2D Materials IF-USP

Transcription

1 2D Materials IF-USP Luis G. Dias da Silva Instituto de Física, Universidade de São Paulo São Paulo, Brazil - luisdias@if.usp.br

2 Today s road map. Graphene: the first 2D material. Impurities in graphene: hiding in plain sight. Beyond graphene: electron transport in 2D topological insulators.

3 Graphene basics: triangular lattices. Direct lattice Reciprocal lattice (k)

4 Graphene: density of states Wallace, P.R., Phys. Rev (1947) (E) E (linear). Phil R. Wallace

5 Dirac cones: massless fermions Analogy: linear dispersion of massless relativistic particles ( E=pc )

6 Impurities in graphene

7 Impurity magnetism in graphene: STM González Herrero et al., Science 352, 437 (2016) Yu Zhang et al., Phys. Rev. Lett. 117, (2016) H adatoms in graphene. Image (left) and STS spectra (right) showing Hubbard peaks. U~20 mev Long-range spin-induced signals. Vacancies in graphene. STS spectra (right) showing Hubbard peaks U~30 mev. Our calculations: U~640 mev V. Miranda, LDS, C.H. Lewenkopf PRB (2016) Editor s Suggestion

8 Does impurity symmetry play a role in electron transport in graphene?

9 Impurities: adatoms and vacancies Adatoms: TOP and HOLLOW SITE (HS) (s or d z2 orbitals) Vacancies: Substitutional (VAC) and reconstructed (REC). Ruiz-Tijerina, LDS PRB (2016)

10 Impurities: coupling function TOP and REC: no nodes in HS and VAC: nodes in : Uchoa, Rappoport, Castro-Neto, PRL 106, (2011) Ruiz-Tijerina, LDS PRB (2016)

11 Kubo formula for the resistivity tensor. However, for HS and VAC, vanishes near the K points at c.n.! Ruiz-Tijerina, LDS PRB (2016) HS and VAC give NO CONTRIBUTION to the =0 resistivity! Invisible J. Duffy et al. PRB (2016)

12 Resistivity for impurity ensembles (U=0). 100% TOP 5% TOP, 95% HS 100% REC 5% REC, 95% VAC Impurity level at the Fermi energy: Power-law divergence in the T 0 resistivity: Signature of a Quantum Phase Transition. Ruiz-Tijerina, LDS PRB (2016)

13 Interacting case: invisibility + Kondo effect. 100% =0 - QPT 0 - Kondo 95% 5% Ruiz-Tijerina, LDS PRB (2017)

14 2D materials beyond graphene: Topological insulators

15 2016 Physics Nobel Prize: Topological matter The Hall conductivity is proportional to a Chern number (Berry-phase-like) Thouless, Kohmoto, Nightingale, den Nijs, Phys. Rev. Lett. 49, 405 (1982) David Thouless System is periodic (BZ is a torus in k-space) - There is an uniform magnetic field in the system. - Fermi energy lies in a gap with N F filled bands. INSULATOR? CONDUCTOR? Neither? = 0,1,2,... : filling factor. Depends only on the topology of the BZ states.

16 2016 Physics Nobel Prize: Topological matter The Hall conductivity is proportional to a Chern number (Berry-phase-like) David Thouless 2016 Thouless, Kohmoto, Nightingale, den Nijs, Phys. Rev. Lett. 49, 405 (1982) = 0,1,2,... : filling factor. Depends only on the topology of the BZ states. nobelprize.org

17 Haldane: Hall conductance with zero flux. -M +M Gap: Duncan Haldane 2016 F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1988) Hall conductance also given by a Chern number: n c = 1: Topological phases

18 Kane and Mele: Quantum Spin Hall effect. -M Gap: +M Topological phase : Chern number i SO Z 2 invariant Charles Kane New ingredients: - Particles with spin s. - Spin-Orbit coupling SO (TRS preserved) - Assuming no Rashba SO. Spin-polarized Edge states C. L. Kane, E. J. Mele Phys. Rev. Lett. 95, (2005) Phys. Rev. Lett. 95, (2005). Topological Non-Topological

19 QSH effect in HgTe QWs: Theory. HgTe quantum wells: inverted band structure. Shoucheng Zhang Gap: Chern number Spin-polarized Edge states Andrei Bernevig Bernevig, Hughes, Zhang, Science 314, 1757 (2006)

20 QSH effect in HgTe QWs: Experiment d<d c Laurens Molenkamp Konig et al, Science 318, 766 (2007) d>d c d>d c Edge state condutance I-L=20 m W=13 m (d< d c ) II-L=20 m W=13 m (d>d c ) III-L=1 m W=1 m (d>d c ) IV-L=1 m W=0.5 m (d>d c )

21 A future Nobel Prize? Physics Frontiers Prize 2013 also: APS Buckley Prize 2012

22 Transport in 2D topological insulators.

23 Transport in 2D topological insulators. Quantum spin Hall effect: HgTe quantum wells Konig et al, Science 318, 766 (2007) Minimal model: BHZ Hamiltonian Bernevig, Hughes, Zhang, Science 314, 1757 (2006)

24 Motivation: HgTe QW n-[p,ti,n ]-n junctions. Gusev et al., Phys. Rev. Lett. 110, (2013) The gate voltage in the central region can be tuned to create n-p-n, n- TI-n and n-n -n junctions in HeTe quantum wells.

25 Local currents: Topological Insulator ribbon. Edge states BHZ parameters: Scharf, Matos-Abiague, Fabian, PRB. 86, (2012)

26 Transport: n-ti-n junction Tight binding calculations: (Paramenters)

27 Non-zero magnetic field: still edge states!

28 Non-zero magnetic field: n-gap-n junction Gap in the spin down: spin polarized edge mode.

29 Non-zero magnetic field: n-p-n junctions

30 Non-zero magnetic field: n-p-n junction

31 Spin oscillations vs gate voltage. Oscillations in the spin down current: spin polarized edge modes.

32 Spin oscillations vs gate voltage. Oscillations in the spin down current: spin polarized edge modes.

33 18th Brazilian Workshop on Semiconductor Physics BWSP thBVVSP August Maresias, São Paulo Invited Speakers 1.Andrew Mitchell University College Dublin. 2. Avik Gosh University of Virginia. 3.Annica Black-Schaffer Uppsala University. 4.Arne Laucht University of New South Wales. 5.Christiano de Matos Universidade Mackenzie. 6.Christoph Deneke LNNano. 7.Dominik Zumbuhl University of Basel. 8.Felix von Oppen Freie Universität Berlin. 9.Gian Salis IBM Zürich. 10.Ingrid Barcelos LNLS. 11.J. Carlos Egues USP S. Carlos. 12.Leandro Malard Moreira UFMG. 13.Paulo V. Santos Paul-Drude-Institut. 14.Sergio Ulloa Ohio University. 15.Sergey A. Dvoretsky ISP Novosibirsk. 16.Sven Höfling Universität Würzburg. 17.Vanessa Sih University of Michigan. 18.Vladimir Falko National Graphene Institute. 19.Werner Wegscheider ETH Zürich.

34 Current Group Members Luis Dias da Silva Professor David Ruiz-Tijerina Post-doc (now at Manchester) Dimy Nanclares Ph.D. student Marcos Medeiros Master s student Jesus Cifuentes Master s student Raphael Levy Master s student Rafael Magaldi Undergrad student

35 Acknowledgements Collaborators (in the BHZ transport project): Caio Lewenkopf (UFF) Leandro Lima (UFF) Caio Lewenkopf Leandro Lima Support: CAPES (PNPD program), FAPESP (2016/ ); CNPq (307107/ and /2014-9); USP-PRP Q-Nano. Thank you for your attention.

36 Vacancies in graphene Lieb s theorem for a bipartite system: NA-NB Tight-binding calculations: single vacancy leads to midgap state Localized state at the vacancy site Typical delocalized state Vacancy tight-binding model:

37 Mid-gap state in the presence of vacancies. Tight-binding calculations: single vacancy leads to midgap state Localized state at the vacancy site Typical delocalized state Vacancy tight-binding model: Agreement with previous results V. M. Pereira et al., PRB (2008)

38 What about the Hubbard U for vacancies? Tight-binding (single orbital) contributions to the charging energy of the localized state 0> give: with Diagonal Off-diagonal V. Miranda, LDS, Caio Lewenkopf PRB (2016)

39 What about the Hubbard U for vacancies? Tight-binding (single orbital) contributions to the charging energy of the localized state 0> give: Diagonal Off-diagonal V. Miranda, LDS, Caio Lewenkopf PRB (2016)

40 What about the Hubbard U for vacancies? Tight-binding (single orbital) contributions to the charging energy of the localized state 0> give: with Diagonal Off-diagonal Our estimates: In the NRG calculations, we use: consistent with recent DFT+cRPA calculations M. Schüler et al., PRL (2013) V. Miranda, LDS, Caio Lewenkopf PRB (2016)