Topology: crystal structure and bands -...

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Topology: crystal structure and bands Claudia Felser © Nature

Transcript of Topology: crystal structure and bands -...

Topology: crystal structure and bands

Claudia Felser

©Nature

Topology in Chemistry Moleculeswithdifferentchirali4escanhave

differentphysicalandchemicalproper4es

Topology in Chemistry

Aroma4ccompounds

•  Aroma4cwith(4n+2)π-electrons•  Thesymmetrycounts

Topology in Chemistry

Magicelectronnumbersofπ-electronsHückel:4n+2aroma;c4nan;aroma;c

Möbius4naroma;c4n+2an;aroma;c

Hückel and Möbius Aromaticity

Graphene

•  Graphene’sconduc4vityexhibitsvaluesclosetotheconduc4vityquantume2/hpercarriertype

•  Graphene’schargecarrierscanbetunedcon4nuouslybetweenelectronsandholesin

concentra4onsn=1013cm

–2

•  Mobili4esμcanexceed15,000cm2V

–1s–1under

ambientcondi4ons

•  InSbhasμ≈77,000cm2V

–1s–1

Geim,A.K.&Novoselov,K.S.Theriseofgraphene.NatureMater.6,183(2007).

Topology – interdisciplinary

LaBii

Chemistry PhysicsRealspace-local Recipro.space-delocalized

Crystals Brillouinzone

Crystalstructure Symmetry Electronicstructure

Posi4onoftheatoms Localsymmetry Bandconnec4vity

Orbitals

Inertpaireffect Rela4vis4ceffects Darwinterm

Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory, Zhijun Wang, C. Felser, M. I. Aroyo, B. Andrei Bernevig, Nature (2017)

TopologicalInsulators

Trivial and Topological Insulators Trivialsemiconductor

CdS

TopologicalInsulator

Withoutspinorbitcoupling

TopologicalInsulator

Withspinorbitcoupling

M.C.Escher

Firstpredic4oningraphenebyKane

KaneandMele,PRL95,146802(2005)

Topological Insulators

λSOC~Z2forvalenceshells

Heavyinsula4ngelements

TopologicalInsulators?

Noinsula4ngelement

Letstake2Elements

Firstpredic4oningraphenebyKane

KaneandMele,PRL95,146802(2005)

Bernevig,etal.,Science314,1757(2006)

Bernevig,S.C.Zhang,PRL96,106802(2006)

König,etal.Science318,766(2007)

Topological Insulators

Heavyinsula4ngbinaries

-

-

Quantum Spin Hall

+

-

3D:Diracconeonthesurface

2D:Diracconeinquantumwell

Bernevig,etal.,Science314,1757(2006)

Bernevig,S.C.Zhang,PRL96,106802(2006)

König,etal.Science318,766(2007)

Inertpaireffect

Bi-SbLegierungen

Bi2Se

3undverwandteStrukturen

MooreandBalents,PRB75,121306(R)(2007)

FuandKane,PRB76,045302(2007)

Murakami,NewJ.Phys.9,356(2007)

Hsieh,etal.,Science323,919(2009)

Xia,etal.,NaturePhys.5,398(2009);Zhang,etal.,NaturePhys.5,438(2009)

Topologische Isolatoren

ClaudiaFelserandXiao-LiangQi,GuestEditors,MRSBull.39(2014)843.

Materials

Tl+1Sn

2+Bi

+3

Inertpaireffect

Inert pair effect

•  1selectronsofheavierelementsmove

atspeeds~speedoflight.

•  electronmass~1/orbitalradius

•  contrac4onofthe1sorbital=decreased

ofshieldingfortheouterselectrons

•  valenceselectronsbehavelikecore

electrons

•  hardertoremoveviaioniza4on

•  Al(III)Al(III)ispreferredoverAl(I)Al(I),

butTl(I)Tl(I)ispreferredoverTl(III)

Topology – interdisciplinary

LaBii

Chemistry PhysicsRealspace-local Recipro.space-delocalized

Crystals Brillouinzone

Crystalstructure Symmetry Electronicstructure

Posi4onoftheatoms Localsymmetry Bandconnec4vity

Orbitals

Inertpaireffect Rela4vis4ceffects Darwinterm

Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory, Zhijun Wang, C. Felser, M. I. Aroyo, B. Andrei Bernevig, Nature (2017)

Crystalstructure

•  Closedpackedstructures– Cubic– Hexagonal

How structures are made

hexagonal close packing

cubic close packing

Anionsalwaysarethe

largestspheres,theybuild

aclosedpackedlarce

Inoxides:O2-

How structures are made: holes Anions always are the largest spheres, they build a closed packed lattice

Cations are stuffed in the holes

Simple ionic structures

K3C60

fcc - lattice 2 fcc – lattices Zincblend structure

3 fcc – lattices MgAgAs structure (F-43m, C1b)

Heusler compounds

Diamond ZnS Heusler XYZ C1b X2YZ L21

2 interpenetrating fcc with half of the tetrahedral

sites filled

3 interpenetrating fcc 4 interpenetrating fcc half of the tetrahedral sites all tetrahedral sites filled and the octahedral sites and the octahedral sites

Diamond ZnS Heusler XYZ C1b

3 + 5 = 8 2 + 6 = 8

Cd Se

ZnLi

1 + 2 + 5 = 8

Sb

RE Pt

3+ 10 + 5 = 18

Bi

Heusler compounds

��

ScNiSb

LaPtBi LaPtBi

S.Chadovetal.,Nat.Mater.9541(2010).H.Linetal.,Nat.Mater.9546(2010).

Predicting topological insulators

Formula Type/fraction of sites occ. HCP CCP

AB All octahedral NiAs Nickel Arsenide

NaCl Rock Salt

Half tetrahedral ZnS Wurtzite

ZnS Zinc Blende

A2B

All tetrahedral Not known CaF2/Mg2Si (Fluorite/Anti-

Fluorite) A3B All octahedral & tetrahedral Not known Li3Bi

AB2

Half octahedral (Alternate layers full/empty)

CdI2

Cadmium Chloride CdCI2

Cadmium Chloride Half octahedral (ordered framework arrangement)

CaCl2

TiO2 (Rutile) TiO2 (Anatase)

AB3

1/3 octahedral Alternate layers 2/3 empty

BiI3

AlCl3

A2B3 2/3 octahedral (Ordered framework)

Al2O3 /FeTiO3

Corundum/Ilmenite

•  Layeredstructure– Removingonelayer

– Anisotropicbonding–lonepairs–inertpair

Layered structure

Layered structure

MoS2 : crystal field

2H 1T 1T’,Td

dz2

dxy,dx2-y

2

dxz,dyz

dz2,dx

2-y

2

dxy,dxz,dyz

dz2

dxz,dyz

dx2-y

2

dxy

Semiconductor Metal Semimetal

Mo4+(d

2)

S2-(s

2p6)

Heuslers go low dimensional

Semiconductor main group

Li2MgSi LiAlSi

Semiconductor transition metal

Fe2VAl TiNiSn

.

NaCuS

Semiconductor heavy

Li2HgPb or Na

Os2ThPb ???

MgHgPb ???

LaPtBi topolo.

PbFCl PbO

Superconductor main group

ZrNi2Ga No inversion NaAlSi Tc=7K

Superconductor transition metal

Pd2ZrAl Tc=5K

LaPtBi Tc=0.6K

LiFeAs Tc=18

FeSe Tc=8K p=9GPa Tc=37K

Raulotetal.,SolidStateSciences4(2002)467

Fe-pnictides

Formula Type/fraction of sites occ. HCP CCP

AB All octahedral NiAs Nickel Arsenide

NaCl Rock Salt

Half tetrahedral ZnS Wurtzite

ZnS Zinc Blende

A2B

All tetrahedral Not known CaF2/Mg2Si (Fluorite/Anti-

Fluorite) A3B All octahedral & tetrahedral Not known Li3Bi

AB2

Half octahedral (Alternate layers full/empty)

CdI2

Cadmium Chloride CdCI2

Cadmium Chloride Half octahedral (ordered framework arrangement)

CaCl2

TiO2 (Rutile) TiO2 (Anatase)

AB3

1/3 octahedral Alternate layers 2/3 empty

BiI3

AlCl3

A2B3 2/3 octahedral (Ordered framework)

Al2O3 /FeTiO3

Corundum/Ilmenite

Electronicstructure

Translation

Topological insulatorTopological semi-metal

Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory, Zhijun Wang, C. Felser, M. I. Aroyo, B. Andrei Bernevig, Nature (2017)

Barry,Andreislide

From Orbitals to Bands

Energybandsinsolidsarisefromoverlappingatomicorbitals

⇒crystalorbitals(whichmakeupthebands)

Recipe:useLCAO(4ghtbinding)approach

Crystal=regularperiodicarray⇒transla4onalsymmetry

Find wavefunction for an electron moving along a circle of atoms Circular symmetry ⇒ apply periodic boundary conditions

HoffmannAngewandte.Chem.26(1987)846

How to find a band structure

HoffmannAngewandte.Chem.26(1987)846

Westartfromatomicorbitals–localizedpicture:linearchainofhydrogenatoms

Usingthetransla4onalsymmetryofthesolidwecansetupthentermsof

symmetrywitha1sbasisfunc4onfortwospecialvaluesofk

Testwitha1sbasisfunc4onfortwospecialvaluesofk

E(k)

k π/a0

E

n

n k = 0

Ψ Σ χ= e πin

n = -1) = - + - .....Σ ( χ χ χ χ χn

n 0 1 2 3 πa

k = πa n

0 n

Band Width

HoffmannAngewandte.Chem.26(1987)846

Animportantfeatureofabandisits

bandwidth(dispersion),i.e.thedifference

betweenhighestandlowestlevel.

Thebandwidthisdeterminedbythe

overlapbetweentheinterac4ngorbitals

How to find a band structure

HoffmannAngewandte.Chem.26(1987)846

E(k)

k π/a0

E

n

n k = 0

Ψ Σ χ= e πin

n = -1) = - + - .....Σ ( χ χ χ χ χn

n 0 1 2 3 πa

k = πa n

0 n

Ψ0 +

= + ..... 0 χ χ χ + χ+ 1 2 3

πa

Ψ + - +

= - .. 0 χ χ χ χ 1 2 3E

π/a

Thetopologyoftheorbitalinterac4ondetermineshowabandruns

k=0bondinginterac4on⇒bandruns„uphill“

k = 0 antibonding interaction ⇒ band runs „downhill“

Square Lattice

Correspondinglarce

inreciprocalspace

EdgesofBrillouinzone:

Special(highsymmetry)

kpoints

Squarelarcewith

onesbasisorbital

BaBiO3

Square nets of electron doped Bi

Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory, Zhijun Wang, C. Felser, M. I. Aroyo, B. Andrei Bernevig, Nature (2017)

Why is Hydrogen a molecule

HoffmannAngewandte.Chem.26(1987)846

PeierlsDistor4on

Graphene

AllPzorbitalsin-phase,Γ,Stronglyπ-bonding

AllPzorbitalsout-of-phase,Γ,Stronglyan4π-bonding

Graphene

pz,π,K:non-bonding pz,π*,Κ:non-bonding

pz, π, Μ: bonding pz,π,Μ�:an4-bonding

Translation

Topological insulatorTopological semi-metal

Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory, Zhijun Wang, C. Felser, M. I. Aroyo, B. Andrei Bernevig, Nature (2017)

Barry,Andreislide

Conetronics in 2D Metal-Organic Frameworks

Wuetal.2DMater.4(2017)015015arXiv:1606.04094

Topology – interdisciplinary

LaBii

Chemistry PhysicsRealspace-local Recipro.space-delocalized

Crystals Brillouinzone

Crystalstructure Symmetry Electronicstructure

Posi4onoftheatoms Localsymmetry Bandconnec4vity

Orbitals

Inertpaireffect Rela4vis4ceffects Darwinterm

Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory, Zhijun Wang, C. Felser, M. I. Aroyo, B. Andrei Bernevig, Nature (2017)

��

ScNiSb

LaPtBi LaPtBi

S.Chadovetal.,Nat.Mater.9541(2010).H.Linetal.,Nat.Mater.9546(2010).

Predicting topological insulators

KHgSb

HgTe

α Ag2Te

LaPtBi

Li2AgSb

AuTlS2

βAg2Te

Structure to Property

Müchler,etal.,AngewandteChemie51(2012)7221.

HgTe

LaPtBi

Li2AgSb

αAg2Te

βAg2Te

AuTlTe2

Graphite

Γ  MK Γ

LiAuTe

KHgSb

Müchler,etal.,AngewandteChemie51(2012)7221.

K ΓA H

N ΓZ X

L Γ X

L Γ XL Γ X

L Γ X

Γ DBA K ΓA H

Struktur und elektronische Struktur

Honeycomb from sp3 to sp2

Zhang,Chadov,Müchler,Yan,Qi,Kübler,Zhang,Felser,Phys.Rev.Lex.106(2011)156402arXiv:1010.2195v1

Bandinversionisfoundintheheavier

compounds

Nosurfacestate?Why?

Interac4onbetweenthetwo

layersintheunitcellandtwoDirac

Cones

a)

b)

Honeycomb: Weak TI

Yan,Müchler,Felser,PhysicalReviewLex.109(2012)116406

Hourglass

Wangetal.Nature2016

unpublishedresults

Dirac-WeylSemimetals

RESEARCH ARTICLE SUMMARY◥

TOPOLOGICAL MATTER

Beyond Dirac and Weyl fermions:Unconventional quasiparticles inconventional crystalsBarry Bradlyn,* Jennifer Cano,* Zhijun Wang,* M. G. Vergniory, C. Felser,R. J. Cava, B. Andrei Bernevig†

INTRODUCTION: Condensed-matter systemshave recently become a fertile ground for thediscovery of fermionic particles and pheno-mena predicted in high-energy physics; exam-ples includeMajorana fermions, aswell asDiracand Weyl semimetals. However, fermions incondensed-matter systems are not constrainedby Poincare symmetry. Instead, they must onlyrespect the crystal symmetry of one of the230 space groups. Hence, there is the po-tential to find and classify free fermionicexcitations in solid-state systems thathave no high-energy counterparts.

RATIONALE: The guiding principle ofour classification is to find irreduciblerepresentations of the little group of lat-tice symmetries at high-symmetry pointsin the Brillouin zone (BZ) for each of the230 space groups (SGs), the dimension ofwhich corresponds to thenumberofbandsthat meet at the high-symmetry point. Be-cause we are interested in systems withspin-orbit coupling, we considered onlythe double-valued representations, wherea 2p rotation gives a minus sign. Further-more, we considered systems with time-reversal symmetry that squares to –1. Foreach unconventional representation, wecomputed the low-energy k · p Hamil-tonian near the band crossings by writ-ing down all terms allowed by the crystalsymmetry. This allows us to further dif-ferentiate the band crossings by thedegeneracy along lines and planes thatemanate from the high-symmetry point,andalso to compute topological invariants.For point degeneracies, we computed themonopole charge of the band-crossing; for linenodes, we computed the Berry phase of loopsencircling the nodes.

RESULTS: We found that three space groupsexhibit symmetry-protected three-band cross-ings. In two cases, this results in a threefolddegenerate point node, whereas the third caseresults in a line node away from the high-

symmetry point. These crossings are requiredto have a nonzero Chern number and hencedisplay surface Fermi arcs. However, upon ap-plying a magnetic field, they have an unusualLandau level structure, which distinguishesthem from single and double Weyl points.Under the action of spatial symmetries, thesefermions transform as spin-1 particles, as a

consequence of the interplay between nonsym-morphic space group symmetries and spin.Additionally, we found that six space groupscan host sixfold degeneracies. Two of theseconsist of two threefold degeneracies withopposite chirality, forced to be degenerate bythe combination of time reversal and inver-sion symmetry, and can be described as “six-fold Dirac points.” The other four are distinct.

Furthermore, seven space groups can hosteightfold degeneracies. In two cases, the eight-fold degeneracies are required; all bands comein groups of eight that cross at a particularpoint in the BZ. These two cases also exhibitfourfold degenerate line nodes, from whichother semimetals can be derived: By addingstrain or a magnetic field, these line nodessplit intoWeyl, Dirac, or line node semimetals.

For all the three-, six-, andeight-band crossings, non-symmorphic symmetriesplay a crucial role in pro-tecting the band crossing.Last,we foundthat seven

space groups may hostfourfold degenerate “spin-3/2” fermions athigh symmetry points. Like their spin-1 coun-terparts, these quasiparticles host Fermi sur-faces with nonzero Chern number. Unlike theother cases we considered, however, these fer-mions can be stabilized by both symmorphicand nonsymmorphic symmetries. Three spacegroups that host these excitations also host un-conventional fermions at other points in the BZ.

We propose nearly 40 candidate mate-rials that realize each type of fermionnear the Fermi level, as verified with abinitio calculations. Seventeen of thesehave been previously synthesized in single-crystal form, whereas others have beenreported in powder form.

CONCLUSION: We have analyzed alltypes of fermions that can occur in spin-orbit coupled crystals with time-reversalsymmetry and explored their topologicalproperties.We found that there are severaldistinct types of such unconventional ex-citations, which are differentiated by theirdegeneracies at and along high-symmetrypoints, lines, and surfaces. We found na-tural generalizations ofWeyl points: three-and four-band crossings described by asimple k · S Hamiltonian, where Si isthe set of spin generators in either thespin-1 or spin-3/2 representations. Thesepoints carry a Chern number and, con-sequently, can exhibit Fermi arc surfacestates. We also found excitations withsix- and eightfold degeneracies. Thesehigher-band crossings create a tunableplatform to realize topological semimetalsby applying an external magnetic field orstrain to the fourfold degenerate line

nodes. Last, we propose realizations for eachspecies of fermion in known materials, many ofwhich are known to exist in single-crystal form.▪

RESEARCH

558 5 AUGUST 2016 • VOL 353 ISSUE 6299 sciencemag.org SCIENCE

The list of author affiliations is available in the full article online.*These authors contributed equally to this work.†Corresponding author. Email: [email protected] this article as B. Bradlyn et al., Science 353, aaf5037(2016). DOI: 10.1126/science.aaf5037

Fermi arcs from a threefold degeneracy. Shown is thesurface density of states as a function of momentum for acrystal in SG 214 with bulk threefold degeneracies thatproject to (0.25, 0.25) and (–0.25, –0.25). Two Fermi arcsemanate from these points, indicating that their monopolecharge is 2. The arcs then merge with the surface projectionof bulk states near the origin.

ON OUR WEBSITE◥

Read the full articleat http://dx.doi.org/10.1126/science.aaf5037..................................................

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New Fermions

Ag3Se

2Au,Ba

4Bi

3

Fermionsincondensed-maxersystemsarenot

constrainedbyPoincaresymmetry.Instead,

theymustonlyrespectthecrystalsymmetryof

oneofthe230spacegroups.Hence,thereis

thepoten4altofindandclassifyfreefermionic

excita4onsinsolid-statesystemsthat

havenohigh-energycounterparts.

Example: PdSb2

6-foldfermionsattheRpoint

SinceSG205containsinversionsymmetry,allbandsaredoubly

degenerate

• non-symmorphic:elementswithfrac%onalla*cetransla%ons

•  Cubiccrystalstructureforhigh

degenera4ons

•  Nonsymmorphicisessen4alforstabilizing

six-andeight-folddegeneratepoints.

Andreislide

Novel Metals• SG I-43d (220) has elementary band representations with 16 branches -> 16fold connected metal if topologically trivial

• Filling -1/8 (lowest filling possible in a material is 1/24)

• Examples in the A15B4 material class (Here Li15Ge4)

Γ H N Γ P H -3

-2

-1

0

1

2

Ener

gy(e

V)

soc

EF

Barry,Andreislide

Topology – interdisciplinary

LaBii

Chemistry PhysicsRealspace-local Recipro.space-delocalized

Crystals Brillouinzone

Crystalstructure Symmetry Electronicstructure

Posi4onoftheatoms Localsymmetry Bandconnec4vity

Orbitals

Inertpaireffect Rela4vis4ceffects Darwinterm

Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory, Zhijun Wang, C. Felser, M. I. Aroyo, B. Andrei Bernevig, Nature (2017)