1. Topology in the solid state sciences José L. Mendoza- Cortés 2011 February 17th

2. Materials Science Physics Chemistry Biology What do they mean by Topology? Why is it important? What can we learn?

3. Main Questions Fundamental question: Given an spectra (e.g. sound), can you tell the shape of the source (e.g. the instrument shape) In other words: Is it possible that two molecules or solids can have the same properties, given the only difference is their topology? Topology is concerned with spatial properties that are preserved under continuous deformations of objects.

4. Familiarity Voronoi-Dirichlet polyhedron Wigner-Seitz cell First Brillouin zone All are example of Voronoi-Dirichlet polyhedron but applied to an specific field

5. Everything we are going to cover today it comes to this!

6. And this: Zeolites

7. From real stuff to abstract stuff Different topologies could be obtained on varying the coordination geometry of the nodes... node rod

8. From real stuff to abstract stuff honeycomb layer

9. Lets see abstract stuff “Topological” Entanglement “Euclidean” Entanglement

10. Borromean links

11. Lets see abstract stuff

12. Models: Lattice hxl/Shubnikov plane net (3,6) Atom coordinates C1 0.00000 0.00000 0.00000 Space Group: P6/mmm Cell Dimensions a=1.0000 b=1.0000 c=10.0000 Crystallographic, not crystallochemical model

13. Models: Net Inherently crystallochemical, but no geometrical properties are analyzed

14. Models: Labeled quotient graph Chung, S.J., Hahn, Th. & Klee, W.E. (1984). Acta Cryst. A40, 42-50. Wrapping NaCl 3D graphNaCl labeled quotient graph

15. Models: Embedded net Diamond (dia) net in the most symmetrical embedding

16. Models: Polyhedral subdivision Voronoi-Dirichlet polyhedron and partition: bcu net K d =0.5

17. Models: Polyhedral subdivision Tilings: dia and bcu nets diabcu ‘Normal’ crystal chemistry -> ‘dual’ crystal chemistry

18. Abstract stuff

22. 4

23. 3-connected graph means that the three vertex are connected with other three vertex (therefore they have three edges)

31. Where can we apply this? Hsieh, D. et al. A tunable topological insulator in the spin helical Dirac transport regime. Nature 460, 1101–1105 (2009).

32. Where can we apply this?

33. world records of Interpenetration 2002 10-fold dia MOF Ag(dodecandinitrile) 2 11-fold dia H-bond [C(ROH) 4 ][Bzq] 2 Class Ia... 18-fold srs H-bond (trimesic acid) 2 (bpetha) 3 Class IIIb 2002 10-fold dia MOF Ag(dodecandinitrile) 2 11-fold dia H-bond [C(ROH) 4 ][Bzq] 2 Class Ia... 18-fold srs H-bond (trimesic acid) 2 (bpetha) 3 Class IIIb

34. 12 interpenetrating nets TIV: [0,1,0] (13.71A) NISE: 2(1)[0,0,1] Zt=6; Zn=2 Class IIIa Z=12[6*2] dia 12f 2008

37. ######################################### 12;RefCode:SOBTUY:C40 H42 Cd2 N12 O21 Pd1 Author(s): Abrahams B.F.,Hoskins B.F.,Robson R. Journal: J.AM.CHEM.SOC. Year: 1991 Volume: 113 Number: Pages: 3606 ######################################### -------------------- Atom Pd1 links with R(A-A) Pd 1 0.5000 -0.5000 1.0000 ( 0-1 1) 19.905A Pd 1 -1.0000 0.0000 -1.5000 (-1 0-2) 17.126A Pd 1 1.0000 0.0000 1.5000 ( 1 0 1) 17.126A Pd 1 -0.5000 0.5000 -1.0000 (-1 0-1) 19.905A ------------------------- Structure consists of 3D framework with Pd (SINGLE NET) Coordination sequences ---------------------- Pd1: 1 2 3 4 5 6 7 8 9 10 Num 4 12 30 58 94 138 190 250 318 394 Cum 5 17 47 105 199 337 527 777 1095 1489 ---------------------- Vertex symbols for selected sublattice ------------------- Pd1 Point/Schlafli symbol:{6^5;8} With circuits:[6.6.6.6.6(2).8(2)] With rings: [6.6.6.6.6(2).*] -------------------------------------- Total Point/Schlafli symbol: {6^5;8} 4-c net; uninodal net Classification of the topological type: cds/CdSO4 {6^5;8} - VS [6.6.6.6.6(2).*] TOPOS OUTPUT

38. O’Keffe & Delgado-Friedrichs SyStRe 3dt Symmetry Structure Realization one can determine without ambiguity whether two nets are isomorphic or not 2002 2003

39. SyStRe

40. 3dt 3D Tiling

41. Thanks to: Delgado-Friedrich, O’Keeffe, Hyde, Blatov, Proserpio.

42. Suplementary slides

43. Self-entanglement

45. dimensionality unchanged increase of dimensionality INTERPENETRATION POLYCATENATION

46. ..\libro_braga\figure\asufig.jpg

47. PolythreadingPolythreading InterpenetrationInterpenetration PolycatenationPolycatenation self-catenationself-catenation Borromean entanglements A new complexity of the solid state “Euclidean” “Topological”

48. Data: Electronic crystallographic databases CSD ~430000 entries ICSD ~100000 entries CrystMet ~100000 entries PDB ~50000 entries

49. Data: Electronic crystallochemical databases RCSR 1620 entries; http://rcsr.anu.edu.au TTD Collection 66833 entries; http://www.topos.ssu.samara.ru TTO Collection 3617 entries; http://www.topos.ssu.samara.ru Atlas of Zeolite Frameworks, 179 entries; http://www.iza-structure.org/databases /

50. Data: Electronic databases of hypothetical nets EPINET 14532 entries; http://epinet.anu.edu.au/ Atlas of Prospective Zeolite Frameworks 2543772 entries; http://www.hypotheticalzeolites.net/

51. History of crystallochemical analysis Mathematical fundamentals J. Hessel, 1830 – 32 geometric crystal classes O. Bravais, 1848 – 14 three-periodic lattices E. Fedorov and A. Shönflies, 1890 – 230 space groups

52. History of crystallochemical analysis Microscopic observations M. Laue, 1912 – diffraction of X-rays in crystals W.G. Bragg and W.L. Bragg, 1913 – first structure determinations

53. History of crystallochemical analysis Experimental technique and methods of X-ray analysis 1920s – 1960s Photomethods and technique First printed manuals on crystal structures First really crystallochemical laws – (L. Pauling, V. Goldschmidt, A. Kitaigorodskii) A.F. Wells, 1954 – graph representation

54. History of crystallochemical analysis Time of automated diffractometers 1960s – present time Rapid accumulation of experimental data Now the number of determined crystal structures exceeds 600,000 and grows faster and faster

55. Algorithms: building adjacency matrix Method of intersecting spheres For inorganic compounds Method of spherical sectors For organic, inorganic and metal-organic compounds DistancesFor all types of compounds, using atomic radii and Voronoi polyhedra Solid AnglesFor artificial nets, intermetallides, noble gases, using Voronoi polyhedra Van der WaalsSpecificValence

56. Algorithms: building adjacency matrix Solid angle of a VDP face is proportional to the bond strength

62. Topological insulators an extremely short explanation y Jose L. Mendoza-Cortes It is an insulator (or a semiconductor) at bulk At the surface, new states appears (The so called surface states) These new states suffer from spin-orbit coupling These surface states determines if they are topological insulators or not. This is that if electrons with a determined energy and momentum can be trapped in the surface. Real SpaceReciprocal space

63. Topological insulators At bulk At the surface new states appears!

64. Topological insulators Topological these two surfaces are equivalent However, the bulk properties of the semiconductor (or isolator) makes the surfaces band to have spin-orbit coupling, so they stop being degenerated.

65. Topological insulators Depending of the properties of the bulk semiconductor (or the insulator), then the surface bands are going to have the topological constrains. Now, what does make a topological insulator one? The fact that one electron with certain energy and momentum would stay in that surface as it would with a conductor. and this is going to be determined by the topology of the surface band! Let’s assume the red dot in the figure above is an electron from diffraction experiment, on the left figure, the electron would bounce with different momentum from the solid. However on figure on the right, the electron would get trapped.

66. Sources Nature Physics 4, 348 - 349 (2008) doi:10.1038/nphys955 Nature 464, 194-198 (11 March 2010) | doi:10.1038/nature08916;