Nanoscale Fabrication EE 235 & NSE 235 Spring 2009 Jeff Neaton, Molecular Foundry, LBNL With thanks to: Leeor Kronik, Weizmann Institute Su Ying Quek & Alexey Zayak, Molecular Foundry, LBNL Steven G. Louie, UC-Berkeley Density Functional Theory & Beyond: Quantum Mechanical Approaches to NanoscienceJeff Neaton, Molecular Foundry, LBNL The Molecular Foundry A national user facility for nanoscience (1 of 5 nationally) A DOE user facility for researchers from any discipline, at any institution, to come, free of charge, to: • use state-of-the-art instruments • learn leading-edge techniques • collaborate with experts in a wide range of nanoscience fieldsJeff Neaton, Molecular Foundry, LBNL Organic & Macromolecular Nanostructures Biological Nanostructures Inorganic Nanostructures Theory of Nanostructured Materials Nanofabrication Imaging & Manipulation of Nanostructures Molecular Foundry Facilities• State-of-the-art instrumentation • Technical support staff • Collaborative scientists to support users • Internal research program • Projects selected through peer review • No charge for non-proprietary research Facility descriptionJeff Neaton, Molecular Foundry, LBNL Theory of Nanostructures Facility • Explain new nanoscale materials and measured phenomena • Help develop new concepts and techniques • Explore and predict new materials and nanoscale behavior • Guide new research directions Theory NanoscienceJeff Neaton, Molecular Foundry, LBNL Fundamental Goals of Nanoscience • How do nanostructures assemble, grow, and evolve? Kinetics and dynamics of self-assembly, growth, stability • What are their properties, behavior, and phenomenology? Electronic, magnetic, thermal, mechanical, optical, etc Structure FunctionJeff Neaton, Molecular Foundry, LBNL Nanoscale Materials: What’s Different? Quantum Confinement, Excitons, & Other Novel Optical Phenomena Discrete Nature of Electron Charge & Importance of Correlation High Density of Active Surfaces & Interfaces CdSe NP Diameter Binary NP Superlattice SWCNT Quantum DotJeff Neaton, Molecular Foundry, LBNL Single-Nanostructure Characterization Single-Molecule Conductance Individual Nanotubes Schuck, et al. PRL 94, 017402 (2005))! Bowtie antenna Nanorods Single-Molecule Raman Huang et al. Nano Lett 5, 1515 (2005))! Trudeau, Sheldon, Altoe, Alivisatos, Nano Lett 8, 1936 (2008))! Conducting AFM Quek et al. Nature Nano 4, 230 (2009))!What do I mean by ‘first-principles’? Gibbs free energy Compute given phase ‘from first-principles’, i.e., knowing only – Identities of the atom – Laws of quantum mechanics and without experimental parameters € G = E + pV − TSJeff Neaton, Molecular Foundry, LBNL Ideal first principles computation – “brute force” solution of the Schrödinger equation. , where and Quantum Mechanics kinetic! ion-e! e-e!Jeff Neaton, Molecular Foundry, LBNL In practice, just a tad difficult … "The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble." P. A. M. Dirac, Proc. Royal Soc. Lond. Ser. A 123, 714 (1929)Density functional theory Walter Kohn (left), receiving the Nobel prize in chemistry in 1998. Energy Hohenberg & Kohn, 1964 Electron density Interacting Non-interacting - - - - - - - - - Kohn & Sham, 1965 € E0= E[n0]Jeff Neaton, Molecular Foundry, LBNL Density as the basic variable! (Hohenberg & Kohn, Phys. Rev., 1964) Electron Density!ion-e!Many-body "wave function!Electron Density!# of electrons!Atomic positions!Atomic numbers!Plausibility argument: (Koch & Holthausen, A Chemist’s guide to DFT, 2001) The Hohenberg-Kohn TheoremJeff Neaton, Molecular Foundry, LBNL Exact mapping to a single particle Schrodinger Eq! (Kohn & Sham, Phys. Rev., 1965) kinetic! ion-e! many body!e-e!Exc[n(r)] Exc is a unique, but complicated and unknown "functional of the charge density. !In principle: Exact ; In practice: approximate The Kohn-Sham EquationJeff Neaton, Molecular Foundry, LBNL Rayleigh-Ritz variational principle: Levy’s constrained search Hohenberg-Kohn variational principle: F[n] is universal !! The Hohenberg-Kohn Variational PrincipleJeff Neaton, Molecular Foundry, LBNL Is there an equivalent non-interacting (Vee=0) electron system that yields the same density as the true one? If so, what external potential, VKS , yields the true density? Is there: Such that: Is exactly equal to the electron density of the real interacting system? Exact Mapping to an Equivalent Non-Interacting SystemJeff Neaton, Molecular Foundry, LBNL Non-interacting kinetic energy Classical Coulomb energy Model system Real System “everything else” Finding VKSJeff Neaton, Molecular Foundry, LBNL Exact mapping to a single particle problem! (Kohn & Sham, Phys. Rev., 1965) kinetic!ion-e! many body!e-e!Exc[n(r)] Exc is a unique, but complicated and unknown "functional of the charge density. !In principle: Exact ; In practice: approximate The Kohn-Sham EquationJeff Neaton, Molecular Foundry, LBNL Principle and practice are the same in principle but not in practice - Relating formal and practical difficulties - No technical issues, only fundamental ones - Only discuss well-established functionals - Physical insight over mathematical rigor How are we doing in practice?Jeff Neaton, Molecular Foundry, LBNL At each point in space, xc energy per particle is given by its value for a homogeneous electron gas Local Density ApproximationJeff Neaton, Molecular Foundry, LBNL - Add info from immediate vicinity via gradient term - Not a gradient expansion! Some quantitative issues with LDA that are mitigated with GGA: - Atomization enrgies - ~0.3 eV versus ~1 eV (Becke, J. Chem. Phys., 1993) - Overbinding (e.g., Na – Kronik et al., J. Chem. Phys., 2001) - Over emphasis of packed structures (e.g., Fe – Zhu et al., Phys. Rev., 1992) … but NOT a qualitative improvement! Generalized Gradient Approximation (GGA)Common Approximations • Exchange-Correlation Functionals!• Local Density Approximation (LDA)"• Generalized Gradient Approximation (GGA)"• Zero-temperature!• Born-Oppenheimer limit!• Pseudopotentials !– e.g. treat only 5d106s for Au (11 e-/atom) or 3s23p2 for Si (4
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