Structural Determination of Multilayer Graphene via Atomic Moir´e Interferometry arXiv:1001.2798v1 [cond-mat.mtrl-sci] 16 Jan 2010 $^{1}$School of Physics, Georgia Institute of Technology, Atlanta GA, 30332 $^{2}$Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg MD, 20899 David L. Miller, 1 Kevin D. Kubista, 1 Gregory M. Rutter, 2 Ming Ruan, 1 Walt A. de Heer, 1 Phillip N. First, 1 and Joseph A. Stroscio 2 Rotational misalignment of two stacked honeycomb lattices produces a moir´e pattern that is observable in scanning tunneling microscopy as a small modulation of the apparent surface height. This is known from experiments on highly-oriented pyrolytic graphite. Here, we observe the combined e GLYPH<11> ect of three-layer moir´e patterns in multilayer graphene grown on SiC (000 ¯ 1). Small-angle rotations between the first and third layer are shown to produce a “double-moir´e” pattern, resulting from the interference of moir´e patterns from the first three layers. These patterns are strongly a GLYPH<11> ected by relative lattice strain between the layers. We model the moir´e patterns as a beat-period of the mismatched reciprocal lattice vectors and show how these patterns can be used to determine the relative strain between lattices, in analogy to strain measurement by optical moir´e interferometry. I. INTRODUCTION Perfect graphene is a single atomic layer of carbon atoms arranged into two interpenetrating triangular sublattices (A & B). It has a unique linear band structure stemming from the unperturbed GLYPH<25> orbitals, which lie above the plane of hy- bridized sp 2 bonds. 1 When placed on a substrate, the band structure may be modified by interaction of the GLYPH<25> orbitals with the substrate. One way to minimize this interaction is to use graphene as its own substrate. In Bernal stacked (AB) bilayer graphene, the low-energy band structure near the Fermi en- ergy is not linear due to a breaking of the sublattice sym- metry; however, if the layers are rotated away from Bernal stacking the sublattice symmetry and linear band dispersion is preserved over the unit cell of the newly formed moir´e superlattice. 2 In addition to a GLYPH<11> ecting the electronic structure, the moir´e pattern resulting from lattice misalignment or mismatch can itself be a useful tool for understanding structural properties. Optical moir´e interferometry has been applied in strain anal- ysis for many years. 3 Because a scanning tunnneling micro- scope (STM) can detect moir´e patterns in rotated graphene layers, 4 analogous methods can be applied to measure local strains between individual sheets of graphene at the nanome- ter scale. Here we show how STM-based “atomic moir´e in- terferometry” can go beyond surface properties to detect lat- tice orientations and strains for depths of several layers. We expect this type of analysis to be beneficial to understanding electronic and transport properties in graphene multilayers. Over the past 20 years, several studies of moir´e patterns occurring on highly-oriented pyrolytic graphite (HOPG) have been conducted. 5–10 These typically occur due to a rota- tion between two layers near the surface, or from an exfoli- ated flake resting on the HOPG surface. 11 Graphene islands grown on transition metal surfaces (e.g., Ir(111), Ru(0001), or Pd(111)) 12–14 also display moir´e-style superlattices due to both rotational misalignment and the inherent size mismatch of the lattices. More recently, it was found that multilayer epi- taxial graphene (MEG) grown on SiC(000 ¯ 1) stacks in such a way that almost every pair of graphene sheets is rotated with respect to its neighbor(s). Graphitic Bernal stacking of the graphene layers appears to be rare in this material, although there are clearly preferred layer orientations. 2,4,15 X-ray di GLYPH<11> raction and low-energy electron di GLYPH<11> raction experi- ments show broad peaks corresponding to rotations of GLYPH<6> 2 : 2 GLYPH<14> and 30 GLYPH<14> to the SiC lattice. 2,4 The rotational disorder of the graphene samples produces moir´e patterns in STM to- pographs, and has been under active investigation. 2,4,16–19 The patterns appear as an additional corrugation in STM imaging; however, the apparent height variation is thought to be dom- inated by a modulation in the local density of states. 7,20 The local stacking changes continuously between regions that re- semble AA, AB, BA and slip stacking. 7 Despite regions of lo- cally AB stacking, the sublattice symmetry is preserved over the full moir´e unit cell. In each unit cell, AB regions are com- plemented by BA regions that restore the inversion symmetry. The result is that the electronic properties of an N -layer film of MEG are nearly equivalent to N independent graphene layers: the linear band dispersion near the charge neutrality (Dirac) points of the Brillouin zone is preserved. 2,15,21 Multilayer graphene is grown on SiC(000 ¯ 1) by a low- vacuum induction furnace technique. 22,23 When heated, the SiC crystal thermally decomposes leaving behind carbon atoms to reform into graphene sheets. The sample studied has an average of 10 GLYPH<6> 1 layers as measured by ellipsometry. It was initially imaged using atomic force microscopy (AFM) and inspected with low energy electron di GLYPH<11> raction (LEED) and Auger electron spectroscopy (AES) to ensure the sam- ple quality. The sample was then placed into a custom built ultra-high vacuum chamber (base pressure < 10 GLYPH<0> 9 Pa), where it was subsequently heated to 1250 $^{GLYPH<14>}$C in order to clean the surface after exposure to air. After cleaning, it was studied at low temperature (4 : 3 K) in a cryogenic STM chamber. 21 Multilayer graphene often yields more complicated moir´e patterns than HOPG, resulting from the successive rotation of layers throughout the depth of the material. Multiple moir´e II. EXPERIMENTAL METHODS III. MULTILAYER MOIR ´ E