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Vladimir Cvetković

Vladimir Cvetković

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Vladimir Cvetković

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  1. Electronic Multicriticality In Bilayer Graphene Vladimir Cvetković National High Magnetic Field Laboratory Florida State University Physics Department Colloquium Colorado School of Mines Golden, CO, October 2, 2012

  2. Superconductivity National High Magnetic Field Laboratory TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

  3. Collaborators Dr. Robert E. Throckmorton Prof. Oskar Vafek NSF Career Grant (O. Vafek): DMR-0955561 V. Cvetkovic, R. Throckmorton, O.Vafek, Phys. Rev. B 86, 075467 (2012)

  4. Graphite Carbon allotrope Greek (γράφω) to write Graphite: a soft, crystalline form of carbon. It is gray to black, opaque, and has a metallic luster. Graphite occurs naturally in metamorphic rocks such as marble, schist, and gneiss. U.S. Geological Survey Mohs scale 1-2

  5. Graphite electronic orbitals • Hexagonal lattice • space group P63/mmc • Orbitals: • sp2 hybridization (in-plane bonds) • pz (layer bonding)

  6. Massless Dirac fermions in graphene Interesting electronic properties p-bond s-bond Strong cohesion (useful mechanical properties)

  7. Massless Dirac fermions in graphene Tight binding Hamiltonian where Spectrum Velocity: vF = t a ~106 m/s Dirac cones: Sufficient conditions: C3v and Time reversal Necessary conditions: Inversion and Time reversal (*if Spin orbit coupling is ignored)

  8. Graphene fabrication Obstacle: Mermin-Wagner theorem Fluctuations disrupt long range crystalline order in 2D at any finite temperature Epitaxially grown graphene on metal substrates (1970): Hybridization between pz and substrate Exfoliation: chemical and mechanical Scotch Tape method (Geim, Novoselov, 2004)

  9. YouTube Graphene Making tutorial (Ozyilmaz' Group)

  10. How to see a single atom layer? P. Blake, et al, Appl. Phys. Lett. 91, 063124 (2007) graphene 300nm SiO2 Si

  11. Ambipolar effect in Graphene A. K. Geim & K. S. Novoselov, Nature Materials 6, 183 (2007) Isd Graphene Vg • Mobility: • m = 5,000 cm2/Vs (SiO2 substrate, this sample = 2007) • m = 30,000 cm2/Vs (SiO2 substrate, current) • m = 230,000 cm2/Vs (suspended)

  12. Graphene in perpendicular magnetic field: QHE Isd Hall bar geometry H Graphene Vg IQHE: Novoselov et al, Nature 2005 Room temperature IQHE: Novoselov et al, Science 2007

  13. Graphene in perpendicular magnetic field: FQHE FQHE: C.R. Dean et al, Nature Physics 7, 693 (2011)

  14. Bilayer Graphene Two layers of graphene Bernal stacking Tight binding Hamiltonian Spectrum

  15. Trigonal warping inBilayer Graphene Parabolic touching is fine tuned (g3 = 0) Tight binding Hamiltonian with g3 : Vorticity:

  16. Bilayer Graphene in perpendicular magnetic field Isd Hall bar geometry H BLG Vg IQHE: Novoselov et al, Nature Physics 2, 177 (2006)

  17. Widely tunable gap inBilayer Graphene Y. Zhang et al, Nature 459, 820 (2009)

  18. Trilayer Graphene ABA and ABC stacking

  19. Band structureABC Trilayer Graphene Tight binding Hamiltonian

  20. Non-interacting phases inABC Trilayer Graphene Spectrum: Phase transitions, even with no interactions D 3- 9- 3+ Dc2 Dc1

  21. Electron interactions(Mean Field) An example: Bardeen-Cooper-Schrieffer Hamiltonian (one band, short range) Superconducting order parameter Decouple the interaction into quadratic part and neglect fluctuations 0 The transition temperature Debye frequency wD = L2/2m Only when g>0 !

  22. Different theories at different scales (RG) What if wD were different? Make a small change in L: How to keep Tc the same? This example shows that the interaction is different at different scales. • The main idea of the renormalization group (RG): • select certain degrees of freedom (e.g., high energy modes, high momenta modes, internal degrees of freedom in a block of spins...) • treat them as a perturbation • the remaining degrees of freedom are described by the same theory, but the parameters (couplings, masses, etc) are changed Our example (BCS): treat high momentum modes perturbatively (one-loop RG) ... but RG is much more powerful and versatile than what is shown here.

  23. Finite temperature RG Revisit our example (BCS) Treat fast modes perturbatively The change in the coupling constant The effective temperature also changes In this simple example we can solve the b-function ... and find the Tc

  24. Electron Interactions inSingle Layer Graphene Rich and open problem, nevertheless in zero magnetic field: Short-range interactions: irrelevant (in the RG sense) when weak. As a consequence, the perturbation theory about the non- interacting state becomes increasingly more accurate at energies near the Dirac point Coulomb interactions: marginally irrelevant (in the RG sense) when weak semimetal* QCP insulator O. Vafek, M.J. Case, Phys. Rev. B 77, 033410 (2008) In either case, a critical strength of e-e interaction must be exceeded for a phase transition into a different phase to occur. Hence, this is strong coupling problem.

  25. Electron Interactions inBilayer Graphene The kinetic part of the action where Short range interactions: marginal by power counting Classified according to IR’s of D3d Fierz identities implemented

  26. Symmetry allowed Dirac bilinears (order parameters) in BLG VC, R.E. Throckmorton, O. Vafek, Phys. Rev. B 86, 075467 (2012)

  27. RG in BilayerGraphene (no spin) Fierz identities reduce no of independent couplings to 4 O. Vafek, K. Yang, Phys. Rev. B 81, 041401(R) (2010) O. Vafek, Phys. Rev. B 82, 205106 (2010) Susceptibilities (leading instabilities, all orders tracked simultaneously) Possible leading instabilities: nematic, quantum anomalous Hall, layer-polarized, Kekule current, superconducting

  28. Experiments on Bilayer Graphene A.S. Mayorov, et al, Science 333, 860 (2011) Low-energy spectrum reconstruction

  29. RG in Bilayer Graphene (spin-1/2) VC, R.E. Throckmorton, O. Vafek, Phys. Rev. B 86, 075467 (2012) Finite temperature RG with trigonal warping … used to be tanh(1/2t) Susceptibilities (determine leading instabilities)

  30. Forward scattering phase diagram in BLG Only

  31. General phase diagram(density-density interaction) Density-density interaction Bare couplings in RG:

  32. Coupling constantsfixed ratios In the limit the ratios of g’s are fixed The leading instability depends on the ratios (stable ray) • Stable flows: • Target plane • Ferromagnet • Quantum anomalous Hall • Loop current state • Electronic density instability • (phase segregation)

  33. RG in Trilayer Graphene Belongs to a different symmetry class Number of independent coupling constants in Hint: 15 Spectrum RG flow

  34. Generic Phase Diagramin Trilayer Graphene

  35. Trilayer Graphene(special interaction cases) Hubbard model (on-site interaction) Forward scattering

  36. Generic Phase Diagramin Trilayer Graphene