Topological Materials

Topological Materials Hsin Lin Institute of Physics, Academia Sinica, Taipei, Taiwan Colloquium : Topological band theory, A. Bansil, H. Lin, T. Das, Reviews of Modern Physics 88, 021004 (2016). May 21, 2018 Donghua University

Institute of Physics, Academia Sinica, Taipei, Taiwan: Cheng-Yi Huang, Liangzi Yao National University of Singapore: Guoqing Chang (Princeton U.), Chuang-Han Hsu, Bahadur Singh Arun Bansil (Northeastern Univ.) BaoKai Wang, Susmita Basak, W. Al-Sawai, Yung Jui Wang M.Z. Hasan s group (Princeton Univ.): Ilya Belopolski, Daniel S. Sanchez, Songtian S. Zhang, Nasser Alidoust, Y. Xia, David Hsieh (Caltech), Guang Bian (Missouri), Madhab Neupane (Central Florida), Hao Zheng (Shanghai Jiao Tong), L. Andrew Wray (NYU), Su-Yang Xu (MIT), Vidya Madhavan (UIUC) Yoshinori Okada (OIST), Ilija Zeljkovic (Boston College) Liang Fu (MIT), Titus Neupert (Zurich) Rajendra Prasad (IITK), Tanmoy Das (Indian Institute of Physics) Feng-Chuan Chuang (NSYSU, Taiwan): Chia-Hsiu Hsu(SUST), Zhi-Quan Huang, Christian Crisostomo Horng-Tay Jeng (NTHU/IOP), Han Hsu (Natl. Central U.), Chung-Yuan Ren(NKNU), Shin-Ming Huang (NSYSU, Taiwan), Tay-Rong Chang (NCKU) : Xiaoting Zhou Gengchiau Liang (NUS ECE) Shengyuan A. Yang (Singapore Univ of Technology and Design)

Outline Introduction: band topology Topological phase transition: TlBi(Se,S) Weyl semimetals: TaAs, LaAlGe New Fermions : RhSi Spin filter and spin separator : Silicence Topological crystalline insulators : SnTe, Ca2As

Energy Topological Insulators Fu & Kane, PRL 100, 096407 (2008) New possibilities for Fundamental physics Spintronics Quantum computing k x k y

Energy Ideal Dirac cone one-to-one spinmomentum locked E=const. k y k x k x k y backward scattering suppressed +k -k

k y Energy backward scattering suppressed k x E=const. k y k x Spin polarized conducting edge states

Energy Band structure One atom Crystal E n (k) E 4 E 3 Energy E 2 E 1 E n k 0 2π/a

Topology Gaussian curvature χ χ=2 χ=0 χ=-2

Energy [ev] Adiabatic transformation χ=2 χ=2 + +? + + EF L Γ X W L Γ X W

Energy [ev] Topological phase transition χ=2 χ=0 Trivial Critical Non-trivial + + 0 + + + EF L Γ X W L Γ X W

Energy Momentum Band inversion trivial critical Non-trivial

Parity analysis (inversion symmetry)

Bulk-boundary Correspondence Theorem Energy Energy Energy Metallic surface/edge states Z 2 :even EF Γ M Z 2 :odd EF Γ M Momentum k x k y time reversal: E(k, )=E(-k, )

Energy Band inversion and edge states (a) m>0 (b) m=0 (c) m<0 band gap (d) m(x)>0 k=k 0 k=k 0 k=k 0 m(x)<0 Momentum k Position x

Bi 2 Se 3, Spin-orbit coupling Y. Xia et al., Nature Physics 5, 398 (2009).

Energy (ev) Surface calculation + surface probe First-principles calculations DFT (KKR, LAPW, Plane Wave) + Angle resolved photoemission (ARPES) Single-Dirac-cone surface states in topological insulator Bi 2 Se 3 Y. Xia et al., Nature Physics 5, 398 (2009).

Finite size slab Ab initio calculations for Electronic Band Structures Wannierization Semi-infinite slab Material-specific Tight-binding Hamiltonians Surface state Calculations Angle-resolved photoemission (ARPES) and Scanning tunneling Spectroscopy (STS)

Weyl semimetal Wan, Turner, Vishwanath, and Savrasov, PRB 2011

Chern numbers and Fermi arcs unpublished

Our Roadmap for 3D New Topological Materials 2 nd Gen, Bi 2 Se 3 /Bi 2 Te 3 : single Dirac cone, large bulk gap, but naturally doped with electrons or holes Nat. Phys. 5, 398 (2009); PRL 103, 146401 (2009); Nature 460, 1101 (2009). Half-Heuslers, Li 2 AgSb, quaternary chalcogenides, famatinites, ternary GeBi 2 Te 4, Bi 2 Te 2 Se families: tunability of lattice/dopants Nat. Mat. 9, 546 (2010); PRB 82, 125208 (2010); arxiv:1007.5111; New J. Phys. 13, 085017 (2011); New J. Phys. 13, 095005 (2011); PRB 85, 235406 (2012); PRB 87, 121202R (2013). TlBiSe 2 family: isolated single Dirac cone, topological phase transition PRL 105, 036404 (2010); Science 332, 560 (2011); PRB 86, 115208 (2012); Nat. Commun. 6, 6870 (2015); PRB (2016) (Pb/Sn)Te family: first topological crystalline insulator, topological phase transition. Nat. Commun. 3, 982 (2012); Nat. Commun. 3, 1192 (2012); PRB 87, 235317 (2013); Science 341, 1496 (2013); Nat. Phys. 10, 572 (2014); PRB 92, 075131 (2015); Nat. Commun. 6, 6559 (2015); Nat. Mat. 14, 318 (2015);

Our Roadmap for 3D New Topological Materials TaAs family: first Weyl semimetal. Nat. Commun. (2015) ; Science (2015); Nat. Phys. (2015); Science Advances (2015); PRB (2015); ACS nano (2016); Nat. Commun. (2016); PRL (2016); SrSi 2 Weyl semimetal: Double Weyl, no mirrors PNAS (2016) (Mo,W)Te 2 : Type II Weyl semimetal Nat. Commun. (2016), Nature Communications 7, 13643 (2016), Phys. Rev. Lett. 117, 266804 (2016), Phys. Rev. B 94, 085127 (2016) PbTaSe 2, TlTaSe 2 : Topological nodal-line semimetals: Nat. Commun. (2016) ; PRB 93, 121113 (2016); PRB 93, 245130 (2016) ; Ta 3 S 2 Weyl semimetal: large separation between Weyl nodes. Science Advances (2016) Co 2 TiSi Heuslers: magnetic Weyl/nodal-line semimetal with high Curie temperature. Scientific Reports (2016)

Our Roadmap for 3D New Topological Materials LaAlGe, CeAlGe: magnetic Type II Weyl semimetal Science Advances (2016), PRB (2018) VAl3 family: Type-II Topological Dirac Semimetals PRL (2017) WC, ZrTe: New Fermion, triply-degenerate Scientific Reports (2017) Kramers theorem-enforced Weyl fermions: Ag3BO3, TlTe2O6, Ag2Se arxiv:1611.07925 ZrPtGe: Saddle-like topological surface states PRB 97, 075125 (2017) RhSi: New Fermion, 6-fold- and 4-fold-degenerate chiral fermions PRL 119, 2016401 (2017) Co 2 MnGa: Hopf and chain link PRL 119, 156401 (2017)

Our Roadmap for 3D New Topological Materials LiOsOo 3 : Cubic Dirac cone PRM 2, 051201R (2018) TlMo 3 Se 3 : a candidate for topological superconductor PRB 97, 014510 (2018) Ca2As: Rotation anomaly topological crystalline insulator arxiv:1805.05215

Pseudo PbTe: TlBiTe 2 1Γ 3X 4L PbTe - - + SnTe - - - Both are Z 2 trivial. Te Tl (Pb) k z + [111] Te z x y Bi (Pb) k x + Γ L X k y H. Lin et al., Physical Review Letters 105, 036404 (2010).

S.-Y. Su et al., Science 332, 560 (2011)

S.-Y. Su et al., Science 332, 560 (2011)

S.-Y. Su et al., Science 332, 560 (2011)

Energy Topological phase transition S.-Y. Su et al., Science 332, 560 (2011)

Weyl semimetal without Breaking Time-reversal Symmetry TaAs class: (Ta,Nb)(As,P) Stoichiometric compound I4 1 md (C 4v ) No inversion symmetry C 4 rotation axis 2 mirror planes, xz & yz Huang et al., Nat. Comm., 6, 7373 (2015) S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia, A. Bansil, H. Lin, and M. Z. Hasan, Nature Commun. 6, 7373 (2015)

band structure of TaAs S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia, A. Bansil, H. Lin, and M. Z. Hasan, Nature Commun. 6, 7373 (2015) No SOC SOC included

Xu, Belopolski, Alidoust, Neupane, Bian, Zhang, Sankar, Chang, Yuan, Lee, Huang, Zheng, Ma, Sanchez, Wang, Bansil, Chou, Shibayev, Lin, Jia, Hasan, Science 349, 613 (2015). Weyl points and topological chiral charge in TaAs: (a) Position of WPS and nodal ringin TaAs. (e) A schematic for projected WPS on (001) surface. (f) spin texture

Surface States of TaAs Xu, Belopolski, Alidoust, Neupane, Bian, Zhang, Sankar, Chang, Yuan, Lee, Huang, Zheng, Ma, Sanchez, Wang, Bansil, Chou, Shibayev, Lin, Jia, Hasan, Science 349, 613 (2015).

LaAlGe Xu, Alidoust, Chang, Lu, Singh, Belopolski, Sanchez, Zhang, Bian, Zheng, Husanu, Bian, Huang, Hsu, Chang, Jeng, Bansil, Strocov, Lin, Jia and Hasan, arxiv:1603.07318

LaAlGe Xu, Alidoust, Chang, Lu, Singh, Belopolski, Sanchez, Zhang, Bian, Zheng, Husanu, Bian, Huang, Hsu, Chang, Jeng, Bansil, Strocov, Lin, Jia and Hasan, arxiv:1603.07318

RhSi SG#198 4-fold 2-fold Γ Guoqing Chang, Su-Yang Xu, Benjamin J. Wieder, Daniel S. Sanchez, Shin-Ming Huang, Ilya Belopolski, Tay-Rong Chang, Songtian Zhang, Arun Bansil, Hsin Lin, and M. Zahid Hasan, PRL (2017). R 6-fold 2-fold

RhSi Guoqing Chang, Su-Yang Xu, Benjamin J. Wieder, Daniel S. Sanchez, Shin-Ming Huang, Ilya Belopolski, Tay-Rong Chang, Songtian Zhang, Arun Bansil, Hsin Lin, and M. Zahid Hasan, PRL (2017).

Summary We have identified many materials families of various topological phases. Metals with nontrivial band topology could be more interesting. Bulk states: Weyl, Dirac, nodal-line, new fermions. Surface states: Fermi arcs, drumhead surface states. RhSi exhibit 6-fold- and 4-fold-degenerate chiral fermions in the bulk and large Fermi arcs on the surface

Surface/edge states 2D quantum spin Hall 3D topological insulator

New 2D material with graphene-like structure Without SO z K x c With SO Compare with graphene: sp 2 sp 3 C.C. Liu et al., PRB 84 (2011) & PRL 107 (2011) K 40

Quantum spin Hall (2D Topological insulator) Fu and Kane s parity analysis:prb 76, 045302 (2007) Parity of each valence band Sign of parity product Gap (mev) Z 2 = 1 Z 2 = 1 8 23 M M K Γ M Z 2 = 1 72 Z 2 = 0 554 The 2D systems of Si, Ge, and Sn are 2D QSH materials, whereas Pb is not C.C. Liu et al., PRB 84 (2011) & PRL 107 (2011)

Under E Field: Inversion Symmetry Breaking Gap evolution z A B x Drummond et al., PRB 85, 12 The gap size reduces linearly as we turn on the electric field: At E z = E c, the gap reduces to zero As E z > E c, the system reopens a gap

E-field tunable topological phase transition E z = 0 0 < E z < E c E z = E c E z > E c AB AB QSH phase QSH phase Critical phase BB AA Band insulator silicene Bi 2 Se 3

A low-energy effective description 1 st term = Hopping term 2 nd term = Intrinsic NNN SOC 3 rd term = Rashba NNN SOC 4 th term = Staggered sublattice potential 5 th term = Zeeman splitting 4 th term turns out driving the transition between QSH and trivial BI C.-C. Liu, H. Jiang and Y. Yao, PRB 84, 195430 (2011). W.-F. Tsai, C.-Y. Huang, T.-R. Chang, H. Lin, H.-T. Jeng, and A. Bansil, Nature Communications 4:1500 (2013).

Quantum spin Hall: spin separator Spin polarized conducting edge states Silicene Spin Separator

Gapless edge states G. Gupta, H. Lin, A. Bansil, M. B. A. Jalil, C.-Y. Huang, W.-F. Tsai, and G.C. Liang, Applied Physics Letters 104,032410 (2014).

High efficiency spin separator G. Gupta, H. Lin, A. Bansil, M. B. A. Jalil, C.-Y. Huang, W.-F. Tsai, and G.C. Liang, Applied Physics Letters 104,032410 (2014).

Field-tunable spin separator G. Gupta, H. Lin, A. Bansil, M. B. A. Jalil, C.-Y. Huang, W.-F. Tsai, and G.C. Liang, Applied Physics Letters 104,032410 (2014).

Field-tunable High Efficiency Spin Filter Basic idea (from bulk property): E z > 0 K K K

Field-tunable High Efficiency Spin Filter Basic idea (from bulk property): E z > 0 h K μ 0 K K

Quantum Point Contact E F μ 0 K K K K *In Rycerz et al., Nat. Phys. 3 (2007), edge state property is used in graphene!

Conductance (e 2 /h) High-efficiency spin filter Iterative Green s function method for two-terminal conductance [T. Ando, PRB 44 (1991)] Spin polarization L s =0 L s =8 μ 0 /t W.-F. Tsai, C.-Y. Huang, T.-R. Chang, H. Lin, H.-T. Jeng, and A. Bansil, Nature Communications 4:1500 (2013).

Robustness against weak (nonmagnetic) disorder Disorder-averaged spin polarization as a function of the maximum strength of the random onsite potential V w. Disorder-averaged spin polarization as a function of the perrcentage of edge vacancies r in the constriction. W.-F. Tsai, C.-Y. Huang, T.-R. Chang, H. Lin, H.-T. Jeng, and A. Bansil, Nature Communications 4:1500 (2013).

Summary At low energies, silicene can be described by two nearly fully spin-polarized (massless/massive) Dirac cones in the presence of the perpendicular E field Quantum spin Hall insulators can be used as a spin separator. It is possible to localize conducting channels anywhere in the 2D silicene by applying an inhomogeneous electric field. The proposed spin filter gives rise to nearly 100% spinpolarized currents via gate control More functional electronic devices based on 2D spin-orbit thin films can be anticipated in the future!

Thank you for your attention!

Energy Crystal symmetry protected Dirac node Metallic surface/edge states Z 2 TI TCI -i +i EF Γ X Γ Momentum X Mirror eigenvalues ±i

Topological crystalline insulator (Pb,Sn)Te 1Γ 3X 4L PbTe - - + SnTe - - - T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, Nature Communications 3, 982 (2012).

Intrinsic band inversion in SnTe T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, Nature Communications 3, 982 (2012).

Surface states and four Dirac cones Y. J. Wang, W.-F. Tsai, H. Lin, S.-Y. Xu, M. Neupane, M. Z. Hasan, and A. Bansil, PRB 87, 235317 (2013). T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, Nature Communications 3, 982 (2012).

Experimental observation Xu, Liu, Alidoust, Qian, Neupane, Denlinger, Wang, Wray, Cava, Lin, Marcinkova, Morosan, Bansil, Hasan, Nature Communications 3, 1192 (2012).

Two coaxial Dirac-cone model Te Sn Sn Sn Te Te Chen Fang, Matthew J. Gilbert, Su-Yang Xu, B. Andrei Bernevig, M. Z. Hasan, PRB 88, 125141 (2013). Junwei Liu, Wenhui Duan, and Liang Fu, Phys. Rev. B 88, 241303(R) (2013). Y. J. Wang, W.-F. Tsai, H. Lin, S.-Y. Xu, M. Neupane, M. Z. Hasan, and A. Bansil, PRB 87, 235317 (2013).

Spin texture Te Sn

Observed spin texture Xu, Liu, Alidoust, Qian, Neupane, Denlinger, Wang, Wray, Cava, Lin, Marcinkova, Morosan, Bansil, Hasan, Nature Communications 3, 1192 (2012).

Topological Crystalline Insulators Including Crystalline Symmetry (Spinless) Spinless Topological Crystalline Insulator Including Crystalline Symmetry (Spinfull) Mirror-protected Topological Crystalline Insulator SnTe Fu, L. Phys. Rev. Lett., 2011, 106, 106802 Hsieh, T. H.; Lin, H.; Liu, J.; Duan, W.; Bansi, A. & Fu, L. Nat. Commun., 3, 982 (2012)

Rotation Anomaly Topological Crystalline Insulators Including Crystalline Symmetry (Spinfull) Opposite Helicity The interaction changes sign after the C2z operation arxiv:1709.01929

Band Structures 6 arxiv:1805.05215

Topological phases Number of Band Inversions Topological Invariants 6 arxiv:1805.05215

arxiv:1805.05215 6 Nontrivial Surface States

6 Breaking Nontrivial Mirror Planes arxiv:1805.05215

arxiv:1805.05215 Topological Phase Transition Between Two TCIs Ca 2 Sb Sr 2 Sb

Topological Phase Transition Between Two TCIs Ca 2 Sb Sr 2 Sb

Summary SnTe Surface states exhibit gapless Dirac cones with Dirac nodes protected by mirror symmetry. -- Spin texture can be understood by a two-coaxial-dirac-cone model. New topological crystalline insulators are predicted in Ca2As crystal family. More topological materials can be anticipated in the future!

Thank you for your attention!