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

100

0012

010

0102

001

1002

001

1002

001

1002

010

0102

010

0102

100

0012

100

0012

100

0012

010

0102

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;