Topology in the solid state sciences
José L. Mendoza-
Cortés
2011 February 17th
Materials SciencePhysics Chemistry
Biology
What do they mean by Topology?
Why is it important? What can we learn?
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.
FamiliarityVoronoi-Dirichlet polyhedron
Wigner-Seitz cell
First Brillouin zone
All are example of Voronoi-Dirichlet polyhedron but applied to an specific
field
Everything we are going to cover today it comes to this!
And this:
Zeolites
From real stuff to abstract stuff
Different topologies could be obtained on varying the coordination geometry of the nodes...
node rod
From real stuff to abstract stuff
honeycomb layerhoneycomb layer
Lets see abstract stuff
““Topological” Topological” EntanglementEntanglement““Topological” Topological” EntanglementEntanglement
““Euclidean” Euclidean” EntanglementEntanglement““Euclidean” Euclidean”
EntanglementEntanglement
Borromean linksBorromean linksBorromean linksBorromean links
Lets see abstract stuff
Models: Lattice
hxl/Shubnikov plane net (3,6)Atom coordinatesC1 0.00000 0.00000 0.00000Space Group: P6/mmmCell Dimensionsa=1.0000 b=1.0000 c=10.0000
Crystallographic, not crystallochemical model
Models: Net
Inherently crystallochemical, but no geometrical properties are analyzed
Models: Labeled quotient graph
001
1002
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0012
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1002
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1002
001
1002
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a b
Chung, S.J., Hahn, Th. & Klee, W.E. (1984). Acta Cryst. A40, 42-50.
Wrapping NaCl 3D graph NaCl labeled quotient graph
Models: Embedded net
Diamond (dia) net in the most symmetrical embedding
Models: Polyhedral subdivision
Voronoi-Dirichlet polyhedron and partition: bcu net
Kd=0.5
Models: Polyhedral subdivision
Tilings: dia and bcu nets
dia bcu
‘Normal’ crystal chemistry -> ‘dual’ crystal chemistry
Abstract stuff
4
3-connected graph means that the three vertex are connected with other three vertex (therefore they have three edges)
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).
Where can we apply this?
world records of world records of InterpenetrationInterpenetrationworld records of world records of InterpenetrationInterpenetration
2002200210-fold 10-fold diadia MOF MOFAg(dodecandinitrile)2
11-fold 11-fold diadia H-bond H-bond[C(ROH)4][Bzq]2
Class IaClass Ia......
18-fold srs H-bond18-fold srs H-bond(trimesic acid)2(bpetha)3
Class IIIb Class IIIb
2002200210-fold 10-fold diadia MOF MOFAg(dodecandinitrile)2
11-fold 11-fold diadia H-bond H-bond[C(ROH)4][Bzq]2
Class IaClass Ia......
18-fold srs H-bond18-fold srs H-bond(trimesic acid)2(bpetha)3
Class IIIb Class IIIb
12 interpenetrating netsTIV: [0,1,0] (13.71A)NISE: 2(1)[0,0,1]Zt=6; Zn=2Class IIIa Z=12[6*2]
dia 12fdia 12f20082008
#########################################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) 19.905APd 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 10Num 4 12 30 58 94 138 190 250 318 394Cum 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 netClassification of the topological type: cds/CdSO4 {6^5;8} - VS [6.6.6.6.6(2).*]
TOPOS OUTPUT
O’Keffe & Delgado-FriedrichsO’Keffe & Delgado-Friedrichs
SyStReSyStRe
3dt3dt
Symmetry Structure RealizationSymmetry Structure Realizationone can determine without ambiguity whether two nets are isomorphic or not
20022002
20032003
SyStReSyStRe
3dt3dt3D Tiling3D Tiling
Thanks to: Delgado-Friedrich, O’Keeffe, Hyde, Blatov, Proserpio.
Suplementary slides
Suplementary slides
Self-entanglementSelf-entanglement
a) 0D 1D
b) 0D 1D
c) 1D 2Dd) 1D 3D
e) 2D 3D f) 2D 3D
a) 0D 1D
b) 0D 1D
c) 1D 2Dd) 1D 3D
e) 2D 3D f) 2D 3D
dimensionalityunchanged
increase of dimensionality
INTERPENETRATIONINTERPENETRATIONPOLYCATENATIONPOLYCATENATION
..\libro_braga\figure\asufig.jpg
PolythreadingPolythreadingPolythreadingPolythreading
InterpenetrationInterpenetrationInterpenetrationInterpenetration
PolycatenationPolycatenationPolycatenationPolycatenation
self-catenationself-catenationself-catenationself-catenation
Borromean Borromean entanglementsentanglements
Borromean Borromean entanglementsentanglements
A new complexity of the solid stateA new complexity of the solid state
“Euclidean”“Topological”
Data: Electronic crystallographic databases
CSD ~430000 entries
ICSD ~100000 entries
CrystMet ~100000 entries
PDB ~50000 entries
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/
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/
History of crystallochemical analysis
Mathematical fundamentals
J. Hessel, 1830 – 32 geometric crystal classesO. Bravais, 1848 – 14 three-periodic latticesE. Fedorov and A. Shönflies, 1890 – 230 space groups
History of crystallochemical analysis
Microscopic observations
M. Laue, 1912 – diffraction of X-rays in crystalsW.G. Bragg and W.L. Bragg, 1913 – first structure determinations
History of crystallochemical analysis
Experimental technique and methods of X-ray analysis
1920s – 1960sPhotomethods and technique First printed manuals on crystal structuresFirst really crystallochemical laws – (L. Pauling, V. Goldschmidt, A. Kitaigorodskii)A.F. Wells, 1954 – graph representation
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
Algorithms: building adjacency matrix
Method of intersecting spheres
For inorganic compounds
Method of spherical sectors
For organic, inorganic and metal-organic compounds
Distances For all types of compounds, using atomic radii and Voronoi polyhedra
Solid Angles For artificial nets, intermetallides, noble gases, using Voronoi polyhedra
Van der Waals Specific Valence Valence Valence
Algorithms: building adjacency matrix
Solid angle of a VDP face is proportional to the bond strength
Topological insulatorsan 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 Space Reciprocal space
Topological insulators
At bulk At the surface new states appears!
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.
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.
Sources
• Nature Physics 4, 348 - 349 (2008) doi:10.1038/nphys955
• Nature 464, 194-198 (11 March 2010) | doi:10.1038/nature08916;