Solution to Exercise 2
|
|
- Grant Barnett
- 5 years ago
- Views:
Transcription
1 Department of physics, NTNU TFY Mesoscopic Physics Spring Solution to xercise Question Apart from an adjustable constant, the nearest neighbour nn) tight binding TB) band structure for the D triangular lattice is with nn lattice vectors k x, k y ) = γ R R R j= = aˆx e ik R j + e ik R ) j, = a ˆx + a ŷ = a ˆx a ŷ Thus, ensuring that ) =, k x, k y ) = γ + γ cosk x a + cos k xa + k y a + cos k xa ) k y a = γ + γ cosk x a + cos k ) x cos ky a = γ + γ cos k xa cos k ) xa + cos ky a Here, we have used the trigonometric identities cosa + b) + cosa b) = cosacosb and cos a = cos a. With b i a j = πδ ij, one finds reciprocal lattice vectors b = π ˆx ) ŷ a b = π a ŷ so the reciprocal lattice is also triangular, with nearest neighbour distance b = b = b = π/ a. The six M points are located at the midpoints of the edges of the hexagonal BZ, one of them is: k M = b = π ŷ. a
2 The six K points are located at the corners of the BZ, one of them is: k K = π a ˆx. Using Matlab, we obtain the following dispersion relation between the Γ point k = ) and the M point, and from Γ to K:.... between Gamma and M..... between Gamma and K In both figures, the energy is given in units of γ = γ, as function of the dimensionless quantities k y a and k x a. The latter figure suggests that the band width equals γ, with min = in the Γ point k = and max = γ in the K points. A D plot of the dispersion relation over the BZ actually, over the rectangular area in k space defined by ±k K and ±k M given above, since I did not take the time to figure out how to limit the plot region to the hexagonal BZ) supports this suggestion:
3 A formal way to find the band width amounts to locating the various stationary points of k), i.e., to find where k =, and insert these k values into k). For the D gradient of to vanish, we must require both / k x = and / k y = simultaneously. Let α = k x a and β = k y a. Then Here, we have several solutions: α = sin α cos α + ) β cos = β = cos α β sin = sinα/) = and sin β/) = yields the energy minimum min = at the Γ point k x = k y =, as well as saddle points M = γ in two of the six M points, k M =, ±π/ a) cosα/) = and cos β/) = yields the remaining four saddle points M = γ in k M = ±π/a, ±π/ a) sin β/) = and cosα/) = /) cos β/) yields the six energy maxima K = γ at the K points k K = ±π/a, ) and k K = ±π/a, ±π/ a). A D plot, viewed down the axis, and with a somewhat extended region of k space, could be illustrative:
4 Question In xercise, Question, we found reciprocal lattice vectors b = π/a)ˆx ŷ) and b = π/a)ˆx+ ŷ) for the hexagonal graphene lattice. Hence, the midpoints M of the edges of the hexagonal BZ are located at k values ±b /, ±b /, and ±π/a)ˆx. The corners K of the BZ are located at ±π/ a)ŷ, ±b xˆx+/)b y ŷ) = ±π/a)ˆx+π/ a)ŷ, and ±b xˆx +/)b y ŷ) = ±π/a)ˆx π/ a)ŷ. See figure in Solution.) Hence, by making a D plot of + and in the range π/ < k x a < π/ and π/ < k y a < π/, we cover the BZ plus a little extra, / of the BZ, actually). We set = and γ = and obtain the following valence band and conduction band +, showing clearly how the VB and the CB touch at the six K points: If we imagine the various energy bands of graphene being constructed from atomic orbitals in carbon, and taking into account that each primitive cell contains two carbon atoms, each giving rise to one band pr atomic orbital, we will end up with the two bands σ and σ from the s state. Next, we have hybridized sp bands, constructed as linear combinations of the atomic s, p x, and p y states of the two carbon atoms. The sp bands are responsible for the strong C C bonds, and the corresponding wave functions are directional, pointing towards the three nearest neighbours of any given carbon atom. As discussed in the lectures, each primitive cell will contribute electronic states to each energy band. Hence, of the electrons in each primitive cell will occupy the σ and σ bands and three sp bands. We are left with electrons pr primitive cell one from each C atom), and these electrons have precisely the energy bands and + of the present problem at their disposal. It is the p z atomic orbital that is the basis for constructing these two energy bands. In chemistry, they are typically referred to as π and π bands. The notion of π bonding is probably well known from molecules like ethylene and benzene, I guess.) At zero temperature, the electrons will be in the ground state of the system, i.e., the valence
5 band will be filled and the conduction band + will be empty. Hence, graphene may be considered a semiconductor, with zero bandgap, g =. Graphene, a single layer of graphite, was first made experimentally in, using simple equipment like Scotch tape. Since then, it has been a hot research topic, both experimentally and theoretically. One important reason is the rather peculiar band structure in the vicinity of the K points. We will come back to graphene in the next xercise. The nearest neighbour tight binding solution of graphene and graphite) was published already more than sixty years ago P. R. Wallace, Phys Rev, )).
Nearly Free Electron Gas model - I
Nearly Free Electron Gas model - I Contents 1 Free electron gas model summary 1 2 Electron effective mass 3 2.1 FEG model for sodium...................... 4 3 Nearly free electron model 5 3.1 Primitive
More informationQuantum Condensed Matter Physics
QCMP-2017/18 Problem sheet 2: Quantum Condensed Matter Physics Band structure 1. Optical absorption of simple metals Sketch the typical energy-wavevector dependence, or dispersion relation, of electrons
More informationwhere a is the lattice constant of the triangular Bravais lattice. reciprocal space is spanned by
Contents 5 Topological States of Matter 1 5.1 Intro.......................................... 1 5.2 Integer Quantum Hall Effect..................... 1 5.3 Graphene......................................
More informationECE 535 Theory of Semiconductors and Semiconductor Devices Fall 2015 Homework # 5 Due Date: 11/17/2015
ECE 535 Theory of Semiconductors and Semiconductor Devices Fall 2015 Homework # 5 Due Date: 11/17/2015 Problem # 1 Two carbon atoms and four hydrogen atoms form and ethane molecule with the chemical formula
More informationGraphene and Planar Dirac Equation
Graphene and Planar Dirac Equation Marina de la Torre Mayado 2016 Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 1 / 48 Outline 1 Introduction 2 The Dirac Model Tight-binding model
More informationRefering to Fig. 1 the lattice vectors can be written as: ~a 2 = a 0. We start with the following Ansatz for the wavefunction:
1 INTRODUCTION 1 Bandstructure of Graphene and Carbon Nanotubes: An Exercise in Condensed Matter Physics developed by Christian Schönenberger, April 1 Introduction This is an example for the application
More informationELECTRONIC ENERGY DISPERSION AND STRUCTURAL PROPERTIES ON GRAPHENE AND CARBON NANOTUBES
ELECTRONIC ENERGY DISPERSION AND STRUCTURAL PROPERTIES ON GRAPHENE AND CARBON NANOTUBES D. RACOLTA, C. ANDRONACHE, D. TODORAN, R. TODORAN Technical University of Cluj Napoca, North University Center of
More informationQuantum Oscillations in Graphene in the Presence of Disorder
WDS'9 Proceedings of Contributed Papers, Part III, 97, 9. ISBN 978-8-778-- MATFYZPRESS Quantum Oscillations in Graphene in the Presence of Disorder D. Iablonskyi Taras Shevchenko National University of
More informationNanoscience quantum transport
Nanoscience quantum transport Janine Splettstößer Applied Quantum Physics, MC2, Chalmers University of Technology Chalmers, November 2 10 Plan/Outline 4 Lectures (1) Introduction to quantum transport (2)
More informationECE 659, PRACTICE EXAM II Actual Exam Friday, Feb.21, 2014, FNY B124, PM CLOSED BOOK. = H nm. Actual Exam will have five questions.
1 ECE 659, PRACTICE EXAM II Actual Exam Friday, Feb.1, 014, FNY B14, 330-40PM CLOSED BOOK Useful relation h( k ) = H nm [ ] e +i k.( r m r n ) Actual Exam will have five questions. The following questions
More information... 3, , = a (1) 3 3 a 2 = a (2) The reciprocal lattice vectors are defined by the condition a b = 2πδ ij, which gives
PHZ646: Fall 013 Problem set # 4: Crystal Structure due Monday, 10/14 at the time of the class Instructor: D. L. Maslov maslov@phys.ufl.edu 39-0513 Rm. 114 Office hours: TR 3 pm-4 pm Please help your instructor
More informationCarbon nanotubes: Models, correlations and the local density of states
Carbon nanotubes: Models, correlations and the local density of states Alexander Struck in collaboration with Sebastián A. Reyes Sebastian Eggert 15. 03. 2010 Outline Carbon structures Modelling of a carbon
More informationCondensed Matter Physics April, 8, LUMS School of Science and Engineering
Condensed Matter Physics April, 8, 0 LUMS School of Science and Engineering PH-33 Solution of assignment 5 April, 8, 0 Interplanar separation Answer: To prove that the reciprocal lattice vector G = h b
More information3-month progress Report
3-month progress Report Graphene Devices and Circuits Supervisor Dr. P.A Childs Table of Content Abstract... 1 1. Introduction... 1 1.1 Graphene gold rush... 1 1.2 Properties of graphene... 3 1.3 Semiconductor
More informationstructure of graphene and carbon nanotubes which forms the basis for many of their proposed applications in electronics.
Chapter Basics of graphene and carbon nanotubes This chapter reviews the theoretical understanding of the geometrical and electronic structure of graphene and carbon nanotubes which forms the basis for
More informationEnergy bands in two limits
M. A. Gusmão IF-UFRGS 1 FIP10601 Text 4 Energy bands in two limits After presenting a general view of the independent-electron approximation, highlighting the importance and consequences of Bloch s theorem,
More informationNotes on Ewald summation techniques
February 3, 011 Notes on Ewald summation techniques Adapted from similar presentation in PHY 71 he total electrostatic potential energy of interaction between point charges {q i } at the positions {r i
More information3a 2. a 1 = 3a. a 2 = 3a
Physics 195 / Applied Physics 195 Assignment #4 Professor: Donhee Ham Teaching Fellows: Brendan Deveney and Laura Adams Date: Oct. 6, 017 Due: 1:45pm + 10 min grace period, Oct. 13, 017 at the dropbox
More informationPhysics 211B : Problem Set #0
Physics 211B : Problem Set #0 These problems provide a cross section of the sort of exercises I would have assigned had I taught 211A. Please take a look at all the problems, and turn in problems 1, 4,
More informationQuantum Condensed Matter Physics Lecture 5
Quantum Condensed Matter Physics Lecture 5 detector sample X-ray source monochromator David Ritchie http://www.sp.phy.cam.ac.uk/drp2/home QCMP Lent/Easter 2019 5.1 Quantum Condensed Matter Physics 1. Classical
More informationElectronic properties of graphene. Jean-Noël Fuchs Laboratoire de Physique des Solides Université Paris-Sud (Orsay)
Electronic properties of graphene Jean-Noël Fuchs Laboratoire de Physique des Solides Université Paris-Sud (Orsay) Cargèse, September 2012 3 one-hour lectures in 2 x 1,5h on electronic properties of graphene
More informationFermi surfaces and Electron
Solid State Theory Physics 545 Fermi Surfaces Fermi surfaces and Electron dynamics Band structure calculations give E(k) E(k) determines the dynamics of the electrons It is E(k) at the Fermi Surface that
More informationCalculating Electronic Structure of Different Carbon Nanotubes and its Affect on Band Gap
Calculating Electronic Structure of Different Carbon Nanotubes and its Affect on Band Gap 1 Rashid Nizam, 2 S. Mahdi A. Rizvi, 3 Ameer Azam 1 Centre of Excellence in Material Science, Applied Physics AMU,
More informationLecture 4: Basic elements of band theory
Phys 769 Selected Topics in Condensed Matter Physics Summer 010 Lecture 4: Basic elements of band theory Lecturer: Anthony J. Leggett TA: Bill Coish 1 Introduction Most matter, in particular most insulating
More informationLecture 6. Tight-binding model
Lecture 6 Tight-binding model In Lecture 3 we discussed the Krönig-Penny model and we have seen that, depending on the strength of the potential barrier inside the unit cell, the electrons can behave like
More informationTight-Binding Approximation. Faculty of Physics UW
Tight-Binding Approximation Faculty of Physics UW Jacek.Szczytko@fuw.edu.pl The band theory of solids. 2017-06-05 2 Periodic potential Bloch theorem φ n,k Ԧr = u n,k Ԧr eik Ԧr Bloch wave, Bloch function
More informationThree Most Important Topics (MIT) Today
Three Most Important Topics (MIT) Today Electrons in periodic potential Energy gap nearly free electron Bloch Theorem Energy gap tight binding Chapter 1 1 Electrons in Periodic Potential We now know the
More informationPROJECT C: ELECTRONIC BAND STRUCTURE IN A MODEL SEMICONDUCTOR
PROJECT C: ELECTRONIC BAND STRUCTURE IN A MODEL SEMICONDUCTOR The aim of this project is to present the student with a perspective on the notion of electronic energy band structures and energy band gaps
More informationNearly Free Electron Gas model - II
Nearly Free Electron Gas model - II Contents 1 Lattice scattering 1 1.1 Bloch waves............................ 2 1.2 Band gap formation........................ 3 1.3 Electron group velocity and effective
More informationMP464: Solid State Physics Problem Sheet
MP464: Solid State Physics Problem Sheet 1 Write down primitive lattice vectors for the -dimensional rectangular lattice, with sides a and b in the x and y-directions respectively, and a face-centred rectangular
More informationLectures Graphene and
Lectures 15-16 Graphene and carbon nanotubes Graphene is atomically thin crystal of carbon which is stronger than steel but flexible, is transparent for light, and conducts electricity (gapless semiconductor).
More informationCHAPTER 6 CHIRALITY AND SIZE EFFECT IN SINGLE WALLED CARBON NANOTUBES
10 CHAPTER 6 CHIRALITY AND SIZE EFFECT IN SINGLE WALLED CARBON NANOTUBES 6.1 PREAMBLE Lot of research work is in progress to investigate the properties of CNTs for possible technological applications.
More informationPHYSICS 220 : GROUP THEORY FINAL EXAMINATION SOLUTIONS
PHYSICS 0 : GROUP THEORY FINAL EXAMINATION SOLUTIONS This exam is due in my office, 5438 Mayer Hall, at 9 am, Monday, June 6. You are allowed to use the course lecture notes, the Lax text, and the character
More informationBonding in solids The interaction of electrons in neighboring atoms of a solid serves the very important function of holding the crystal together.
Bonding in solids The interaction of electrons in neighboring atoms of a solid serves the very important function of holding the crystal together. For example Nacl In the Nacl lattice, each Na atom is
More informationGraphite, graphene and relativistic electrons
Graphite, graphene and relativistic electrons Introduction Physics of E. graphene Y. Andrei Experiments Rutgers University Transport electric field effect Quantum Hall Effect chiral fermions STM Dirac
More informationIntroduction to Condensed Matter Physics
Introduction to Condensed Matter Physics The Reciprocal Lattice M.P. Vaughan Overview Overview of the reciprocal lattice Periodic functions Reciprocal lattice vectors Bloch functions k-space Dispersion
More informationFrom Graphene to Nanotubes
From Graphene to Nanotubes Zone Folding and Quantum Confinement at the Example of the Electronic Band Structure Christian Krumnow christian.krumnow@fu-berlin.de Freie Universität Berlin June 6, Zone folding
More informationTight binding models from band representations
Tight binding models from band representations Jennifer Cano Stony Brook University and Flatiron Institute for Computational Quantum Physics Part 0: Review of band representations BANDREP http://www.cryst.ehu.es/cgi-bin/cryst/programs/bandrep.pl
More informationQuantum Condensed Matter Physics Lecture 9
Quantum Condensed Matter Physics Lecture 9 David Ritchie QCMP Lent/Easter 2018 http://www.sp.phy.cam.ac.uk/drp2/home 9.1 Quantum Condensed Matter Physics 1. Classical and Semi-classical models for electrons
More informationHomework 4 Due 25 October 2018 The numbers following each question give the approximate percentage of marks allocated to that question.
Name: Homework 4 Due 25 October 218 The numbers following each question give the approximate percentage of marks allocated to that question. 1. Use the reciprocal metric tensor again to calculate the angle
More informationCovalent Bonds: overlap of orbitals σ-bond π-bond Molecular Orbitals
Covalent Bonding What is covalent bonding? Covalent Bonds: overlap of orbitals σ-bond π-bond Molecular Orbitals Hybrid Orbital Formation Shapes of Hybrid Orbitals Hybrid orbitals and Multiple Bonds resonance
More informationPH575 Spring Lecture #28 Nanoscience: the case study of graphene and carbon nanotubes.
PH575 Spring 2014 Lecture #28 Nanoscience: the case study of graphene and carbon nanotubes. Nanoscience scale 1-100 nm "Artificial atoms" Small size => discrete states Large surface to volume ratio Bottom-up
More informationELECTRONS AND HOLES Lecture 21
Solid State Physics ELECTRONS AND HOLES Lecture 21 A.H. Harker Physics and Astronomy UCL Electrons and Holes 8 Electrons and Holes 8.1 Equations of motion In one dimension, an electron with wave-vector
More informationGraphene and Carbon Nanotubes
Graphene and Carbon Nanotubes 1 atom thick films of graphite atomic chicken wire Novoselov et al - Science 306, 666 (004) 100μm Geim s group at Manchester Novoselov et al - Nature 438, 197 (005) Kim-Stormer
More informationThe calculation of energy gaps in small single-walled carbon nanotubes within a symmetry-adapted tight-binding model
The calculation of energy gaps in small single-walled carbon nanotubes within a symmetry-adapted tight-binding model Yang Jie( ) a), Dong Quan-Li( ) a), Jiang Zhao-Tan( ) b), and Zhang Jie( ) a) a) Beijing
More informationHeterostructures and sub-bands
Heterostructures and sub-bands (Read Datta 6.1, 6.2; Davies 4.1-4.5) Quantum Wells In a quantum well, electrons are confined in one of three dimensions to exist within a region of length L z. If the barriers
More informationThe many forms of carbon
The many forms of carbon Carbon is not only the basis of life, it also provides an enormous variety of structures for nanotechnology. This versatility is connected to the ability of carbon to form two
More informationLecture. Ref. Ihn Ch. 3, Yu&Cardona Ch. 2
Lecture Review of quantum mechanics, statistical physics, and solid state Band structure of materials Semiconductor band structure Semiconductor nanostructures Ref. Ihn Ch. 3, Yu&Cardona Ch. 2 Reminder
More informationPhysics 541: Condensed Matter Physics
Physics 541: Condensed Matter Physics Final Exam Monday, December 17, 2012 / 14:00 17:00 / CCIS 4-285 Student s Name: Instructions There are 24 questions. You should attempt all of them. Mark your response
More informationCarbon nanotubes and Graphene
16 October, 2008 Solid State Physics Seminar Main points 1 History and discovery of Graphene and Carbon nanotubes 2 Tight-binding approximation Dynamics of electrons near the Dirac-points 3 Properties
More informationDraft of solution Exam TFY4220, Solid State Physics, 29. May 2015.
Draft of solution Exam TY40, Solid State Physics, 9. May 05. Problem (5%) Introductory questions (answers can be found in the books) a) Small Ewald sphere, not many reflections in Bragg with a single crystal.
More informationSo why is sodium a metal? Tungsten Half-filled 5d band & half-filled 6s band. Insulators. Interaction of metals with light?
Bonding in Solids: Metals, Insulators, & CHEM 107 T. Hughbanks Delocalized bonding in Solids Think of a pure solid as a single, very large molecule. Use our bonding pictures to try to understand properties.
More informationTwo Dimensional Chern Insulators, the Qi-Wu-Zhang and Haldane Models
Two Dimensional Chern Insulators, the Qi-Wu-Zhang and Haldane Models Matthew Brooks, Introduction to Topological Insulators Seminar, Universität Konstanz Contents QWZ Model of Chern Insulators Haldane
More informationChapter 1. Crystal structure. 1.1 Crystal lattices
Chapter 1 Crystal structure 1.1 Crystal lattices We will concentrate as stated in the introduction, on perfect crystals, i.e. on arrays of atoms, where a given arrangement is repeated forming a periodic
More informationElectrons in a periodic potential
Chapter 3 Electrons in a periodic potential 3.1 Bloch s theorem. We consider in this chapter electrons under the influence of a static, periodic potential V (x), i.e. such that it fulfills V (x) = V (x
More informationCalculation of Cutting Lines of Single-Walled Carbon Nanotubes
65 C.Ü. Fen Fakültesi Fen Bilimleri Dergisi, Cilt 33, No. 1 (2012) Calculation of Cutting Lines of Single-Walled Carbon Nanotubes Erdem UZUN 1* 1 Karamanoğlu Mehmetbey Üniversitesi, Fen Fakültesi, Fizik
More informationQuantum mechanics can be used to calculate any property of a molecule. The energy E of a wavefunction Ψ evaluated for the Hamiltonian H is,
Chapter : Molecules Quantum mechanics can be used to calculate any property of a molecule The energy E of a wavefunction Ψ evaluated for the Hamiltonian H is, E = Ψ H Ψ Ψ Ψ 1) At first this seems like
More informationHybridization of Atomic Orbitals. (Chapter 1 in the Klein text)
Hybridization of Atomic Orbitals (Chapter 1 in the Klein text) Basic Ideas The atomic structures, from the Periodic Table, of atoms such as C, N, and O do not adequately explain how these atoms use orbitals
More informationSolid State Physics 460- Lecture 5 Diffraction and the Reciprocal Lattice Continued (Kittel Ch. 2)
Solid State Physics 460- Lecture 5 Diffraction and the Reciprocal Lattice Continued (Kittel Ch. 2) Ewald Construction 2θ k out k in G Physics 460 F 2006 Lect 5 1 Recall from previous lectures Definition
More informationBasic Semiconductor Physics
6 Basic Semiconductor Physics 6.1 Introduction With this chapter we start with the discussion of some important concepts from semiconductor physics, which are required to understand the operation of solar
More informationBeautiful Graphene, Photonic Crystals, Schrödinger and Dirac Billiards and Their Spectral Properties
Beautiful Graphene, Photonic Crystals, Schrödinger and Dirac Billiards and Their Spectral Properties Cocoyoc 2012 Something about graphene and microwave billiards Dirac spectrum in a photonic crystal Experimental
More informationEnergy band of graphene ribbons under the tensile force
Energy band of graphene ribbons under the tensile force Yong Wei Guo-Ping Tong * and Sheng Li Institute of Theoretical Physics Zhejiang Normal University Jinhua 004 Zhejiang China ccording to the tight-binding
More informationBand Structure of Isolated and Bundled Nanotubes
Chapter 5 Band Structure of Isolated and Bundled Nanotubes The electronic structure of carbon nanotubes is characterized by a series of bands (sub- or minibands) arising from the confinement around the
More informationAn Extended Hückel Theory based Atomistic Model for Graphene Nanoelectronics
Journal of Computational Electronics X: YYY-ZZZ,? 6 Springer Science Business Media, Inc. Manufactured in The Netherlands An Extended Hückel Theory based Atomistic Model for Graphene Nanoelectronics HASSAN
More information3.14. The model of Haldane on a honeycomb lattice
4 Phys60.n..7. Marginal case: 4 t Dirac points at k=(,). Not an insulator. No topological index...8. case IV: 4 t All the four special points has z 0. We just use u I for the whole BZ. No singularity.
More informationLecture 16 C1403 October 31, Molecular orbital theory: molecular orbitals and diatomic molecules
Lecture 16 C1403 October 31, 2005 18.1 Molecular orbital theory: molecular orbitals and diatomic molecules 18.2 Valence bond theory: hybridized orbitals and polyatomic molecules Bond order, bond lengths,
More informationPage III-8-1 / Chapter Eight Lecture Notes MAR. Two s orbitals overlap. One s & one p. overlap. Two p orbitals. overlap MAR
Bonding and Molecular Structure: Orbital ybridization and Molecular Orbitals Chapter 8 Page III-8-1 / Chapter Eight Lecture Notes Advanced Theories of Chemical Bonding Chemistry 222 Professor Michael Russell
More informationThe Solid State. Phase diagrams Crystals and symmetry Unit cells and packing Types of solid
The Solid State Phase diagrams Crystals and symmetry Unit cells and packing Types of solid Learning objectives Apply phase diagrams to prediction of phase behaviour Describe distinguishing features of
More informationLecture 15 From molecules to solids
Lecture 15 From molecules to solids Background In the last two lectures, we explored quantum mechanics of multi-electron atoms the subject of atomic physics. In this lecture, we will explore how these
More informationWe can model covalent bonding in molecules in essentially two ways:
CHEM 2060 Lecture 22: VB Theory L22-1 PART FIVE: The Covalent Bond We can model covalent bonding in molecules in essentially two ways: 1. Localized Bonds (retains electron pair concept of Lewis Structures)
More informationMath 5BI: Problem Set 6 Gradient dynamical systems
Math 5BI: Problem Set 6 Gradient dynamical systems April 25, 2007 Recall that if f(x) = f(x 1, x 2,..., x n ) is a smooth function of n variables, the gradient of f is the vector field f(x) = ( f)(x 1,
More informationSolids. properties & structure
Solids properties & structure Determining Crystal Structure crystalline solids have a very regular geometric arrangement of their particles the arrangement of the particles and distances between them is
More informationHomework 2 - Solutions
Homework 2 - Solutions Carbon Nanotubes Part 4) We can write the wavefunction of the grahene sheet within the tight-binding model in the usual way ( ~ R)= e i~ k ~R Then, to imose eriodic boundary condition
More informationGraphene: massless electrons in flatland.
Graphene: massless electrons in flatland. Enrico Rossi Work supported by: University of Chile. Oct. 24th 2008 Collaorators CMTC, University of Maryland Sankar Das Sarma Shaffique Adam Euyuong Hwang Roman
More informationTheoretical Concepts of Spin-Orbit Splitting
Chapter 9 Theoretical Concepts of Spin-Orbit Splitting 9.1 Free-electron model In order to understand the basic origin of spin-orbit coupling at the surface of a crystal, it is a natural starting point
More informationMP464: Solid State Physics Problem Sheet
MP464: Solid State Physics Problem Sheet 1) Write down primitive lattice vectors for the -dimensional rectangular lattice, with sides a and b in the x and y-directions respectively, and a face-centred
More informationConductance of Graphene Nanoribbon Junctions and the Tight Binding Model
Wu and Childs Nanoscale es Lett, 6:6 http://www.nanoscalereslett.com/content/6//6 NANO EXPE Open Access Conductance of Graphene Nanoribbon Junctions and the Tight Binding Model Y Wu, PA Childs * Abstract
More informationFrom graphene to graphite: Electronic structure around the K point
PHYSICL REVIEW 74, 075404 2006 From graphene to graphite: Electronic structure around the K point. Partoens* and F. M. Peeters Universiteit ntwerpen, Departement Fysica, Groenenborgerlaan 171, -2020 ntwerpen,
More informationChapter Molecules are 3D. Shapes and Bonds. Chapter 9 1. Chemical Bonding and Molecular Structure
Chapter 9 Chemical Bonding and Molecular Structure 1 Shape 9.1 Molecules are 3D Angle Linear 180 Planar triangular (trigonal planar) 120 Tetrahedral 109.5 2 Shapes and Bonds Imagine a molecule where the
More informationQuantum Condensed Matter Physics Lecture 4
Quantum Condensed Matter Physics Lecture 4 David Ritchie QCMP Lent/Easter 2019 http://www.sp.phy.cam.ac.uk/drp2/home 4.1 Quantum Condensed Matter Physics 1. Classical and Semi-classical models for electrons
More informationLecture 3b. Bonding Model and Dopants. Reading: (Cont d) Notes and Anderson 2 sections
Lecture 3b Bonding Model and Dopants Reading: (Cont d) Notes and Anderson 2 sections 2.3-2.7 The need for more control over carrier concentration Without help the total number of carriers (electrons and
More informationSupplementary Figure S1. STM image of monolayer graphene grown on Rh (111). The lattice
Supplementary Figure S1. STM image of monolayer graphene grown on Rh (111). The lattice mismatch between graphene (0.246 nm) and Rh (111) (0.269 nm) leads to hexagonal moiré superstructures with the expected
More informationCarbon nanomaterials. Gavin Lawes Wayne State University.
Carbon nanomaterials Gavin Lawes Wayne State University glawes@wayne.edu Outline 1. Carbon structures 2. Carbon nanostructures 3. Potential applications for Carbon nanostructures Periodic table from bpc.edu
More informationTheoretical Modeling of Tunneling Barriers in Carbon-based Molecular Electronic Junctions
Second Revised version, jp-2014-09838e Supporting Information Theoretical Modeling of Tunneling Barriers in Carbon-based Molecular Electronic Junctions Mykola Kondratenko 1,2, Stanislav R. Stoyanov 1,3,4,
More informationZero-bias conductance peak in detached flakes of superconducting 2H-TaS2 probed by STS
Zero-bias conductance peak in detached flakes of superconducting 2H-TaS2 probed by STS J. A. Galvis, L. C., I. Guillamon, S. Vieira, E. Navarro-Moratalla, E. Coronado, H. Suderow, F. Guinea Laboratorio
More informationSupplementary Figures
Supplementary Figures 8 6 Energy (ev 4 2 2 4 Γ M K Γ Supplementary Figure : Energy bands of antimonene along a high-symmetry path in the Brillouin zone, including spin-orbit coupling effects. Empty circles
More informationIntroduction to Solid State Physics or the study of physical properties of matter in a solid phase
Introduction to Solid State Physics or the study of physical properties of matter in a solid phase Prof. Germar Hoffmann 1. Crystal Structures 2. Reciprocal Lattice 3. Crystal Binding and Elastic Constants
More informationPhysics 140A WINTER 2013
PROBLEM SET 1 Solutions Physics 140A WINTER 2013 [1.] Sidebottom Problem 1.1 Show that the volume of the primitive cell of a BCC crystal lattice is a 3 /2 where a is the lattice constant of the conventional
More informationLecture Notes D: Molecular Orbital Theory
Lecture Notes D: Molecular Orbital Theory Orbital plotting applet: http://www.mpcfaculty.net/mark_bishop/hybrid_frame.htm Images below from: http://employees.oneonta.edu/viningwj/chem111/hybrids%20and%20pi%20bonding.jpg
More informationLecture 14 Chemistry 362 M. Darensbourg 2017 Spring term. Molecular orbitals for diatomics
Lecture 14 Chemistry 362 M. Darensbourg 2017 Spring term Molecular orbitals for diatomics Molecular Orbital Theory of the Chemical Bond Simplest example - H 2 : two H atoms H A and H B Only two a.o.'s
More informationMolecular Shape and Molecular Polarity. Molecular Shape and Molecular Polarity. Molecular Shape and Molecular Polarity
Molecular Shape and Molecular Polarity When there is a difference in electronegativity between two atoms, then the bond between them is polar. It is possible for a molecule to contain polar bonds, but
More informationLecture 11 - Phonons II - Thermal Prop. Continued
Phonons II - hermal Properties - Continued (Kittel Ch. 5) Low High Outline Anharmonicity Crucial for hermal expansion other changes with pressure temperature Gruneisen Constant hermal Heat ransport Phonon
More informationGraphene and chiral fermions
Graphene and chiral fermions Michael Creutz BNL & U. Mainz Extending graphene structure to four dimensions gives a two-flavor lattice fermion action one exact chiral symmetry protects mass renormalization
More informationLecture 4 Symmetry in the solid state -
Lecture 4 Symmetry in the solid state - Part IV: Brillouin zones and the symmetry of the band structure. 1 Symmetry in Reciprocal Space the Wigner-Seitz construction and the Brillouin zones Non-periodic
More informationLecture 4 Symmetry in the solid state -
Lecture 4 Symmetry in the solid state - Part IV: Brillouin zones and the symmetry of the band structure. 1 Symmetry in Reciprocal Space the Wigner-Seitz construction and the Brillouin zones Non-periodic
More informationOptical properties of single-layer, double-layer, and bulk MoS2
Optical properties of single-layer, double-layer, and bulk MoS Alejandro Molina-Sánchez, Ludger Wirtz, Davide Sangalli, Andrea Marini, Kerstin Hummer Single-layer semiconductors From graphene to a new
More information14.1 Shapes of molecules and ions (HL)
14.1 Shapes of molecules and ions (HL) The octet is the most common electron arrangement because of its stability. Exceptions: a) Fewer electrons (incomplete octet) if the central atom is a small atoms,
More informationLecture 16, February 25, 2015 Metallic bonding
Lecture 16, February 25, 2015 Metallic bonding Elements of Quantum Chemistry with Applications to Chemical Bonding and Properties of Molecules and Solids Course number: Ch125a; Room 115 BI Hours: 11-11:50am
More informationTransversal electric field effect in multilayer graphene nanoribbon
Transversal electric field effect in multilayer graphene nanoribbon S. Bala kumar and Jing Guo a) Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida 32608, USA
More informationDiamond. There are four types of solid: -Hard Structure - Tetrahedral atomic arrangement. What hybrid state do you think the carbon has?
Bonding in Solids Bonding in Solids There are four types of solid: 1. Molecular (formed from molecules) - usually soft with low melting points and poor conductivity. 2. Covalent network - very hard with
More information