Modelling Surface Adrian Wander Computational Materials Group CCLRC Daresbury Laboratory

1. Modelling Surface Adrian Wander Computational Materials Group CCLRC Daresbury Laboratory Daresbury, Warrington, Cheshire, WA4 4AD, UK MSSC2004

2. Overview • Why Surfaces? • Surface Notation - Miller Indices • Slabs and Supercells • Surface Terminations • Convergence • Unit Cells - Matrix and Wood Notation • Stoichiometry • Classification of Ionic Surface Structures • Stability of Polar Oxide Surfaces MSSC2004

3. Why Surfaces? • Surfaces dominate many activities: • Heterogeneous Catalysis • Gas Sensors • Corrosion • Adhesion • Lubrication • Nano-technology MSSC2004

4. Miller Indices • Consider a cubic lattice: • Construct required plane • Take reciprocals of intercepts on axes • Multiply to give whole numbers • Shaded plane intercepts at • 1, infinity, infinity • Reciprocals give 1,0,0 • Plane is (100) MSSC2004

5. Miller Indices II The (110) plane (left) and (111) plane (below) MSSC2004

6. Miller Indices III Many metals occur in the face centred cubic lattice The (110) plane is shown above. The (110) surface is produced by cutting along the (110) plane. The result is shown right MSSC2004

7. Lattice and Basis The MgO Structure is illustrated (left) A single layer of MgO in the (110) direction contains two inequivalent atoms MSSC2004

8. Terminations The Miller indices do not fully define the surface structure Both images show the MgO(111) surface MSSC2004

9. Slabs Surfaces represent semi-infinite systems Not practical - surfaces in CRYSTAL are represented as slabs of material Periodic boundary conditions are applied in the plane of the material (Bloch Condition). Perpendicular to the slab the BC states that the wave function is zero at infinity MSSC2004

10. Supercells Not the only approach Plane wave codes use periodicity in all three dimensions Surfaces are represented using supercells MSSC2004

11. Slabs and Supercells A 4 layer MgO(100) Slab MSSC2004

12. Running Calculations We are now in a position to run a surface calculation using CRYSTAL 1. Construct bulk unit cell 2. Specify Miller indices 3. Specify termination 4. Specify slab thickness Don’t forget - you will need to optimise the bulk structure before studying surfaces MSSC2004

13. CRYSTAL Input MGO SURFACE CRYSTAL 0 0 0 225 4.23134 2 12 0. 0. 0. 8 0.5 0.5 0.5 SLAB 1 0 0 1 8 Standard bulk definition Surface construction MSSC2004

14. Properties The most basic property is the surface energy For a slab containing m formulae units the surface energy is defined as Esurf = (Eslab - mEbulk)/2A where A is the area of the surface unit cell NB: Usually surface energies are expressed in joules per square metre - numbers from CRYSTAL will provide hartrees per square Bohr MSSC2004

15. How Good is that Calc? MgO(100) Surface Energy 1 layer slab 0.007745 2 layer slab 0.007586 3 layer slab 0.007589 Charges Mg O Bulk 10.09114 9.90885 3 layer 10.09270 9.90836 4 layer 10.09191 9.90861 5 layer 10.09112 9.90886 MSSC2004

16. Supercells and Slabs • For a slab calculation we have to worry about • Convergence wrt slab thickness • For a supercell we have the same problem + • Convergence wrt vacuum gap Need to have a thick enough slab to converge the property of interest MSSC2004

17. Supercells (again?!) Many adsorbates do not completely cover the surface - how do we define these? MSSC2004

18. Supercell Notation The primitive lattice is defined by two vectors a1 = A11x + A12y a2 = A21x + A22y which defines a matrix A. Similarly the overlayer is defined by a new set of vectors b1 = B11x + B12y b2 = B21x + B22y and an associated matrix B The surface is defined by a matrix M = BA-1 MSSC2004

19. Matrix Notation The full notation for the surface is S(hkl) - M - A where hkl are the miller indices of a substrate of composition S M is the matrix just defined, and the overlayer contains  adspecies of type A MSSC2004

20. a2 b2 a1 b1 Example MSSC2004

21. Properties of M • General • Can describe all surface structure • M defines the lattice of the overlayer relative to the substrate • detM is the ratio of the overlayer unit cellarea to the substrate area • Therefore /detM gives the coverage CRYSTAL accepts the supercell (matrix) notation for surfaces…. MSSC2004

22. Wood’s Notation Unfortunately surface scientists don’t! MSSC2004

23. 1 1 1 -1 Matrix Most Common Notation M(100) - c(2x2) - A Example Relative lengths 2 M(100) - (2 x 2)R45o - A MSSC2004

24. FCC(111) Surfaces There is no Wood’s Notation for the primitive unit cell M(111) - c(4 x 2) - A M(111) - (2 x 3)rect - A MSSC2004

25. Stoichiometry This slab is symmetric… but non-stoichiometric (Mg3O4) How do we define a surface energy? Esurf = (Eslab - mEbulk)/2A (Actually we can do this - we see how in the next lecture!) MSSC2004

26. Surfaces of Ionic Materials Type I Surface: Each plane is electrically neutral e.g. MgO(100) MSSC2004

27. Type II Surfaces M2O3 Corundum type structures The (0001) Surface is type II: The layers are not neutral but the repeat unit has zero net dipole moment MSSC2004

28. Type III Surfaces MgO(111) Stoichiometric slab Type III Surfaces: Layers are not electrically neutral and the repeat unit has a net dipole moment MSSC2004

29. Stability I The Octopolar Reconstruction MSSC2004

30. ZnO Polar Surface (0001) Cleavage energy ~ 5 Jm-2comparable to the non-polar surface MSSC2004

31. Surface Stability II -0.2 |e| The polar surfaces are stabilised by charge transfer creating ‘metallic’ surface states MSSC2004

32. Surface Stability III Dissociative adsorption of H2O leads to a symmetric slab But that’s Surface Chemistry which is the next lecture MSSC2004