Topology in Chemistry
Magicelectronnumbersofπ-electronsHückel:4n+2aroma;c4nan;aroma;c
Möbius4naroma;c4n+2an;aroma;c
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)
Trivial and Topological Insulators Trivialsemiconductor
CdS
TopologicalInsulator
Withoutspinorbitcoupling
TopologicalInsulator
Withspinorbitcoupling
M.C.Escher
Firstpredic4oningraphenebyKane
KaneandMele,PRL95,146802(2005)
Topological Insulators
λSOC~Z2forvalenceshells
Heavyinsula4ngelements
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)
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
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
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
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
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)
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
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
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..................................................
on Se
ptemb
er 5,
2016
http:/
/scien
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ience
mag.o
rg/Do
wnloa
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Science,353,6299,(2016)
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)
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