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) Energy Image potential Energy (eV) Energy −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

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 (eV) Energy 0.5 1.0 50 Å {111}−faceted step 50 Å Distance from (Å) the surface 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 ödinger equation Born-Oppenheimer approximation Atoms 10 4 to 10 5 heavier than electrons Nonrelativistic Schr ödinger 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 ödinger 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 ödinger 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 indices ~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 ödin- 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~ ′). (27) veff (~r ) = veff (~r + R~), (25)

~ i~k R~ Left panel: fcc crystal with 100 faces and one 111 face, ~ ci(R) = e · . (28) with R any Bravais lattice vector. ⇒ right panel: fcc crystal with 100 faces and one 110 face. Eigenfunction ψi(~r ) characterized by the crystal-momentum ~k

i~k ~r ψ~k(~r ) = e · u~k(~r ) (29) with the periodic function ~ u~k(~r ) = u~k(~r + R) (30)

Bravais Lattices in 2 Dimensions Ideal, Relaxed and Reconstructed (110) Surfaces

2D lattice sketch examples Ideal Relaxed Reconstructed Square ~a fcc(100) 2D lattice sketch examples -2 Structure h h h ~a 1 = ~a 2 ~a 1 ϕ Centered ~a 2- bcc(110) 1 1 1 1 2 1 Changed surface periodicity: | | | | ? hA hA h h × × × h h h rectangular A ϕ = 90 ◦ ~a 1 AU A unit cell spanned by h h h s s h h h ϕ a = ma1 and a = na2: ~a 1 = ~a 2 Top view 1 2 | | 6 | | h h h h Hexagonal ~a 2- fcc(111) ϕ = 90 ◦ surface labelled by (hkl )( m n) ~a 1 = ~a 2 ~a 1 hϕ h h × or (hkl )p(m n), p primitive | | | | ¡ ¡ h h h × ϕ = 120 ◦ Oblique ~a 2- fcc(210) centered structures: ¢ h ¢ h h h h h ~a ¢ ~a 1 = ~a 2 1 ¢ ϕ¢ Side view (hkl )c(m n) ¢ ¢ | | 6 | | h h h × Rectangular ~a 2- fcc(110) ~a = ~a ~a h h h ϕ = 90 ◦ 1 2 1 ϕ h h h | | 6 | | ? 6 h h h ϕ = 90 ◦ h h h Ideal surface: interatomic distances the same as in the bulk Relaxed surface: rearrangement preserving surface symmetry Reconstructed surface: symmetry of surface changed Vicinal Surfaces Defects at Surfaces

Vicinal surface: A surface that is only slightly misaligned from a low index plane adatom island kink @ @ step with (100)-oriented ledge step [111] @ @@R terrace [100] ©

? vacancy

adatom Defected (755) = 5(111) (100) surface × Study of defects important: steps, kinks and other surface irregularities often dominate surface processes such as surface reactions, oxidation, corrosion, crystal growth or surface melting.

A stepped (911) = 5(100) (111) vicinal surface. × Vicinal surfaces also denoted by n(hkl ) (h′k′l′), × (hkl ) and ( h′k′l′) Miller indices of the terraces and of the ledges, n width of the terraces in number of atomic rows parallel to the ledges.

Electronic Structure methods Quantum mechanical approach for the electronic structure problem Time-independent electronic Schr ödinger equation Rayleigh-Ritz variational principle Independent electrons Hel Ψ( ~r ) = Eel Ψ( ~r ). (31) Ground state energy E0, Assumption: N electrons in the effective Parametric dependence on the nuclear coordinates omitted Ψ arbitrary guess for the true ground external potential state| i wave function Ψ : Z e2 | 0i v (~r ) = I (33) ext − ~ Mathematically Schr ödinger equation corresponds to a partial differential equation in 3 N I ~r RI unknows with N commonly larger than 100 completely intractable to solve Ψ H Ψ X | − | E0 h | | i. (32) One-particle Schr ödinger equation ⇒ ≤ Ψ Ψ ⇒ Way out: h | i Rayleigh-Ritz variational principle ~2 expand the electronic wave function in some suitable, but necessarily finite basis set 2 + v (~r ) ψ (~r ) = εo ψ (~r ). 2m ext i i i Conversion of the partial differential equation into a set of algebraic equations that are − ∇ provides a route to come closer to the (34) ⇒ much easier to handle. true ground state energy by improving Product wave function solution for the guesses for Ψ , preferentially in a many-body⇒ wave function systematic way. | i

ΨH(~r 1, . . . , ~r N) = ψ1(~r 1) . . . ψN(~r N) · · (35) Hartree Approximation I Hartree Approximation II

Assume product wave function as a first guess for the many-body wave function Variation leads to the Hartree equations Expection value of the electronic Hamiltonian: ⇒ ~2 N 2 2 e 2 + vext (~r ) + d~r ′ ψj(~r ′) ψk(~r ) = εk ψk(~r ). (38) N −2m∇ ~r ~r | |  ~2 j=1 ′ 2  Z | − |  Ψ H Ψ = d~rψ ∗(~r ) + v (~r ) ψ (~r ) X h H| | Hi i −2m∇ ext i i=1 Mean-field approximation: Hartree equations describe an electron embedded in the X Z   N 2 electrostatic field of all electrons including the particular electron itself . 1 e 2 2 self interaction + d~rd~r ′ ψi(~r ) ψj(~r ′) + Vnucl nucl . (36) This causes the which is erroneously contained in the Hartree equations. 2 ~r ~r | | | | − i,j =1 Z | − ′| X Define electron density n(~r ) and Hartree potential vH: N 2 2 e Minimize the expectation value (36) with respect to more suitable single-particle functions n(~r ) = ψi(~r ) , v H(~r ) = d~r ′ n(~r ′) (39) | | ~r ~r ψ (~r ) under the constraint that the wave functions are normalized: i=1 ′ i X Z | − | δ N Hartree equations: Ψ H Ψ ε ( ψ ψ 1) = 0 . (37) ⇒ δψ h H| | Hi − { i h i| ii − } ~2 k∗ " i=1 # 2 + v (~r ) + v (~r ) ψ (~r ) = ε ψ (~r ). (40) X −2m∇ ext H i i i εi: Lagrange multipliers ensuring the normalisation of the eigen functions.

Hartree Approximation III Self-consistent ﬁeld solution

Hartree equations: Effective one-particle Hartree-Fock Hamiltonians contain solution: Self-consistent iteration scheme ~2 2 ⇒ + v (~r ) + v (~r ) ψ (~r ) = ε ψ (~r ). (41) Initial guess: −2m∇ ext H i i i 0 0 n (~r ) veff (~r ) Solutions ψ (~r ) of the Hartree equations enter the effective one-particle Hamiltionan: −→ i ? Hartree equations can only be solved in an iterative fashion Solve Schr ödinger equations: ⇒ ~2 Hartree approximation “self-consistent field approximation”. − 2 j (j+1) (j+1) + veff (~r ) ψi (~r ) = εiψi (~r ) ⇒ ≡ ( 2m ∇ ) Expectation value of the total energy in the Hartree approximation EH: ? N Mixing scheme: 2 Determine new density: 1 e n(~r ) n(~r ′) (j+1) j ΨH H ΨH = εi d~rd~r ′ + Vnucl nucl N veff (~r ) = αv eff (~r ) (j+1) (j+1) 2 h | | i − 2 ~r ~r ′ − n (~r ) = ψ (~r ) , new i=1 Z | − | | i | + (1 α)veff (~r ) XN i=1 − X new with α > 0.9 and new effective potential veff (~r ) 6 = εi VH + Vnucl nucl = EH (42) − − ? i=1 X -Yes Do vnew (~r ) and vj (~r ) differ by more than ε 1? Hartree energy VH enters the Hartree eigenvalues twice has to be subtracted. eff eff ≪ Reflects the interaction between particles in a self-consiste⇒ nt scheme. ? No

? Ready Flow-chart diagram of a self-consistent ﬁeld solution scheme Hartree-Fock Approximation I Hartree-Fock Approximation II

Hartree ansatz only partially obeys the Pauli principle Follow same procedure as in the Hartree ansatz: ﬁrst write down expectation value by populating each electronic state once, but it does not take into account the anti-symmetry of the wave function. N ~2 2 Construction of a Slater determinant from the single-particle functions ψ(~rσ ), Ψ H Ψ = d~rψ ∗(~r ) + v (~r ) ψ (~r ) ⇒ h HF | | HF i i −2m∇ ext i where σ denotes the spin: i=1 X Z N 2 1 e 2 2 + d~rd~r ′ ψi(~r ) ψj(~r ′) ψ1(~r 1σ1) ψ1(~r 2σ2) ... ψ 1(~r NσN) 2 ~r ~r | | | | i,j =1 ′ 1 ψ2(~r 1σ1) ψ2(~r 2σ2) ... ψ 2(~r NσN) X Z | − | ΨHF (~r 1σ1, . . . , ~r NσN) = . . . . (43) √ ...... N 2 N! 1 e ψN(~r 1σ1) ψN(~r 2σ2) ... ψ N(~r NσN) d~rd~r ′ δσiσjψi∗(~r )ψi(~r ′)ψj∗(~r ′)ψj(~r ). −2 ~r ~r ′ i,j =1 Z | − | X + Vnucl nucl . (44) −

σi spin of the state ψi(~r )

Hartree-Fock Approximation III Homogeneous electron gas

Minimize expectation value of total energy with respect to the ψi∗ under the constraint of Homogeneous electron gas: n(~r ) uniform. normalization: Assumption: electrostatic potential of the electrons ~2 compensated by a positive charge background. 2 + v (~r ) + v (~r ) ψ (~r ) −2m∇ ext H i “jellium model”: ⇒ N 2 e vext (~r ) + vH(~r ) = 0 . (47) d~r ′ ψ∗(~r ′)ψ (~r ′)ψ (~r )δ = ε ψ (~r ). (45) − ~r ~r j i j σiσj i i j=1 Z | − ′| X Eigenfunctions of the Hartree equations for the homogeneous electron gas (periodic boundary conditions) : Additional term, exchange term , integral operator: V (~r, ~r ′)ψ(~r ′)d~r ′ Total energy in the Hartree-Fock approximation:R 1 ~ iki ~r ψi(~r ) = e · . (48) N √V ΨHF H ΨHF = EHF = εi VH Ex + Vnucl nucl (46) h | | i − − − i=1 Plane waves X Exchange in the homogeneous electron gas One-particle energies

Total energy: Hartree term VH exactly cancels with Vnucl nucl One-particle energies for the homogeneous electron gas in the Hartree and the Hartree-Fock − approximation: N ~2~k2 E = ε(~k ) = 2 . (49) H i 2m i=1 ~ k

2 1 1 x 1 + x k/k F F (x) = + − ln . (52) 2 4x 1 x

−

Total energy in the Hartree-Fock approximation Exchange hole distribution

i Total energy for the homogeneous electron gas: Exchange-hole density nx(~r, ~r ′) around the i-th electron: ~2~k2 e2 k N i ψj∗(~r ′)ψi(~r ′)ψj(~r ) EHF = 2 kF F nx(~r, ~r ′) = δσiσj . (56) 2m − π kF − ψi(~r ) k

Mean exchange-hole density in the homogeneous electron gas: Hartree-Fock: electrons of same spin are correlated, but not electrons of opposite spin

N i Further reduction in energy, if electrons of opposite spin are also correlated ψi∗(~r ′)nx(~r, ~r ′)ψi(~r ) nx(~r, ~r ′) = δσ σ ; ~r ~r ′ =r ¯ n(~r ) i j | − | Exchange and exchange-correlation energy per particle i=1 ′ X 2 in the homogeneous electron gas: 9N k r¯cos( k r¯) sin( k r¯) = n (¯r) = F F − F . (59) x 2 V (k r¯)3 0 F

-2 exchange energy 1,0 exchange-correlation energy

-4 V/N ⋅ energy (eV) 0,5 -6 density

average density n + -8 average density n exchange-hole density nx 0 10 20 30 2 -3 0,0 electron density x 10 ( Å ) 0 2 4 6 distance r/k F 2 3 Typical valence electron densities of metals are in the range of 5 - 10 10 (A˚− ). ×

Quantum chemistry methods I Quantum chemistry methods II Cluster Quantum chemistry Hartree-Fock Requirements Usually Hartree-Fock theory good star- Historically, chemists were the first to be intere- 1 kcal/mol 0.04 eV (“chemical accu- • ting point for a theoretical description: • sted in the theoretical description of surfaces and • racy”) for energy≈ differences required exact total energy of a molecule is of- processes at surfaces ten reproduced to up to 99% Quantum chemists have developped Theoretical tools used by chemists are made to • a whole machinery of theoretical me- Unfortunately, the missing part, name- • describe finite systems such as molecules thods that treat the electron correlati- • ly the electron correlation energy, is on at various levels of sophistication: rather important for a reliable descrip- MP2, CI, CISD, FCI, CCSD, CCSD(T), In the chemistry approach surfaces are regarded tion of chemical bond formation. •as ⇒ big molecules and modelled by finite clusters MCSCF, CASSCF, . . . Si 9H12 cluster used to model a Si(100) Correlation energy per electron can ea- substrate Methods can be devided into to This ansatz is guided by the idea that bonding • sily be more than 1 eV ( free electron • • two categories: the so-called single- on surfaces is a local process. It depends on the gas) → localisation of the electronic orbitals. reference and the multiple-reference methods. Single reference methods Møller-Plesset theory (MP)

Starting point: Slater determinant Regard the Hartree-Fock Hamiltonian as the unperturbed Hamiltonian and the diﬀerence H′ to the true many-body Hamiltonian as the perturbation Hartree-Fock: correlation between electrons of opposite spin neglected

Ansatz: introduce correlation eﬀects by considering excited states. H′ = H H . (60) el − HF Generate excited states by replacing occupied orbitals in the Slater determinant by unoccupied ones Second-order expression (MP2): By replacing one, two, three, four or more states, single, double, triple, quadruple or higher 2 excitations can be created. (2) Ψl H′ Ψ0 E = EHF + Ψ0 H′ Ψ0 + |h | | i| . (61) h | | i E0 El l=0 Møller-Plesset theory: excitations are treated perturbatively X6 −

Higher-order corrections: MP3, MP4, . . . MP2 rather popular, but due to the perturbative treatment of electron correlation its applicability is still limited

Conﬁguration Interaction (CI) Size consistency

Instead of perturbative treatment include excitations explicitly in the wave: CISD approach does not fulﬁl size consistency or size extensivity

(1) (1) (2) (2) Size extensitivity: linear scaling of the energy with the number of electrons ΨCISD = Ψ HF + ci Ψi + ci Ψi . (62) Infinitely separated systems: additive energies of the separated components X X (j) Optimum wave function found by varying the coefficients ci This property is not only important for large systems, but even for small molecules Different configurations (or Slater determinants) are included in (62) method called configuration interaction (CI) ⇒ Product of two fragment CISD wave functions contains triple and quadruple excitations and is therefore no CISD function If only single (S) and double (D) excitations are included: CISD no size-consistency If all possible excitations are included: Full CI (FCI) ⇒ Coupled Cluster Theory (CC) Multiple reference methods

Recover size extensitivity by exponentiating single and double excitations operator: Single-reference methods very reliable in the vicinity of equilibrium configurations Often they are not adequate to describe a bond-breaking process: ΨCCSD = exp( T1 + T2) Ψ HF . (63) Dissociation products should be described by a linear combination of two Slater determinants Coupled cluster (CC) theory taking into account the proper spin state Limitation to single and double excitations: CCSD One Slater determinant plus excited states derived from this determinant not sufficient ⇒ Triple excitations included: CCSDT Accuracy can be systematically improved by considering more and more configurations Full configuration interaction (FCI) CCSDT has an eight-power dependence on the size of the system ⇒ Because of the large required computational effort, FCI calculations are limited to rather Perturbative treatment of triple excitations: CCSD(T) small systems. CCSD(T) calculations are method of choice for very accurate determination of molecular Approximate multi-reference methods are needed equilibrium structures

SCF methods Basis set nomenclature

Multiconfigurational self-consistent field (MCSCF) approach: Preferred basis set in quantum chemistry: Gaussian functions integral can be evaluated analytically A relatively small number of configurations selected and both the orbitals and the ⇒ configuration interactions coefficients are determined variationally Nomenclature: Chemical insight needed for the selection of the configurations One atomic orbital per valence state: “minimal basis set” or “single zeta” ⇒ Complete active space (CASSCF) approach: Two or more orbitals: “double zeta”(DZ), “triple zeta” (TZ) and so on A set of active orbitals identified and all excitations within this active space are included. Often polarization functions added (one or more sets of d functions on first row atoms): “P” added to the acronym of the basis, e.g., DZ2P Method computationally very costly Delocalized states (anionic or Rydberg excited states): further diffuse functions added Density functional theory (DFT) Variational principle

Wave function based methods limited to a small number of atoms due to unfavorable scaling Variational principle for the energy functional: Electronic structure methods based on density more eﬃcient: DFT Etot = min E[n] = min (T [n] + Vext [n] + VH[n] + Exc [n]) . (64) DFT based upon the Hohenberg-Kohn theorem: n(~r ) n(~r ) The ground-state density n(~r ) of a system of interacting electrons in an external potential Vext [n] functional of the external potential uniquely determines this potential U[n] functional of the classical electrostatic interaction energy (Hartree energy) Proof simple! T [n] is the kinetic energy functional for non-interacting electrons Density n(~r ) uniquely related to the external potential and the number N of electrons via All quantum mechanical many-body eﬀects are contained in the so-called xc N = n(~r )d~r exchange-correlation functional E [n] This non-local functional is not known in general n(~r ) determinesR the full Hamiltonian ⇒ Important property: Exc [n] universal functional of the electron density In principle it determines all quantities that can be derived from the Hamiltonian such as, e.g., the electronic excitation spectrum Advantage: Instead of using the many-body quantum wave function which depends on 3 N coordinates now only a function of three coordinates has to be varied. Unfortunately this has no practical consequences since the dependence is only implicit

Kohn-Sham equations Ground state energy in DFT

In practice no direct variation of the density is performed because, e.g., the kinetic energy Ground state energy functional T [n] is not well-known N Express density as a sum over single-particle states E = ε + E [n] v (~r )n(~r ) d~r V (68) i xc − xc − H N i=1 Z n(~r ) = ψ (~r ) 2, (65) X | i | i=1 If the exchange-correlation terms E and v are neglected Hartree approximation X xc xc ⇒ Minimize E[n] with respect to the single particle states under the constraint of Diﬀerence to Hartree and the Hartree-Fock approximation: normalization: Kohn-Sham equations ⇒ ~2 Ground-state energy (102) is in principle exact 2 + v (~r ) + v (~r ) + v (~r ) ψ (~r ) = ε ψ (~r ) . (66) −2m∇ ext H xc i i i Reliability of any practical implementation of DFT depends crucially on the accuracy of the expression for the exchange-correlation functional Exchange-correlation potential vxc (~r ): δE [n] v (~r ) = xc . (67) xc δn

Again: “single-particle energies” εi enter the formalism just as Lagrange multipliers. Exchange-correlation functional Local Density Approximation (LDA)

Exchange-correlation functional Exc [n]: Exchange-correlation energy per particle εxc [n]( ~r ) not known in general

Exc [n] = d~rn (~r ) εxc [n]( ~r ), (69) Local density approximation: use exchange-correlation energy for the homogeneous electron Z gas also for non-homogeneous situations

Introduce exchange-correlation hole distribution LDA 3 LDA Exc [n] = d ~rn (~r ) εxc (n(~r )) (74) n (~r, ~r ′) = g(~r, ~r ′) n(~r ′) , (70) xc − Z In a wide range of bulk and surface problems LDA surprisingly successful g(~r, ~r ′) conditional density to find an electron at ~r ′ if there is already an electron at ~r Sum rules: Reasons still not fully understood, probably: Cancellation of opposing errors in the LDA exchange and correlation d~r ′ n (~r, ~r ′) = 1 (71) xc − Z LDA satisfies the sum rules For chemical reactions in the gas phase and at surfaces, however, the LDA results are not nxc (~r, ~r ′) 0 . (72) ~r −→~r sufficiently accurate. | − ′|→∞ Usually LDA shows over-binding , i.e. binding and cohesive energies are too large and lattice nxc (~r, ~r ′) 1 constants and bond lengths are too small d~r ′ (73) ~r ~r ~r −→ − ~r Z | − ′| | |→∞ | |

Generalized Gradient Approximation (GGA) Problematic systems within the GGA

Straightforward Taylor expansion of the exchange-correlation energy εxc [n]: There are important systems where GGA functionals do not give satisfactory results: Sum rules violated O2 binding energy for diﬀerent exchange-correlation functionals: Generalized gradient approximation: modify dependence on the gradient in such a way as to (B. Hammer, L.B. Hansen, and J.K. Nørskov, Phys. Rev. B 59 , 7413 (1999)) satisfy sum rules functional LDA PW91 PBE revPBE RPBE Exp. Binding energy (eV) 7.30 6.06 5.99 5.63 5.59 5.23 GGA 3 GGA Exc [n] = d ~rn (~r ) εxc (n(~r ), n(~r ) , (75) |∇ | Due to the error in O binding energies, O adsorption energies in dissociative adsorption will Z 2 2 be unreliable, too. DFT calculations in the GGA achieve chemical accuracy (error 0.1 eV) for many chemical General problem: non-locality of the exchange-correlation functional reactions; Example: ≤ approximation uncontrolled, i.e., there is no systematic way of improving the functionals Dissociation barrier for H 2/Cu(111) since there is no expansion in some controllable parameter LDA: 0.05 eV PW91-GGA: 0.6 eV Exp.: 0.5 - 0.7 eV Pseudopotential Norm-conserving pseudopotentials

Chemical properties of most atoms are determined by their valence electrons. Construction scheme Example Pseudopotentials ⇒ Asymptotically pseudopotential should des- Schr ödinger equation with pseudopotential cribe the long-range interaction of the core 0,1 Outside of a core radius rc the pseudo- ~2 wavefunction should coincide with the full 0 2 + v ψ = ε ψ . (76) −2m∇ ps | ps i v | ps i wavefunction -0,1 3s pseudo-wave function all-electron 3s valence wave function all-electron ionic potential Inside of this radius both the pseudopo- pseudopotential with tential and the wavefunction should be as Energy (arb. units) Amplitude units) (arb. -0,2 r smooth as possible ( softness) 0 c 1 2 3 → Radial distance r (Å) v = v (~r ) + (ε ε ) ψ ψ . (77) Schematic illustration of the difference between the all- ps eff v − ci | cii h ci| The requirement of norm conservation au- i tomatically ensures that it has the same electron (solid line) and pseudo 3s wave function (dashed X line) and their corresponding potentials. The core radius one-particle energy εv as the true valence r is indicated by the vertical line. vps non-local and energy-dependent wave function c ψ is not normalized | ps i

Ultrasoft Pseudopotentials Plane wave codes

Norm-conserving constraint removed DFT calculations ﬁrst successfully applied to solid-state problems: lattice constants, cohesive energies, bulk modulus, . . . Constructed by a generalized orthonormality condition Natural basis for periodic structures: plane waves

In order to recover the correct charge density, augmentation charges are introduced in the G~ 1 i(~k +G~ ) ~r ψ (~r ) = e i · (78) core region i √V

The electron density subdivided in a delocalized smooth part and a localized hard part in the G~ reciprocal lattice vector core regions Plane waves are already of the form required by the Bloch theorem Dramatic reduction in the neccessary size of a plane-wave basis set ⇒ Only plane waves that diﬀer by a reciprocal lattice vector do couple:

ψ~ h ψ~ = 0 for ~k = ~k′ + G~ . (79) h k| | k′i 6 Plane wave codes II Plane wave codes for surface problems

For periodic, inﬁnite systems, the band structure energy in the total energy expression has to Plane wave basis Super cell be replaced by an integral over the ﬁrst Brillouin zone:

V Expansion of wave functions in plane waves ε d~k ε (~k) (80) computationally very eﬃcient i −→ (2 π)3 i X bandsX BZZ Requires three-dimensional periodicity

~ Surfaces modelled in the super cell ap- Integral can be approximated rather accurately by a sum over a finite set of k-points, either proach by using equally spaced ~k-points or so-called special ~k-points Convergence checks: Parameter characterizing the size of the plane wave basis: cutoff energy Slab and vacuum thickness ~2(~k + G~ )2 Typical values: Ecutoff = max (81) slab at least four layers thick G~ 2m vacuum layer 10 A˚ ≥ Norm-conserving Troullier-Martins pseudopotentials: cutoff energy 10 Ryd for semiconductors and 50 Ry for transition metals ∼ ≥ Ultrasoft pseudopotentials: cutoff energy for transition metals reduced to about 20 Ryd 250 eV. ≈

Augmented plane waves Augmented plane waves II

Electronic structure method for extended systems that take into account the core electrons: Main idea: Different basis set for core and for interstitial region. 1 i~k ~r √ e · interstitial region First implementation: muffin-tin potential V φ~k,ε (~r ) =  , (84) alm(~k, ε ) ul(ri, ε ) Ylm(θ, φ ) ri = ~r R~ i < r MT  lm | − | 0 interstitial region P veff (~r ) = , (82)  v ( ~r R~ ) ~r R~ < r u (r, ε ) solutions of the radial Schr ödinger equation ( MT | − i| | − i| MT l ~2 d2 ~2 l(l + 1) + + v (r) ε ru (r, ε ) = 0 . (85) Basis set: Augmented plane waves (APW) −2mdr 2 2m r2 MT − l 1 ei~k ~r interstitial region ~ √V · Important: there is no constraint relating k and ε φ~k,ε (~r ) =  , (83) alm(~k, ε ) ul(ri, ε ) Ylm(θ, φ ) ri = ~r R~ i < r MT Correct solution expanded as  lm | − | P ψ~k(~r ) = cG~ φ~k+G,ε~ (~r ) , (86)  XG~ Problem: basis functions depend on energy ε Linearized augmented plane waves Linearized augmented plane waves II

Problem of energy-depended basis functions avoided by the concept of linearized augmented LAPW basis functions: plane waves (LAPW): O.K. Andersen (1975) 1 ei~k ~r interstitial region √V · Idea: expand radial function around ﬁxed energy εl:  a (~k) u (r , ε ) + φ~k(~r ) =  lm l i l (89)  lm ul(r, ε ) = ul(r, ε l) +u ˙ l(r, ε l) ( ε εl) + ... , (87)  h − Pb (~k)u ˙ (r , ε ) Y (θ, φ ) ~r R~ < r lm l i l lm | − i| MT  u˙ (r, ε ) energy derivative  i l l  alm(~k) and blm(~k) determined through the continuity requirement of both the wave function du (r, ε ) and its ﬁrst derivative. u˙ (r, ε ) = l . (88) l l dε ε=εl FP-LAPW: no restriction to spherical symmetric potentials in the core region

~ iG~ ~r εl should be in the middle of the corresponding energy band with l character. veff (G) e · interstitial region v (~r ) = G~ . (90) eff  P lm ~  veff (ri) Ylm(θ, φ ) ~r Ri < r MT  lm | − | P 

Tight-Binding Tight-Binding II

Ab initio electronic structure methods still computationally expensive Bloch sums are used to determine Hamiltonian matrix elements: There is a need for a approximative method that retains the quantum nature of bonding: h (~k) = φ h φ Tight-binding αβ h α~k| | β~ki ~ ~ ~ ~ ik Rj ik Rj ~ Tight-binding: expansion of the eigenstates of the eﬀective one-particle Hamiltonian in an = e · φ0α h φjβ = e · hαβ (Rj) . (93) h | | i atomic-like basis set and replacing the exact many-body Hamiltonian with parametrized j j X X Hamiltonian matrix elements ~ Formally, one starts with a basis of atomic functions Two-center approximation for hαβ (Rj): ~ ~ ~ φ (~r ) = φ (~r R~ ) , (91) hαβ (R) function of R = R and of the direction cosines k, l, m of R. iα α − i | | Eigenfunctions χ~k of the one-particle Hamiltonian i lattice site, α type of the atomic orbital Periodic functions constructed by forming Bloch sums ~ χ~k(~r ) = cα(k) φα~k(~r ) . (94) 1 i~k R~ α φ (~r ) = e · i φ (~r R~ ) , (92) X α~k √ α − i N i X Nonorthogonal tight-binding Orthogonal tight-binding

Eigenfunctions χ~k of the one-particle Hamiltonian Further simpliﬁcation: Assume that the atomic orbitals are already orthogonalized according to the L ¨owdin scheme ~ χ~k(~r ) = cα(k) φα~k(~r ) . (95) 1/2 − α ψiα (~r ) = Siαjβ φjβ (~r ) , (98) X Xjβ Energies determined by eigenvalue equation

Dispersion curves ε(~k) obtained by the diagonalization of h(~k). h(~k) c(~k) = S(~k) ε(~k) c(~k), (96) ⇒ Nc atoms in the unit cell and l atomic orbitals per atoms ~ N l N l matrix has to be diagonalized S(k) overlap matrix ⇒ c × c Atomic basis function do not explicitly appear in tight-binding i~k R~ i~k R~ S (~k) = e · j φ φ = e · j S (R~ ) . (97) parametrization scheme determines the reliability and accuracy of any tight-binding αβ h 0α| jβ i αβ j ⇒ j j calculation X X

1 Dispersion curves ε(~k) obtained by the diagonalization of S(~k)− h(~k). ⇒ Nonorthogonal tight-binding

Total energies in tight-binding Structure and energetics of clean surfaces

Total energies: usually a repulsive term written as a sum of pair terms is added to the band Surface energies Experiment energy (can be derived from DFT) Surfaces can be assumed to be created by Experimental determination of surface cleaving an infinite solid energies non-trivial Etot = Eband + Erep This requires energies, otherwise crystal Usually surface energies are not direct- = ε(~k) + U . (99) ij would cleave spontaneously ly measured, but only relative to other ~ ij Xk X Calculated surface energies of a monoato- surfaces mic solid at 0K: Tight-binding useful scheme to understand qualitative trends. Example: dispersion for a s-band in a metal under the assumption of only nearest neighbors 1 Many reported surface energies are deri- (nn) interaction γ = (Eslab N Ebulk ) (101) ved from surface tension measurements 2A − · in the liquid phase and extrapolated to ε(~k) = β + γ cos ~k R~ , (100) zero temperature · nn Eslab total energy of the slab per supercell, X Ebulk is the bulk cohesive energy per atom, These surface energies correspond to β = φ h φ on-site term, γ = φ h φ A is the surface area in the supercell. an⇒ average over different orientations h 0s| | 0si h 0s| | (nn )si fcc crystal: s-band width = 12 γ proportional to overlap integrals → Electronic states of surfaces and adsorbates Density of states Ground state energy in density functional theory: N Local density of states (LDOS) useful tool to discuss the reactivity of surfaces 3 E = εi + Exc [n] vxc (~r )n(~r ) d ~r VH + Vnucl nucl (102) − − − i=1 Z n(~r, ε ) = φ (~r ) 2 δ(ε ε ) (107) X | i | − i i Energy contributions Discussion X Band structure term: Recall that fundamental Hamiltonian only Derived quantities Example N • contains kinetic and electrostatic energy 4 Ebs = εi (103) Total density of states: Layer 1 ⇒ 3 i=1 3 X Exchange-correlation and Hartree energy n(ε) = n(~r, ε ) d ~r (108) 2 Exchange-correlation energy: • also included in the Kohn-Sham eigenva- Z 1 D 0 Total Electron density: 4 lues εi (double counting) eV Layer 2 3 ⇒ E [n] v (~r )n(~r ) d ~r (104) 3 xc xc n(~r ) = n(~r, ε ) dε (109) − states 2 3 @ Z Correction Exc [n] vxc (~r )n(~r )d ~r VH Z 1 Hartree energy: • reflects the interaction− of the particles− Band structure term: LDOS 0 ⇒ 4 Layer 3 2 R N 1 e n(~r )n(~r ′) 3 3 3 VH = d ~rd ~r ′ . (105) The following discussion will mainly focus εi = n(ε) ε dε (110) 2 2 ~r ~r ′ • Z | − | on the band structure term i=1 Z 1 Electostatic nucleus-nucleus interaction: X 0 Projected density of states: -8 -6 -4 -2 0 - 2 E EF @eV D 1 ZIZJe Vnucl nucl = . (106) d-LDOS of the three top Pd(210) layers (M. Lischka and A. − 2 ~ ~ 2 I=J RI RJ na(ε) = φi φa δ(ε εi) (111) Groß, Phys. Rev. B 65, 075420 (2002)) X6 | − | |h | i| − i X Jellium model Work function

Model description Charge density Deﬁnition Jellium results

Positive ion charges replaced by a uni- Work function Φ: minimum energy that form charge background: rs = 5 must be done to remove an electron Jellium model 1.0 from a solid at 0K. 5 Ion lattice model Au Zn Experiment n,¯ z 0 − Al Cu Positive Ag n+(~r ) = ≤ . (112) Φ = φ( ) + EN 1 EN 4 ( 0, z > 0 background ∞ − − Φ = ∆ φ µ¯ Pb Mg − 3 Electrons spill out into the vacuum 0.5 = φ( ) εF , (113) Li ∞ − Na Dipole layer (eV) Work function K ⇒ 2 Rb Cs

Electron density (1/n) density Electron r = 2 Screening of the positive background s

φ( 8 ) by the electrons incomplete since only 2 3 4 5 6 Wigner−Seitz radius r (a.u.) electrons with wave vectors up to the Φ ∆φ s 0 ε Jellium model surprisingly good for the descrip- Fermi wave vector kF are available −1.0 −0.50 0.5 1.0 Energy F µ− φ(z) tion of simple sp -bonded metals, but fails for Friedel oscillations with wavelength Distance from the surface (1/k F )

φ( ) 8 ⇒ − d-metals such as Au, Ag, Cu, . . . π/k F Charge density at a surface in the jellium model Distance from the surface z Realistic surface structure has to be taken into⇒ account Electronic band structure at surfaces Electronic surface states

At surfaces,symmetry broken in z-direction DFT surface band structure of Cu(111) Discussion

kz no longer a good quantum number ⇒ 5 Cu(111): Bandgap and parabolic surface ¯ Projected bulk band structure Discussion band at the Γ point Eﬀective mass: m∗ = 0 .37 me ε (k) · Electronic band structure along kz projected corresponds to Shockley state in the onto kxky-plane ε sp -band gap 0 F In gaps, localized surface states can exist: Surface state 1.5 eV below Fermi level: Shockley surface states , Tamm surface state that is pushed out of whose existence can be derived within the the d band one-dimensional nearly-free electron model: Caused by the strong perturbation of the

V (z) = V + W Θ( z) + V cos( Gz ) Θ( z) . (eV) Energy −5 local atomic orbitals 0 0 − G True surface state since it is orthogonal to the sp -continuum a W0 |V | Surface resonances emanating from the (0,0,0) (k ,k ,0) G x y V 0 −10 Tamm surface band Potential energy V(z) M Σ Γ Σ M 0 Distance from the surface z Surface states that join the bulk continuum: A. Euceda, D.M. Bylander, and L. Kleinman, Phys. Rev. B 28 , 528 (1983) (0,0,k z ) Surface resonances

Tamm surface states in tight-binding Cu(111) surface states

Consider an equidistant linear chain of identical atoms having only one s orbital φ at each l Don Eigler, IBM Almaden site l. Use the orthogonal tight-binding approximation with only nearest-neighbor| interactions. The on-site and hopping matrix elements are given by Quantum Corral Quantum Mirage

φl h φm = βδ l,m + γδ l,m 1 . (114) | | ±

“Bulk” band structure can be determined for an infinite chain of atoms. Consider a semi-infinite chain of atoms with sites n = 0 , 1,... as a model for a crystal with a surface. The on-site term for the surface atom differs from the bulk value, i.e.

φ h φ = β′ = β . (115) 0| | 0 6

For a strong perturbation of the on-site term at the surface β′ β > γ , new states above and below the bulk continuum appear, the so-called Tamm| − surface| | states| , which are localized at the surface. Standing surface waves Kondo resonance eﬀect 48 Fe atoms deposited on Cu(111) in a Elliptical Quantum Corral with a Co atom circle of diameter 71 A˚ in the focal point of the ellipse M.F. Crommie et al. , Science 262 , 218 (1993). H.C. Manoharan et al. , Nature 403 , 512 (2000). Image potential states on Cu(100) Trends in the energetics and structure of Cu surfaces

Charge density of Cu(100) Image potential Surface energies Relaxations The more densely packed a certain lattice 6[110] plane, the less bonds have to be broken 2 upon cleavage. d12 8 d23 d 0 n=1 Low-index surfaces: 34 n=2 γ(111) < γ (100) < γ (110) −2 6 n=3 ε ((2 n 1) , 1, 1) surfaces: in fact −4 vac n(100)− (111) surfaces [001] direction (Å) [001] direction 4 ×

Energy (eV) Energy At metal surfaces, the smoothening of −6 Image potential Large n: γ((2 n 1) ,1,1) γ(100) the electron density usually leads to a V(z)=−e2 /4z surf. energy decreases− with→ increasing n. −4 −2 0 2 4 2 ⇒ contractive relaxation of the ﬁrst layer. [100] direction (Å) ε From the sequence of ((2 n 1) , 1, 1) sur- Inward relaxation causes a charge accu- GGA-DFT calculations, S. Sakong (TUM) F face energies the step formation− energy can mulation at the second layer expan- Cu(100): surface band structure similar to 20 40 be derived. sion ⇒ Cu(111), but band gap at the Γ¯ point Distance from the surface z(Å) extends above the vacuum level Rydberg-image potential states detected Electrons with small parallel momentum with time-resolved photoelectron spectros- can not penetrate into the bulk copy U. H ¨ofer, I.L. Shumay, Ch. Reuß, U. Thomann, W. Wallauer, Image potential states ⇒ and Th. Fauster, Science 277 , 1480 (1997)

Smoluchowski smoothing of the electron density Qualitative picture of ﬁrst-layer relaxation of metal surfaces Cu(100) surface Schematic illustration Electron smoothing Explanation + + + + − − − − − [001] 2 Assumption: n(~r ) constant in the metal 6 + + + + − − − − − 0 Driving force: positively charge atomic z ( A)˚ core rearranges so that the electric ﬁeld -2

(~r ~r ′) -4 ~ E(~r ) = d~r ′ n(~r ′) − 3 (116) Wigner-Seitz (WS) cells at the (100) surface of a fcc ~r ~r Wigner-Seitz (WS) cells at the (100) surface of a fcc crystal Z | − ′| -6 crystal -4 -2 0 2 4 Smoluchowski smoothing of electron outside of the WS cells is small x ( A)˚ - [100] density atomic core will be at the site where the⇒ electric ﬁeld of the electron charge Kinetic energy of electrons at the surface reduced by the smoothing distribution vanishes, i.e. where ~r d~rn (~r ) = 0 (117) r3 WZ S inward relaxation: (111) 1.6%, (100) 4.6%, (110) 16 % ⇒ − − − Surface energies and relaxations of Cu surfaces Calculated surface energies of 4d surfaces

4 4 2 ˚ Surface Method γ (J/m ) ∆d12 ∆d23 ∆d34 d0(hkl) ( A) fcc(111) a) b) ) Cu(111) Theory 1.30 -0.9 -0.3 2.10 3 fcc(100) 2 3 fcc(110) Cu(111) Exp. 1.79 -0.7 2.08 ∼ Cu(100) Theory 1.45 -2.6 1.5 1.821 2 2 Cu(100) Exp. 1.79 -2.1 0.4 0.1 1.807 ∼ fcc(111) Cu(110) Theory 1.53 -10.8 5.3 0.1 1.29 fcc(100)

1 surface energy (J/m 1 fcc(110) Cu(110) Exp. 1.79 -8.5 2.3 1.28 surface energy (eV/atom) ∼ Cu(311) Theory 1.82 -15.0 4.0 -0.6 1.10 0 0 Cu(311) Exp. -11.9 1.8 1.09 Y Zr Nb Mo Tc Ru Rh Pd Ag Y Zr Nb Mo Tc Ru Rh Pd Ag Cu(511) Theory 1.68 -11.1 -16.4 8.4 0.70 DFT-LDA surface energies for the 4d transition metals (Methfessel et al. , 1992) Cu(511) Exp. -14.2 -5.2 5.2 0.69 Bond-cutting model: surface energy in tight-binding picture estimated by

bulk surf Nc Nc σ = − E′ , (118) bulk coh p Ncp bulk surf Nc and Nc coordinationp numbers, Ecoh′ cohesive energy

Semiconductor surfaces Semiconductor surfaces II

Surface energies Si(111)-(7 7) Surface electronic properties Si(100)-(2 2) Semiconductors are characterized by direc- × Further driving force for reconstructions: × tional covalent bonds semiconductor surfaces avoid to become metallic Upon cleavage unsaturated highly unstable bonds, the dangling bonds are created Example: Si(100)-(2 2) × Surface tries to lower its energy by minimi- After dimerization still two half-filled dan- zing the number of dangling gling bonds at the Si(100)-(2 1) surface × Most prominent example: Si(111)-(7 7) The two half-filled dangling bond lead to a × metallic surface state DAS model(Dimer-Adatom Stacking-fault): twelve top-layer adatoms, six rest atoms, Charge transfer: one dangling bond be- a stacking fault in one of the two triangular comes⇒ completely filled, the other one em- subunits of the second layer, pty, causing asymmetric buckled dimers nine dimers and a deep corner hole Si(100) (2 2) surface structure with alternating asym- Mechanical surface stress released by alter- × nating buckled dimers: (2 2) structure metric buckled dimers. × Compound semiconductor surfaces GaAs(100)

If the substrate consists of more than one element, then the surface energy depends on the GaAs(100) surface structures: Discussion specific thermodynamic conditions, i.e. the reservoir with which the atoms are exchanged in Filled circles: As atoms, (a) (1 1) (b) α(2 4) a structural transition. × × empty circles: Ga atoms Most stable surface structure determined by the minimum of the free energy. α(2 4) and β2(2 4): As-terminated, β2(4× 2): Ga-terminated× Zero temperature: × Ideal (1 1) structure highly unstable 1 because× of the dangling bonds γ = (E µ N ) . (119) A surf − i i i (c) β2(2 4) (d) β2(4 2) Electron-counting principle: X × × Anion dangling bonds filled and cation dangling bonds empty Surface energy of GaAs: written as a function of a single variable such as, e.g., µAs : Charge transfer from Ga to As 1 Filled⇒ p orbitals of As favor perpendicu- γ = [E µ N µ (N N )] . (120) 2 A surf − GaAs Ga − As As − Ga lar bonds, sp -like hybridization of Ga favors planar configuration

GaAs(100): Stoichiometry GaAs(100): Surface energies

GaAs(100) surface structures: Stoichiometry GaAs(100) surface energies: Discussion Stoichiometry ∆N = NAs NGa : (a) (1 1) (b) α(2 4) − 1 (1 1) As-terminated surface ∆N = 2 1,6 GaAs tries to minimize the number of dan- × × × 1 (1 1) Ga-terminated surf. ∆N = 2 gling bonds by dimerization

× − ) 1,4 Derivation symmetric slab 2 → Further driving force: electrostatics α(2 4) structure: ∆N = 0 1,2 β2(4x2) × 1 α(2x4) β2 structures minimize the electrostatic re- β2(2 4): ∆N = 4 per 1 1 unit cell × × 1,0 β2(2x4) pulsion β2(4 2): ∆N = 1 per 1 1 unit cell (c) β2(2 4) (d) β2(4 2) × −4 × surface energy (J/m 0,8 Planar Ga-terminated configuration energe- × × Stoichiometry and surface energies not c(4x4) tically more costly than uotward relaxation obvious, if the slab has two inequivalent 0,6 -0,6 -0,4 -0,2 0 of As-terminated structures surfaces as, e.g., the (111) and ( 1¯1¯1¯) µ µ(As bulk) Ga rich As - As (eV) As rich surfaces of zinc-blende-structures Ga-terminated β2(4 2) surface is only Surface energies of different GaAs(100) reconstructions in ⇒ × Energy density formulation (Chetty 2 stable under extrem Ga rich conditions ⇒ J/m as a function of the difference of the chemical potential and Martin) of As and bulk As (Moll et al , 1996). Insulator surfaces Polar ionic oxide surface: Al 2O3(0001)

NaCl structure Ionic surfaces α-Al 2O3(0001) surface important substrate for very high frequency microelectronic devices due to its insulating character NaCl structure: Two fcc sublattices trans- lated by a/ 2( ~e + ~e + ~e ) x y z Al 2O3(0001) surface structure: Al 2O3 Electrostatic potential outside of a slab α-Al O (sapphire) cystallizes in structure: 2 3 the corundum structure

2π z a 1: Al Rhombohedral unit cell: 2 Al O φ(z ) = φG e− + 2 πσ (121) → 2 3 → ∞ ⊥ 2: O 3 → formula units 3: Al 4: Al → σ : dipole moment perpendicular to the → Hexagonal unit cell: 12 Al atoms ⊥ 5: O plane per unit area 3 → and 18 O atoms 6: Al Polar surface highly unstable → O atoms stacked in slightly dis- Sodium chloride structure Nonpolar surface: electrostatic potential torted hcp structure falls oﬀ very rapidly, surface termination Al atoms occupy 2/3 of octahe- almost ideal Side and top view of the Al-terminated α-Al 2O3(0001) surface. dral holes in O sublattice

Al 2O3(0001): Surface energies Al 2O3(0001) surface relaxations

Interlayer- Theory a Theory b Theory c Theory d Exp. e Exp. f Al 2O3(0001) surface energies: Discussion GGA LDA LDA LDA (1 1) structures stable, no reconstructions Al-O 3 1-2 -86 -87 -85 -77 -51 +30 8 × O O3-Al 2-3 +6 +3 +3 +11 +16 +6 3 AlAl 3 Stoichiometric AlO 3Al-termination stable )

2 Al-Al 3-4 -49 -42 -45 -34 -29 -55 6 AlAlO over the entire range of oxygen chemical O AlAl potentials Al-O 3 4-5 +22 +19 +20 +19 +20 – AlAl 2 O 1 O -Al 5-6 +6 +6 – +1 – – 4 Non-stoichiometric surfaces metallic 3 AlO Al 3 Oxygen-terminated surface only by hydro- a Wang et al. (GGA), b Verdozzi et al. (LDA), c Di Felice et al. (LDA), e Guenard et al. , d Batyrev et al. , f Toofan et al. . 2 H surface energy (J/m 3 O genation stable 3 AlAl Significant first-layer contraction in order to minimize surface dipole 0 -6 -5 -4 -3 -2 -1 0 O Exp. f) Probably hydrogenated O-terminated surface µ µ 2 Al rich O - O (eV) O rich Surface energies of different Al O (0001) (1 1) structure in 2 3 × J/m 2 as a function of the difference of the oxygen chemical potential (Wang et al (2000), Batyrev et al. (1999)). Al 2O3(0001) surface reconstructions at higher Ultrathin Al 2O3(0001) films on NiAl(110) temperatures STM: conducting substrate needed Above T = 1350 K, oxygen evaporates from the top layers Deposite ultrathin oxide films on metal substrate Al O film ⇒ → 10 13 Complex Al-rich structures addressed with empirical potentials ⇒ hexagonal and square arrangement of Al atoms in top layer

(111)−like

(100)−like

(111)−like

Structure of ultrathin Al 2O3 on NiAl(110): STM and DFT calculations (3√3 3√3)r30 ◦ reconstructed α-Al 2O3(0001) surface obtained by simulations using empirical potentials × G. Kresse et al. , Science 308 , 1440 (2005) I. Vilfan et al. , Surf. Sci. 505 , L215 (2002) Strong Ni-Al interaction plus hexagonal and square Al arrangement in top layer

Scanning tunneling microscope (STM) Interpretation of STM pictures

Principle Animation Graphite(0001) Discussion

STM constant height modus: Magnitude of tunneling current is monitored

Atoms are not directly imaged, but current distribution⇒

Graphite: Honeycomb net not reproduced in STM pictures, every second atom is missing

Electronic structure calculations are needed in⇒ order to understand STM pictures

HOPG: Highly Oriented Puriﬁed Graphite

Michael Schmid, TU Wien Michael Schmid, TU Wien Tersoﬀ-Hamann picture Perturbation approach

Tunneling current Simulated STM picture Also known as Bardeen approach (Bardeen, 1961) Tunneling current Simulated STM picture 2πe I = f(Eµ)[1 f(Eν)] ~ − Sample and tip treated as separate entities Maleic anhydride (C 4H2O3) on Si(100) µν X 2 Mµν δ(Eµ (Eν + eV )) (122) ×| | − 4πe 2 I = χ∗ ψ ψ χ∗ ~ µ∇ ν − ν∇ µ Tersoﬀ-Hamann ( ~r t; tip position): µν ZS X

δ(Eµ (Eν + eV )) (124) I ψ(~r ) 2δ(E E ) = ρ(~r , E ) × − ∝ | t | µ − ν t F ν χ : eigenstates of the STM tip X (123) µ Graphite(0001) (Courtesy of T. Markert) ψν: eigenstates of the surface

S: separation surface between tip and surface W. Hofer et al. , Chem. Phys. Lett. 355 , 3437 (2002). Tunneling current proportional to overlap of wave functions ⇒ Simulation necessary for interpretation of STM picture