University of Pennsylvania University of Pennsylvania
ScholarlyCommons ScholarlyCommons
Department of Physics Papers Department of Physics
4-12-2010
Commensuration and Interlayer Coherence in Twisted Bilayer Commensuration and Interlayer Coherence in Twisted Bilayer
Graphene Graphene
Eugene J. Mele
University of Pennsylvania
, mele@physics.upenn.edu
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Suggested Citation:
Mele, E.J. (2010). "Commensuration and interlayer coherence in twisted bilayer graphene."
Physical Review B.
81,
161405(R).
© The American Physical Society
http://dx.doi.org/10.1103/PhysRevB.81.161405
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Commensuration and Interlayer Coherence in Twisted Bilayer Graphene Commensuration and Interlayer Coherence in Twisted Bilayer Graphene
Abstract Abstract
The low-energy electronic spectra of rotationally faulted graphene bilayers are studied using a
longwavelength theory applicable to general commensurate fault angles. Lattice commensuration
requires lowenergy electronic coherence across a fault and pre-empts massless Dirac behavior near the
neutrality point. Sublattice exchange symmetry distinguishes two families of commensurate faults that
have distinct low-energy spectra which can be interpreted as energy-renormalized forms of the spectra
for the limiting Bernal and
AA
stacked structures. Sublattice-symmetric faults are generically fully gapped
systems due to a pseudospin-orbit coupling appearing in their effective low-energy Hamiltonians.
Disciplines Disciplines
Physical Sciences and Mathematics | Physics
Comments Comments
Suggested Citation:
Mele, E.J. (2010). "Commensuration and interlayer coherence in twisted bilayer graphene."
Physical
Review B.
81, 161405(R).
© The American Physical Society
http://dx.doi.org/10.1103/PhysRevB.81.161405
This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/12
Commensuration and interlayer coherence in twisted bilayer graphene
E. J. Mele
*
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
共Received 22 March 2010; published 12 April 2010
兲
The low-energy electronic spectra of rotationally faulted graphene bilayers are studied using a long-
wavelength theory applicable to general commensurate fault angles. Lattice commensuration requires low-
energy electronic coherence across a fault and pre-empts massless Dirac behavior near the neutrality point.
Sublattice exchange symmetry distinguishes two families of commensurate faults that have distinct low-energy
spectra which can be interpreted as energy-renormalized forms of the spectra for the limiting Bernal and AA
stacked structures. Sublattice-symmetric faults are generically fully gapped systems due to a pseudospin-orbit
coupling appearing in their effective low-energy Hamiltonians.
DOI: 10.1103/PhysRevB.81.161405 PACS number共s兲: 73.20.⫺r
Coherent interlayer electronic motion in multilayer
graphenes play a crucial role in their low-energy properties.
1
This physics is well understood for stacked structures with
neighboring crystallographic axes rotated by multiples of
/ 3, including AB 共Bernal兲, AA, ABC stackings, and their
related polymorphs.
2
Here the interlayer coupling scale typi-
cally exceeds 0.5 eV and pre-empts the massless Dirac phys-
ics of an isolated graphene sheet. Indeed experimental work
on Bernal stacked bilayers
3–6
demonstrates that their elec-
tronic properties are radically different from those of a single
layer.
7,8
Yet, recent experimental work has revealed a family
of multilayer graphenes that show only weak 共if any兲 effects
of their interlayer interaction. These include graphenes
grown epitaxially on the SiC共0001
¯
兲 surface,
9–11
mechanically
exfoliated folded graphene bilayers,
12
and graphene flakes
deposited on graphite.
13
A common structural attribute of
these systems is the rotational misorientation of their neigh-
boring layers at angles
⫽ n
/ 3. A continuum theoretic
model has suggested that misorientation by an arbitrary fault
angle induces a momentum mismatch between the tips of the
Dirac cones in neighboring layers suppressing coherent in-
terlayer motion at low energy.
14
In this interpretation, the
Dirac points of neighboring layers remain quantum mechani-
cally decoupled across a rotational fault
11,14–18
accessing
two-dimensional physics in a family of three-dimensional
materials.
This Rapid Communication presents a long-wavelength
theory of electronic motion in graphene bilayers containing
rotational faults at arbitrary commensurate angles. I find that
the Dirac nodes of these structures are directly coupled
across any commensurate rotational fault, producing unex-
pectedly rich physics near their charge neutrality points. The
theory generalizes previous approximate analyses
14
by treat-
ing the lateral modulation of the interlayer coupling between
rotated layers which is essential for understanding the low-
energy physics. Importantly, commensurate rotational faults
occur in two distinct forms distinguished by their sublattice
parity. Structures that are even under sublattice exchange
共SE兲 are generically gapped 共nonconducting兲 materials while
those that break SE symmetry have two massive 共curved兲
bands contacted at discrete Fermi points. Both these behav-
iors derive from the spectral properties of AA and Bernal
stacked structures, and can be understood as energy-
renormalized versions of these limiting cases. The gap in the
faulted sublattice-symmetric states appears as a new feature
specific to the faulted structures due to a pseudospin depen-
dence of the transmission amplitude across a twisted bilayer.
These results provide the appropriate low-energy Hamilto-
nian共s兲 for rotationally faulted bilayers superseding the mass-
less Dirac model of an isolated sheet.
The crystal structure of two-dimensional graphene 共Fig.
1兲 has a Bravais lattice spanned by two primitive translations
t
1
=e
−i
/6
and t
2
=e
i
/6
with sublattice sites at
A共B兲
=0共1 /
冑
3兲.
We consider rotational stacking faults that fix overlapping
A-sublattice sites at the origin and rotate one layer through
angle
with respect to the other with translation vectors
共t
1
⬘
,t
2
⬘
兲=e
i
共t
1
,t
2
兲 and basis
A共B兲
⬘
=e
i
A共B兲
. A commensurate
rotation occurs when T
m,n
=mt
1
+nt
2
=m
⬘
t
1
⬘
+n
⬘
t
2
⬘
=T
m
⬘
,n
⬘
⬘
which is generically satisfied by angles indexed by two inte-
gers m and n with
共m , n兲= arg关共me
−i
/6
+ne
i
/6
兲/ 共ne
−i
/6
+me
i
/6
兲兴. In this notation AA stacking 共all sites in neighbor-
ing layers eclipsed兲 has
=0 and Bernal stacking has
=
/ 3. Small angular deviations from the Bernal structure
have indices m = 1 and large n. The
冑
13⫻
冑
13 structures with
=30°⫾2.204° structures observed by electron diffraction
from epitaxial graphene on the Si共0001
¯
兲 face correspond to
共m , n兲= 共1,3兲 and 共m , n兲=共2,5兲.
19
Commensurate faults occur in two families determined by
their SE symmetry. With the A -sublattice sites at the origin, a
commensuration is SE symmetric if B-sublattice sites are
coincident at some other lattice position in the primitive cell.
FIG. 1. 共Color online兲共Left兲 Lattice structure of graphene with
two sites in the primitive cell 共A and B兲 and primitive translations t
1
and t
2
. 共Right兲 Brillouin zones for the two layers in a rotational
fault: the Brillouin-zone corners labeled K
m
and K
m
⬘
are rotated by
angle
to the points K
m
共
兲 and K
m
⬘
共
兲 in the neighboring layer.
PHYSICAL REVIEW B 81, 161405共R兲共2010兲
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This occurs when
B
+T
p,q
=
B
⬘
+T
p
⬘
,q
⬘
⬘
for integers 共p,q兲 and
共p
⬘
,q
⬘
兲, requiring integer solutions to p = 共m−n+3mq兲/ 共3n兲.
This occurs only when m−n is divisible by 3 and then the
coincident B共B
⬘
兲 sublattice site occurs at one of the three
possible threefold-symmetric positions of the cell 共e.g.,
T
m,n
/ 3 in Fig. 2兲. The remaining threefold positions are oc-
cupied by the A共A
⬘
兲-sublattice sites 共the origin兲 and by over-
lapping hexagon centers 共H , H
⬘
兲. When m−n is not divisible
by 3 the only coincident site is the A site at the origin and the
remaining two threefold-symmetric positions are occupied
by B-sublattice atoms of one layer aligned with the hexagon
centers 共H
⬘
兲 of its neighbor. Rotational faults at angles
¯
=
/ 3−
form commensuration partners with primitive cells
of equal areas but opposite sublattice parities. Figure 2 illus-
trates this situation for two partner commensurations at
30° –8.213° 关共m , n兲=共1,2兲兴 共left兲 and 30°+8.213° 关共m,n兲
=共1,4兲兴 共right兲. The limiting cases of Bernal 共odd兲 and AA
共even兲 stackings form the shortest period commensuration
pair.
Because of the rotation, the Brillouin zones of the two
layers have different orientations 关Fig. 1共b兲兴 shifting their
zone corners 共K
m
,K
m
⬘
兲 to rotated counterparts
关K
m
共
兲,K
m
⬘
共
兲兴. The low-energy electronic bands of the de-
coupled layers have isotropic conical dispersions near each
of these points with E共q兲= ⫾ប
v
F
兩q兩, where q is the crystal
momentum measured relative to the corner and
v
F
is the
Fermi velocity. These spectra are described by a pair of
massless Dirac Hamiltonians for the K and K
⬘
points of the
two layers.
21
Interlayer coupling is studied using a long-
wavelength theory that represents the low-energy states as
spatially modulated versions of the orthogonal zone corner
Bloch states of the two layers, i.e., ⌿共r
ជ
兲=兺
␣
K,
␣
共r
ជ
兲u
␣
共r
ជ
兲.By
retaining the reciprocal-lattice vectors that constrain the sum
in
K,
␣
to the three equivalent corners of the Brillouin zone
we obtain the Bloch states
K,
␣
=共1 /
冑
3兲兺
m
e
iK
ជ
m
·共r
ជ
−
ជ
␣
兲
. The
coupling between layers is derived from an interaction func-
tional which correlates the amplitudes and phases of the
Bloch states in neighboring layers
U = 共1/2兲
冕
d
2
rT
ᐉ
共r
ជ
兲兩⌿
1
共r
ជ
兲 − ⌿
2
共r
ជ
兲兩
2
, 共1兲
where T
ᐉ
is a supercell-periodic modulation of the coupling
due to the lattice structure of the commensuration cell. Using
Eq. 共1兲 one finds that the interlayer coupling is expressed in
terms of the Fourier transforms of the smooth fields u
␣
,
U
int
N
=
A
o
24
2
冕
d
2
q
兺
␣
,

兺
m;m
⬘
e
iK
ជ
m
·
ជ
␣
e
−iK
ជ
m
⬘
·
ជ

⬘
⫻
兺
G
ជ
关t共G
ជ
兲u
1,
␣
ⴱ
共q
ជ
兲u
2,

共q
ជ
+ K
ជ
m
− K
ជ
m
⬘
− G
ជ
兲 + c.c.兴,
共2兲
where N is the system size, A
o
is the area of a graphene
primitive cell, and t共G
ជ
兲 is the Fourier transform of the inter-
layer potential T
ᐉ
共r
ជ
兲 on the reciprocal lattice of the commen-
suration cell G
ជ
.
The continuum theory of Ref. 14 is recovered from Eq.
共2兲 by retaining only its G
ជ
=0 terms, thus treating the inter-
layer coupling as spatially uniform. In this approximation the
states near the tip of the Dirac cone in one layer are coupled
to three pairs of states at energies ⫾W
ⴱ
= ⫾ប
v
F
兩K
m
−K
m
共
兲兩
in its neighbor. At low energies this coupling can be treated
perturbatively, preserving the Dirac nodes of two isotropic
velocity-renormalized layer-decoupled Hamiltonians.
Different physics arises from the G
ជ
⫽ 0 contributions to
Eq. 共2兲 which mediate a direct coupling between the Dirac
nodes of neighboring layers and prevent massless low-
energy behavior. To study it, note that the reciprocal lattice
of the bilayer is spanned by momenta with four integer indi-
ces G
ជ
= pG
ជ
1
+qG
ជ
2
+ p
⬘
G
ជ
1
⬘
+q
⬘
G
ជ
2
⬘
.
20
Momentum conserving
couplings between K points in neighboring layers occur
when K
m
−K
m
⬘
共
兲=G
ជ
共p , q , p
⬘
,q
⬘
兲 with the angle
共m , n兲
specified. This is an interlayer umklapp process where the
spatial modulation of T
ᐉ
共r
ជ
兲 provides precisely the transverse
momentum required to transport an electron between the
Dirac nodes of neighboring layers. The momentum matching
condition requires integer p solutions to p=共m − n 兲/ 3n
+qm/ n and occurs only for supercommensurate structures
with nonzero mod共m ,3兲= mod共n ,3兲. Importantly if this con-
dition is not satisfied, momentum-conserving interlayer cou-
plings still occur, but instead through the analogous interval-
ley umklapp process, i.e., K
m
⬘
−K
m
⬘
共
兲=G
ជ
共p , q , p
⬘
,q
⬘
兲. These
two possibilities are complementary and mutually exclusive:
one or the other must occur if the rotational fault is commen-
surate. These two criteria distinguish SE-even and SE-odd
structures so that the SE-even structures require direct K
−K共
兲 coupling and SE-odd structures K − K
⬘
共
兲 coupling.
To understand the consequences of the interlayer interac-
tion one requires a theory for the Fourier coefficients t共G
ជ
兲 in
Eq. 共2兲. These can be calculated from atomistic models but
their relevant properties are determined by symmetry. Note
that the coupling function T
ᐉ
共r
ជ
兲 is a real periodic function
FIG. 2. 共Color online兲 Geometry of the commensuration cells
for bilayers faulted at 21.787° 共left兲 and 38.213° 共right兲. Red and
blue dots denote the atom positions in the two layers. The white
rhombuses denote primitive commensuration cells with the same
area for these two structures. The dashed yellow rhombus denotes a
冑
3⫻
冑
3 nonprimitive cell. The left-hand structure is SE odd, with
coincident atomic sites only on the A共A
⬘
兲 sublattice at the origin,
the right-hand structure is SE even with coincident sites on the
A共A
⬘
兲 and B共B
⬘
兲 sublattices at threefold-symmetric positions in the
primitive cell and overlapping hexagon centers 共H , H
⬘
兲. The density
plot gives the magnitude of the interlayer hopping potential dis-
cussed in the text.
E. J. MELE PHYSICAL REVIEW B 81, 161405共R兲共2010兲
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with the translational symmetry of the commensuration cell.
The structure function for the
th layer, n
共r
ជ
兲
=兺
m苸关1兴
兺
␣
e
iG
ជ
,m
·共r
ជ
−
ជ
,
␣
兲
superposes the six plane waves of the
lowest star of reciprocal-lattice vectors G
ជ
,m
producing a
standing wave with maxima on atom sites and minima in
hexagon centers. A useful model for the interlayer coupling
potential is T
ᐉ
共r
ជ
兲=C
0
exp关C
1
n共r
ជ
兲兴, where n共r
ជ
兲=n
1
+n
2
and
C
0
and C
1
are constants; T
ᐉ
is a superlattice-periodic func-
tion with maxima for coincident sites and with exponential
suppression in regions that are out of interlayer registry. The
grayscale plot in Fig. 2 show the spatial distribution of T
ᐉ
共r
ជ
兲,
where C
1
is determined by matching the decay of the hop-
ping amplitude between neighboring layer atoms as a func-
tion of small lateral offsets. This density plot shows that the
interlayer amplitudes between rotated layers have coherent
structures in the forms of fivefold rings 共from overlapping
misaligned hexagons兲 arranged to form two-dimensional
space-filling patterns. SE-odd structures are symmetric under
threefold rotations while the SE-even structures retain a six-
fold symmetry. The separable form T
ᐉ
共r
ជ
兲= f
1
共r
ជ
兲f
2
共r
ជ
兲 allows
one to deduce a scaling rule for the Fourier coefficients:
t共G
ជ
兲⬇共ae
−b/N
c
/ N
c
兲兺
苸关1兴
f
2
共r
ជ
兲e
−iG
ជ
·r
ជ
, where the sum is over
atomic sites in layer 1, a and b are constants, and N
c
is the
number of graphene cells 共per layer兲 in the commensuration
cell. For large N
c
the prefactor decays as a power law of the
cell size reflecting the fraction of atomic sites in good inter-
layer registry while the sum decays quickly as a function of
N
c
because of canceling phases in its argument.
The interlayer Hamiltonian can be expressed by a 3⫻3
array of scattering amplitudes derived from the t共G
ជ
兲’s giving
the allowed transitions K
m
→ K
m
⬘
共
兲. Threefold symmetry re-
quires this matrix to have the form
V
ˆ
ps
=
冢
V
0
V
1
V
2
V
2
V
0
V
1
V
1
V
2
V
0
冣
,
where the pseudopotential coefficients V
i
are matrix elements
of T
ᐉ
. Completing the sum in Eq. 共2兲 transforms this to the
sublattice 共pseudospin兲 basis and gives the 2⫻2 interlayer
transition matrices H
ˆ
int
seen by the Dirac fermions. The low-
energy Hamiltonian for an SE-even bilayer is expressed as a
4⫻4 matrix 共acting on the two sublattice and two layer de-
grees of freedom兲,
H
ˆ
even
=
冉
− iប
v
˜
F
1
· ⵜ
H
ˆ
int
+
共H
ˆ
int
+
兲
†
− iប
v
˜
F
2
· ⵜ
冊
共3兲
and for the SE-odd bilayer
H
ˆ
odd
=
冉
− iប
v
˜
F
1
· ⵜ
H
ˆ
int
−
共H
ˆ
int
−
兲
†
iប
v
˜
F
2
ⴱ
· ⵜ
冊
, 共4兲
where
n
are Pauli matrices acting in the sublattice pseu-
dospin basis of the nth layer and
v
˜
F
is the renormalized
Fermi velocity. The interlayer matrices H
ˆ
int
⫾
are
H
ˆ
int
+
= Ve
i
冉
e
i
/2
0
0
e
−i
/2
冊
, H
ˆ
int
−
= Ve
i
冉
1
0
00
冊
. 共5兲
H
ˆ
int
+
shows that interlayer motion of an electron for SE-even
faults involves an intravalley transition with a unitary trans-
formation of its 共A , B兲 sublattice amplitudes represented as
an xy rotation of its pseudospin through angle
. This angle
is not defined geometrically by the fault angle
but rather is
determined by the relative magnitudes of the three pseudo-
potential matrix elements V
i
. By contrast interlayer motion
across a sublattice asymmetric fault requires an intervalley
transition through only the amplitudes on its dominant
共eclipsed兲 sublattice. The continuum model of Ref. 14 is re-
covered by setting H
ˆ
int
=0.
In either case, below an energy scale V the electronic
spectra deviate from the massless Dirac form and inherit cur-
vature from the interlayer coupling as shown in Fig. 3. V
⬃10 meV for commensurations at
=30°⫾8.213° with
N
c
=7 graphene cells per layer in their commensuration cells.
Nevertheless the forms of these spectra apply generally to
any pair of commensuration partners. SE-odd faults mix the
degenerate Dirac bands gapping one pair on the scale V,
leaving a second pair of massive 共curved兲 bands in contact at
E=0. By contrast, SE even structures are fully gapped where
the gap arises entirely from the pseudospin rotation in Eq.
共5兲. Indeed for
=0 these spectra consist of a pair of Dirac
cones offset in energy by a bonding-antibonding splitting and
FIG. 3. Low-energy electronic spectra for SE-odd and SE-even
faulted bilayers are illustrated using partner commensurations at
rotation angles
=21.787° 共odd, left兲 and
=38.213° 共even, right兲.
These spectra are symmetric under rotations in momentum space.
SE-odd faults have massive bands that contact at Fermi points 共left兲
and SE-even faults are gapped 共right兲. The lower row gives the
spectrum for a Bernal bilayer 共left兲 and for an AA bilayer, which
show related spectral properties.
COMMENSURATION AND INTERLAYER COHERENCE IN… PHYSICAL REVIEW B 81, 161405共R兲共2010兲
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