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Dirac fermions with zero effective mass in condensed matter: new perspectives

Dirac fermions with zero effective mass in condensed matter: new perspectives. Lara Benfatto* Centro Studi e Ricerche “Enrico Fermi” and University of Rome “La Sapienza”. *e-mail: lara.benfatto@roma1.infn.it www: http://www.roma1.infn.it/~lbenfat/. 29-30 Novembre Conferenza di Progetto.

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Dirac fermions with zero effective mass in condensed matter: new perspectives

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  1. Dirac fermions with zero effective mass in condensed matter: new perspectives Lara Benfatto* Centro Studi e Ricerche “Enrico Fermi” and University of Rome “La Sapienza” *e-mail: lara.benfatto@roma1.infn.it www: http://www.roma1.infn.it/~lbenfat/ 29-30 Novembre Conferenza di Progetto

  2. Outline • Why Dirac fermions? Common denominator in emerging INTERESTING new materials • Dirac fermions from lattice effect: the case of graphene • Bilayer graphene: “protected” optical sum rule • Dirac fermions from interactions: d-wave superconductivity • Collective phase fluctuations: Kosterlitz-Thouless vortex physics • Acknowledgments: C. Castellani, Rome, Italy T.Giamarchi, Geneva, Switzerland S. Sharapov, Macomb (Illinois), USA J. Carbotte, Hamilton (Ontario), Canada

  3. Basic understanding of many electrons in a solid • k values are quantized • Pauli principle: N electrons cannot occupy the same quantum level • Fermi-Diracstatistic:all level up to the Fermi level are occupied • Excitations: unoccupied levels Quadratic energy-momentum dispersion

  4. Effect of the lattice • Allowed electronic states forms energybands

  5. Effect of the lattice • Allowed electronic states forms energybands and have an “effective mass” Quadratic energy-momentum dispersion Semiconductor physics!!

  6. Dirac fermions from lattice effects: graphene • One layer of Carbon atoms

  7. Au contacts SiO2 GRAPHENE Si Dirac fermions from lattice effects: graphene • One layer of Carbon atoms • Graphene: a 2D metal controlled by electric-field effect Vg

  8. Dirac fermions from lattice effects: graphene • Carbon atoms: many allotropes • Graphene: a 2D metal controlled by electric-field effect • In momentum space

  9. Dirac fermions from lattice effects: graphene • Carbon atoms: many allotropes • Graphene: a 2D metal controlled by electric-field effect • In momentum space: Dirac cone

  10. Universal conductivity • Despite the fact that at the Dirac point there are no carriers the system has a finite and (almost) universal conductivity!! Dirac fermions are “protected” against disorder Deviations: charged impurities, self-doping, Coulomb interactions, vertex corrections

  11. Bilayer graphene: tunable-gap semiconductor Oostinga et al. arXiv:0707.2487 (2007) LARGE gap (a fraction of the Fermi energy) Does it affect the total spectral weight of the system? Ohta et al. Science 313, 951 (2006) ≈

  12. Bilayer graphene: tunable-gap semiconductor Oostinga et al. arXiv:0707.2487 (2007) LARGE gap (a fraction of the Fermi energy) Does it affect the total spectral weight of the system? Ohta et al. Science 313, 951 (2006) Analogous problem in oxides: electron correlations decrease considerably the carrier spectral weight

  13. “Protected” optical sum rule The optical sum rule is almost constant despite the large gap opening: large redistribution of spectral weight is expected (a prediction to be tested experimentally) ≈ L.Benfatto, S.Sharapov and J. Carbotte, preprint (2007)

  14. Cooper pair Dirac fermions from interactions: d-wave superconductors • Example of High-Tc superconductor La1-xSrxCu2O4: • quasi two-dimensional in nature • CuO2layers are the key ingredient • LaandSrsupply “doping” • Superconductivity: formation of Cooper pairs which “Bose” condense High Tc: not explained within standard BCS theory for “conventional” low-Tc superconductors New quasiparticle excitations! New “collective” excitations!

  15. Dirac fermions from interactions: d-wave superconductors s-wave Conventional s-wave SC: Δ=const over the Fermi surface Gapped excitations

  16. massless Dirac fermions vF vD Dirac fermions from interactions: d-wave superconductors s-wave ¹ d-wave Conventional s-wave SC: Δ=const over the Fermi surface Gapped excitations High-Tc d-wave SC: Δ vanishes at nodal points Gapless Dirac excitations

  17. vF vD Measuring Dirac excitations Gomes et al. Nature 447, 569 (2007) Dirac fermions are “protecetd” against disorder Low-energy part does not depend on the position High-energy part is affected by position, disorder, etc.

  18. Collective phase fluctuations: vortices! • In BCS superconductors superconductivity disappears when |Δ| 0 at Tc: standard paradigm applies • In HTSC superconductivity is destroyed by phase fluctuations where |Δ| remains finite Crucial role of vortices water vortex

  19. Collective phase fluctuations: vortices! • In BCS superconductors superconductivity disappears when |Δ| ->0 at Tc: standard paradigm applies • In HTSC superconductivity is destroyed by phase fluctuations where |Δ| remains finite Crucial role of vortices Kosterlitz-Thouless like physics J.M.K. and D.J.T. J. Phys. C (1973, 1974) Superconducting hc/2e vortex Superconducting vortex is a topological defect in phase .  winds by 2π around the vortex core

  20. Understanding Kosterlitz-Thouless physics

  21. Understanding Kosterlitz-Thouless physics • Need of a new theoretical approach to the Kosterlitz-Thouless transition • Mapping to the sine-Gordon model • Crucial role of the vortex-core energy • “Non-universal” jump of the superfluid density L.Benfatto, C.Castellani and T.Giamarchi, PRL 98, 117008 (07) L.Benfatto, C.Castellani and T.Giamarchi, in preparation

  22. Understanding Kosterlitz-Thouless physics • Need of a new theoretical approach to the Kosterlitz-Thouless transition • Mapping to the sine-Gordon model • Crucial role of the vortex-core energy • “Non-universal” jump of the superfluid density L.Benfatto, C.Castellani and T.Giamarchi, PRL 98, 117008 (07) L.Benfatto, C.Castellani and T.Giamarchi, in preparation • Non-linear field-induced magnetization L.Benfatto, C.Castellani and T.Giamarchi, PRL 99, 207002 (07)

  23. The absence of the superfluid-density jump • In pure 2D superfluid/superconductors Js jumps discontinuously to zero, with an universal relation to TKT 4He films McQueeney et al. PRL 52, 1325 (84)

  24. YBCO D.Broun et al, cond-mat/0509223 The absence of the superfluid-density jump • In pure 2D superfluid/superconductors Js jumps discontinuously to zero, with an universal relation to TKT

  25. The absence of the superfluid-density jump L.Benfatto, C. Castellani and T. Giamarchi, PRL 98, 117008 (07)

  26. Non-linear magnetization effects • Field-induced magnetization is due to vortices but one does not recover the LINEAR regime as T approaches Tc Tc Correlation length (diverges at Tc) M=-a H L. Li et al, EPL 72, 451 (2005)

  27. Magnetization above TKT L.B. et al, PRL (2007) ξ diverges at Tc! No linear M in the range of fields accessible experimentally

  28. Magnetization above TBKT L.B. et al, PRL (2007) ξ diverges at Tc! No linear M in the range of fields accessible experimentally

  29. Conclusions • New effects in emerging low-dimensional materials • Need for new theoretical paradigms: quantum field theory for condensed matter borrows concepts and methods from high-energy physics Dirac cone!! Einstein cone

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