Theoretical Methods for Surface Science part I Johan

Theoretical Methods for Surface Science part I Johan M. Carlsson Theory Department Fritz-Haber-Institut der Max-Planck-Gesellschaft Faradayweg 4 -6, 14195 Berlin International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 1

Bulk International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 2

Surfaces A I II The surface break the 3 D-periodicity of the bulk crystal Total energy of the system: GI+II=GI+GII+DGsurface International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 3

Surface effects • Surface energy • Atomic structure relaxation • Charge redistribution • Work function • Surface states • Adsorption Lang and Kohn, PRB 1, 4555 (1970) International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 4

Real world problems are complex L[m] 1 10 -3 10 -6 macroscopic regime Eka. Nobel, Bohus, Sweden mesoscopic regime 10 -9 microscopic regime 10 -12 10 -9 10 -6 10 -3 1 International Max-Planck Research School t[s] Theoretical Methods for Surface Science Part I Slide 5

Surface Science methods AB+C=>AC+B Experimental surface science International Max-Planck Research School Computational surface science Construction of models Theoretical Methods for Surface Science Part I Slide 6

The Multi scale approach L[m] 1 macroscopic regime 10 -3 10 -6 mesoscopic regime Classical mechanics kinetic Monte Carlo simulations 10 -9 electronic structure 10 -12 10 -9 10 -6 10 -3 Thermodynamics Electron structure methods 1 t[s] It is necessary to combine different methods in order to tackle realistic problems International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 7

Total energy methods ab-initio techniques - Hartree-Fock - Density functional theory (DFT) Accuracy Intermediate methods - Tight-binding, - many-body potentials: EAM Accuracy Number of atoms treated Simple empirical potentials - Force fields, pair potentials… Beyond DFT: - GW - Quantum Monte Carlo - Quantum Chemical: CI International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 8

Density Functional Theory (DFT) DFT is nowadays an established method DFT is capable of treating a few hundred atoms with very good accuracy DFT-Properties: • Charge density • Total energy • Forces • Structure determination • Phonons • Electronic structure Walter Kohn received the Nobel prize in 1998 for the development of DFT. International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 9

Hohenberg-Kohn theorem Hamiltonian for a many-electron system: Variational principle: < Y | H |Y > < Y 0 | H |Y 0>= E 0 Hohenberg-Kohn theorem: Phys. Rev. 136, B 864 (1964) The total energy of the electron system is a functional of the electron density n(r): International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 10

Kohn-Sham equations Phys. Rev. 140, A 1133 (1965) Minimize the total energy with the constraint to conserve the number of electrons N: N= n(r) dr where the electron density con be calculated from This gives the Lagrange equation This equation can be identified with a Schrödinger like equation =Kohn-Sham equation for non-interacting electrons in an effective potential: International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 11

Kohn-Sham equations The effective potential contains three contributions: The electron density appears in the effective potential which means that the Kohn-Sham equations needs to be solved selfconsistently. This means that the total energy of the electron system can be obtained by solving the Kohn-Sham equations. Add the ion-ion interaction EII to get the full total energy E[n(r)]=EII+ Ee. I[n(r)]+{ Ekin[n(r)]+ EH[n(r)]+ Exc[n(r)]} International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 12

The self-consistent scheme Payne et al. , Rev. Mod. Phys. 64, 1045 (1992). International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 13

Exchange-Correlation functionals Local density approximation (LDA): Assume that the exchange-correlation is the same as the value for a homogeneous electron gas with the same density. Generalized Gradient Approximation (GGA): Take also the density variations into account by defining the exchange-correlation as a function of both the density and its gradients. International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 14

K-point sampling Bloch’s theorem states that the wave function in a periodic crystal can be described as: where the wave vector k is located in the first Brillouin zone (BZ). It is therefore necessary to sample the wave function at multiple k-points in BZ to get a correct description of the electron density and effective potential. ky IBZ kx Using symmetry lowers the number of necessary k-point to the ones in the Irreducible Brillouin zone (IBZ). International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 15

Basis set The wave functions are fourier expanded in a basis set. Ex: such that the Kohn-Sham equations are transformed from a set of differential equations into a set of algebraic equations. Ex: International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 16

Basis sets Two common basis sets are: Plane waves: Localized orbitals: + Complete basis set +Only a few basis functions needed per atom + Systematic way of improving the accuracy -Many plane wave are needed to accurately describe localized wave functions -Periodic boundary conditions necessary International Max-Planck Research School + Hamiltonian matrix is sparse + Periodic boundary conditions not necessary - No systematic way of improving accuracy Theoretical Methods for Surface Science Part I Slide 17

Ion-electron interaction All-electron, full potential method: The true Coulomb potential from the ions is used and all electrons are treated explicitly. +All electrons are treated on the same footing + Very accurate - Very expensive E Ri Rj valence electrons x core electrons International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 18

Jellium model Smear out the potential from the ions as a constant positive background. +Very easy to treat mathematically +Can anyway give qualitative results -Can at most give a crude description of the ionelectron interaction, since all corrugation is removed. International Max-Planck Research School Lang and Kohn, PRB 1, 4555(1970) Theoretical Methods for Surface Science Part I Slide 19

Pseudo potential method Remove the core electrons and replace the ion potential by a smooth pseudo potential +Much cheaper than the Full potential method, since only the valence electrons are treated explicitly, but much more accurate the jellium model. -The interaction between the core and valence electrons is treated statically, since the core electrons are frozen into the pseudo potential. International Max-Planck Research School Hamann et al. , PRL 43, 1494 (1979) Theoretical Methods for Surface Science Part I Slide 20

Ab-initio Pseudo potentials Start with an all-electron atom calculation Immitate the effective potential felt by the valence electrons by screening the potential from the ion nucleus by the core electrons Hamann et al. proposed four constraints: I. II. Vps=VAE, r>rc eips= ei. AE III. Norm conservation: IV. Maintain scattering properties: Hamann et al. , PRL 43, 1494 (1979) International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 21

Modeling your system Build your supercell Check for convergence of basis set and k-point sampling Calculate the bulk properties using the Murnaghan equation of state. Calculate the electronic structure, Density of states(DOS) and bandstructure International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 22

Bulk properties a 0=equilibrium lattice parameter and V 0 equilibrium volume Ecoh=cohesive energy B=Bulk modulus= V d 2 E d. V 2 The bulk properties can be determined using the Murnaghan equation of state: E 0=total energy at equilibrium lattice constant, B 0=Bulk modulus, B 1=first derivative of B 0 with respect to pressure Murnhagan, Proc. Nat. Acad. Sci. USA 30, 244 (1944) International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 23

Calculating Bulk properties Vary the lattice parameter and calculate the total energy. Make a curve fit of the total energy values to the Murnaghan equation of state: x x x Unit cell for Cu x International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 24

Band structure The dispersion relation between the wave vector and the energy eigenvalues Free electron model: In general are the eigenvalues a complicated function of k: E(k)=f(k) International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 25

How to Calculate DFT Band structure I. Solve the Kohn-Sham equations self-consistently to determine the effective potential using an even kpoint sampling. II. Use the effective potential while solving the Kohn-Sham equations non self-consistently along high symmetry lines in the Brillouin zone Bouckaert et al. , Phys. Rev 50, 58 (1938). International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 26

Example: Band structure of Cu Cu has FCC structure. Bouckaert et al. , Phys. Rev 50, 58 (1938). High symmetry points in the Brillouin zone: G=center of the Brillouin zone L=mid point on the zone boundary plane in the {111}-directions W=corner point on the hexagon of the {kikj}-plane K=mid point on the edge between two hexagons {110}-direction X= mid point on the zone boundary plane in the {100}-direction International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 27

Band structure of Cu Electronic configuration of Cu: 3 d 94 s 2 Bouckaert et al. , Phys. Rev 50, 58 (1938). International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 28

Density of states (DOS) Method: Calculate the Kohn-Sham eigenvalues with a very dense k-point mesh. Use a Gaussian or Lorentzian broadening function for the delta function. Perform the summation of the states over the Brillouin zone. International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 29

Projected density of states (PDOS) Method: Calculate the Kohn-Sham eigenvalues ei and wave functions yi. Calculate the overlap between the Kohn-Sham wave functions yi and atomic wave functions fal Use a Gaussian or Lorentzian broadening function for the delta function. International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 30

DOS for Graphene Brillouin Zone K DOS e-e. F [e. V] G M K G M k[Å-1] K G van Hove singularities International Max-Planck Research School e-e. F [e. V] Theoretical Methods for Surface Science Part I Slide 31

PDOS for Graphene Brillouin Zone G M K DOS PDOS px, py s pz e-e. F [e. V] International Max-Planck Research School e-e. F [e. V] Theoretical Methods for Surface Science Part I Slide 32

Surfaces International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 33

Surface energy A I II Gibbs free energy: G(T, p) = E-TS + p. V= Sj. Njmj where the chemical potential is defined Surface energy g = Energy cost to create a surfaces g= 1 (GI+II(T, p)-Si. Nimi]) A Solids (low T): G(T, p) ~ G(0, 0) ~ Etot g= 1 (Esurf -Ebulk) A International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 34

Modeling Surfaces The jellium model Fridell oscillations in the electron density near the surface electrons spill out from the surface Lang and Kohn, PRB 1, 4555(1970) International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 35

Modeling Surfaces Lang and Kohn, PRB 1, 4555(1970) The surface energy diverges for metals with high electron density when the Jellium model is used! International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 36

Modeling Surfaces Supercell geometries: + proper surface electronic structure + good convergence with slab thickness + suitable for plane wave basis sets artificial lateral periodicity: “ordered arrays” inherently expensive Payne et al. , Rev. Mod. Phys. 64, 1045 (1992). Cluster geometries: + very cheap for small clusters (local basis sets) + ideal for local aspects (defects etc. ) slow convergence with cluster size (embedding etc. ) International Max-Planck Research School Healy et al, PRL 87, 016105 (2001) Theoretical Methods for Surface Science Part I Slide 37

Convergence of slab models The slab should be thick enough that the middle layers obtain bulk properties and that the two surfaces do not interact with each other through the slab. vacuum ts slab The vacuum region should be thick enough that the two surfaces do not interact with each other through the vacuum region. International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 38

Quantum size effects The electronic states in the slab are quantized perpendicular to the surface. Boettger, PRB 53, 13133 (1996) International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 39

Atomic Relaxation It is necessary to relax the forces on the atoms in order to find the lowest energy ground state of the crystal. Calculate the forces on the atoms: The ions are so heavy that they can be considered classical Move the atoms according to the discretized version of Newton’s second law: International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 40

Atomic Relaxation To get a rapid convergence it is necessary to have a good choice of the step length. Local minima Global minima However, the system might get trapped in a local minima, so it is sometimes necessary check different reconstructions and compare the surface energies! International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 41

Surface relaxations at metal surfaces Smoluchowski smoothing at metal surfaces, Finnis and Heine, J. Phys. Chem. B 105, L 37 (1973) The charge density will be redistributed at the surface such the charge is moved from the regions directly above the atom cores to the regions between the atoms. The atoms in the surface layer experience a charge imbalance. This give rise to an inward electrostatic force which leads to a compression of the separation between the surface layers. International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 42

Surface relaxation of Cu surfaces the charge density is smoothened at the surface All-electron LCGO DFT-calculations for Cu(111)-surface. Euceda et al. , PRB 28, 528 (1983) International Max-Planck Research School Gross, Theoretical Surface Science Theoretical Methods for Surface Science Part I Slide 43

Surface relaxation at semiconductor surfaces Basic principle: The observed surface structure has the lowest freeenergy among the kinetically accessible structures under the paricular preparation conditions. Principle 1: A surface tend to minimize the number of dangling bonds by the formation of new bonds. The remaining dangling bonds tend to be saturated. Principle 2: A surface tend to compensate charges. Principle 3: A semiconductor surface tend to be insulating. International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 44

Ga. As is a compound material: Remove the chemical potential for Ga and express the surface energy as function of As and Ga. As The limits for chemical potential of As is given by International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 45

Surface reconstruction of Ga. As(100) Moll et al. , PRB 54, 8844 (1996). International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 46

Summary The foundations of the DFT How to calculate bulk properties and electronic structure How to model surfaces Surface structures Next lecture: Electronic structure at surfaces Adsorption International Max-Planck Research School Theoretical Methods for Surface Science Part I Slide 47