Commensuration and Interlayer Coherence in Twisted Bilayer Graphene

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 Physical Sciences and Mathematics | Physics 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: http://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 Received 22 March 2010; published 12 April 2010 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. DOI: 10.1103/PhysRevB.81.161405 PACS number s : 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 Bernal , AA, ABC stackings, and their related polymorphs.2 Here the interlayer coupling scale typically exceeds 0.5 eV and pre-empts the massless Dirac physics of an isolated graphene sheet. Indeed experimental work on Bernal stacked bilayers3–6 demonstrates that their electronic 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 if any effects of their interlayer interaction. These include graphenes grown epitaxially on the SiC 0001̄ 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 neighboring 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 interlayer motion at low energy.14 In this interpretation, the Dirac points of neighboring layers remain quantum mechanically decoupled across a rotational fault11,14–18 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 unexpectedly rich physics near their charge neutrality points. The theory generalizes previous approximate analyses14 by treating the lateral modulation of the interlayer coupling between rotated layers which is essential for understanding the lowenergy physics. Importantly, commensurate rotational faults occur in two distinct forms distinguished by their sublattice parity. Structures that are even under sublattice exchange SE are generically gapped nonconducting materials while those that break SE symmetry have two massive curved bands contacted at discrete Fermi points. Both these behaviors derive from the spectral properties of AA and Bernal stacked structures, and can be understood as energyrenormalized 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 dependence of the transmission amplitude across a twisted bilayer. These results provide the appropriate low-energy Hamiltonian s for rotationally faulted bilayers superseding the massless Dirac model of an isolated sheet. The crystal structure of two-dimensional graphene Fig. 1 has a Bravais lattice spanned by two primitive translations t1=e −i /6 and t2=e i /6 with sublattice sites at A B =0 1 / 3 . 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 t1 , t2 =e i t1 , t2 and basis A B =e i A B . A commensurate rotation occurs when Tm,n=mt1+nt2=m t1 +n t2 =Tm ,n which is generically satisfied by angles indexed by two integers m and n with m ,n =arg me−i /6+nei /6 / ne−i /6 +mei /6 . In this notation AA stacking all sites in neighboring layers eclipsed has =0 and Bernal stacking has = /3. Small angular deviations from the Bernal structure have indices m=1 and large n. The 13 13 structures with =30° 2.204° structures observed by electron diffraction from epitaxial graphene on the Si 0001̄ face correspond to m ,n = 1,3 and m ,n = 2,5 .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. Color online Left Lattice structure of graphene with two sites in the primitive cell A and B and primitive translations t1 and t2. Right Brillouin zones for the two layers in a rotational fault: the Brillouin-zone corners labeled Km and Km are rotated by angle to the points Km and Km in the neighboring layer. PHYSICAL REVIEW B 81, 161405 R 2010 RAPID COMMUNICATIONS 1098-0121/2010/81 16 /161405 4 ©2010 The American Physical Society 161405-1 This occurs when B+Tp,q= B +Tp ,q for integers p ,q and p ,q , requiring integer solutions to p= m−n+3mq / 3n . This occurs only when m−n is divisible by 3 and then the coincident B B sublattice site occurs at one of the three possible threefold-symmetric positions of the cell e.g., Tm,n /3 in Fig. 2 . The remaining threefold positions are occupied by the A A -sublattice sites the origin and by overlapping hexagon centers H ,H . 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 H of its neighbor. Rotational faults at angles ̄ = /3− form commensuration partners with primitive cells of equal areas but opposite sublattice parities. Figure 2 illustrates this situation for two partner commensurations at 30° –8.213° m ,n = 1,2 left and 30°+8.213° m ,n = 1,4 right . The limiting cases of Bernal odd and AA even stackings form the shortest period commensuration pair. Because of the rotation, the Brillouin zones of the two layers have different orientations Fig. 1 b shifting their zone corners Km ,Km to rotated counterparts Km ,Km . The low-energy electronic bands of the decoupled layers have isotropic conical dispersions near each of these points with E q = vF q , where q is the crystal momentum measured relative to the corner and vF 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 longwavelength theory that represents the low-energy states as spatially modulated versions of the orthogonal zone corner Bloch states of the two layers, i.e., r = K, r u r . 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, = 1 / 3 me m· r − . The coupling between layers is derived from an interaction functional which correlates the amplitudes and phases of the Bloch states in neighboring layers U = 1/2 drT r 1 r − 2 r , 1 where T is a supercell-periodic modulation of the coupling due to the lattice structure of the commensuration cell. Using Eq. 1 one finds that the interlayer coupling is expressed in terms of the Fourier transforms of the smooth fields u ,