DFT – Practice Surface Science [based on Chapter 4, Sholl & Steckel]

1. DFT – PracticeSurface Science[based on Chapter 4, Sholl & Steckel] Recap Supercells for surfaces Surface relaxation, Surface energy and Surface reconstruction More advanced topics Also see following article:

2. Required input in typical DFT calculations VASP input files • Initial guesses for the unit cell vectors (a1, a2, a3) and positions of all atoms (R1, R2, …, RM) • k-point mesh to “sample” the Brillouin zone • Pseudopotential for each atom type • Basis function information (e.g., plane wave cut-off energy, Ecut) • Level of theory (e.g., LDA, GGA, etc.) • Other details (e.g., type of optimization and algorithms, precision, whether spins have to be explicitly treated, etc.) POSCAR KPOINTS POTCAR INCAR

3. The DFT prescription for the total energy(including geometry optimization) Guess ψik(r) for all the electrons Self-consistent field (SCF) loop Solve! No Is new n(r) close to old n(r) ? Yes Geometry optimization loop Calculate total energy E(a1,a2,a3,R1,R2,…RM) = Eelec(n(r); {a1,a2,a3,R1,R2,…RM}) + Enucl Calculate forces on each atom, and stress in unit cell Move atoms; change unit cell shape/size No Are forces and stresses zero? Yes DONE!!!

4. Approximations Approximation 1: finite number of k-points Approximation 2: representation of wave functions Approximation 3: pseudopotentials Approximation 4: exchange-correlation

5. The general “supercell” • Initial geometry specified by the periodically repeating unit  “Supercell”, specified by 3 vectors {a1, a2, a3} • Each supercell vector specified by 3 numbers • Atoms within the supercell specified by coordinates {R1, R2, …, RM} a3 = a3xi + a3yj + a3zk a1 = a1xi + a1yj + a1zk a2 = a2xi + a2yj + a2zk

6. More on supercells (in 2-d) Primitive cell Wigner-Seitz cell

7. The simple cubic “supercell” • Applicable to real simple cubic systems, and molecules • May be specified in terms of the lattice parameter a a3 = ak a1 = ai a2 = aj

8. The FCC “supercell” • The primitive lattice vectors are not orthogonal • In the case of simple metallic systems, e.g., Cu  only one atom per primitive unit cell • Again, in terms of the lattice parameter a a3 = 0.5(i + k) a2 = 0.5a(j + k) a1 = 0.5a(i + j)

9. Supercell for surface calculations • (001) slab • Note: periodicity along x, y, and z directions • Two (001) surfaces • Vacuum and slab thicknesses have to be large enough to minimize interaction between 2 adjacent surfaces

10. Side view of slab supercell Slab Supercell Vacuum

11. Yet another view

12. Atomic coordinates in supercell • The atomic positions, in terms of “fractional coordinates”, i.e., in the units of the lattice vector lengths are • k-point mesh: M x M x 1 (where M is determined from bulk calculations) • The lattice vectors are fixed (only atomic positions within the supercell are optimized) • Lattice parameter along surface plane fixed at DFT bulk value

13. Other surfaces Top views

14. Surface unit cells • Smallest possible surface unit cells preferred, but gives atoms less “freedom” Smaller unit cell

15. Surface unit cells

16. Surface relaxation • Once the initial slab geometry is set, the system is then subjected to geometry optimization, i.e., the atoms within the supercell are allowed to adjust their positions such that the atomic forces are close to zero • Surface relaxation: a general phenomenon, in which the interplanar distances normal to the free surface change with respect to the bulk value. How? And, why?

17. Surface relaxation • Results have to be converged with respect to the number of layers • Remember, larger the number of layers, more accurate the result, but longer is the computational time

18. Surface relaxation – Convergence • Relaxation: change in the interplanar spacing normal to the surface plane with respect to the corresponding bulk value • Note the convergence of interplanar spacings as the number of layers is increased • Also note the “oscillations” in the sign of the change in the interplanar spacing with respect to bulk

19. Asymmetric vs. symmetric slabs • If symmetry is exploited, symmetric slabs are better • The bottom or central layers are fixed to ensure a bulk-like region • The lattice vectors are fixed (only atomic positions within the supercell are optimized) • Lattice parameter along surface plane fixed at DFT bulk value

20. Surface energy • Energy needed to create unit area of a surface from the bulk material • The surface energy is an anisotropic quantity, being smaller for the more stable closer-packed surfaces • Can be computed as Units: eV/A2 = 16.02 J/m2 Two surfaces per supercell Energy per atom of bulk material Energy of entire supercell containing n atoms Surface unit cell area

21. Surface energy • Note the quicker convergence with respect to the number of layers • Experimental value is an average over a number of surfaces; also, experimental value is surface free energy, while DFT value is the surface internal energy (i.e., DFT results are at 0 K and entropic effects are not taken into account)

22. Surface energy

23. Surface energy – the Wulff constructionThe surface energy as a polar plot

24. Surface reconstruction • Relaxation: movement of atoms normal to the surface plane • Reconstruction: movement of atoms along the surface plane (what do we need to do to allow this?)

25. The unreconstructed Si (001) surface Surface unit cell

26. The 2x1 reconstruction Reconstructed (001) surface Unreconstructed (001) surface • To see this reconstruction, the surface unit cell has to be twice as large as the primitive cell • Why does this reconstruction happen? To “passivate” dangling bonds

27. The (7x7) Si(111) reconstruction When heated to high temperatures in ultra high vacuum the surface atoms of the Si (111) surface rearrange to form the 7x7 reconstructed surface

28. Multi-element systems

29. CdSe surfaces • The {0001} family of surfaces are polar (i.e., surface plane does not have bulk stoichiometry) • Most of the other surfaces are nonpolar

30. CdSe nonpolar surfaces Reconstructions & relaxation • The already stable nonpolar surfaces undergo a lot of reconstruction, and become even more stable Top view Side view Top view Side view Before reconstruction After reconstruction

31. CdSe polar surfaces • 4 types of {0001} surfaces: • (0001) Cd-terminated • (0001) Se-terminated • (000-1) Cd-terminated • (000-1) Se-terminated • Display hardly any relaxation or reconstruction Top view Side view

32. Complications: Surface energy • The fundamental difficulty: If a surface plane does not have the same stoichiometry of the bulk material (e.g., polar surfaces), its surface energy cannot be uniquely determined! Why? • The above formula is inadequate, as slab will either not have an integer number of CdSe units, or will not have identical top and bottom surfaces • However, the following formula will work, but the surface energy will be dependent on the elemental chemical potential

33. CdSe surface energies Bare surfaces O covered surfaces

34. CdSe surface energies • O passivation  only the 2 (0001) surfaces are unstable and hence prone to growth  nanorods

35. Rock salt (NaCl) crystal structure for all alloy compositions Ti N (or C)

36. TiCxN1-x alloy surfaces • Surfaces may be polar depending on orientation and composition <001> TiC0.5N0.5 TiN TiN TiC0.5N0.5

37. TiCxN1-x alloy surface energies • As with CdSe, surface energies are dependent on elemental (C and N) chemical potentials • Most stable surface for a given choice of C and N chemical potentials can be determined • Moreover, the “allowed” values of C and N chemical potentials to maintain a stable bulk alloy may be determined (hatched regions) Conclusion: (001) surfaces are the most stable, regardless of alloy composition

38. Key Dates/Lectures • Oct 12 – Lecture • Oct 19 – No class • Oct 26 – Midterm Exam • Nov 2 – Lecture • Nov 9 – Lecture • Nov 16 – Guest Lectures • Dec 7 – In-class term paper presentations