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