Theoretical Methods for Surface Science part I

1. Theoretical Methods for Surface Sciencepart I Johan M. Carlsson Theory Department Fritz-Haber-Institut der Max-Planck-Gesellschaft Faradayweg 4-6, 14195 Berlin

2. Bulk

3. Surfaces A I II The surface break the 3D-periodicity of the bulk crystal Total energy of the system: GI+II=GI+GII+DGsurface

4. Surface effects • Surface energy • Atomic structure relaxation • Charge redistribution • Work function • Surface states • Adsorption Lang and Kohn, PRB 1, 4555 (1970)

5. L[m] 1 10-3 10-6 macroscopic regime 10-9 mesoscopic regime t[s] 1 10-12 10-9 10-6 10-3 microscopic regime Real world problems are complex EkaNobel, Bohus, Sweden

6.  Computational surface science Construction of models Experimental surface science Surface Science methods AB+C=>AC+B

7. L[m] 1 macroscopic regime 10-3 mesoscopic regime 10-6 10-9 electronic structure t[s] 10-12 10-9 10-6 10-3 1 The Multi scale approach Thermodynamics Classical mechanics kinetic Monte Carlo simulations Electron structure methods It is necessary to combine different methods in order to tackle realistic problems

8. Total energy methods Simple empirical potentials - Force fields, pair potentials… Intermediate methods - Tight-binding, - many-body potentials: EAM ab-initio techniques - Hartree-Fock - Density functional theory (DFT) Beyond DFT: - GW - Quantum Monte Carlo - Quantum Chemical: CI Accuracy Accuracy Number of atoms treated

9. 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.

10. Hohenberg-Kohn theorem: Phys. Rev. 136, B864 (1964) The total energy of the electron system is a functional of the electron density n(r): Hohenberg-Kohn theorem Hamiltonian for a many-electron system: Variational principle: < Y | H |Y >  < Y0 | H |Y0>= E0

11. 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: Kohn-Sham equations Phys. Rev. 140, A1133 (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

12. 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 self-consistently. 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+ EeI[n(r)]+{ Ekin[n(r)]+ EH[n(r)]+ Exc[n(r)]}

13. The self-consistent scheme Payne et al., Rev. Mod. Phys. 64, 1045 (1992).

14. 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.

15. ky kx 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). IBZ 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. Using symmetry lowers the number of necessary k-point to the ones in the Irreducible Brillouin zone (IBZ).

16. 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:

17. Basis sets Two common basis sets are: • Plane waves: • + Complete basis set • + Systematic way of improving the accuracy • Many plane wave are needed to accurately describe localized wave functions • Periodic boundary conditions necessary Localized orbitals: +Only a few basis functions needed per atom + Hamiltonian matrix is sparse + Periodic boundary conditions not necessary - No systematic way of improving accuracy

18. 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 x valence electrons core electrons

19. 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 ion-electron interaction, since all corrugation is removed. Lang and Kohn, PRB 1,4555(1970)

20. 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. Hamann et al., PRL 43, 1494 (1979)

21. III. Norm conservation: IV. Maintain scattering properties: Hamann et al., PRL 43, 1494 (1979) 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: • Vps=VAE, r>rc II. eips= eiAE

22. 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

23. a0=equilibrium lattice parameter and V0 equilibrium volume Ecoh=cohesive energy B=Bulk modulus= V d2E dV2 Bulk properties The bulk properties can be determined using the Murnaghan equation of state: E0=total energy at equilibrium lattice constant, B0=Bulk modulus, B1=first derivative of B0 with respect to pressure Murnhagan, Proc. Nat. Acad. Sci. USA 30, 244 (1944)

24. x x x x x x x 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: Unit cell for Cu

25. Free electron model: Band structure The dispersion relation between the wave vector and the energy eigenvalues In general are the eigenvalues a complicated function of k: E(k)=f(k)

26. Bouckaert et al., Phys. Rev 50, 58 (1938). How to Calculate DFT Band structure I. Solve the Kohn-Sham equations self-consistently to determine the effective potential using an even k-point sampling. II. Use the effective potential while solving the Kohn-Sham equations non self-consistently along high symmetry lines in the Brillouin zone

27. Example: Band structure of Cu Bouckaert et al., Phys. Rev 50, 58 (1938). Cu has FCC structure. 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

28. Bouckaert et al., Phys. Rev 50, 58 (1938). Band structure of Cu Electronic configuration of Cu: 3d94s2

29. 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.

30. 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.

31. M G K DOS for Graphene Brillouin Zone e-eF [eV] DOS K G M K G e-eF [eV] k[Å-1] van Hove singularities

32. M G K PDOS for Graphene Brillouin Zone px,py PDOS s DOS pz e-eF [eV] e-eF [eV]

33. Surfaces

34. A g= 1 (GI+II(T,p)-SiNimi]) g= 1 (Esurf -Ebulk) A A Surface energy I II Gibbs free energy: G(T,p) = E-TS + pV=SjNjmj where the chemical potential is defined Surface energy g = Energy cost to create a surfaces Solids (low T): G(T,p) ~ G(0,0) ~ Etot

35. 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)

36. 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!

37. Healy et al, PRL 87, 016105 (2001) 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 Cluster geometries: + very cheap for small clusters (local basis sets) + ideal for local aspects (defects etc.)  slow convergence with cluster size (embedding etc.) Payne et al., Rev. Mod. Phys. 64, 1045 (1992).

38. vacuum ts slab 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. The vacuum region should be thick enough that the two surfaces do not interact with each other through the vacuum region.

39. Quantum size effects The electronic states in the slab are quantized perpendicular to the surface. Boettger, PRB 53, 13133 (1996)

40. 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:

41. 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!

42. Surface relaxations at metal surfaces Smoluchowski smoothing at metal surfaces, Finnis and Heine, J. Phys. Chem. B 105, L37 (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.

43. All-electron LCGO DFT-calculations for Cu(111)-surface. Euceda et al., PRB 28,528 (1983) Surface relaxation of Cu surfaces the charge density is smoothened at the surface Gross, Theoretical Surface Science

44. Surface relaxation at semiconductor surfaces Basic principle: The observed surface structure has the lowest free-energy 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.

45. GaAs GaAs is a compound material: Remove the chemical potential for Ga and express the surface energy as function of As and GaAs The limits for chemical potential of As is given by

46. Surface reconstruction of GaAs(100) Moll et al., PRB 54, 8844 (1996).

47. 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