction of the spin with a possible applied external field. σ is a vector constituted by the Pauli matrices, Ψ is the electronic wave function; ŵ is the Coulomb interaction while ˆV I accounts for the ionic potential. Finally the operator takes into account the kinetic energy of the electrons. It is important to observe that some of the terms discarded in Eq. (1.1) does not have in practice a role in the physical process we are interested in. The reason is that the energy scales involved are so different that the corresponding dynamics can be neglected. This is the case of the nuclear forces and of the gravitational force. Some other terms instead are discarded as they are usually very small although they could be needed for the description of some physical phenomena. This is the case of relativistic corrections 2 which are needed in the description of materials 2 The terms due to the presence of a magnetic field included in Eq. (1.1) are already relativistic corrections to Ĥext. Here we consider these terms as in this thesis we are interested in the de- 1.1 Looking for the ground state 5 composed of heavy nuclei; the dynamics of the nuclei, which is neglected in the BO approximation, is relevant for example for the description of the superconducting phase of some materials. Many development of the state of the art are devoted to overcome such approximations. On the other hand the idea of describing solid state devices as an infinite repetition of the same fundamental building blocks can be applied to the description of specific kind of materials only. As predicted by Feynman times ago, objects at the nanometric scale can display very peculiar properties which are completely different from the case of molecules or of bulk systems. In these direction the state-of-the-art tools need to be pushed beyond their present limits. From one side the common approximations involved, which are often based on physical intuition, cease to be valid and new approximations are needed. From the other side the lack of a repetitive structure calls for the need of instruments able to describe systems with hundred, thousand and even more interacting electron. Only thanks to the very recent increase of computational power and, in the same time, to recent developments of the techniques, it has become possible to tackle, at least in some cases, the description of such system ab-initio and so to test the prediction of more simple theoretical models. 1.1 Looking for the ground state The solution to Eq. (1.1) is the main goal of the many body physics. The first objective is to solve such equation with Ĥext = 0. That is to find out the ground state of the system. An exact analytical solution can be obtained only in oversimplified systems such as the hydrogen atom. For any realistic system this is far beyond our possibilities. A computational solution can be obtained at the price of a computational time, that grows exponentially with the number of electrons, only for very small systems. This is why approximations are needed. The problematic part of the Hamiltonian is the interaction term w( r 1 r 2 ) for which different possible strategies are available. scription of the Aharonov Bohm effect in CNTs. The relativistic corrections to the Ĥint part of the Hamiltonian, that is the magnetic field generated by the electronic current (and spin) and the magnetic field due to the nuclei (the spin orbit interaction mainly) are neglected here. 6 Many Body Systems In this work we will tackle the problem using Many Body Perturbations Theory (MBPT) and the Density Functional Theory (DFT). Both methods start from the consideration that the many body wave function contains much more information than really needed. So instead of the exact wave function, which is a function of 3N variables, where N is the number of electrons, one can look for simpler quantities which contains only the informations needed to give a quantitative description of experiments. In quantum mechanics any measurable quantity is related to an Hermitian operator. In particular one body operators can be written in second quantization as: Â = d 3 x 1 d 3 x 2 dt 1 dt 2 A σ1 σ 2 (x 1 t 1,x 2 t 2 ) ˆψ σ 1 (x 1,t 1 ) ˆψ σ2 (x 2,t 2 ), (1.2) σ 1,σ 2 We introduce here the notations 1 = (1,t) = (x,t,σ) which will be used from now on. So for example Eq. (1.2) can be written in the two forms: Â = d1d2 A(1, 2) ˆψ (1) ˆψ(2), (1.3) Â = d1d2 A(1t

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