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Surface States Theoretical Surface Science 1. Introduction Wintersemester 2007/08 Surfaces Theoretical approach Axel Groß Processes on surfaces play an enormous- Some decades ago: phenomenological • • Universit ¨at Ulm, D-89069 Ulm, Germany ly important technological role thermodynamic approach prevalent http://www.uni-ulm.de/theochem Harmful processes: Nowadays: microscopic approach Outline • • 1. Rust, corrosion 1. Introduction 2. Wear Theoretical surface science no longer li- • mited to explanatory purposes 2. The Hamiltonian 3. Electronic Structure Methods and Total Energies Advantageous processes: • Many surface processes can indeed be 4. Approximate Interatomic Potentials 1. Production of chemicals • described from first pricinples, i.e., with- 5. Dynamics of Processes on Surfaces 2. Conversion of hazardous waste out invoking any empirical parameter 6. Kinetic Modelling of Processes on Surfaces Here: microscopic perspective of theoretical surface science 7. Electronically non-adiabatic Processes Experiment: a wealth of microscopic information available: 8. Outlook STM, REMPI, QLEED, HREELS, . Surface and image potential states Cu(111) surface states Cu(111) DFT surface band structure Cu(100): Image potential states Don Eigler, IBM Almaden 5 Quantum Corral Quantum Mirage 8 ε n=1 0 F n=2 6 n=3 ε vac 4 Energy (eV) Image potential Energy (eV) −5 V(z)=−e2 /4z 2 ε F 48 Fe atoms on Cu(111) placed in a circle Elliptival Quantum Corral with a Co atom −10 20 40 with diameter 71 A˚ Distance from the surface z(Å) at the focus of the ellipse Σ Γ Σ M.F. Crommie et al. , Science 262 , 218 (1993). M M H.C. Manoharan et al. , Nature 403 , 512 (2000). Rydberg image potential states observable Cu(111): Band gas and parabolic surface ¯ with time-resolved photo electron spectros- band at the Γ-Point copy A. Euceda, D.M. Bylander, and L. Kleinman, U. H ¨ofer, I.L. Shumay, Ch. Reuß, U. Thomann, W. Wallauer, Phys. Rev. B 28 , 528 (1983) and Th. Fauster, Science 277 , 1480 (1997) Diffusion on surfaces: oxygen atoms on Ru(0001) Non-linear phenomenon on surfaces STM images Oxygen diffusion Theoretical Modelling Diffusion barriers of oxygen on Ru(0001) rather Reaction scheme for the description of CO oxi- 10 µ m high so that there are only few jumps to adjacent dation on Pt(110) sites at room temperature. CO + CO (ads) , (1) 50 consecutive images of ∗ ↔ O + 2O (ads) , (2) oxygen atoms on Ru(0001) 2 ∗ → at room temperature on 13% coverage O(ads) + CO (ads) 2 + CO ,(3) → ∗ 2 Scan rate: 8 images per second 1 2 1 1 . (4) × ↔ × Joost Wintterlin et.al., FHI Berlin http://w3.rz-berlin.mpg.de/pc/stm/WS-fstm.html Kinetik: u = θCO , v = θO, w = θ1 1 × u˙ = s p k u k uv CO CO − 2 − 3 + D 2u, (5) ∇ Photo- emission electronmicroscopy-(PEEM) image of a v˙ = sO2 pO2 k3uv, (6) 4 5 − Pt(110) surface at 4 10 − mbar O 2 and 4.3 10 − mbar CO × × w˙ = k5 [f(u) w], (7) partial pressures at a temperature of T = 448 K (A. Nettes- − heim et al. , JCP 98 , 9977 (1993)). Manipulation of reaction fronts by laser Hydrogen dimer vacancy on hydrogen-covered Pd(111) J. Wolff, A.G. Papathanasiou, I.F. Kevrekidis, H.H. Rotermund, G. Ertl, Science 294 , 134 (2001) Salmeron et al. , Nature 422 , 705 (2003); PRL 93 , 146103 (2004). Reaction fronts locally manipulated by heating through a laser spot (diameter 80 µm). H.H. Rotermund et.al., FHI Berlin http://w3.rz-berlin.mpg.de/ rotermun/science/index.html Dimer vacancies appear as triangles because of the rapid diffusion of the hydrogen atoms at the rim No adsorption events into dimer vacancy found Adsorption of H /(3 3)7H/Pd(100) Dissociation dynamics of H /(6 6)Pd(100) 2 × 2 × Ab initio MD simulation of H /Pd(100) for a kinetic energy of 100 meV Ab initio MD simulation of H /(6 6)Pd(100) for a kinetic energy of 200 meV 2 2 × Dissociation Trapping Trajectories Animation 12 10 8 6 4 2 y-coordinate (Å) 0 -2 -4 Significant energy transfer to pre-adsorbed hydrogen atoms upon the dissociative adsorption -8 -6 -4 -2 0 2 4 6 8 x-coordinate (Å) Atoms can stay relatively close to each other Molecular adsorption: O 2/Pt(111) KMC simulation of the growth of Al(111) Simulation (1/8 of simulation area = 600 600 array) × Ekin = 1 .1 eV Energy redistribution 50 Å 50 Å O2 lateral and internal kinetic energy O perpendicular kinetic energy 1.5 2 3.0 kinetic energy of Pt atoms O2 distance from the surface {100}−faceted step T = 80 K T = 130 K 1.0 2.0 − − <110> <110> Energy Energy (eV) 0.5 1.0 50 Å {111}−faceted step 50 Å from the surface (Å) Distance Calculated TBMD trajectory with an initial kinetic energy 2 0.0 0.0 O of Ekin = 0 .2 eV 0 250 500 750 1000 1250 1500 Run time (fs) T = 210 K T = 250 K − − Energy redistribution during the run time of the trajectory <110> <110> Energy transfer of the impinging molecule into lateral and internal degrees of freedom Trapping into a dynamical precursor state ⇒ Deposition flux 0.08 ML/s, coverage θ = 0.08 ML P. Ruggerone et al. , Prog. Surf. Sci. 54 , 331 (1997) Growth of a compound semiconductor: GaAs 2. Hamiltonian Solid-state physics and chemistry: 31 processes (barriers) explicitly considered by DFT as an input for KMC simulations, Only electrostatic interaction considered Hamiltonian: 13 1 prefactor: 10 s− ⇒ P. Kratzer and M. Scheffler, Phys. Rev. Lett. 88 , 036102 (2002). H = Tnucl + Tel + Vnucl nucl + Vnucl el + Vel el (8) − − − Front view Top view 2 1 ZI ZJ e Vnucl nucl = , (11) − 2 ~ ~ I=J RI RJ L ~ 2 X6 | − | PI Tnucl = , (9) 2MI I=1 X 2 N 2 ZI e ~p i Vnucl el = , (12) Tel = , (10) − − ~ 2m i,I ~r i RI i=1 X | − | X 1 e2 Vel el = . (13) − 2 ~r i ~r j i=j X6 | − | Magnetic effects could be explicitly included Schr ¨odinger equation Born-Oppenheimer approximation Atoms 10 4 to 10 5 heavier than electrons Nonrelativistic Schr ¨odinger equation: (except for hydrogen and helium) electrons are 10 2 to 10 3 times faster than the nuclei ~ ~ ⇒ H Φ( R, ~r ) = E Φ( R, ~r ). (14) Born-Oppenheimer of adiabatic approximation: electrons follow motion of the nuclei instantaneously In principle we are ready here, however Practical implementation: Define electronic Hamiltonian Hel for fixed nuclear coordinates R~ solution of Schr ¨odinger equation in closed form not possible { } Hel ( R~ ) = Tel + Vnucl nucl + Vnucl el + Vel el . (15) { } − − − Hierarchy of approximations Nuclear coordinates R~ do not act as variables but as parameters ⇒ { } The Schr ¨odinger equation for the electrons H ( R~ ) Ψ( ~r, R~ ) = E ( R~ ) Ψ( ~r, R~ ). (16) el { } { } el { } { } Born-Oppenheimer approximation II Born-Oppenheimer approximation (BOA) III Schr ¨odinger equation for the electrons In the BOA electronic transitions neglected H ( R~ ) Ψ( ~r, R~ ) = E ( R~ ) Ψ( ~r, R~ ). (17) el { } { } el { } { } Exact derivation: Expansion of Schr ¨odinger equation in the small parameter m/M E ( R~ ) Born-Oppenheimer energy surface: potential for the nuclear motion: BOA very successful, but still its validity hardly directly obvious el { } T + E (R~) χ(R~) = E χ(R~). (18) Physical arguments { nucl el } nucl If quantum effects negligible: classical equation of motion Systems with a band gap: electronic transitions improbable ∂2 ∂ Metals: electronic system strongly coupled M R~ = E ( R~ ) . (19) I ∂t 2 I − ~ el { } short lifetimes and fast quenchening of electronic excitations ∂RI ⇒ Forces evaluated according to the Hellmann-Feynman theorem ∂ ∂ FI = Eel ( R~ ) = Ψ( ~r, R~ ) Hel ( R~ ) Ψ( ~r, R~ ) (20) −∂R~ I { } h { } |∂R~ I { } | { } i Structure of the Hamiltonian in Surface Science Three-dimensional periodic systems Interaction of a system with few degrees of freedom, atoms or molecules, with a system, the 3D crystal lattice Miller indices surface or substrate, that has in principle infinitely many degrees of freedom. Bravais lattice: Lattice planes described by the shortest ~ ~ ~ quantum chemistry solid-state methods. reciprocal lattice vector hb1 + kb2 + hb3 ↔ perpendicular to this plane Symmetries Exploitation of symmetries R~ = n1~a 1 + n2~a 2 + n3~a 3. (21) Integer coefficients hkl : Miller indices. T operator of a symmetry transformation First step: Determine symmetries of a • the Hamiltonian H and T commute, i.e. [H, T ] = 0 . i linearly independent unit vectors, ni Lattice planes: ( hkl ) ⇒ integer ψ H ψ = 0 , Family of lattice planes: hkl ⇒ h i| | ji { } Group theory if ψ and ψ are eigenfunctions of T 14 Bravais lattices in three dimensions • ⇒ | ii | ji Directions: [ hkl ] belonging to different eigenvalues Ti = Tj. Provides exact results 6 Dual space to real space: Reciprocal space Face-centered cubic (fcc) and body- • Only functions with the same centered cubic (bcc) crystals: ⇒ ~ ~a 2 ~a 3 symmetry couple in the Hamiltonian b1 = 2 π × (22) Miller indices are usually related to the Reduces computationally cost drama- ~a 1(~a 2 ~a 3) • tically | × | underlying simple cubic lattice ~b2 and ~b3: cyclic permutation of the indi- ces ~a ~b = 2 π δ , (23) i · j ij Bloch Theorem Structure of Surfaces Bloch theorem illustrates power of group theory Surface: created by just cleaving an infinite crystal along one surface plane. Hamiltonian Exploitation of group theory Consider effective one-particle Schr ¨odin- Translations TR~ form an Abelian group ger equation 1-D representation, eigenfunctions: ⇒ ~2 2 + veff (~r ) ψi(~r ) = εiψi(~r ). T ψ (~r ) = ψ (~r + R~) = c (R~)ψ (~r ) (26) −2m∇ R~ i i i i (24) Effective one-particle potential veff (~r ) satisfies translational symmetry: with ci(R~)ci(R~ ′) = ci(R~ + R~ ′).
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