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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with some (yet unpublished) results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