Modelowanie Nanostruktur

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Chair of Condensed Matter Physics Institute of Theoretical Physics Faculty of Physics, Universityof Warsaw Semester Zimowy 011/01 Wykład Modelowanie Nanostruktur Jacek A.

pl Modelowanie Nanostruktur, 011/01 Jacek A.

1 Chair of Condensed Matter Physics Institute of Theoretical Physics Faculty of Physics, Universityof Warsaw Semester Zimowy 011/01 Wykład Modelowanie Nanostruktur Jacek A. Majewski Kohn Sham realization of the Density Functional Theory key to computational materials science modeling of complex nanostructures Modelowanie Nanostruktur, 011/01 Jacek A. Majewski Wykład 3 18 X 011 Teoria Fukcjonału Gȩstości Coarse-graining Metoda ciasnego wiązania (Tight-binding method) Zastosowanie dla grafenu Materials Science Basic Problem: N electrons in an external potential {} r i Materials are composed of nuclei { Z, M, R } and electrons the interactions are known 1 1 i Ze Z Z e e H,,, i m i R ri i j ri r j R R Kinetic of electrons Electron-Nucleus interaction External potential Electron-Electron interaction Vext v ext ( r i ) i Ze α v ext ( r ) r R α Nucleus-Nucleus interaction α Lecture 3 1

2 Density Functional Theory (DFT) One particle density determines the ground state of the system for arbitrary external potential E[ρ] d 3 r ρ( r )υ ext ( r ) F [ ρ] Total functional E[ρ ] E 0 0 ground state density ground state E[ρ] drυ ( r )ρ( r ) T [ ρ] U [ ρ] E [ ρ] E [ ρ] ext S x c External Kinetic Classic Coulomb Exchange Correlation unknown!!! DFT- The Kohn- Sham Method System of interacting electrons with density ( r ) Real system W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965) ρ( r ) T[ρ] S System of non-interacting electrons with the same density ( r ) Fictitious or Kohn-Sham reference system T[ρ] ρ S( r ) ρ( r ) E[ρ] drυ ( r )ρ( r ) T [ ρ] U [ ρ] E [ ρ] E [ ρ] ext S x c N ρ( r ) θ* i ( r )θ i( r ) T[ρ] drθ ( r ) θ ( r ) unknown!!! i1 N * s i i m i 1 * 1 θ * j( r )θ j( r') x i i i j r r' E [ρ] drdr' θ ( r ) θ ( r') Density Functional Theory (DFT) in Kohn-Sham realization Density Functional Theory (DFT) Interacting particles Independent particles Ψ( x, x,, x ) θ( x 1)θ( x ) θ( x 3 ) Idea: consider electrons as independent particles moving in an effective potential Approximations to the exchange & correlation E[ρ] drυ ( r )ρ( r ) T [ ρ] U [ ρ] E [ ρ] E [ ρ] ext S x c unknown!!! Local Density Approximation (LDA) LDA hom Exc [ ] dr( r ) xc ( ( r )) This reduction is rigorously possible! Generalized Gradient Approximation (GGA) GGA xc E [ ] dr f ( ( r ), ( r )) xc Lecture 3

3 υ ( r ) υ ( r ) υ ( r ) υ ( r ) θ ( r ) ε θ ( r ) ext H x c i i i m The Kohn-Sham Method Kohn-Sham equations with local potential δu ρ( r') υ H ( r ) dr' δρ r r' υ ( r ) e ext s δe x [ ρ] δρ Z rη R s,n s n N ρ( r ) θ* i ( r )θ i( r ) i1 δe c [ ρ] δρ These equation are nonlinear and must be solved iteratively (self-consistently) Solution of the Kohn-Sham Equations - Methods Expansion of the Kohn-Sham orbitals in a basis { ( r k )} ( ) (, ) ( ) nk r c n k k r Plane waves and pseudopotential Linearized Muffin Tin Orbital (LMTO) Method Linearized Augnented Plane Wave (LAPW) Method Projector Augnented Waves (PAW) Method takes into account core electrons and keeps simultaneously valence functions smooth state-of-the-art in the pseudopotential based methods The Kohn- Sham Method Aufbau principle How to calculate one particle density? The Kohn-Sham Method Total Energy & Forces N * E[ρ] drθ i ( r ) θ i( r ) U [ ρ] E x [ ρ] E c [ ρ] drυ ext( r )ρ( r ) m i 1 LUMO HOMO ε ε N 1 N unoccupied total electronic in the field of ions 1 ZαZβ Eion α,β Rα R β Energy of ions Total E ({ R }) E ({ R }) E ({ R }) tot α el α ion α ε ε 1 occupied Forces on ions Equilibrium: αf 0 Dynamics: MαRα F α α F α E R tot α Rα E tot Lecture 3 3

4 Energy structure [Ry/atom] Calculated [ev] GaAs Ge AlAs GaN Si AlN SiC C Solution of the Kohn-Sham Equations Direct methods on a mesh in r-space Expansion of the Kohn-Sham orbitals in a basis ( ) (, ) ( ) nk r c n k k r ' Eigenvalueproblem Bandstructure { ( r k )} KS ( r ) ( ) ' n k c ' '( n, k ) 0 k m k k k Hamiltonian matrix elements [ H '( k) n( k) S '( k)] c '( n, k) 0 det[ H ( k) ( k) S ( k)] 0 ' n( k ) Overlap integrals n ' 8 6 Cohesive energies in semiconductors Comparison of LDA, EXX, and HF Methods Cohesive Energy 4 LDA EXX(GGA) EXX HF Experimental [ev] Cohesive = Energy of free (separated) atoms Energy of solid Cohesive = -- Binding LDA overestimates (LDA world is smaller than the real one) HF -- underestimates, EXX gives excellent cohesive energies in semiconductors First (convincing) LDA Calculations: Stability of crystals and pressure induced phase transitions fcc bcc hcp SC Silicon Tin DIAMOND Hexagonal Diamond Volume M. T. Yin & M. L. Cohen, Phys. Rev. B 6, 5668 (198) (1) (1) () () Etot ( Vt ) Etot ( Vt ) Ptransition () (1) Vt Vt E P V Interesting prediction: Under high hydrostatic pressure and in low temperature silicon becomes superconducting! Later on confirmed experimentally DFT Calculation of the equation of state and elastic constants Equation of state - V 0 B 0 B' 0 Elastic constants E tot (V ) Energy of the strained system or P(V ) 1 1 ρ0 E( η ) c η η c η η η! 3! ij i j ijk i j j i, j 1, 6 i, j,k 1, 6 Second order Elastic constants Third order Elastic constants Lecture 3 4

5 Energy [ev] Energy [ev] Calc. band gaps [ev] LDA calculations in semiconductors Fundamental band gaps in semiconductors: Local Density Approximation & Exact Exchange Valence bands for GaAs as determined from angle-resolved photoemission experiments and pseudopotential theory 0 L L L L GaAs Experiment Pseudopotential theory X K X Wave vector X 5 X LDA gives very good description of the occupied s-p valence bands (4s & 4p) in semiconductors Various methods of solving Kohn-Sham equations give very similar results Fundamental Band Gaps 6 LDA AlN C 5 EXX Ge Si GaAs SiC AlAs GaN Exp. band gaps [ev] EXX Method leads to Kohn-Sham gaps that agree very well with experiment Large part of the error in the fundamental gaps is connected to the approximated functionals (LDA, GGA) LDA calculations in semiconductors Energy gap E GAP Band structure of diamond silicon Si Wave vector E GAP = E LUMO - E HOMO Too small by factor of For all semiconductors and insulators, LDA (GGA) give gaps that are 40%-70% of experimental gaps Kohn-Sham gap KS E Gap KS cbb KS vbt KS KS E Gap 1 ( ) KS N N N ( N ) Relation of the Kohn-Sham gap to the quasi-particle (change of system caused by adding a particle)? Is the Kohn-Sham gap generally wrong, for description of one particle excitations? Does the error is caused by the approximation of the functionals? The band gap problem DFT - further developements required Density functional theory has revolutionized the way scientists approach the electronic structure of atoms, molecules,and solid materials in physics, chemistry, and materials science We are not at the end of this way! Lecture 3 5

Coarse-graining & effective approaches Large scale modeling - Coarse-Graining For large scale modeling, one may introduce alternative approaches using simplified coarse-grained models (lattice gas

Question: how to provide a link between atomistic calculations (ab initio, classical potentials) and the potential parameters suitable for coarse-grained models. What to do with large systems?

6 Coarse-graining & effective approaches Large scale modeling - Coarse-Graining For large scale modeling, one may introduce alternative approaches using simplified coarse-grained models (lattice gas models) These models can be treated with the methods used commonly in statistical mechanics such as mean-field theory, the cluster variation method (CVM), Monte Carlo methods. Question: how to provide a link between atomistic calculations (ab initio, classical potentials) and the potential parameters suitable for coarse-grained models. What to do with large systems? Coarse-grained model for ions around DNA Polyelectrolyte problem: ions around DNA All-atom model Coarse-grained model Atomistic DFT calculations not really possible to sample distances A from DNA Na + Water molecule Na + Lecture 3 6

Integral charge Coarse-grained model for ions around DNA Lipid bilayer in water Na + Ions interacting with DNA by effective solvent-mediated potentials Different potentials for various parts of DNA

7 Integral charge Coarse-grained model for ions around DNA Lipid bilayer in water Na + Ions interacting with DNA by effective solvent-mediated potentials Different potentials for various parts of DNA No explicit water density profile and integral charge Li + Na + Na * K + * Cs + 0. PB r (Å) Picture contains atoms Minimal, reasonable piece of bilayer for atomistic calculations 7 lipids + at least 0 H O per lipid = atoms Another example: Coarse-grained lipid model Tight-binding method All-atom model 118 atoms Coarse-grained model 10 sites Lecture 3 7

8 υ ( r ) υ ( r ) υ ( r ) υ ( r ) θ ( r ) ε θ ( r ) ext H x c i i i m The Kohn- Sham Method The Kohn-Sham Equations Schrödinger-like equations with local potential δu ρ( r') υ H ( r ) dr' δρ r r' υ ( r ) e ext s δe x [ ρ] δρ Z rη R s,n s n N ρ( r ) θ* i ( r )θ i( r ) i1 δe c [ ρ] δρ These equation are nonlinear and must be solved iteratively (self-consistently) Tight-Binding Formalism i n n i, i ( r ) c ( r ) index of orbital αi iα index of atom αi βj n n n n n iα jβ H ε iα iα t iα jβ αi,βj iα, jβ { χ αi }orthogonal set ε θ H θ ( c )*c iα H jβ Tight binding Hamiltonian on-site of functions NOT ATOMIC ORBITALS! hoping TB-parameters n LCAO (Linear Combination of Atomic Orbitals) at n ( r ) c ( r X ) Kohn-Sham orbitals in periodic systems ( r ) e l ( r Rl s( )) k { } k All atoms in the system Rl ikr at - fulfill Bloch s theorem at - Atomic Kohn-Sham orbitals: ˆ p at at at at KS [ ] m ( ) (, ) ( ) nk r c n k k r { s} Basis atoms { R l } Primitive translations ik R ( ) n n ( ) k r R e k r Problems Minimal Basis, i.e., one orbital per electron, is not sufficient Results depend on the chosen basis LCAO - Semiempirical Tight-Binding Method Hamiltonian matrix elements ˆ H ' H ' and overlap integrals ' S ' are treated as empirical parameters Mostly, orthonormality of orbitals is assumed, Huge spectrum of applications S ' Qualitative insight into physics and semi-quantitative results W. A. Harrison, Electronic Structure and the Properties of Solids The Physics of the Chemical Bond, Freeman & Comp. (1980) ' Lecture 3 8

9 αi Tight-Binding Hamiltonian iα iα iα iα, jβ iα jβ αi,βj H ε c c t c c creation & anihilation operators On-site energies are not atomic eigenenergies They include on average the effects of neighbors Problem: Transferability E.g., Si in diamond lattice (4 nearest neighbors) & in fcc lattice (1 nearest neighbors) Dependence of the hopping energies on the distance between atoms Tight-Binding Formalism Parameters The TB parameters: on-site, hopping (overlap integrals) are usually determined empirically by fitting TB energies (eigenvalues) to the ab initio (experimantal) ones. One could also try to calculate them directly by performing the same calculation for a localized basis set exactly e.g., F. Liu, Phys. Rev. B 5, (1995) Simple version of the TB method universal parameters W. Harrison, Electronic structure and the properties of solids (Dover, New York, 1980) not very transferable and not accurate enough allow to extract qualitative physics Tight-Binding Formalism Overlap Integrals In the general case orbitals will not be an orthonormal set and we define the overlap integrals as In orthogonal-tb schemes S reduces to the unit matrix. One needs to solve Tight-Binding Formalism Parameters Fitting the ab initio band structure LDA ( ) and tight-binding ( ) band structures for GaAs in the zinc-blende structure Parameters to be determined Semiempirical Tight-Binding Method Wave vector Lecture 3 9

10 Tight-Binding Formalism Physical meaning of the on-site energies αi iα iα iα iα, jβ iα jβ αi,βj H ε c c t c c Anderson has shown that there exists a pseudoatomic Hamiltonian that has as its eigenstates the basis states iα, but this Hamiltonian is not an atomic one and depends yet again on neighboring atoms. H iα (T V V jβ jβ V ) iα ε iα i i j j iα ji,β Similar procedure to the construction of the pseudopotential P. W. Anderson, Phys. Rev. Lett. 1, 13 (1968) Phys. Rev. 181, 5 (1969) Tight-Binding Formalism Dependence of the hopping integrals on atomic distance Calculations for systems with distorted lattice The dependence of the hopping integrals on the inter-atomic distance Harrison s ~d - dependence Exponential dependence t t0 exp( βr ) t αβ ( R ij ) t ( r ) f ( R αβ 0 ij ) n nc nc r r r 0 exp n 0 r r f ( r ) r 1 rc rc 3 c0 c 1( r r 1 ) c ( r r ) c 3( r r ) r r1 C. Xu et al., J. Phys. Condens. Matter 4, 6047 (199) Tight-Binding Formalism Physical meaning of the on-site energies The expression for the pseudo-hamiltonian of atom i H iα (T V V jβ jβ V ) iα ε iα i i j j iα ji,β Tight-Binding Formalism Dependence of the hopping integrals on atomic distance In the pseudopotential one projects out core states Here one projects out the states of the neighboring atoms which overlap with the atomic basis function. This is where the dependence on the environment comes from in this atomic pseudo-hamiltonian LDA ( ) and tight-binding ( ) band structures for GaAs in the zinc-blende structure for two different bond lengths Lecture 3 10

11 Tight-Binding Formalism Dependence of the hopping integrals on atomic distance Minimal sp basis used Tight-binding hopping integrals with the functional dependence t t exp( βr ) 0 (lines) as functions of the interatomic distance for GaAs in the zinc-blende structure. Optimum fits of the LDA band structure at selected nearest-neighbour distances are given by the data points. Y. P. Feng, C. K. Ong, H. C. Poon and D. Tomanek, J. Phys.: Condens. Matter 9, 4345 (1997). Tight-Binding Formalism Band Energy This term is called the band, and is usually attractive and responsible for the cohesion. In fact, if atoms get closer their overlap increases, the range of the eigenvalues increases and, since only the lowest states are occupied, the decreases (bonding increases). Real description of solids requires repulsive term (to prevent colaps) Tight-Binding Formalism Dependence of the hopping integrals on atomic distance Highly optimized tight-binding model of silicon Matrix elements h ssζ ( r ), h spζ ( r ), h ppζ ( r ) & h ppπ ( r ) Tight-Binding Formalism The Total Energy However, the TB formalism shown above describes only bonding due to the outer electrons. 1st nd 3rd 4th neighbor The matrix elements are defined whenever r is greater than 1.5 Å. The distances corresponding to the first four neighbor shells in the diamond structure are marked by short vertical lines; each matrix element goes smoothly to zero between the third and fourth neighbor shells. T. J. Lenosky et al., Phys. Rev. B 55, 158 (1996) If one brings two atoms close together, inner shells will start to overlap and bring additional (in the form of a strong repulsion) that is not included in the band term. The total will therefore be given as an empirical repulsive term Lecture 3 11

12 Tight-Binding Formalism The Repulsive Energy In most cases this is modeled simply as a sum of two-body repulsive potentials between atoms but many-body expressions such as Tight-Binding Formalism Parametrization of the repulsive term Using the interpolated hopping integrals, the tightbinding band-structure can be obtained for any geometry and inter-atomic distance. We then define the repulsive as the difference between the exact binding, obtained using ab initio calculations, and the tight-binding band-structure Several crystallographic phases of a material are usually used (where g is a non-linear embedding function, which can be fitted by a polynomial) have also been proposed. Φ has similar dependence on the R ij as hopping integrals Structure independent parametrization of the repulsive terms Tight-Binding Formalism The Total Energy Tight-Binding Formalism The Total Energy Band structure Repulsive Charge transfer The total repulsive contains ion ion repulsion, exchange correlation, and accounts for the double counting of electron electron interactions in the bandstructure term. The last term imposes an penalty on large inter-atomic charge transfers Charge transfer Typical U ~ 1eV The binding (E coh ), repulsive (E rep ), and band-structure (E bs ) for GaAs in the zinc-blende structure, as functions of the interatomic distance. Lecture 3 1

13 Tight-binding method Application: graphene Graphene: a sheet of carbon atoms What is graphene? Bottom-up & Top-down nanosystems -dimensional hexagonal lattice of carbon sp hybridized carbon atoms Among strongest bonds in nature fullerene (dots - 0D) Carbon nanotubes (q. wires 1D) Graphene (D systems) Basis for: C-60 (bucky balls) nanotubes graphite Lecture 3 13

(later Mermin) showed that thermodynamics prevents

Melting temperature of thin films decreases rapidly

004, experimental discovery of graphene - high

!! a) Atomic force microscopy b) Transmission

microscope image Andrey Geim (Manchester) Nobel Prize

nm 800 nm The Nobel Prize in Physics 010 was awarded

14 Does graphene exist? Samples of graphene In the 1930s, Landau and Peierls (later Mermin) showed that thermodynamics prevents the existence of -d crystals in free state. Melting temperature of thin films decreases rapidly with temperature -> monolayers generally unstable In 004, experimental discovery of graphene - high quality -d crystal!!! a) Atomic force microscopy b) Transmission electron microscopy image c) Scanning electron microscope image Andrey Geim (Manchester) Nobel Prize in Physics 010 Andre Geim Konstantin Novoselov What stabilizes graphene? Possibly, 3-d rippling stabilizes crystal 800 nm 800 nm 800 nm The Nobel Prize in Physics 010 was awarded jointly to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the twodimensional material graphene" Crumpling of graphene sheet the main source of disorder Lecture 3 14

y Graphene a single sheet of C atoms x Two unit-cell vectors: 3 1 a1 a(, ) 3 1 a a(, ) Two

graphene σ bonds not considered in this model π bonds considered One p z orbital pro atom Only

Covalent bonds between carbons k y Γ P Q k x sp 3 and sp hybrids Brillouin Zone Q Γ P Q The band

15 y Graphene a single sheet of C atoms x Two unit-cell vectors: 3 1 a1 a(, ) 3 1 a a(, ) Two non-equivalent atoms A and B in the unit cell (two sublattices) Tight-binding description of graphene σ bonds not considered in this model π bonds considered One p z orbital pro atom Only couplings between nearest neighbors taken into account Electronic band structure of graphene Covalent bonds between carbons k y Γ P Q k x sp 3 and sp hybrids Brillouin Zone Q Γ P Q The band structure was calculated with a first-principles method M. Machon, et al., Phys. Rev. B 66, (00) Lecture 3 15

16 Tight-binding description of graphene Band structure of graphene T-B metod () ε p ε( k ) H AB( k ) 0 H ( k ) ε ε( k ) * AB p ε( k ) H AB( k ) H H ε AA BB p H ( k ) exp( ik R ) θ ( η ) H ˆ θ ( η R ) AB n A A B B n R n H ( k ) t [ 1 exp( ik a ) exp( ik a )] p AB ε 0 (zero of ) AB 1 1/ ε( k ) t H ( k )H ( k ) * AB Band structure of graphene T-B metod (1) Band structure of graphene T-B metod (3) Lecture 3 16

Nearest-neighbors tight-binding electronic structure of

graphene - summary Graphene is semi-metallic Fermi is

Close to corner points, relativistic dispersion (light

lattice potential of graphene, not from carriers moving

Behavior ONLY present in monolayer graphene; disappears

Tight-binding band structure of graphene 1/ k ya ka k x

17 Nearest-neighbors tight-binding electronic structure of graphene T-B Ab-initio k y Γ P Q k x Band structure of graphene - summary Graphene is semi-metallic Fermi is zero, no closed Fermi surface, only isolated Fermi points Close to corner points, relativistic dispersion (light cone), up to ev scales Brillouin Zone Q Γ P Q Hopping parameter t = -.7 ev Relativistic behavior comes from interaction with lattice potential of graphene, not from carriers moving near speed of light. Behavior ONLY present in monolayer graphene; disappears with or more layers. Tight-binding band structure of graphene 1/ k ya ka k x ya 3 ε( k ) t 1 4cos 4cos cos Massless D Dirac Fermions Graphene is semi-metallic Energy gap is equal zero only in one k-point (P-point) light cone Intriguing Physics of Graphene Lecture 3 17

Modelowanie Nanostruktur. On theoretical concepts for modeling nanostructures the messages to take home. Examples of Nanostructures

Modelowanie Nanostruktur. On theoretical concepts for modeling nanostructures the messages to take home. Examples of Nanostructures Chair of Condensed Matter Physics stitute of Theoretical Physics Faculty of Physics, University of Warsaw Examples of Nanostructures Si(111)7 7 Surface TEM image of a As/GaAs dot Semester Zimowy 2011/2012