Topological Insulators Charles L. Kane, University of Pennsylvania E k=La k=Lb E k=La k=Lb Trieste, July 4, 2013

Thanks! Mentors and Role Models : Matthew Fisher Steve Girvin Dung Hai Lee Patrick Lee Tom Lubensky Allan MacDonald Eugene Mele Naoto Nagaosa Nick Read Tom Rosenbaum Collaborators on Topological Insulators: Zahid Hasan Matthew Fisher Liang Jiang Max Metlitski Eugene Mele Joel Moore Tony Pantev John Preskill Andrew Rappe Students and Post Docs : Sugata Chowdhury Liang Fu Jeffrey Teo Ben Wieder Steve Young Saad Zaheer Fan Zhang Financial Support :

Thanks! Mentors and Role Models : Matthew Fisher Steve Girvin Dung Hai Lee Patrick Lee Tom Lubensky Allan MacDonald Eugene Mele Naoto Nagaosa Nick Read Tom Rosenbaum Collaborators on Topological Insulators: Zahid Hasan Matthew Fisher Liang Jiang Max Metlitski Eugene Mele Joel Moore Tony Pantev John Preskill Andrew Rappe Students and Post Docs : Sugata Chowdhury Liang Fu Jeffrey Teo Ben Wieder Steve Young Saad Zaheer Fan Zhang Financial Support :

Topological Insulators Introduction : Topological Band Theory Quantum Spin Hall Effect in 2D Topological insulators in 3D Proximity effects and heterostructure devices

The Insulating State Characterized by energy gap: absence of low energy electronic excitations Covalent Insulator Atomic Insulator The vacuum e.g. intrinsic semiconductor e.g. solid Ar electron 4s Dirac Vacuum Egap ~ 10 eV Egap = 2 mec2 ~ 106 eV 3p Egap ~ 1 eV positron ~ hole Silicon

The Integer Quantum Hall State 2D Cyclotron Motion, Landau Levels sxy = e2/h E IQHE with zero net magnetic field Haldane model : Graphene with a periodic magnetic field B(r) (Haldane PRL 1988) Band structure k + - B(r) = 0 Zero gap, Dirac point B(r) ≠ 0 Energy gap sxy = e2/h Eg

Topological Band Theory The distinction between a conventional insulator and the quantum Hall state is a topological property of the band structure Classified by the TKNN (or Chern) topological invariant (Thouless et al, 1984) Insulator : n = 0 IQHE state: sxy = n e2/h Bulk - Boundary correspondence: Conducting states occur at interfaces where n changes Edge States at a domain wall Gapless Chiral Dirac Fermions QHE state n=1 Vacuum n=0 E Eg Haldane Model K’ K k

Fall 2004 : Graphene Intrinsic spin orbit interaction : 2lSO Gene Mele : Did you see the paper by Geim and Novoselov et al.? We should run with graphene. CLK : I have an idea: Graphene actually has an energy gap! Graphene: 2D Dirac fermions with 2 valleys (tz = ±1) and 2 spins (sz = ±1) E p Intrinsic spin orbit interaction : K’ K Respects all symmetries, so it must* be present. Two copies of integer quantum Hall state 2lSO

Quantum Spin Hall Effect in Graphene Simplest model: |Haldane|2 (conserves Sz) J↑ J↓ E Bulk energy gap, but gapless edge states p/a E ↑ ↓ ↓ ↑ V Two terminal Conductance: G = 2e2/h Edge states form a unique 1D conductor Half an ordinary 1D electron gas Protected by time reversal symmetry even when spin conservation is violated.

time reversal invariant momenta Time Reversal Invariant 2 Topological Insulator 2D Time reversal invariant band structures are characterized by a Z2 topological invariant, n = 0,1 n=0 : Conventional Insulator n=1 : Topological Insulator E k*=0 k*=p/a Edge States E k*=0 k*=p/a Kramers degenerate at time reversal invariant momenta k* = -k* + G Stability to Interactions and Disorder: E Kramers parity switching: T2=+1 → T2=-1 F ↑ ↑↓ p Edge states are impossible to localize and remain in presence of interactions

Quantum Spin Hall Insulator in HgTe quantum wells Theory: Bernevig, Hughes and Zhang, Science 2006 HgTe HgxCd1-xTe d Predict inversion of conduction and valence bands for d>6.3 nm → QSHI Expt: Konig, Wiedmann, Brune, Roth, Buhmann, Molenkamp, Qi, Zhang Science 2007 d< 6.3 nm normal band order conventional insulator d> 6.3nm inverted band order QSH insulator G=2e2/h Laurens Molenkamp Measured conductance 2e2/h independent of W for short samples (L<Lin)

Summer 2006: Race to three dimensions Early July : Liang Fu : I think there are multiple Z2 invariants in 3D. CLK : I think there’s just one, which specifies an even or odd number of surface Dirac points. July 12 : Joel Moore and Leon Balents “Topological invariants of time-reversal invariant band structures” - Homotopy theory for 4 Z2 invariants in 3D - Coined the name “Topological Insulators” July 20 : Rahul Roy “Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect” July 26 : Fu, Kane, Mele “Topological Insulators in Three Dimensions” Characterized surface states Strong vs weak TI’s Explicit model

3D Topological Insulators 3D insulators are characterized by 4 Z2 topological invariants : (n0 ; n1n2n3 ) n0 = 1 : Strong Topological Insulator Surface Fermi circle encloses odd number of Dirac points on all faces ky EF Topological Metal on the surface : 1/4 graphene Spin and momentum correlated. Protected by time reversal symmetry Robust to disorder: impossible to localize or passivate. kx Described by 2D Massless Dirac Hamiltonian n0 = 0 : Weak Topological Insulator (n1n2n3 ) Layered 2D topological insulator : surface states on the sides, but less robust surface states

semiconductor Egap ~ 30 meV Bi1-xSbx Pure Bismuth semimetal Alloy : .09<x<.18 semiconductor Egap ~ 30 meV Pure Antimony semimetal EF La EF Ls La Egap La Ls Ls E EF T L T L T L k Inversion symmetry  Predict: Bi1-xSbx is a strong topological insulator: (1 ; 111). Other predicted strong TI’s : strained HgTe, strained PbxSn1-xTe

Bi1-xSbx Bi2 Se3 ARPES on 3D Topological Insulators EF 5 surface state Hsieh et al. Nature ’08 5 surface state bands cross EF between G and M Zahid Hasan Bi2 Se3 ARPES Experiment + Band theory : Y. Xia et al., Nature Phys. ‘09 Band Theory : H. Zhang et. al, Nature Phys. ‘09 Energy gap: D ~ .3 eV : A room temperature topological insulator Simple surface state structure : A textbook Dirac cone EF

Surface Quantum Hall Effect Fu, Kane ’07; Qi, Hughes, Zhang ’08, Essen, Moore, Vanderbilt ‘09 Broken time reversal symmetry allows a surface gap Orbital QHE: E=0 Landau Level for Dirac fermions 2 1 B n=1 chiral edge state -1 -2 “fractional” integer quantum Hall effect impossible in 2D Allowed on surface of TI because a surface has no boundary. Anomalous QHE: Induce a surface gap by depositing magnetic material Gap due to Exchange field EF M↑ M↓ Chiral Edge State at Domain Wall TI

Superconducting Proximity Effect A route to engineering topological superconductivity Fu, Kane ‘08 s wave superconductor Topological insulator Similar to 2D spinless px+ipy topological superconductor, except T invariant Supports Majorana fermion bound states D -D E SC h/2e TI S.C. M QSHI

In Search of Majorana B SC Kitaev (2006) : Zero energy Majorana modes associated with topological superconductivity offer A topologically protected memory for quantum information Prospects for a (rudimentary) topological quantum computer Ettore Majorana 1906 – 1938? Superconductor – semiconductor nanowire structure Theory: Sau, Lutchyn, Tewari, Das Sarma ’09 Oreg, von Oppen, Alicea, Fisher ’10 Expt : Mourik, …, Kouwenhoven, Nature ‘12 SC InSb nanowire Majorana end state B Zero bias peak in tunneling conductance observed. Attributed to 0D Majorana end state.

Conclusion Topological Insulators have taught us that new phases with useful and/or interesting and/or beautiful properties can be found if you know what is possible and you know where to look. The road ahead: Perfect the materials and phases that we have found. Many materials challenges remain. There has been much recent progress on what is (in principle) possible : - States with fractionalized or non-Abelian excitations. - Symmetry protected topological states (beyond free fermions) - Exotic surface terminations of more conventional states. There has been less progress on “where to look” - May require new theoretical or computational methods - There is much room for good ideas