Vladimir Cvetković

Physics Department ColloquiumColorado School of Mines

Golden, CO, October 2, 2012

Electronic MulticriticalityIn Bilayer Graphene

National High Magnetic Field LaboratoryFlorida State University

National High Magnetic Field Laboratory

Superconductivityhttp://www.magnet.fsu.edu/mediacenter/seminars/winterschool2013/

Collaborators

Dr. Robert E. Throckmorton Prof. Oskar Vafek

V. Cvetkovic, R. Throckmorton, O.Vafek, Phys. Rev. B 86, 075467 (2012)

NSF Career Grant (O. Vafek): DMR-0955561

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

Orbitals:• sp2 hybridization (in-plane bonds)• pz (layer bonding)

Hexagonal lattice• space group P63/mmc

Massless Dirac fermions in graphene

bond

Strong cohesion (useful mechanical properties)

bond

Interesting electronic properties

Massless Dirac fermions in graphene

Sufficient conditions: C3v and Time reversalNecessary conditions: Inversion and Time reversal(*if Spin orbit coupling is ignored)

Dirac cones:

Tight binding Hamiltonian

where

Spectrum

Velocity: vF = t a ~106 m/s

Graphene fabrication

Obstacle: Mermin-Wagner theoremFluctuations 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?

Si

SiO2300nm

graphene

P. Blake, et al, Appl. Phys. Lett. 91, 063124 (2007)

Ambipolar effect in Graphene

A. K. Geim & K. S. Novoselov, Nature Materials 6, 183 (2007)

Isd

Vg

Graphene

Mobility:• = 5,000 cm2/Vs (SiO2 substrate, this sample = 2007)• = 30,000 cm2/Vs (SiO2 substrate, current)• = 230,000 cm2/Vs (suspended)

Graphene in perpendicular magnetic field: QHE

Isd

Vg

Graphene

H

Hall bar geometry

IQHE: Novoselov et al, Nature 2005Room 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 grapheneBernal stacking

K K'

t

3t

t

3t

Ek

Tight binding Hamiltonian

Spectrum

Trigonal warping inBilayer Graphene

Parabolic touching is fine tuned (3 = 0)

2:4meV1:6meV

0:8meV

Tight binding Hamiltonian with 3 :

Vorticity:

Bilayer Graphene in perpendicular magnetic field

Isd

Vg

BLG

H

Hall bar geometry

IQHE: Novoselov et al, Nature Physics 2, 177 (2006)

Widely tunable gap inBilayer Graphene

Y. Zhang et al, Nature 459, 820 (2009)

Trilayer Graphene

ABA and ABC stacking

Band structureABC Trilayer Graphene

Tight binding Hamiltonian

Non-interacting phases inABC Trilayer Graphene

Phase transitions, even with no interactions

Spectrum:

3+9-3-

c2 c1

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

The transition temperature

Debye frequency D = 2/2m Only when g>0 !

0

Different theories at different scales (RG)

What if D were different? Make a small change in :

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.

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 -function

... and find the Tc

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* insulatorQCP

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 inBilayer Graphene

Short range interactions: marginal by power counting

Classified according to IR’s of D3d

The kinetic part of the action

where

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 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

Experiments on Bilayer Graphene

A.S. Mayorov, et al, Science 333, 860 (2011)

2:4meV1:6meV

0:8meV

Low-energy spectrum reconstruction

RG in Bilayer Graphene (spin-1/2)

Finite temperature RG with trigonal warping

VC, R.E. Throckmorton, O. Vafek, Phys. Rev. B 86, 075467 (2012)

Susceptibilities (determine leading instabilities)

… used to be tanh(1/2t)

Forward scattering phase diagram in BLG

Only

General phase diagram(density-density interaction)

Density-density interaction

Bare couplings in RG:

Coupling constantsfixed ratios

In the limitthe 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 Diagramin Trilayer Graphene

Trilayer Graphene(special interaction cases)

Forward scatteringHubbard model(on-site interaction)

Generic Phase Diagramin Trilayer Graphene