Commensuration and Interlayer Coherence in Commensuration and Interlayer Coherence in Twisted Twisted BilayerBilayer GrapheneGraphene

Gene MeleDepartment of Physics and Astronomy

University of Pennsylvania

Three things to discuss:Three things to discuss:Review prior results1

1Phys. Rev. B 161405(R) (2010)

with some (yet unpublished) results

and some brand new results

Family behavior of commensurate rotational faults

Scaling of interlayer coherence energy

Bilayer Green Functions: Pseudospin Coherence in A(k,),Fermi velocities, Layer-polarized chiral modes

The interlayer coherence scale is largeThe interlayer coherence scale is largefor a Bernal for a Bernal bilayerbilayer

Single layer Bernal Bilayer

but is small inbut is small infaulted faulted multilayersmultilayerse.g. M. Sprinkle et al.PRL 103, 226803 (2009) (ARPES)

and some theoryand some theoryDecoupling via Momentum Mismatch

J.M.B. Lopes dos Santos, N.M.R. Peres and A.H. Castro Neto“Graphene Bilayer with a Twist: Electronic Structure”Phys. Rev. Lett. 99, 256802 (2007)

S. Shallcross, S. Sharma and O.A. Pankratov:“Quantum Interference at the Twist Boundary in Graphene”Phys. Rev. Lett. 101, 056803 (2008)

Platform for Excitonic Insulator?

H. Min et al.: “Room temperature superfluidity in graphene bilayers”. Phys. Rev. B 78, 121401 (2008)

Review prior results1

1Phys. Rev. B 161405(R) (2010)

Family behavior of commensurate rotational faults

Highest Symmetry Highest Symmetry BilayersBilayers

AB (Bernal) Stacking AA Stacking

Rotationally Faulted Rotationally Faulted BilayersBilayers

32.204o 27.796o

Faults are indexed by a 2D graphene translation vector (m,n) andoccur in complementary partners with the same commensurationarea but opposite sublattice exchange symmetry

Conventional interpretationConventional interpretation: : Layer decoupling fromthe momentum mismatch across a rotational fault(Ref. 1: J.M.B. Lopes dos Santos, N.M.R. Peres and A.H. Castro Neto

Physical Review Letters 99, 256802 (2007))

eff FH u r i v u r( ) ( )

pseudo-spinor wf

New ideaNew idea:: Momentum-conserving interlayer motion is selected by the lattice commensuration of the fault.(this is missing from Ref. 1)

mismatch

Low energy theory with a latticeLow energy theory with a lattice

Kr r u r

,( ) ( ) ( ) ● Bloch wave = Basis * Dirac Envelope

221 2

1 ( ) ( ) ( )2

U d r T r r r

, ,

1

( )( ) m

m

iG r

G G

n r e

10 1 2( ) exp ( ) ( )T r n r n rC C

strength range

● Symmetry constrains the form of T(r):

Interlayer PotentialsInterlayer Potentials

Bernal AA stacking

3030oo ±± 8.2138.213oo

SE odd SE even

More Interlayer PotentialsMore Interlayer PotentialsSESE--even near 30even near 30oo

Interlayer potential for aInterlayer potential for asmall angle fault (SEsmall angle fault (SE--even)even)

AB

AA

“BA”

Interlayer potential @ 30Interlayer potential @ 30oo::quasiperiodicquasiperiodic

Model for Interlayer PotentialModel for Interlayer Potential

, ,

1

( )( ) m

m

iG r

G G

n r e

10 1 2( ) exp ( ) ( )T r n r n rC C

Two parameters determined by maximum value and curvature ofinterlayer coupling potential.

Separable form (permits scaling theory)

Nonlinear mixing of overlapping lattices: all G_s present.

(real, lattice-symmetric field)

BilayerBilayer Hamiltonian in the Hamiltonian in the pseudospinpseudospin basisbasis

1 int†

int 2

ˆˆ

ˆF

evenF

i v HH

H i v

1 int† *

int 2

ˆˆ

ˆF

oddF

i v HH

H i v

SE even

SE odd

Interlayer &Intra-valley(two copies)

/ 2

int / 2

0ˆ0

ii

i

eH V e

e

int

1 0ˆ0 0

iH V e

Transport on two sublatticesvia x-y pseudospin rotation

Transport on the dominant (eclipsed) sublattice

Interlayer &Inter-valley(two copies)

Low energy spectraLow energy spectra

Lifts degeneracyon A (eclipsed) sublattice

Quadratic near CNP

Fully gapped!

Gap from pseudospin-orbit coupling

cohe

renc

e sc

ale

Scaling of Interlayer CoherenceScaling of Interlayer Coherence

The separable form for T(r) produces a simple scalingrule for the coherence scale

0 1 1 2( ) exp ( ) ( )T r C C n r n r

1 1 1

2 1 2

2( ) ( )2

1 2 2

( ) ' ( ')2

( ) [ ](2 )

[ ' ]

i G q r C n r

i G q r C n r

d qt K G G d r e e

d r e e

1 2 1 1 2 2; ( ) ( ) ( )G

q G t G G F G G F G G

“coincident reciprocal lattice”

2| |2 ''

Q Gn

c

etA

Commensuration in reciprocal spaceCommensuration in reciprocal space

(Area)-1

phase cancellation

Review prior results1

1Phys. Rev. B 161405(R) (2010)

with some (yet unpublished) results

and some brand new results

Family behavior of commenurate rotational faults

Scaling properties of coherence scale

Bilayer Green Functions: Pseudospin Coherence,Fermi velocities, Layer-polarized chiral modes

2 22 2 †2

†21 21 1' ( ) ( , ') ( ') ' ( ) ( , ') ( ')d r d d r d r r hr r h r rH r rr r

11

2

( )( , )

( )E h Tk

G k EE hT k

1 1 11( , ) ( ) ( , )G k E E Eh k k

Bilayer Green Functions

Show results for spatially uniform T

2 2 †2 2 2

2 2 †1 1

222

1

1

' ( ) ( ,' ( )

( )

( , ') ( '') ) ( ')

( ) ( )

d r d r r h r rd r d r r h r r

d r r

H

r

r

T

r

r

Constant Energy Surfaces(Charge Neutrality Points)

q_x a

q_y

a

K

K()K+G= K() +G()

Constant Energy Surfaces

q_x a

q_y

a

Constant Energy Surfaces(Charge Neutrality Points)

q_x a

q_y

a

Partner Commensurations

21.787o 38.213o

KK’K K

Partner Commensurations

21.787o 38.213o

K KK

K’

q_par q_par

Quasiparticle Dispersion for Partners

21.787o 38.213o

Pseudospin coherence factor suppresses interlayer coupling

Ener

gy

q_par q_par

Layer Polarized One-Way Modes in AB/BA Domain Wall

top layer bottom layer

qa

Ene

rgy

Is the Fermi velocity renormalized?

single layer

7o faulted BL

seen here

but heals here

Fv K

Change of order 1/K in energy window K

With new structure at very small fault angles

Fv K t

Fv K t

SummarySummary

● Two structural families distinguished by their sublattice exchange parity

Lattice Commensuration Essential for LowEnergy Physics of Twisted BLG’s

● on a small but accessible coherence scale

● which controls the low energy mass structure(this preempts massless Dirac physics)

Theory CollaboratorsTheory Collaborators::

Seungchul KimAndy RappeCharlie Kane

Motivation for Studying Motivation for Studying Twisted Twisted GraphenesGraphenes

● They are made experimentally

● They pose an interesting math problem

● They may provide a platform for interestingmany body physics

Motivation for Studying Motivation for Studying Twisted Twisted GraphenesGraphenes

● They are made experimentally

● They pose an interesting math problem

● They may provide a platform for interestingmany body physics