From graphene to Z2 topological insulator

Transcription

1 From graphene to Z2 topological insulator single Dirac topological AL mass U U valley WL ordinary mass or ripples WL U WL AL AL U AL WL Rashba Ken-Ichiro Imura Condensed-Matter Theory / Tohoku Univ.

2 Dirac electrons topological order relativistic condensedmatter physics graphene Z2 topological insulator cf. Landau theory, symmetry breaking Chern number, topological invariants - mathematics Dirac electron engineering spin-orbit interaction spin Hall effect vs. quantum Hall effect anomaly, lattice fermions, etc. high-energy physics Concepts of diverse background related to this research - fusion of ideas from different fields at various levels!

3 Structure of this talk : 1- Focus on Dirac electrons - graphene i) weak localization properties ii) Klein tunneling in graphene absence of backscattering Ando, Nakanishi, Saito, JPSJ Focus on gapless edge modes of a Z2 topological insulator topological insulator = metallic on its surface Structure of our team : Condensed-matter theory group Prof. Yoshio Kuramoto Students : Ai Yamakage, Akira Hotta, Shijun Mao Ken-Ichiro Imura -> Hiroshima Univ. (March ) Kentaro Nomura (Shibata G.) Wataru Izumida (Saito G.) Collaboration: Jerome Cayssol (Bordeaux)

4 What is graphene?

5 Dirac cones in graphene - real space - reciprocal space B A - tight-binding model with only nearest neighbor (NN) hopping K K valley spin = (K, K ) Dirac Hamiltonian sublattice spin = (A,B)

6 Dirac electrons show in the presence of a mass gap (spontaneous) half-integer quantum Hall effect! Kubo formula parity (chiral) anomaly Fermi energy E Deser, Jackiew,Templeton, PRL 82 However, (on a lattice) Dirac cones appear in pairs. Nielsen, Ninomiya, Phys. Lett. 81; NPB 81 cf. in graphene (K and K ) anomaly cancellation

7 Nevertheless, there is a way out! Kane-Mele model : a three story house 1- NN hopping: 2- NNN imaginary hopping: 3- Rashba spinorbit interaction: In the continuum limit, ideal graphene topological mass quantized spin Hall effect (QSH) Rashba Kane-Mele, PRL 06

8 Weak localization properties

9 Backward scattering amplitudes : Time reversal operation : anti-unitary : Quantum interference between A and B = A+B : constructive : destructive WL : peak in MR AL : dip in MR MR : magnetoresistance

10 Effective time-reversal operation defined in the subspace of activated spins : Rashb a SR S Rashb a SR S KI, Y. Kuramoto, Ken Nomura, Europhys. Lett. 89, (2010); Phys. Rev. B 80, (2009).

11 System s WL property under doping determined by the parity of the number Ns =1,2,3 of activated spins if Ns is even if Ns is odd WL AL KI, Y. Kuramoto, Ken Nomura, Europhys. Lett. 89, (2010); Phys. Rev. B 80, (2009).

12 The weak localization phase diagram topological mass U WL WL U AL valley WL AL spins - system s WL property under ordinary mass or ripples WL U single Dirac (massless) AL U AL Rashba - three types of (pseudo) spins : AB, KK & real doping counting of the number Ns=1,2,3 of activated spins KI, Y. Kuramoto, Ken Nomura, Europhys. Lett. 89, (2010); Phys. Rev. B 80, (2009).

13 Klein tunneling

14 Klein paradox/tunneling E 0

15 Klein tunneling in monolayer graphene: Case of normal incidence Incident wave: Reflected wave: orthogonal Berry phase - continuity condition: perfect transmission! Katsnelson et al., Nature Physics, 06

16 bi-layer graphene quadratic, but gapless Pseudo-spin eigenstates: Klein tunneling - normal incidence: - reflected wave: Berry phase perfect reflection!

17 Klein tunneling in Kane-Mele model 1- Normal incidence - crossover from perfect reflection to perfect transmission - Rashba SOI drives the crossover controllable by the gate voltage 2- Angular dependence 3- conductance, Fano comparison with experiments A. Yamakage, KI, J. Cayssol, Y. Kuramoto, Europhys. Lett. 87, (2009).

18 Characterizing Z2 topological insulator by a pair of gapless edge modes

19 Bulk/edge correspondence nature of the edge modes of topological insulators Bulk : non-trivial topological order e.g., Edge : gapless mode In Z2 topological insulators, Only a single pair of gapless edge mode is protected by time-reversal symmetry. Remark: Such gapless modes were also known in lattice gauge theory (QCD). Creutz, Horvath, PRD, 94; Creutz, Rev. Mod. Phys., 01

20 BHZ tight-binding Hamiltonian : Bernevig, Hugues, Zhang, Science 06 where - relevance to experiments in HgTe/CdTe quantum well Koenig et al., Science 07 - Edge spectrum in the straight edge geometry: trivial vs. topological phases straight edge geometry trivial phase: topological phase:

21 BHZ model in the zigzag edge geometry E KI, A. Yamakage, Shijun Mao, A. Hotta, Y. Kuramoto, in preparation. E k k zigzag edge geometry 1- completely flat edge spectrum at cf. edge modes of graphene in the zigzag geometry 2- edge modes reentrant in k-space edge solution in the strip geometry: 1 with k edge bulk edge bulk edge analytic solution available based on the conjecture inspired by numerical tests reentrant edge modes at reentrance!

22 Concluding remarks In the study of Dirac electron systems (= graphene + spin-orbit interaction) 1- Contemporary viewpoint to the theory of weak localization simpler understanding of the WL phase diagram 2- Klein tunneling realized in a semi-conductor p-n junction Characterizing a Z2 topological insulator (BHZ model) from the viewpoint of gapless edge modes - comparison of edge modes in i) QSH, ii) QHE and iii) graphene - edge modes of BHZ model shows specific features in the zigzag edge geometry: 1- completely flat edge spectrum 2- reentrance of edge modes in k-space

23 Future of the GCOE program, future of our Dirac electron project ~ beyond the interdisciplinarity ~ our future condensed-matter physics high-energy physics, mathematics present (we are here) Physics looks simpler and more suggestive from the viewpoint of Dirac electrons!