Structural Calculations phase stability, surfaces, interfaces etc
|
|
- Elfrieda Gaines
- 5 years ago
- Views:
Transcription
1 Structural Calculations phase stability, surfaces, interfaces etc Keith Refson STFC Rutherford Appleton Laboratory September 19, 2007 Phase Equilibrium 2 Energy-Volume curves Model equations of State Birch-Murnaghan Equation of State First-order phase transitions More degrees of freedom Technical aspects, convergence Zero-point energy Zero-point energy and the quasi-harmonic approximation Elastic constants Elastic properties of CaO CASTEPs use of symmetry Surfaces 14 Surface modelling with slab geometry Slab models I Surface Energy calculations Polarization in slabs Defect calculations 19 Defect models
2 Phase Equilibrium 2 / 20 Energy-Volume curves Consider simple structure such as rocksalt (B1) one of simplest calculations is to compute E as function of V B1 rocksalt every atom is on a crystallographic high-symmetry site. No geometry optimisation needed! Equation of state (Really PV curve, which is commonly measured experimental quantity.) V(A 3 ) CaO Equation of State P (GPa) Energy (ev) CaO E vs V V (A 3 ) CASTEP Workshop 2007: York 3 / 20 Model equations of State Real EOS are PVT relations, but this lecture restricted to easy case, T=0. Model EOS for fluids, eg Van der Waals, Redlich-Kwong There are several common models of EOS for solids based on elasticity theory Birch Murnaghan (Phys. Rev (1947)) is most commonly used EOS E(V ) = E 0 + 9V 0B 0 16 j h` V0 V 2 i B 0 + h` V0 V 2 3 1i 2 h 6 4 ` V 0 V 2 3 i ff Two definitions of bulk modulus B 0; B 0 = V 0 d 2 P dv 2 and as a fit to B-M or other EOS. These are not equivalent. Compression/EOS experiments use a B-M fit whereas ultrasound methods measure 2nd derivative CASTEP Workshop 2007: York 4 / 20 2
3 Birch-Murnaghan Equation of State Birch-Murnaghan fits to CaO EOS Effect of fitting range Castep calculation V 0 =110.2, B 0 =28.4 GPa, B 0 =4.16 V 0 =110.3, B 0 =27.9 GPa, B 0 =4.23 range of data changes fit result. Only V <= V 0 accessible to experiment. V > V 0 also accessible to calculation. B 0 should be compared using fit over same or similar range to expt. Need to understand how B 0 was measured to make valid comparison. B 0 from ultrasonic expt. should be compared to V 0 d 2 P Energy (ev) dv 2 V (A 3 ) CASTEP Workshop 2007: York 5 / 20 First-order phase transitions Phase equilibrium occurs when 2 criteria met simultaneously Pressures of 2 phases are equal, ie P = de 1 = de 2 dv Enthalpies are equal E 1 + PV 1 = E 2 + PV 2 E 2 E 1 = P (V 2 V 1). dv. Then P eq is gradient of common tangent CaO B1/B2 transition 4 40 CaO Equation of State 3 30 Energy (ev) 2 1 V(A 3 ) V (A 3 ) P (GPa) Alternatively use enthalpy-pressure plot, where most stable phase is that of lowest enthalpy. Need accurate calculation of pressures for this approach to work, so higher degree of PW convergence needed. CASTEP Workshop 2007: York 6 / 20 3
4 More degrees of freedom Eg TiO2 - Tetragonal rutile phase has 3 parameters a, c and internal coord u. Geometry optimization at range of P cheaper than exploring 3d energy surface. Complex phases with many internal degrees of freedom can be handled the same way, eg high-pressure Mg 2SiO 4 polymorphs (right). Olivine-Wadsleyite and Wadsleyite-Ringwoodite transitions detectable as seismic discontinuities in Earth s Mantle due to different elastic properties. CASTEP geometry optimiser has (unique?) capability of optimising cell at fixed volume. Lattice Energy (ev) Forsterite P6 3 mc P - 3m1 Ringwoodite Wadsleyite P-3m Volume (A 3 ) CASTEP Workshop 2007: York 7 / 20 Technical aspects, convergence K-point integration accuracy varies with volume (not usually serious for insulators) K-point sampling - No error cancellation between different phases need absolute convergence with k-point. Gradient of common tangent sensitive to small absolute errors. Need high-accuracy calculations. Enthalpy minimum calculations equally sensitive, but pressure harder to calculate accurately than energy, even with FSBC. Does not account for quantum nature of nuclear motion. Need to include zero-point energy correction even at 0K. Care needed using symmetry - constraint restricts search space If using cell optimisation fixed-basis calculation gives systematic basis-set error with volume. Use fixed-cutoff option. No prescription for determining which phases to try. Crystal structure prediction is very hard problem. (But recent work is making progress). CASTEP Workshop 2007: York 8 / 20 4
5 Zero-point energy Zero-point energy contribution P β ω i can be significant and is in general different for different phases 2 Can calculate if vibrational phonon spectrum or DOS known. Use lattice dynamics calculation or (less satisfactory) experimental data. Harmonic Free R energy is ` F = E + 1 β g(ω)log `2sinh β ω 2 dω where g(ω) is the phonon density of states (DOS). To calculate DOS need phonon frequencies at all q in BZ. MgH2 phonon DOS g(ω ω (cm -1 ) CASTEP Workshop 2007: York 9 / 20 Zero-point energy and the quasi-harmonic approximation Energy (ev) E (athermal) E(0K) E(20K) E(300K) Quasi-harmonic approximation assumes that phonon frequencies depend only on the cell parameters. Ignores intrinsic anharmonic thermal effects on DOS. Works for relatively harmonic systems Valid for T 0.5 T m. QHA is cheapest way of extending ab-initio to TT > 0 In Zero Static Internal Stress Approximation geometry optimise coordinates at each volume (or strain) Quasi-Harmonic Free energy geometry optimization possible with empirical force fields but not yet ab-initio Pressure (GPa) CASTEP Workshop 2007: York 10 / 20 5
6 Elastic constants Elastic strain theory gives E(ǫ) = E 0 + V/2 6X C ijǫ iǫ j + O(3) Programmed strains ǫ may be used to extract individual elastic constants, e.g. if 2 3 ǫ = 1 0 δ δ 4δ 0 δ5 then 2 δ δ 0 ij E = E V C44δ2 + O(3) for a cubic crystal. See book by Nye on elastic theory for symmetry-adapted strains. For low symmetry crystals more efficient to compute stress and use σ αβ = C αβγδ ǫ γǫ δ. But need very well converged stress. No automated calculation built into CASTEP; available in Materials Studio CASTEP Workshop 2007: York 11 / 20 Elastic properties of CaO Theory Expt(1) Expt (2) a ± K ± ± 0.6 K ± ± ± 0.1 C s 83.1 ± ± ± 0.3 C ± ± ± 0.6 C ± ± ± 0.7 C ± ± ± Dragoo and Spain, J. Phys. Chem. Solids (1977) 38, Chang and Graham J. Phys. Chem. Solids (1977) 38, 1355 CASTEP Workshop 2007: York 12 / 20 6
7 CASTEPs use of symmetry CASTEP uses symmetry to optimize BZ integration. K-point grid is reduced to Irreducible BZ using a weighted sum. Geometry optimization preserves initial symmetry of atomic co-ordinates and cell vectors. Warning 1. The result of a symmetry-constrained SPE or Geom. Opt. calculation may be mechanically unstable to a symmetry-breaking perturbation. System sits on saddle point of energy hypersurface. Forces are still zero at saddle point. A phonon calculation at q = 0 will give an imaginary frequency at a saddle point diagnostic test. Alternative test is to break symmetry and re-optimize. Warning 2. The size of unit cell may still be a constraint. e.g. cell-doubling may arise from imaginary gamma-point phonon at BZ boundary. CASTEP Workshop 2007: York 13 / 20 7
8 Surfaces 14 / 20 Surface modelling with slab geometry Surfaces can be modelled as a slab cleaved from bulk crystal. Can calculate Surface energy or free energy Energies of steps Adsorption energies and structures of adsorbates Surface chemical reaction energies. CASTEP Workshop 2007: York 15 / 20 8
9 Slab models I Choice of slab: usually need to make 2 surfaces identical. Surfaces related by symmetry operation are more easily geometry optimised. Simulation cell should not be optimised. (use fix all cell =T). In-plane cell parameters usually fixed at values from relaxed bulk crystal. CASTEP Workshop 2007: York 16 / 20 9
10 Surface Energy calculations Surface free energy defined as E surf = (E slab E bulk )/A A is total area of both surfaces. E surf is sensitive quantity requiring well-converged total energies. Can sometimes gain some k-point error cancellation between slab and bulk calcs by using non-primitive bulk cell with same in-plane vectors as slab. (not CaCO shown). Need only 1 k-point in direction perpendicular to slab. Any dispersion in bands is error due to insufficient vacuum gap - no point in calculating accurately! Need to test convergence with both slab thickness and vacuum gap CASTEP Workshop 2007: York 17 / 20 10
11 Polarization in slabs Electric dipoles perpendicular to surface raise theoretical difficulties Energy does not converge with slab thickness. P.W. Tasker (Surf. Sci 87, 315 (1979) described 3 classes of surface. In classical charge model, Type III unstable and must always reconstruct. In ab initio calculation, surfaces can instead become metallic. CASTEP Workshop 2007: York 18 / 20 Defect calculations 19 / 20 Defect models Point- and extended- defects may be modelled using supercell approach. Several tricky convergence issues with supercells, to converge defect-defect interaction to zero. See M. I. J. Probert and M. C. Payne Phys. Rev. B (2003). Only need to converge energy to a few mev, but still need accuracy in forces to correctly describe strain relaxation. Strain relaxation. Local strain around defect decreases as 1/R. Can model long-range strain relaxation using classical models if suitable potential exists. Charged defects can be modelled using periodic interaction correction terms (M. Leslie and M. Gillan, J. Phys. Cond. Mat. 18, 973 (1985)) Corrections for higher multipoles also available G. Makov and M. C. Payne Phys. Rev. B51, 4104 (1995), L. N. Kantorovich Phys. Rev. B60, (1999)). CASTEP Workshop 2007: York 20 / 20 11
Practical calculations using first-principles QM Convergence, convergence, convergence
Practical calculations using first-principles QM, convergence, convergence Keith Refson STFC Rutherford Appleton Laboratory September 20, 2012 - CASTEP Workshop: Frankfurt 2012 1 / 21 First-principles
More informationConvergence: How to obtain reliable results
Convergence: How to obtain reliable results Keith Refson Royal Holloway, University of London August 7, 2015 Results of First-Principles Simulations................................................................................
More informationPractical calculations using first-principles QM Convergence, convergence, convergence
Practical calculations using first-principles QM Convergence, convergence, convergence Keith Refson STFC Rutherford Appleton Laboratory September 18, 2007 Results of First-Principles Simulations..........................................................
More informationThe electronic structure of materials 1
Quantum mechanics 2 - Lecture 9 December 18, 2013 1 An overview 2 Literature Contents 1 An overview 2 Literature Electronic ground state Ground state cohesive energy equilibrium crystal structure phase
More informationSupplementary Information
Supplementary Information Supplementary Figure 1: After structural optimization of the CH 3 NH 3 PbI 3 unit cell, the resulting relaxed volumes for three distinct orientation of the MA molecules are shown.
More informationPotentials, periodicity
Potentials, periodicity Lecture 2 1/23/18 1 Survey responses 2 Topic requests DFT (10), Molecular dynamics (7), Monte Carlo (5) Machine Learning (4), High-throughput, Databases (4) NEB, phonons, Non-equilibrium
More informationStrain-related Tensorial Properties: Elasticity, Piezoelectricity and Photoelasticity
Strain-related Tensorial Properties: Elasticity, Piezoelectricity and Photoelasticity Torino, Italy, September 4-9, 2016 Alessandro Erba Dipartimento di Chimica, Università di Torino (Italy) alessandro.erba@unito.it
More informationVibrational Spectroscopy
Vibrational Spectroscopy Keith Refson STFC Rutherford Appleton Laboratory August 28, 2009 Density Functional Methods for Experimental Spectroscopy 2009: Oxford 1 / 22 Two similar structures Zincblende
More informationPhonon Dispersion, Interatomic Force Constants Thermodynamic Quantities
Phonon Dispersion, Interatomic Force Constants Thermodynamic Quantities Umesh V. Waghmare Theoretical Sciences Unit J N C A S R Bangalore ICMR OUTLINE Vibrations and interatomic force constants (IFC) Extended
More informationGeometry Optimisation
Geometry Optimisation Matt Probert Condensed Matter Dynamics Group Department of Physics, University of York, UK http://www.cmt.york.ac.uk/cmd http://www.castep.org Motivation Overview of Talk Background
More informationStructural, vibrational and thermodynamic properties of Mg 2 SiO 4 and MgSiO 3 minerals from first-principles simulations
1 2 3 4 5 6 7 8 Structural, vibrational and thermodynamic properties of Mg 2 SiO 4 and MgSiO 3 minerals from first-principles simulations E. R. Hernández,a, J. Brodholt b, D. Alfè b a Instituto de Ciencia
More informationab initio Lattice Vibrations: Calculating the Thermal Expansion Coeffcient Felix Hanke & Martin Fuchs June 30, 2009 This afternoon s plan
ab initio Lattice Vibrations: Calculating the Thermal Expansion Coeffcient Felix Hanke & Martin Fuchs June 3, 29 This afternoon s plan introductory talk Phonons: harmonic vibrations for solids Phonons:
More informationOn Dynamic and Elastic Stability of Lanthanum Carbide
Journal of Physics: Conference Series On Dynamic and Elastic Stability of Lanthanum Carbide To cite this article: B D Sahoo et al 212 J. Phys.: Conf. Ser. 377 1287 Recent citations - Theoretical prediction
More informationMustafa Uludogan 1, Tahir Cagin, William A. Goddard, III Materials and Process Simulation Center, Caltech, Pasadena, CA 91125, U.S.A.
Ab Initio Studies On Phase Behavior of Barium Titanate Mustafa Uludogan 1, Tahir Cagin, William A. Goddard, III Materials and Process Simulation Center, Caltech, Pasadena, CA 91125, U.S.A. 1 Physics Department,
More informationSTRUCTURAL AND MECHANICAL PROPERTIES OF AMORPHOUS SILICON: AB-INITIO AND CLASSICAL MOLECULAR DYNAMICS STUDY
STRUCTURAL AND MECHANICAL PROPERTIES OF AMORPHOUS SILICON: AB-INITIO AND CLASSICAL MOLECULAR DYNAMICS STUDY S. Hara, T. Kumagai, S. Izumi and S. Sakai Department of mechanical engineering, University of
More informationGround-state Structure and Dynamics
Ground-state Structure and Dynamics Jonathan Yates jonathan.yates@materials.ox.ac.uk Materials Modelling Laboratory, Oxford Materials For a given set of atomic positions the ions will experience a force
More informationTUTORIAL 6: PHONONS, LATTICE EXPANSION, AND BAND-GAP RENORMALIZATION
Hands-On Tutorial Workshop, July 29 th 2014 TUTORIAL 6: PHONONS, LATTICE EXPANSION, AND BAND-GAP RENORMALIZATION Christian Carbogno & Manuel Schöttler Fritz-Haber-Institut der Max-Planck-Gesellschaft,
More informationThermodynamics of Solids: Harmonic and Quasi-harmonic Approximations
Thermodynamics of Solids: Harmonic and Quasi-harmonic Approximations, USA, July 9-14, 2017 Alessandro Erba Dipartimento di Chimica, Università di Torino (Italy) alessandro.erba@unito.it 2017 Outline -
More informationPrerequisites for reliable modeling with first-principles methods. P. Kratzer Fritz-Haber-Institut der MPG D Berlin-Dahlem, Germany
Prerequisites for reliable modeling with first-principles methods P. Kratzer Fritz-Haber-Institut der MPG D-14195 Berlin-Dahlem, Germany Prerequisites for modeling (I) Issues to consider when applying
More informationMatSci 331 Homework 4 Molecular Dynamics and Monte Carlo: Stress, heat capacity, quantum nuclear effects, and simulated annealing
MatSci 331 Homework 4 Molecular Dynamics and Monte Carlo: Stress, heat capacity, quantum nuclear effects, and simulated annealing Due Thursday Feb. 21 at 5pm in Durand 110. Evan Reed In this homework,
More informationRoger Johnson Structure and Dynamics: Displacive phase transition Lecture 9
9.1. Summary In this Lecture we will consider structural phase transitions characterised by atomic displacements, which result in a low temperature structure that is distorted compared to a higher temperature,
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 informationInternational Journal of Quantum Chemistry
International Journal of Quantum Chemistry First-principles calculation of second-order elastic constants and equations of state for Lithium Azide, LiN, and Lead Azide, Pb(N ) Journal: International Journal
More informationInterpreting Geophysical Data for Mantle Dynamics. Wendy Panero University of Michigan
Interpreting Geophysical Data for Mantle Dynamics Wendy Panero University of Michigan Chemical Constraints on Density Distribution Atomic Fraction 1.0 0.8 0.6 0.4 opx cpx C2/c garnet il olivine wadsleyite
More informationHigh Temperature High Pressure Properties of Silica From Quantum Monte Carlo
High Temperature High Pressure Properties of Silica From Quantum Monte Carlo K.P. Driver, R.E. Cohen, Z. Wu, B. Militzer, P. Lopez Rios, M. Towler, R. Needs, and J.W. Wilkins Funding: NSF, DOE; Computation:
More informationElectronic Supplementary Information
Electronic Supplementary Information Xenon as a mediator of chemical reactions? ( ) by Dominik Kurzydłowski and Wojciech Grochala Contents. 1. Unit cell vectors, fractional atomic coordinates and enthalpy
More informationSurface stress and relaxation in metals
J. Phys.: Condens. Matter 12 (2000) 5541 5550. Printed in the UK PII: S0953-8984(00)11386-4 Surface stress and relaxation in metals P M Marcus, Xianghong Qian and Wolfgang Hübner IBM Research Center, Yorktown
More informationTUTORIAL 8: PHONONS, LATTICE EXPANSION, AND BAND-GAP RENORMALIZATION
TUTORIAL 8: PHONONS, LATTICE EXPANSION, AND BAND-GAP RENORMALIZATION 1 INVESTIGATED SYSTEM: Silicon, diamond structure Electronic and 0K properties see W. Huhn, Tutorial 2, Wednesday August 2 2 THE HARMONIC
More information6: Plane waves, unit cells, k- points and all that
The Nuts and Bolts of First-Principles Simulation 6: Plane waves, unit cells, k- points and all that Durham, 6th- 13th December 2001 CASTEP Developers Group with support from the ESF ψ k Network Overview
More informationGeometry Optimization
Geometry Optimization Matt Probert Condensed Matter Dynamics Group Department of Physics, University of York, U.K. http://www-users.york.ac.uk/~mijp1 Overview of lecture n Background Theory n Hellman-Feynman
More informationCrystal Relaxation, Elasticity, and Lattice Dynamics
http://exciting-code.org Crystal Relaxation, Elasticity, and Lattice Dynamics Pasquale Pavone Humboldt-Universität zu Berlin http://exciting-code.org PART I: Structure Optimization Pasquale Pavone Humboldt-Universität
More informationENERGETICS AND DYNAMICS OF CAGED Zn 4 IN i-sczn
ENERGETICS AND DYNAMICS OF CAGED Zn 4 IN i-sczn Marek Mihalkovič and Christopher L. Henley [Support: U.S. Dept. of Energy] ICQ11 Poster P02-3, June 14, 2010, Sapporo Japan 1 Introduction Recall: cluster
More informationThe Power of FirstPrinciples Simulation
The Power of FirstPrinciples Simulation From electronic structure to real materials Keith Refson Scientific Computing Department STFC Rutherford Appleton Laboratory Computer Simulation Supercomputer Laws
More informationExploring deep Earth minerals with accurate theory
Exploring deep Earth minerals with accurate theory K.P. Driver, R.E. Cohen, Z. Wu, B. Militzer, P. Lopez Rios, M. Towler, R. Needs, and J.W. Wilkins Funding: NSF, DOE; Computation: NCAR, TeraGrid, NCSA,
More informationTinselenidene: a Two-dimensional Auxetic Material with Ultralow Lattice Thermal Conductivity and Ultrahigh Hole Mobility
Tinselenidene: a Two-dimensional Auxetic Material with Ultralow Lattice Thermal Conductivity and Ultrahigh Hole Mobility Li-Chuan Zhang, Guangzhao Qin, Wu-Zhang Fang, Hui-Juan Cui, Qing-Rong Zheng, Qing-Bo
More information3.091 Introduction to Solid State Chemistry. Lecture Notes No. 5a ELASTIC BEHAVIOR OF SOLIDS
3.091 Introduction to Solid State Chemistry Lecture Notes No. 5a ELASTIC BEHAVIOR OF SOLIDS 1. INTRODUCTION Crystals are held together by interatomic or intermolecular bonds. The bonds can be covalent,
More informationElectronic Structure Theory for Periodic Systems: The Concepts. Christian Ratsch
Electronic Structure Theory for Periodic Systems: The Concepts Christian Ratsch Institute for Pure and Applied Mathematics and Department of Mathematics, UCLA Motivation There are 10 20 atoms in 1 mm 3
More informationFirst-principles studies of the structural, electronic, and optical properties of a novel thorium compound Rb 2 Th 7 Se 15
First-principles studies of the structural, electronic, and optical properties of a novel thorium compound Rb 2 Th 7 Se 15 M.G. Brik 1 Institute of Physics, University of Tartu, Riia 142, Tartu 5114, Estonia
More informationFirst-Principles Vibrational spectroscopy and lattice dynamics of materials in the solid state
First-Principles Vibrational spectroscopy and lattice dynamics of materials in the solid state Keith Refson Computational Science and Engineering Department STFC Rutherford Appleton Laboratory First principles
More informationAb initio statistical mechanics of surface adsorption and desorption. I. H 2 O on MgO 001 at low coverage
THE JOURNAL OF CHEMICAL PHYSICS 127, 114709 2007 Ab initio statistical mechanics of surface adsorption and desorption. I. H 2 O on MgO 001 at low coverage D. Alfè London Centre for Nanotechnology, University
More informationRustam Z. Khaliullin University of Zürich
Rustam Z. Khaliullin University of Zürich Molecular dynamics (MD) MD is a computational method for simulating time evolution of a collection of interacting atoms by numerically integrating Newton s equation
More informationEELS, Surface Plasmon and Adsorbate Vibrations
EELS, Surface Plasmon and Adsorbate Vibrations Ao Teng 2010.10.11 Outline I. Electron Energy Loss Spectroscopy(EELS) and High Resolution EELS (HREELS) II. Surface Plasmon III. Adsorbate Vibrations Surface
More informationFirst-principles calculations of structural, electronic and optical properties of HfZn 2
~ 1 ~ First-principles calculations of structural, electronic and optical properties of HfZn 2 Md. Atikur Rahman *1, Md. Afjalur Rahman 2, Md. Zahidur Rahaman 3 1, 2, 3 Department of Physics, Pabna University
More informationVibrational Spectroscopy Methods
Vibrational Spectroscopy Methods Keith Refson STFC Rutherford Appleton Laboratory September 10, 2012 Phonons and Spectroscopy: Workshop: Frankfurt 2012 1 / 39 Motivations from experimental spectroscopy:
More informationEquilibrium state of a metal slab and surface stress
PHYSICAL REVIEW B VOLUME 60, NUMBER 23 15 DECEMBER 1999-I Equilibrium state of a metal slab and surface stress P. M. Marcus IBM Research Division, T. J. Watson Research Center, Yorktown Heights, New York
More informationand strong interlayer quantum confinement
Supporting Information GeP3: A small indirect band gap 2D crystal with high carrier mobility and strong interlayer quantum confinement Yu Jing 1,3, Yandong Ma 1, Yafei Li 2, *, Thomas Heine 1,3 * 1 Wilhelm-Ostwald-Institute
More informationarxiv:cond-mat/ v1 10 Jun 1994 K. M. Rabe
October 2, 2018 Phase transitions in BaTiO 3 from first principles W. Zhong and David Vanderbilt Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849 arxiv:cond-mat/9406049v1
More informationSupporting Information
Electronic Supplementary Material (ESI) for Nanoscale. This journal is The Royal Society of Chemistry 2015 Supporting Information Single Layer Lead Iodide: Computational Exploration of Structural, Electronic
More information10: Testing Testing. Basic procedure to validate calculations
The Nuts and Bolts of First-Principles Simulation 10: Testing Testing. Basic procedure to validate calculations Durham, 6th-13th December 2001 CASTEP Developers Group with support from the ESF ψ k Network
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 informationTight-binding molecular dynamics study of palladium
PHYSICAL REVIEW B 79, 064107 2009 Tight-binding molecular dynamics study of palladium A. Shabaev and D. A. Papaconstantopoulos George Mason University, Fairfax, Virginia 22030, USA Received 24 September
More informationAb Initio modelling of structural and electronic. Matt Probert University of York
Ab Initio modelling of structural and electronic properties of semiconductors Matt Probert University of York http://www-users.york.ac.uk/~mijp1 Overview of Talk What is Ab Initio? What can we model? How
More information22 Path Optimisation Methods
22 Path Optimisation Methods 204 22 Path Optimisation Methods Many interesting chemical and physical processes involve transitions from one state to another. Typical examples are migration paths for defects
More informationBasics of DFT applications to solids and surfaces
Basics of DFT applications to solids and surfaces Peter Kratzer Physics Department, University Duisburg-Essen, Duisburg, Germany E-mail: Peter.Kratzer@uni-duisburg-essen.de Periodicity in real space and
More informationFinite-temperature equation of state. T ln 2sinh h" '-
Finite-temperature equation of state * $ F(V,T) = U 0 + k B T ln 2sinh h" '- #, & )/ + % 2k B T (. 0 # Compute vibrational modes, frequencies Evaluate at a given volume V Compute F at various temperatures
More informationSupplementary Information for. Universal elastic-hardening-driven mechanical instability in α-quartz and quartz. homeotypes under pressure
Supplementary Information for Universal elastic-hardening-driven mechanical instability in α-quartz and quartz homeotypes under pressure Juncai Dong, Hailiang Zhu, and Dongliang Chen * Beijing Synchrotron
More informationα phase In the lower mantle, dominant mineralogy is perovskite [(Mg,Fe)SiO 3 ] The pyrolite mantle consists of: 60% olivine and 40% pyroxene.
Summary of Dan Shim s lecture on 3/1/05 Phase transitions in the Earth s mantle In this lecture, we focused on phase transitions associated with the transition zone 1. 410 km alpha olivine beta wadsleyite
More informationproperties Michele Catti Dipartimento di Scienza dei Materiali Università di Milano Bicocca, Italy
Elastic and piezoelectric tensorial properties Michele Catti Dipartimento di Scienza dei Materiali Università di Milano Bicocca, Italy (catti@mater.unimib.it) 1 Tensorial physical properties of crystals
More informationLecture 11: Periodic systems and Phonons
Lecture 11: Periodic systems and Phonons Aims: Mainly: Vibrations in a periodic solid Complete the discussion of the electron-gas Astrophysical electrons Degeneracy pressure White dwarf stars Compressibility/bulk
More informationSupporting Information. Potential semiconducting and superconducting metastable Si 3 C. structures under pressure
Supporting Information Potential semiconducting and superconducting metastable Si 3 C structures under pressure Guoying Gao 1,3,* Xiaowei Liang, 1 Neil W. Ashcroft 2 and Roald Hoffmann 3,* 1 State Key
More informationComplementary approaches to high T- high p crystal structure stability and melting!
Complementary approaches to high T- high p crystal structure stability and melting! Dario ALFÈ Department of Earth Sciences & Department of Physics and Astronomy, Thomas Young Centre@UCL & London Centre
More informationT. Interface Energy of Metal-Ceramic Interface Co-WC Using ab initio Thermodynamics
Application Note T. Using ab initio Thermodynamics Introduction In many metal-ceramic composites the interface between the metallic and ceramic phases determines the mechanical properties of the material.
More informationPhonons I - Crystal Vibrations (Kittel Ch. 4)
Phonons I - Crystal Vibrations (Kittel Ch. 4) Displacements of Atoms Positions of atoms in their perfect lattice positions are given by: R 0 (n 1, n 2, n 3 ) = n 10 x + n 20 y + n 30 z For simplicity here
More informationGeneration of Thermal Scattering Laws for YH 2 using Ab Initio Methods
Generation of Thermal Scattering Laws for YH 2 using Ab Initio Methods Michael L. Zerkle Bettis Atomic Power Laboratory WPEC SG42 Meeting May 18, 2015 May 18-19, 2015 WPEC SG42 Slide 1 Outline Motivation
More information6.730 Physics for Solid State Applications
6.730 Physics for Solid State Applications Lecture 5: Specific Heat of Lattice Waves Outline Review Lecture 4 3-D Elastic Continuum 3-D Lattice Waves Lattice Density of Modes Specific Heat of Lattice Specific
More informationIntroduction to Density Functional Theory with Applications to Graphene Branislav K. Nikolić
Introduction to Density Functional Theory with Applications to Graphene Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, U.S.A. http://wiki.physics.udel.edu/phys824
More informationConcepts in Surface Physics
M.-C. Desjonqueres D. Spanjaard Concepts in Surface Physics Second Edition With 257 Figures Springer 1. Introduction................................. 1 2. Thermodynamical and Statistical Properties of
More informationPhonon wavefunctions and electron phonon interactions in semiconductors
Phonon wavefunctions and electron phonon interactions in semiconductors Bartomeu Monserrat bm418@cam.ac.uk University of Cambridge Quantum Monte Carlo in the Apuan Alps VII QMC in the Apuan Alps VII Bartomeu
More informationOutline. Introduction: graphene. Adsorption on graphene: - Chemisorption - Physisorption. Summary
Outline Introduction: graphene Adsorption on graphene: - Chemisorption - Physisorption Summary 1 Electronic band structure: Electronic properties K Γ M v F = 10 6 ms -1 = c/300 massless Dirac particles!
More informationAb initio phonon calculations in mixed systems
Ab initio phonon calculations in mixed systems Andrei Postnikov apostnik@uos.de Outline: Experiment vs. ab initio theory Ways of theory: linear response and frozen phonon approaches Applications: Be x
More informationCO Adsorption Site Preference on Platinum: Charge Is the Essence
Supporting Information CO Adsorption Site Preference on Platinum: Charge Is the Essence G.T. Kasun Kalhara Gunasooriya, and Mark Saeys *, Laboratory for Chemical Technology, Ghent University, Technologiepark
More informationEffect of water on the spinel-postspinel transformation in Mg 2 SiO 4
Effect of water on the spinel-postspinel transformation in Mg 2 SiO 4 * Pressures for spinel postspinel phase boundary has been subject of debate - XRD measurements indicates that the transition pressure
More informationElectron energy loss spectroscopy (EELS)
Electron energy loss spectroscopy (EELS) Phil Hasnip Condensed Matter Dynamics Group Department of Physics, University of York, U.K. http://www-users.york.ac.uk/~pjh503 Many slides courtesy of Jonathan
More information2) Atom manipulation. Xe / Ni(110) Model: Experiment:
2) Atom manipulation D. Eigler & E. Schweizer, Nature 344, 524 (1990) Xe / Ni(110) Model: Experiment: G.Meyer, et al. Applied Physics A 68, 125 (1999) First the tip is approached close to the adsorbate
More informationA molecular dynamics study of structural and dynamical correlations of CaTiO 3
Available online at www.sciencedirect.com Acta Materialia 59 (2011) 1409 1423 www.elsevier.com/locate/actamat A molecular dynamics study of structural and dynamical correlations of CaTiO 3 J.A. Souza,
More informationStructural and Electronic Effects on the Properties of Fe 2 (dobdc) upon Oxidation with N 2 O
Supporting information for paper in Inorganic Chemistry, April 11, 016, page S-1 Structural and Electronic Effects on the Properties of Fe (dobdc) upon Oxidation with N O oshua Borycz, 1, oachim Paier,
More informationSupporting Information
Supporting Information Controlled Growth of Ceria Nanoarrays on Anatase Titania Powder: A Bottom-up Physical Picture Hyun You Kim 1, Mark S. Hybertsen 2*, and Ping Liu 2* 1 Department of Materials Science
More informationElectric field dependent sound velocity change in Ba 1 x Ca x TiO 3 ferroelectric perovskites
Indian Journal of Pure & Applied Physics Vol. 49, February 2011, pp. 132-136 Electric field dependent sound velocity change in Ba 1 x Ca x TiO 3 ferroelectric perovskites Dushyant Pradeep, U C Naithani
More informationSupporting Online Material (1)
Supporting Online Material The density functional theory (DFT) calculations were carried out using the dacapo code (http://www.fysik.dtu.dk/campos), and the RPBE (1) generalized gradient correction (GGA)
More informationLecture 1 - Electrons, Photons and Phonons. September 4, 2002
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2002 Lecture 1-1 Lecture 1 - Electrons, Photons and Phonons Contents: September 4, 2002 1. Electronic structure of semiconductors 2. Electron statistics
More informationDFT modeling of novel materials for hydrogen storage
DFT modeling of novel materials for hydrogen storage Tejs Vegge 1, J Voss 1,2, Q Shi 1, HS Jacobsen 1, JS Hummelshøj 1,2, AS Pedersen 1, JK Nørskov 2 1 Materials Research Department, Risø National Laboratory,
More informationLuigi Paolasini
Luigi Paolasini paolasini@esrf.fr LECTURE 7: Magnetic excitations - Phase transitions and the Landau mean-field theory. - Heisenberg and Ising models. - Magnetic excitations. External parameter, as for
More informationJeremy Kua Materials and Process Simulation Center (139-74), California Institute of Technology, Pasadena, California 91125
JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 12 22 SEPTEMBER 2001 Direct comparisons of rates for low temperature diffusion of hydrogen and deuterium on Cu 001 from quantum mechanical calculations and
More informationAccuracy and transferability of GAP models for tungsten
Accuracy and transferability of GAP models for tungsten Wojciech J. Szlachta Albert P. Bartók Gábor Csányi Engineering Laboratory University of Cambridge 5 November 214 Motivation Number of atoms 1 1 2
More informationLattice Vibrations. Chris J. Pickard. ω (cm -1 ) 200 W L Γ X W K K W
Lattice Vibrations Chris J. Pickard 500 400 300 ω (cm -1 ) 200 100 L K W X 0 W L Γ X W K The Breakdown of the Static Lattice Model The free electron model was refined by introducing a crystalline external
More informationSupporting Information
Supporting Information Conversion of multilayer graphene into continuous ultrathin sp 3 - bonded carbon films on metal surfaces Dorj Odkhuu 1, Dongbin Shin 2, Rodney S. Ruoff 3, and Noejung Park 1,2 1
More informationFirst Principles Investigation of Structural, Electronic and Optical Properties of MgRh Intermetallic Compound
American Journal of Modern Physics 2016; 5(3): 25-29 http://www.sciencepublishinggroup.com/j/ajmp doi: 10.11648/j.ajmp.20160503.11 ISSN: 2326-8867 (Print); ISSN: 2326-8891 (Online) First Principles Investigation
More informationPHASE-FIELD SIMULATION OF DOMAIN STRUCTURE EVOLUTION IN FERROELECTRIC THIN FILMS
Mat. Res. Soc. Symp. Proc. Vol. 652 2001 Materials Research Society PHASE-FIELD SIMULATION OF DOMAIN STRUCTURE EVOLUTION IN FERROELECTRIC THIN FILMS Y. L. Li, S. Y. Hu, Z. K. Liu, and L. Q. Chen Department
More informationSupplementary Figures
Supplementary Figures Supplementary Figure S1: Calculated band structure for slabs of (a) 14 blocks EuRh2Si2/Eu, (b) 10 blocks SrRh2Si2/Sr, (c) 8 blocks YbRh2Si2/Si, and (d) 14 blocks EuRh2Si2/Si slab;
More informationGeometry optimization of solids
The Minnesota Workshop on ab Initio Modeling in Solid State Chemistry with CRYSTAL Minneapolis, MN(U.S.A.) 9-14 July 2017 7 1 Geometry optimization of solids Bartolomeo Civalleri Dip. di Chimica IFM, Via
More informationAb initio calculations of f-orbital electron-phonon interaction in laser cooling
Ab initio calculations of f-orbital electron-phonon interaction in laser cooling Jedo Kim and Massoud Kaviany* Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2125,
More informationSupport Information. For. Theoretical study of water adsorption and dissociation on Ta 3 N 5 (100) surfaces
Support Information For Theoretical study of water adsorption and dissociation on Ta 3 N 5 (100) surfaces Submitted to Physical Chemistry Chemical Physics by Jiajia Wang a, Wenjun Luo a, Jianyong Feng
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 information4/14/11. Chapter 12 Static equilibrium and Elasticity Lecture 2. Condition for static equilibrium. Stability An object is in equilibrium:
About Midterm Exam 3 When and where Thurs April 21 th, 5:45-7:00 pm Rooms: Same as Exam I and II, See course webpage. Your TA will give a brief review during the discussion session. Coverage: Chapts 9
More informationMaterials Studio 5.0: Spectroscopy Methods in CASTEP
Materials Studio 5.0: Spectroscopy Methods in Dr. Keith Refson, Science & Technology Facilities Council Dr. Stewart Clark, University of Durham Webinar Series: Materials Studio 5.0 December 2 nd, 2009
More informationSupplementary Information for Topological phase transition and quantum spin Hall edge states of antimony few layers
1 Supplementary Information for Topological phase transition and quantum spin Hall edge states of antimony few layers Sung Hwan Kim, 1, 2 Kyung-Hwan Jin, 2 Joonbum Park, 2 Jun Sung Kim, 2 Seung-Hoon Jhi,
More informationSelectivity in the initial C-H bond cleavage of n-butane on PdO(101)
Supporting Information for Selectivity in the initial C-H bond cleavage of n-butane on PdO(101) Can Hakanoglu (a), Feng Zhang (a), Abbin Antony (a), Aravind Asthagiri (b) and Jason F. Weaver (a) * (a)
More informationMSE 201A Thermodynamics and Phase Transformations Fall, 2008 Problem Set No. 8. Given Gibbs generals condition of equilibrium as derived in the notes,
MSE 201A hermodynamics and Phase ransformations Fall, 2008 Problem Set No. 8 Problem 1: (a) Let a homogeneous fluid with composition, {x}, be surrounded by an impermeable wall and in contact with a reservoir
More information3.091 Introduction to Solid State Chemistry. Lecture Notes No. 6a BONDING AND SURFACES
3.091 Introduction to Solid State Chemistry Lecture Notes No. 6a BONDING AND SURFACES 1. INTRODUCTION Surfaces have increasing importance in technology today. Surfaces become more important as the size
More informationDFT EXERCISES. FELIPE CERVANTES SODI January 2006
DFT EXERCISES FELIPE CERVANTES SODI January 2006 http://www.csanyi.net/wiki/space/dftexercises Dr. Gábor Csányi 1 Hydrogen atom Place a single H atom in the middle of a largish unit cell (start with a
More information