Modern Electronic Structure: Modern Electronic Structure: many- -body physics body physics many in nano nano- -world world in Liviu Chioncel Chioncel Liviu Graz University of Technology Graz University of Technology Austria Austria Universitatea din Oradea Universitatea din Oradea Romania Romania 1 1
Content Content NANO scale NANO scale – More than just length and size More than just length and size – Electronic structure, Electronic structure, – Density Functional Theory, implementations Density Functional Theory, implementations – Many- -body physics and electronic structure body physics and electronic structure Many – Dynamical Mean Field Theory (DMFT) Dynamical Mean Field Theory (DMFT) – Magnetism in nanosystems Magnetism in nanosystems – Molecular magnets Molecular magnets – – Surface states vs. Kondo effect Surface states vs. Kondo effect – – Correlated adatom on a metal surface Correlated adatom on a metal surface – 2 2
Nano- -scales in every day life scales in every day life Nano Aircraft Carrier Boeing 747 Car 1 m Humans Laptop Butterfly Microprocessor 1 mm Resolving power of the eye ~ 0.2 mm Micromachines electron Biological cell Nucleus of a cell 1 µ m neutron Visible Light Microelectronic chips Proteins proton Nanostr. & Quantum Devices Width of DNA 1 nm Size of an atom 3 3
Nano- -scale in numbers scale in numbers Nano Water molecules – 3 atoms H H thousands O Protein molecule milions Carbon Molecule nanotube of DNA 4 4
Development of Nanotechnology Development of Nanotechnology Fundamental Understanding Characterization Modeling and and Simulation Experimentation Synthesis and Integration Nano to Macro Inorganic and Organic Optical with Mechanical with Electrical with Magnetic with … 5 5
Nanostructures Nanostructures (At least one dimension is between 1 At least one dimension is between 1 - - 100 nm) 100 nm) ( 2- -D structures (1 D structures (1- -D confinement): D confinement): 2 – – Thin films Thin films – Planar quantum wells – Planar quantum wells – Superlattices – Superlattices 1- -D structures (2 D structures (2- -D confinement): D confinement): 1 2 µ µ m µ µ – Nanowires – Nanowires – Quantum wires Si Nanowire Array – Quantum wires – Nanorods – Nanorods – Nanotubes – Nanotubes Multi-wall carbon 0- -D structures (3 D structures (3- -D confinement): D confinement): nanotube 0 – – Nanoparticles Nanoparticles – Quantum dots – Quantum dots Si 0.76 Ge 0.24 / Si 0.84 Ge 0.16 superlattice 6 6
NaNo.... NaNo.... “The art of understanding / developing The art of understanding / developing “ materials on an atomic or molecular scale materials on an atomic or molecular scale with the aim of building devices.” ” with the aim of building devices. 7 7
Condensed matter physics Condensed matter physics Quantum theory & Electronic structure Quantum theory & Electronic structure ‘59 Pseudopotential ‘26 Schrodinger 1998 Nobel Prize Kohn 1900 Planck ’75 LMTO, LAPW ’85 Car-Parrinello ‘28 Dirac ‘64 DFT ’37 APW 2007 ‘51 Slater X ’30 Hartree 1900 ’27 Fermi ’72 LSDA ’91 LDA+U ’86 GGA ‘96 DMFT 8 8
Electrons in solids - Effective potential Atom Solid - Bloch states FERMI sea - Pauli principle 1964 Density Functional Theory (DFT) Density Functional Theory (DFT) - Effective one Effective one- -particle states particle states - - Local Density Approximation (LDA) Local Density Approximation (LDA) - ? Behaviour at different dimensions 9 9
Density Functional Theory (DFT) Density Functional Theory (DFT) many- -particle interacting system particle interacting system ----- ----- non non- -interacting reference system interacting reference system many Hohenberg Hohenberg- -Kohn theorems: Kohn theorems: [ ] [ ] [ ] ρ = ρ + ρ F ( r ) F ( r ) F ( r ) Existence of the single particle H xc 1. Existence of the single particle 1. [ ] ρ = ρ ρ density of a non- density of a non -degenerate degenerate 1 − ∫ F ( r ) drdr ' ( r ) V ( r ' ) H e e ground state of an interacting ground state of an interacting 2 electron system electron system [ ] [ ] 1 ρ = α ρ 1 − α ∫ ∫ F ( r ) drdr ' V d g , r , r ' xc e e 2 0 Local density approximation Local density approximation 2. Variational principle Variational principle The total The total 2. energy of the N- energy of the N -electron system is electron system is [ ] ρ = − ρ − ρ minimized by the ground state minimized by the ground state α g , r , r ' ( n ( r ) ( r ))( n ( r ' ) ( r ' )) electron density electron density δ [ ] ρ = α F α δρδρ g , r , r ' ' rr ' [ ] ρ = ρ ε ρ LDA 1 LDA ∫ F ( r ) dr ( r ) [ ( r )] xc xc 2 10 10
DFT implementations: choices of the methods DFT implementations: choices of the methods All-electron full potential DMFT All-electron muffin-tin GW All-electron PAW LDA+U Pseudopotential Fully-relativistic Beyond LDA SIC Semi-relativistic GGA(generalized gradient) Non-relativistic LDA(local density) − ∇ + + Ψ = ε Ψ 1 2 k k k ( ) ( ) V r V xc r i i i 2 Atomic orbitals (Gaussian, Slater, numerical) Plane waves Periodic Augmentation (FLAPW, LMTO,ASW) Non-periodic Numerical Spin-polarized Non-spin-polarized 11 11
How well performs the DFT- -LSDA LSDA How well performs the DFT The Fermi Liquid Theory (1957- -59): 59): The Fermi Liquid Theory (1957 – Quasiparticles Quasiparticles - - weak interactions weak interactions – – Interactions Interactions - - slowly switched on slowly switched on – – Energy levels Energy levels - - modified modified – – Eigenstate Eigenstate - - given by occupation number given by occupation number – = ε + Lev Landau 1 σ σ σ σ σ ∑ ∑ E n ( k ) ( k ) f ( k , k ' ) n ( k ) n ( k ' ) , ' ' σ σ σ 2 , k , ' , k , k ' DFT- -LSDA LSDA - - fails for correlated electrons fails for correlated electrons… … DFT – Mott insulators (long range order) V2O3 Mott insulators (long range order) V2O3 – – High High Tc Tc superconductors (quasi 2D) superconductors (quasi 2D) – – Organic conductors (quasi 1D) Organic conductors (quasi 1D) – – Quantumdots (0D) Quantumdots (0D) – 12 12
Correlated electrons on lattices Correlated electrons on lattices ( ) = + µδ ⋅ + + σ σ H ∑ t c c U ∑ n n ↑ ↓ ij ij i j i i ij i U/t U/t U Chemical potential Chemical potential t John Hubbard 13 13
Solving the correlated electrons problem Solving the correlated electrons problem Impurity embedded in a fermionic bath P. Weiss U P.W.Anderson τ τ ’ G.Kotliar Local quantum fluctuations = dynamics Mean Field Theory Dynamical Mean Field Theory 14 14
Dynamical Mean- -Field: Cavity construction Field: Cavity construction Dynamical Mean Effective medium characterized by the action: β β β ( ) ( ) ( ) ( ) ( ) + − = − τ τ τ τ − τ τ + τ τ τ 1 S ∫ d ∫ d ' c σ G σ ' c σ ' U ∫ d n n ↑ ↓ eff bath , 0 0 0 Single impurity in the effective medium: ( ) ( ) ( ) τ − τ = − τ + τ G σ ' T τ c σ c σ ' U S eff τ τ ’ SCF condition connect the impurity solution with the effective medium − = ω + µ − − σ 1 2 G bath σ i t G σ H , W.Metzner, D. Vollhardt, PRL 62, ….(1989) A.Georges et.al. Rev. Mod. Phys 68,13 (1996) 15 15
Analogy with conventional MF Analogy with conventional MF 16 16
DMFT solution for the Hubbard model DMFT solution for the Hubbard model ( ) + = + µδ ⋅ + H t c σ c σ U n n ∑ ∑ ↑ ↓ ij ij i j i i t ij i U METAL Quasi-particle peak Hans Bethe Lower Upper HB HB ~U INSULATOR 17 17
Diagrammatic: Iterated Perturbation Theory Diagrammatic: Iterated Perturbation Theory = − − Yoshida & Yamada H I U ( n 1 / 2 )( n 1 / 2 ) ↑ ↓ Prog. Theor. Phys. 46, 244, 1970 G 0 U U Σ ) = Compute Self-energy in the second G 0 ( 2 order perturbation theory G 0 ρ ε ε ( ) 0 ω → Σ ω = τ → = ( ) d 2 3 G ( ) ( ) FT U G ( ) G ∫ ω − − Σ ω 0 0 i U / 2 ( ) − − = + Σ 1 1 G G 0 18 18
Metal to Insulator transition Metal to Insulator transition IPT- -solution solution IPT Georges & Kotliar PRB 45, 6479, 1992 Zhang, Rozemberg, Kotliar PRL 70, 1666, 1993 19 19
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