Tutorial 1 - Graphene
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Effective Hamiltonians Derived from Equation-Of-Motion Coupled-Cluster
Effective Hamiltonians derived from equation-of-motion coupled-cluster wave-functions: Theory and application to the Hubbard and Heisenberg Hamiltonians Pavel Pokhilkoa and Anna I. Krylova a Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482 Effective Hamiltonians, which are commonly used for fitting experimental observ- ables, provide a coarse-grained representation of exact many-electron states obtained in quantum chemistry calculations; however, the mapping between the two is not triv- ial. In this contribution, we apply Bloch's formalism to equation-of-motion coupled- cluster (EOM-CC) wave functions to rigorously derive effective Hamiltonians in the Bloch's and des Cloizeaux's forms. We report the key equations and illustrate the theory by examples of systems with electronic states of covalent and ionic characters. We show that the Hubbard and Heisenberg Hamiltonians are extracted directly from the so-obtained effective Hamiltonians. By making quantitative connections between many-body states and simple models, the approach also facilitates the analysis of the correlated wave functions. Artifacts affecting the quality of electronic structure calculations such as spin contamination are also discussed. I. INTRODUCTION Coarse graining is commonly used in computational chemistry and physics. It is ex- ploited in a number of classic models serving as a foundation of modern solid-state physics: tight binding[1, 2], Drude{Sommerfeld's model[3{5], Hubbard's [6] and Heisenberg's[7{9] Hamiltonians. These models explain macroscopic properties of materials through effective interactions whose strengths are treated as model parameters. The values of these pa- rameters are determined either from more sophisticated theoretical models or by fitting to experimental observables. -
Lecture Notes
Solid State Physics PHYS 40352 by Mike Godfrey Spring 2012 Last changed on May 22, 2017 ii Contents Preface v 1 Crystal structure 1 1.1 Lattice and basis . .1 1.1.1 Unit cells . .2 1.1.2 Crystal symmetry . .3 1.1.3 Two-dimensional lattices . .4 1.1.4 Three-dimensional lattices . .7 1.1.5 Some cubic crystal structures ................................ 10 1.2 X-ray crystallography . 11 1.2.1 Diffraction by a crystal . 11 1.2.2 The reciprocal lattice . 12 1.2.3 Reciprocal lattice vectors and lattice planes . 13 1.2.4 The Bragg construction . 14 1.2.5 Structure factor . 15 1.2.6 Further geometry of diffraction . 17 2 Electrons in crystals 19 2.1 Summary of free-electron theory, etc. 19 2.2 Electrons in a periodic potential . 19 2.2.1 Bloch’s theorem . 19 2.2.2 Brillouin zones . 21 2.2.3 Schrodinger’s¨ equation in k-space . 22 2.2.4 Weak periodic potential: Nearly-free electrons . 23 2.2.5 Metals and insulators . 25 2.2.6 Band overlap in a nearly-free-electron divalent metal . 26 2.2.7 Tight-binding method . 29 2.3 Semiclassical dynamics of Bloch electrons . 32 2.3.1 Electron velocities . 33 2.3.2 Motion in an applied field . 33 2.3.3 Effective mass of an electron . 34 2.4 Free-electron bands and crystal structure . 35 2.4.1 Construction of the reciprocal lattice for FCC . 35 2.4.2 Group IV elements: Jones theory . 36 2.4.3 Binding energy of metals . -
Introduction to the Physical Properties of Graphene
Introduction to the Physical Properties of Graphene Jean-No¨el FUCHS Mark Oliver GOERBIG Lecture Notes 2008 ii Contents 1 Introduction to Carbon Materials 1 1.1 TheCarbonAtomanditsHybridisations . 3 1.1.1 sp1 hybridisation ..................... 4 1.1.2 sp2 hybridisation – graphitic allotopes . 6 1.1.3 sp3 hybridisation – diamonds . 9 1.2 Crystal StructureofGrapheneand Graphite . 10 1.2.1 Graphene’s honeycomb lattice . 10 1.2.2 Graphene stacking – the different forms of graphite . 13 1.3 FabricationofGraphene . 16 1.3.1 Exfoliatedgraphene. 16 1.3.2 Epitaxialgraphene . 18 2 Electronic Band Structure of Graphene 21 2.1 Tight-Binding Model for Electrons on the Honeycomb Lattice 22 2.1.1 Bloch’stheorem. 23 2.1.2 Lattice with several atoms per unit cell . 24 2.1.3 Solution for graphene with nearest-neighbour and next- nearest-neighour hopping . 27 2.2 ContinuumLimit ......................... 33 2.3 Experimental Characterisation of the Electronic Band Structure 41 3 The Dirac Equation for Relativistic Fermions 45 3.1 RelativisticWaveEquations . 46 3.1.1 Relativistic Schr¨odinger/Klein-Gordon equation . ... 47 3.1.2 Diracequation ...................... 49 3.2 The2DDiracEquation. 53 3.2.1 Eigenstates of the 2D Dirac Hamiltonian . 54 3.2.2 Symmetries and Lorentz transformations . 55 iii iv 3.3 Physical Consequences of the Dirac Equation . 61 3.3.1 Minimal length for the localisation of a relativistic par- ticle ............................ 61 3.3.2 Velocity operator and “Zitterbewegung” . 61 3.3.3 Klein tunneling and the absence of backscattering . 61 Chapter 1 Introduction to Carbon Materials The experimental and theoretical study of graphene, two-dimensional (2D) graphite, is an extremely rapidly growing field of today’s condensed matter research. -
Phys 446: Solid State Physics / Optical Properties
Phys 446: Solid State Physics / Optical Properties Fall 2015 Lecture 2 Andrei Sirenko, NJIT 1 Solid State Physics Lecture 2 (Ch. 2.1-2.3, 2.6-2.7) Last week: • Crystals, • Crystal Lattice, • Reciprocal Lattice Today: • Types of bonds in crystals Diffraction from crystals • Importance of the reciprocal lattice concept Lecture 2 Andrei Sirenko, NJIT 2 1 (3) The Hexagonal Closed-packed (HCP) structure Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, Gd,Tb, Py, Ho, Er, Tm, Lu, Hf, Re, Os, Tl • The HCP structure is made up of stacking spheres in a ABABAB… configuration • The HCP structure has the primitive cell of the hexagonal lattice, with a basis of two identical atoms • Atom positions: 000, 2/3 1/3 ½ (remember, the unit axes are not all perpendicular) • The number of nearest-neighbours is 12 • The ideal ratio of c/a for Rotated this packing is (8/3)1/2 = 1.633 three times . Lecture 2 Andrei Sirenko, NJITConventional HCP unit 3 cell Crystal Lattice http://www.matter.org.uk/diffraction/geometry/reciprocal_lattice_exercises.htm Lecture 2 Andrei Sirenko, NJIT 4 2 Reciprocal Lattice Lecture 2 Andrei Sirenko, NJIT 5 Some examples of reciprocal lattices 1. Reciprocal lattice to simple cubic lattice 3 a1 = ax, a2 = ay, a3 = az V = a1·(a2a3) = a b1 = (2/a)x, b2 = (2/a)y, b3 = (2/a)z reciprocal lattice is also cubic with lattice constant 2/a 2. Reciprocal lattice to bcc lattice 1 1 a a x y z a2 ax y z 1 2 2 1 1 a ax y z V a a a a3 3 2 1 2 3 2 2 2 2 b y z b x z b x y 1 a 2 a 3 a Lecture 2 Andrei Sirenko, NJIT 6 3 2 got b y z 1 a 2 b x z 2 a 2 b x y 3 a but these are primitive vectors of fcc lattice So, the reciprocal lattice to bcc is fcc. -
Part 1: Supplementary Material Lectures 1-5
Crystal Structure and Dynamics Paolo G. Radaelli, Michaelmas Term 2014 Part 1: Supplementary Material Lectures 1-5 Web Site: http://www2.physics.ox.ac.uk/students/course-materials/c3-condensed-matter-major-option Contents 1 Frieze patterns and frieze groups 3 2 Symbols for frieze groups 4 2.1 A few new concepts from frieze groups . .5 2.2 Frieze groups in the ITC . .7 3 Wallpaper groups 8 3.1 A few new concepts for Wallpaper Groups . .8 3.2 Lattices and the “translation set” . .9 3.3 Bravais lattices in 2D . .9 3.3.1 Oblique system . 10 3.3.2 Rectangular system . 10 3.3.3 Square system . 10 3.3.4 Hexagonal system . 11 3.4 Primitive, asymmetric and conventional Unit cells in 2D . 11 3.5 The 17 wallpaper groups . 12 3.6 Analyzing wallpaper and other 2D art using wallpaper groups . 12 1 4 Fourier transform of lattice functions 13 5 Point groups in 3D 14 5.1 The new generalized (proper & improper) rotations in 3D . 14 5.2 The 3D point groups with a 2D projection . 15 5.3 The other 3D point groups: the 5 cubic groups . 15 6 The 14 Bravais lattices in 3D 19 7 Notation for 3D point groups 21 7.0.1 Notation for ”projective” 3D point groups . 21 8 Glide planes in 3D 22 9 “Real” crystal structures 23 9.1 Cohesive forces in crystals — atomic radii . 23 9.2 Close-packed structures . 24 9.3 Packing spheres of different radii . 25 9.4 Framework structures . 26 9.5 Layered structures . -
Valleytronics: Opportunities, Challenges, and Paths Forward
REVIEW Valleytronics www.small-journal.com Valleytronics: Opportunities, Challenges, and Paths Forward Steven A. Vitale,* Daniel Nezich, Joseph O. Varghese, Philip Kim, Nuh Gedik, Pablo Jarillo-Herrero, Di Xiao, and Mordechai Rothschild The workshop gathered the leading A lack of inversion symmetry coupled with the presence of time-reversal researchers in the field to present their symmetry endows 2D transition metal dichalcogenides with individually latest work and to participate in honest and addressable valleys in momentum space at the K and K′ points in the first open discussion about the opportunities Brillouin zone. This valley addressability opens up the possibility of using the and challenges of developing applications of valleytronic technology. Three interactive momentum state of electrons, holes, or excitons as a completely new para- working sessions were held, which tackled digm in information processing. The opportunities and challenges associated difficult topics ranging from potential with manipulation of the valley degree of freedom for practical quantum and applications in information processing and classical information processing applications were analyzed during the 2017 optoelectronic devices to identifying the Workshop on Valleytronic Materials, Architectures, and Devices; this Review most important unresolved physics ques- presents the major findings of the workshop. tions. The primary product of the work- shop is this article that aims to inform the reader on potential benefits of valleytronic 1. Background devices, on the state-of-the-art in valleytronics research, and on the challenges to be overcome. We are hopeful this document The Valleytronics Materials, Architectures, and Devices Work- will also serve to focus future government-sponsored research shop, sponsored by the MIT Lincoln Laboratory Technology programs in fruitful directions. -
Electronic Structure Theory for Periodic Systems: the Concepts
Electronic Structure Theory for Periodic Systems: The Concepts Christian Ratsch Institute for Pure and Applied Mathematics and Department of Mathematics, UCLA Motivation • There are 1020 atoms in 1 mm3 of a solid. • It is impossible to track all of them. • But for most solids, atoms are arranged periodically Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Outline Part 1 Part 2 • Periodicity in real space • The supercell approach • Primitive cell and Wigner Seitz cell • Application: Adatom on a surface (to calculate for example a diffusion barrier) • 14 Bravais lattices • Surfaces and Surface Energies • Periodicity in reciprocal space • Surface energies for systems with multiple • Brillouin zone, and irreducible Brillouin zone species, and variations of stoichiometry • Bandstructures • Ab-Initio thermodynamics • Bloch theorem • K-points Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Primitive Cell and Wigner Seitz Cell a2 a1 a2 a1 primitive cell Non-primitive cell Wigner-Seitz cell = + 푹 푁1풂1 푁2풂2 Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Primitive Cell and Wigner Seitz Cell Face-center cubic (fcc) in 3D: Primitive cell Wigner-Seitz cell = + + 1 1 2 2 3 3 Christian푹 Ratsch, UCLA푁 풂 푁 풂 푁 풂 IPAM, DFT Hands-On Workshop, July 22, 2014 How Do Atoms arrange in Solid Atoms with no valence electrons: Atoms with s valence electrons: Atoms with s and p valence electrons: (noble elements) (alkalis) (group IV elements) • all electronic shells are filled • s orbitals are spherically • Linear combination between s and p • weak interactions symmetric orbitals form directional orbitals, • arrangement is closest packing • no particular preference for which lead to directional, covalent of hard shells, i.e., fcc or hcp orientation bonds. -
Electronic and Structural Properties of Silicene and Graphene Layered Structures
Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2012 Electronic and Structural Properties of Silicene and Graphene Layered Structures Patrick B. Benasutti Wright State University Follow this and additional works at: https://corescholar.libraries.wright.edu/etd_all Part of the Physics Commons Repository Citation Benasutti, Patrick B., "Electronic and Structural Properties of Silicene and Graphene Layered Structures" (2012). Browse all Theses and Dissertations. 1339. https://corescholar.libraries.wright.edu/etd_all/1339 This Thesis is brought to you for free and open access by the Theses and Dissertations at CORE Scholar. It has been accepted for inclusion in Browse all Theses and Dissertations by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. Electronic and Structural Properties of Silicene and Graphene Layered Structures A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science by Patrick B. Benasutti B.S.C.E. and B.S., Case Western Reserve and Wheeling Jesuit, 2009 2012 Wright State University Wright State University SCHOOL OF GRADUATE STUDIES August 31, 2012 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVI- SION BY Patrick B. Benasutti ENTITLED Electronic and Structural Properties of Silicene and Graphene Layered Structures BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in Physics. Dr. Lok C. Lew Yan Voon Thesis Director Dr. Douglas Petkie Department Chair Committee on Final Examination Dr. Lok C. Lew Yan Voon Dr. Brent Foy Dr. Gregory Kozlowski Dr. Andrew Hsu, Ph.D., MD Dean, Graduate School ABSTRACT Benasutti, Patrick. -
Arxiv:1601.03499V1 [Quant-Ph] 14 Jan 2016 Networks [23]
Non-Hermitian tight-binding network engineering Stefano Longhi Dipartimento di Fisica, Politecnico di Milano and Istituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche, Piazza L. da Vinci 32, I-20133 Milano, Italy We suggest a simple method to engineer a tight-binding quantum network based on proper coupling to an auxiliary non-Hermitian cluster. In particular, it is shown that effective complex non-Hermitian hopping rates can be realized with only complex on-site energies in the network. Three applications of the Hamiltonian engineering method are presented: the synthesis of a nearly transparent defect in an Hermitian linear lattice; the realization of the Fano-Anderson model with complex coupling; and the synthesis of a PT -symmetric tight-binding lattice with a bound state in the continuum. PACS numbers: 03.65.-w, 11.30.Er, 72.20.Ee, 42.82.Et I. INTRODUCTION self-sustained emission in semi-infinite non-Hermitian systems at the exceptional point [29], optical simulation of PT -symmetric quantum field theories in the ghost Hamiltonian engineering is a powerful technique regime [30, 31], invisible defects in tight-binding lattices to control classical and quantum phenomena with [32], non-Hermitian bound states in the continuum [33], important applications in many areas of physics such and Bloch oscillations with trajectories in complex plane as quantum control [1{4], quantum state transfer and [34]. Previous proposals to implement complex hopping quantum information processing [5{10], quantum simu- amplitudes are based on fast temporal modulations of lation [11{13], and topological phases of matter [14{19]. complex on-site energies [31, 32], however such methods In quantum systems described by a tight-binding are rather challenging in practice and, as a matter of Hamiltonian, quantum engineering is usually aimed at fact, to date there is not any experimental demonstration tailoring and controlling hopping rates and site energies, of non-Hermitian complex couplings in tight-binding using either static or dynamic methods. -
CP2K: Atomistic Simulations of Condensed Matter Systems
Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2014 CP2K: Atomistic simulations of condensed matter systems Hutter, Juerg; Iannuzzi, Marcella; Schiffmann, Florian; VandeVondele, Joost Abstract: cp2k has become a versatile open-source tool for the simulation of complex systems on the nanometer scale. It allows for sampling and exploring potential energy surfaces that can be computed using a variety of empirical and first principles models. Excellent performance for electronic structure calculations is achieved using novel algorithms implemented for modern and massively parallel hardware. This review briefly summarizes the main capabilities and illustrates with recent applications thescience cp2k has enabled in the field of atomistic simulation. DOI: https://doi.org/10.1002/wcms.1159 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-89998 Journal Article Accepted Version Originally published at: Hutter, Juerg; Iannuzzi, Marcella; Schiffmann, Florian; VandeVondele, Joost (2014). CP2K: Atomistic simulations of condensed matter systems. Wiley Interdisciplinary Reviews. Computational Molecular Science, 4(1):15-25. DOI: https://doi.org/10.1002/wcms.1159 cp2k: Atomistic Simulations of Condensed Matter Systems J¨urgHutter and Marcella Iannuzzi Physical Chemistry Institute University of Zurich Winterthurerstrasse 190 CH-8057 Zurich, Switzerland Florian Schiffmann and Joost VandeVondele Nanoscale Simulations ETH Zurich Wolfgang-Pauli-Strasse 27 CH-8093 Zurich, Switzerland December 17, 2012 Abstract cp2k has become a versatile open source tool for the simulation of complex systems on the nanometer scale. It allows for sampling and exploring potential energy surfaces that can be computed using a variety of empirical and first principles models. -
Pushing Back the Limit of Ab-Initio Quantum Transport Simulations On
1 Pushing Back the Limit of Ab-initio Quantum Transport Simulations on Hybrid Supercomputers Mauro Calderara∗, Sascha Bruck¨ ∗, Andreas Pedersen∗, Mohammad H. Bani-Hashemian†, Joost VandeVondele†, and Mathieu Luisier∗ ∗Integrated Systems Laboratory, ETH Zurich,¨ 8092 Zurich,¨ Switzerland †Nanoscale Simulations, ETH Zurich,¨ 8093 Zurich,¨ Switzerland ABSTRACT The capabilities of CP2K, a density-functional theory package and OMEN, a nano-device simulator, are combined to study transport phenomena from first-principles in unprecedentedly large nanostructures. Based on the Hamil- tonian and overlap matrices generated by CP2K for a given system, OMEN solves the Schr¨odinger equation with open boundary conditions (OBCs) for all possible electron momenta and energies. To accelerate this core operation a robust algorithm called SplitSolve has been developed. It allows to simultaneously treat the OBCs on CPUs and the Schr¨odinger equation on GPUs, taking advantage of hybrid nodes. Our key achievements on the Cray-XK7 Titan are (i) a reduction in time-to-solution by more than one order of magnitude as compared to standard methods, enabling the simulation of structures with more than 50000 atoms, (ii) a parallel efficiency of 97% when scaling from 756 up to 18564 nodes, and (iii) a sustained performance of 15 DP-PFlop/s. arXiv:1812.01396v1 [physics.comp-ph] 4 Dec 2018 1. INTRODUCTION The fabrication of nanostructures has considerably improved over the last couple of years and is rapidly ap- proaching the point where individual atoms are reliably manipulated and assembled according to desired patterns. There is still a long way to go before such processes enter mass production, but exciting applications have already been demonstrated: a low-temperature single-atom transistor [1], atomically precise graphene nanoribbons [2], or van der Waals heterostructures based on metal-dichalcogenides [3] were recently synthesized. -
An Efficient Magnetic Tight-Binding Method for Transition
An efficient magnetic tight-binding method for transition metals and alloys Cyrille Barreteau, Daniel Spanjaard, Marie-Catherine Desjonquères To cite this version: Cyrille Barreteau, Daniel Spanjaard, Marie-Catherine Desjonquères. An efficient magnetic tight- binding method for transition metals and alloys. Comptes Rendus Physique, Centre Mersenne, 2016, 17 (3-4), pp.406 - 429. 10.1016/j.crhy.2015.12.014. cea-01384597 HAL Id: cea-01384597 https://hal-cea.archives-ouvertes.fr/cea-01384597 Submitted on 20 Oct 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. C. R. Physique 17 (2016) 406–429 Contents lists available at ScienceDirect Comptes Rendus Physique www.sciencedirect.com Condensed matter physics in the 21st century: The legacy of Jacques Friedel An efficient magnetic tight-binding method for transition metals and alloys Un modèle de liaisons fortes magnétique pour les métaux de transition et leurs alliages ∗ Cyrille Barreteau a,b, , Daniel Spanjaard c, Marie-Catherine Desjonquères a a SPEC, CEA, CNRS, Université Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France b DTU NANOTECH, Technical University of Denmark, Ørsteds Plads 344, DK-2800 Kgs. Lyngby, Denmark c Laboratoire de physique des solides, Université Paris-Sud, bâtiment 510, 91405 Orsay cedex, France a r t i c l e i n f o a b s t r a c t Article history: An efficient parameterized self-consistent tight-binding model for transition metals using s, Available online 21 December 2015 p and d valence atomic orbitals as a basis set is presented.