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Theoretical Methods for Surface Science part II

Theoretical Methods for Surface Science part II. Johan M. Carlsson Theory Department Fritz-Haber-Institut der Max-Planck-Gesellschaft Faradayweg 4-6, 14195 Berlin. Summary. Last lecture: The foundations of the DFT How to calculate bulk properties and electronic structure

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Theoretical Methods for Surface Science part II

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

  2. Summary Last lecture: The foundations of the DFT How to calculate bulk properties and electronic structure How to model surfaces Surface structures This lecture: Electronic structure at surfaces Adsorption

  3. All-electron LCGO DFT-calculations for Cu(111)-surface. Euceda et al., PRB 28,528 (1983) Charge distribution at Surfaces electrons spill out from the surface Jellium model Lang and Kohn, PRB 1,4555(1970)

  4. Work function Work function F surface dipole d + d - Potential difference Df=f ()-f (-)=4pd Jellium model Lang and Kohn, PRB 1,4555(1970)

  5. Work function Work function F Chemical potential of the electrons m=E(N+1)-E(N)=EF Work function F=f ()-m = Df-m Potential difference Df=f ()-f (-)=4pd Lang and Kohn, PRB 1,4555(1970)

  6. Band gap opens at the zone boundaries V0 2Vg The energies and wave functions: Nearly Free electron model (NFE) Periodic potential V(z) = -Vo+2Vgcos(gz)

  7. Surface states The solution for imaginary values of k is also possible at the surface:

  8. Surface states Matching the two solutions at a/2 leads to a Schockley surface state. *This state has a large amplitude in the surface region, but decay rapidly into the bulk and into the vacuum region. *Its energy is located in the band gap. Schockley, Phys. Rev. 56, 317, (1939)

  9. 2x2 1x1 DFT bandstructure for Cu(111)

  10. Bandstructure of Cu(111) 6-layer slab 18-layer slab Euceda et al., PRB 28,528 (1983)

  11. kz k kx k k Projected Bulk bandstructures Bertel, Surf. Sci. 331, 1136 (1995) There is a range of k-vectors with a k-component along the perpendicular rod for each k-point in the surface plane.

  12. kz k kx k k Projected Bulk bandstructures Calculate the bands along the perpendicular rod. The values between the lowest and highest values correspond to regions of bulk states. Surface states can occur outside the bulk regions.

  13. Surface BZ M Schockley surface state G K Tamm state Bandstructure of Cu(111)

  14. Adsorption

  15. Adsorption Energy Activation barrier Ediss z Physisorption well Eads Chemisorption well

  16. Thermodynamics for adsorption a ma Host Definition of adsorbate energy: Eads=DG=G[host+ads]-{G[host]+Nama} where G(T,p)= E-TS + pV=F+pV Ftrans, Frot, pV negligible for solids, but not in the gas phase The adsorbates vibrate at the surface: Fvib(T,w)=Evib (T,w)-TSvib (T,w) This gives the adsorption energy Eads={E[host+defect]+Fvib(T,w)}-{E[host]+Nama}

  17. Thermodynamics for adsorption Convert the energy values of the chemical potential into T and p-dependence of the gas phase reservoir mi(T,pi)=mDFT+DG(T,p0)+ kTln(pi /p0) InterpolateDG(T,p0) from tables. Reuter and Scheffler, PRB 65, 035406 (2002). Eads(T,p)={E[host+defect]+Fvib(T)}-{E[host]+ma(T,pa)} The adsorbate concentration can be estimated in the dilute limit C=N exp(-Eads/kT) where N is the number of adsorbtion sites

  18. Phase diagram Reuter and Scheffler, PRB 68, 045407 (2003)

  19. metal r’ r z z + - - + Taylor expand in terms of 1/z: Physisorption The electrostatic energy:

  20. van der Waals interaction Cohesive energy for graphite as function of a- and c-lattice parameters. Calculated with GGA XC-functional Rydberg et al., Surf. Sci. 532, 606 (2003).

  21. Physisorption of O2 on graphite h=3.4 Å DFT-GGA: Eads=0.04 eV/O2 TPD-experiment: Eads=0.12 eV/O2 Ulbricht et al.,PRB 66, 075404 (2002)

  22. Chemisorption

  23. Adsorption sites Top site Bridge site Hollow FCC-site Hollow HCP-site T B B F H H F T Close packed (111)-surface

  24. Finding the adsorption site Adsorption system with a barrier: Locate the transition state at the barrier Need to start the atomic relaxation inside the barrier Adsorption without a barrier: Non-activated adsorption: can start the atomic relaxation anywhere Calculation the Potential Energy Surface (PES) barrier chemisorption sites

  25. Potential energy surface O2 on Pt(111), Gross et al., Surf. Sci., 539, L542 (2003).

  26. Newns-Anderson model Anderson, Phys. Rev. 124, 41 (1961) Newns, Phys. Rev. 178, 1123 (1969) Consider an adsorbate atom with a valence level |a > interacting with a metal which has a continuum of states | k >. where is the overlap interaction between the adsorbate atom and the substrate levels | k >. e k | a >

  27. Green’s function techniques The Green’s function Gs(e) is the solution to the equation The Green’s function describe the response of the system to a perturbation and poles gives the excitation energies.

  28. The self energy describes the interactions in the system The real part L(e) leads to a shift of the energy eigenvalues, the imaginary part D(e) gives a broadening Green’s function techniques The imaginary part of the Green’s function is called the spectral function it is equivalent to the projected density of states.

  29. as and identify the self-energy components: Newns-Anderson model continued Calculate the Green’s function for the Hamiltonian

  30. e sp-band Weak chemisorption limit If the interaction between the substrate and the adsorbate is weak, i.e. Vak is small compared to the bandwidth of the substrate band. Ex for a sp-band. D is then independent of energy which means that L =0. The projected density of states for the adsorbate atom is then a Lorentzian with a width D, centered around ea | a > D

  31. Strong chemisorption limit When the adsorbate interacts with a narrow d-band, then the ek can be approximated by center value ec such that the denominator in the Green’s function becomes: Solving this equation gives two roots corresponding to bonding and anti bonding levels of the absorbate system. e | a > d-band

  32. Charge transfer • Gurney suggested that the atomic levels of a adsorbate atom would broaden and that there would be a charge transfer between the substrate and the adsorbate atom. • Charge would be donated to the substrate if the atom has low ionization energy and • charge would be attracted from the substrate if the atom has a high ionization energy. Gurney, Phys Rev. 47, 479 (1933)

  33. Chemisorption on a metal surface Na/Cu(111)

  34. Adsorbate induced work function change DF[eV] Tang et al., Surf. Sci. Lett. 255, L497 (1991).

  35. charge depletion charge accumulation - + adsorbate induced dipole Charge transfer for Na/Cu(111)

  36. Properties for Na/Cu(111)

  37. Quantum well state for Na/Cu(111) Carlsson and Hellsing, PRB 61, 13973 (2000)

  38. Tasker’s rules(J. Phys. C 12, 4977 (1977)) Surface types in ionic crystals Type I Crystals with neutral planes parallel to the surface ex MgO{100}-surfaces Type II charged planes where the repeat unit is neutral Layered materials with stacking -1 +2 -1 -1 +2 ... Type III charged planes leading to a net dipole moment ex MgO{111}-surfaces Type III is unstable unless surface charges set up an opposing surface dipole which quench the internal dipole moment.

  39. Qs=aQ1, where Harding’s compensating surface charge Qs(Surf. Sci. 422, 87 (1999))

  40. Ex: Properties of ZnO • Ground state structure for ZnO: Wurtzite structure • High pressure structure: Rock salt

  41. Electronic structure of ZnO EgapExp=3.4 eV EgapDFT-GGA=0.8eV Under estimation of the bandgap in semi-conductors is a common problem in DFT-calculations with LDA or GGA exchange-correlation functional.

  42. O-terminated [0001]-surface The polar ZnO{0001}-surface Zn-terminated [0001]-surface [0001]

  43. The polar ZnO{0001}-surface B A Carlsson, Comp. Mat. Sci. 22, 24 (2001)

  44. The polar ZnO{0001}-surface Carlsson, Comp. Mat. Sci. 22, 24 (2001)

  45. STM of ZnO[0001]-surfaceDulub et al.,PRL 90, 016102 (2003) Triangular islands Step height=2.7 Å=c/2 n=O-edge atoms b) Triangle # of O-atoms = n(n+1)/2 # of Zn-atoms =n(n-1)/2 Q=#Zn / #O =3/4 => n=7 L = (n-2)*a = 16.25 Å c) Triangle with internal triangle # of O-atoms = 3n(n+1)/2-3 # of Zn-atoms = 3n(n-1)/2 Q=#Zn / #O =3/4 => n=6 L = (2(n-1)-1)*a = 29.25 Å

  46. Surface Phase diagram of ZnO[0001] Kresse et al., PRB 68, 245409 (2003)

  47. Summary • Surface energy • Atomic structure relaxation • Charge redistribution • Work function • Surface states • Adsorption

  48. Literature Review article about DFT implementations: Payne et al., Rev. Mod. Phys. 64, 1045 (1992). A. Zangwill, Physics at Surfaces, Cambridge University Press A. Gross, Theoretical Surface Science A microscopic perspective, Springer Verlag F. Bechstedt, Principles of Surface Physics, Springer Verlag

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