#### Transcript Electronic Multicriticality In Bilayer Graphene

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 http://www.magnet.fsu.edu/mediacenter/seminars/winterschool2013/ Superconductivity National High Magnetic Field Laboratory 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) 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 Graphite electronic orbitals Hexagonal lattice • space group P63/mmc Orbitals: • sp2 hybridization (in-plane bonds) • pz (layer bonding) Massless Dirac fermions in graphene Interesting electronic properties pbond sbond Strong cohesion (useful mechanical properties) 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) 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) YouTube Graphene Making tutorial (Ozyilmaz' Group) How to see a single atom layer? P. Blake, et al, Appl. Phys. Lett. 91, 063124 (2007) graphene 300nm SiO2 Si Ambipolar effect in Graphene A. K. Geim & K. S. Novoselov, Nature Materials 6, 183 (2007) Isd 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) Graphene in perpendicular magnetic field: QHE Hall bar geometry H Isd Vg IQHE: Novoselov et al, Nature 2005 Room temperature IQHE: Novoselov et al, Science 2007 Graphene in perpendicular magnetic field: FQHE FQHE: C.R. Dean et al, Nature Physics 7, 693 (2011) Bilayer Graphene Two layers of graphene Bernal stacking E k 3t t t Tight binding Hamiltonian 3t Spectrum K K' Trigonal warping in Bilayer Graphene Parabolic touching is fine tuned (g3 = 0) Tight binding Hamiltonian with g3 : 0:8meV 2:4meV 1:6meV Vorticity: Bilayer Graphene in perpendicular magnetic field Hall bar geometry H Isd Vg IQHE: Novoselov et al, Nature Physics 2, 177 (2006) Widely tunable gap in Bilayer Graphene Y. Zhang et al, Nature 459, 820 (2009) Trilayer Graphene ABA and ABC stacking Band structure ABC Trilayer Graphene Tight binding Hamiltonian Non-interacting phases in ABC Trilayer Graphene Spectrum: Phase transitions, even with no interactions 3- 9- Dc2 3+ Dc1 D 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 ! 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 (oneloop RG) ... but RG is much more powerful and versatile than what is shown here. 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 Electron Interactions in Single 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 noninteracting 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. Electron Interactions in Bilayer 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 Symmetry allowed Dirac bilinears (order parameters) in BLG VC, R.E. Throckmorton, O. Vafek, Phys. Rev. B 86, 075467 (2012) RG in Bilayer Graphene (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, layerpolarized, Kekule current, superconducting Experiments on Bilayer Graphene A.S. Mayorov, et al, Science 333, 860 (2011) 0:8meV 2:4meV 1:6meV Low-energy spectrum reconstruction 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) Forward scattering phase diagram in BLG Only General phase diagram (density-density interaction) Density-density interaction Bare couplings in RG: Coupling constants fixed 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) RG in Trilayer Graphene Belongs to a different symmetry class Number of independent coupling constants in Hint: 15 Spectrum RG flow Generic Phase Diagram in Trilayer Graphene Trilayer Graphene (special interaction cases) Forward scattering Hubbard model (on-site interaction) Generic Phase Diagram in Trilayer Graphene