Topological properties in solids probed by Experiment - Quantum - PowerPoint PPT Presentation

Markus Morgenstern Topological properties in solids probed by Experiment - Quantum Hall effect - 2D TIs - Weak Topological Insulators - Strong topological insulators - … Vienna, 04.08.2014

What is Topology ? Wikipedia: Topology is the study of continuity and connectivity … homeomorphism …. fiber bundles … ….. Pontryagin classes Haussdorf dimension … Massey product 1/35

What is Topology ? Idea: distinguish geometrical objects by integer numbers Example: 2. Path Winding number of closed path coordinate t  [0,1) 1. Define Complex 3. Find plane topological Invariant 4. Proof correctness How often does the path wind around P, which is never touched ? 2/35

Topology in Solids: quantum Hall effect v. Klitzing et al., PRL 45, 494 (80 ), … 2D system - accurate mesurement of Wikipedia: Topology is the study of continuity and connectivity h/e 2 (precision: 10 -10 ) Thouless et al., PRL 49, 405 (82) Chern number is a distinct integer, … homeomorphism …. if the system is gapped, fiber bundles … i.e. a band is either completely graph theory occupied or completely empty 3/35

4/35 Chern number = integer: the argument 1) Define magnetic Brilloun zone (MBZ) by integer number of flux quanta inside each unit cell  wave function has zeros inside the unit cell (Aharonov Bohm phase  0) 2) Combination with required periodicity of MBZ requires a phase mismatch around the zero for a particular real space x phase of u k (x 0 ) Münks, Master thesis. MSU C=0 C=1 3) Integral of the gradient of the phase mismatch along the interface has to be single valued: 0, 2  , 4  … Kohmoto, Ann. Phys. 160, 343 (85) 4) By Stokes theorem, Requirement: the band must this is identical to be full, such that the MBZ is the Chern number densely occupied

5/35 Chern number = integer: filling the band adding disorder Band degeneracy: eB/h ordered 30 Landau disordered level 2 25 Landau level 1 Energy [meV] 20 15 E F E F 10 Landau C=1 E F 5 level 0 E F 0 C=1 0 2 4 6 8 10 magnetic field [T] E F E F C=1 E F 30  xy =B/en Prange, The QHE 25 Ground h/e 2 state 20  xy [k  ] Wave xy 15 function boundary phase factors h/2e 2 10 Conductance with E F at localized states does not h/4e 2 h/3e 2 5 depend on boundary conditions  0 0 2 4 6 8 10 xy B field [T]

Quantum Hall winding number in real space 2D LDOS at B= 12 T, 0.3 K Corbino-Geometry STM tip 12 T 2 DES x B 50nm 0 PRL 101, 256802 (08) Extended state probed by STM (C=1) 0 0 0 0 0 0 0 0 0 0 0 0 0 Prediction: one more flux quantum = one node encircles the flux = 0 0 winding number of zeros 0 0 0 0 0 One node (0) 0 per flux quantum 0 0 6/35 Arovas et al. PRL 60, 619 (85) in extended state 0

Bulk edge corespondance Where is the charge of the quantized Corbino (insulating) Hall voltage (bulk insulating) ? disordered ordered C=0 C=2 e - e - x B C=2 E F 1 1 Prange, The QHE Laughlins argument: one more flux Answer: at the topological phase moves charge from inner to outer rim boundary, where different Chern without energy cost (WF identical) numbers clash  ….  one chiral edge state per Chern number 7/35

Seeing the edge state Edge state = „metallic“ area of high compressibility Scanning SET image (2.2 T) Scanning capacitance image Edge state Suddards et al. NJP 14, 083015 (12) Edge state 15   15 m courtesy A. Yacoby (Harvard) SET - Local potential changes SET conductance - Metallic edge state screens backgate potential for SET 8/35

Part II 2D Topological Insulators (B = 0 T)

Topology in 2D at B = 0 T Make a band gap in 2D by mixing two bands d-vector in 1st Brillouin zone with different parity Inverted bands + k-mixing M from Parity s + spin orbit M E F - Splitting p inverted from k p k-space Formally: Spin 1 Bernevig et al, Science 314, 1757 (05) no backscattering Spin 2 Pauli Skyrmion topology matrix for s,p Nodal line in k-space topological number for one „ spin “ Δ  xy = +/- 1 for 0 < M < 4B (+: Spin 1, - Spin2) 9/35

Experiment: non-trivial topology at B = 0 T Band structure HgTe (DFT) Tuning sign of M by z-confinement seeing the edge current Scanning SQUID (3 K) calculated M > 0 width of edge state: d d within z 200 nm M > 0 M < 0 30  m gap 1 st transport: Ong et al. PRB 28, 2289(83) Zhou et al., PRL 101, 4-point resistance 246807 (08) 500 nA/  m M > 0 Confined bands in 2D HgTe trivial M < 0 Inverted at Γ (M < 0) within (M > 0) bulk con- M >0 duc- tion band Nowack et al., Nature Mat. 12, 787 (13) Büttner et al., Nature Phys 7, 418 (11) König et al., Science 318, 767 (07) 10/35

Scanning tunneling spectroscopy ? (LDOS with high resolution) Heterostructure 2DES ca . 100 nm 2D TI tunneling current: 10 -50 A tunneling current: 10 -10 A STS-Resolution: 100 nm STS-Resolution < 0.1 nm 11/35

Stacked 2D topological insulators = weak 3D topological insulators Graphene Dirac cone Invert by Spin-orbit Gap by confinement (interlayer interaction) Kane et al., Yoshimura et al., PRB 88, 045408 (13) PRL 95, 226801 (05) First experimental weak TI: Bi 14 Rh 3 I 9 cleaved at the dark side ARPES Heavy Graphene lattice Inverted bulk gap spacer Rasche et al., Nature Mat. 12, 422 (13) 12/35

Probing spin transport in 2D TI E F in Non local 2D TI topol. gap voltage meas. E F in current (2D HgTe) conduc- tion band 1.8 K TI/TI bulk/TI bulk/bulk Strong signal if both areas TI bulk/TI small signal, if one area =TI TI/bulk one area = bulk bulk/bulk Brüne et al., Nature Mat. 19/35 8, 485 (12)

Quantum anomolous Hall effect 20/35

21/35 Quantum anomolous Hall effect (Exp.) A ferromagnetic 2D TI 5 quintuple layers SrTiO Chang et al., Science 340, 167 (13)

Part III 3D Topological Insulators (B = 0 T)

2D/3D Topological Insulators Fu Kane Mele 22/35 PRL 98, 106803 (07) PRB 74, 195312 (07) Topological surface state Kramers pair movement in 2D ribbon Physical realization for k y movement Edge state = bulk band property Spin moved from left to right with band gap in bulk = Strong TI 3D Weak TI spin pol. edge state required at E F (- · -)  (- · +) - + - - Dark surface States important for movement (Pfaffian vs. Determinant at TRIM)

3D Topological Insulators Strong TI E(k) dispersion 3D Weak TI TRIM k y k x Dark surface + spin required surface states at E F , all spin polarized and time reversal invariant only relative Bloch wave function phases at TRIMs matter Bulk inversion symmetry of crystal  Sign at TRIM = product of parities of all states below the gap upper surface TRIM = Time reversal invariant momenta (k = -k) 23/35

Materials: 3D Topological Insulators Bulk inversion symmetry of crystal  Sign at TRIM = product of parities of all states below the gap  Band inversion (= exchanged parity) at 1 TRIM (typically Γ ) Energy levels at Γ (Bi 2 Se 3 ) Conduction band Good means to invert bands Spin-orbit interaction electron-electron interaction … valence band Li et al., Rev. Mod. Phys 83, 1057 (11) 24/35

Exp. proof: 3D Topological Insulators 25/35 Our contribution Zhang et al., Nature Phys. 5, 438 (09) Spin polarization Sb 2 Te 3 (disentangled): 85 % PRB 86, 235106 (12) Surfac state Science 325, 178 (09) Bi 2 Se 3 ARPES Bi 2 Te 3 E (k) with spins

26/35 Y. Ando, J. Phys. Soc. Jap. 82, 102001 (13) Materials: topological insulators

27/35 Detecting prohibited backscattering Backscattering prohibited dI/dV at by destructive interference Fourier transform Berry phase: -  /2 STM map of standing Berry phase: +  /2 electron waves Li et al., Rev. Mod. Phys 83, 1057 (11) Joint DOS from ARPES Experiment Joint DOS without backscatter  k k           ( )/ 2 ik x ik x i k x k x 1 2 2 cos( ) e e e x 1 2 1 2 1 2 2  Roushan k k       2 2 1 2 et al. | | cos ( ) x 1 2 Nature 460, 2 1106 (09)

28/35 3D TI: tuning E F = E D Mixing Bi 2 Se 3 and Sb 2 Te 3 strongly p-doped Bi 1.75 Sb 0.25 Bi 2 Te 2 Se 1 Bi 1 Sb 1 Te 1 Se 2 Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 Sb 2 Te 3 strongly n-doped E F Te 1.85 Se 1.15 conduction Bi 2 Se 3 band E F surface state E D valence band not useful for electrical transport and devices  towards devices ARPES data Y. Ando, J. Phys. Soc. Jap. 82, 102001 (13)

3D TI: magnetotransport B  2D type transport Topological surface state transport Bi 2 Se 3 Bi 2 Te 2 Se 1 phase factor of oscillations due to Berry phase 29/35 Y. Ando, J. Phys. Soc. Jap. 82, 102001 (13)

Melnik et al., Nature 3D TI spin transport 511, 449 (14) analysis of ferromagnetic resonance symmetric part antisymmetric part Spins Spin accumulation at interface induces torque  on ferromagnet fit to equation torque by spin accumulation spin torque per current density 30/35