Session T11: Focus Session: Graphene Structure, Stacking, Interactions: Twisted Layers

Continuum Model of the twisted graphene bilayer

The electronic structure of the twisted bilayer was first considered [1] in the context of a continuum description of the two layers, coupled by a spatially modulated hopping. The model's predictions were subsequently confirmed by experiments [2,3], including a scanning tunneling spectroscopy finding of two low energy Van-Hove peaks in the density of states [4], and by band structure calculations [5,6]. We discuss the extension of the model in several directions: the two families of commensurate structures discovered by Mele [7], will be characterized by elementary geometrical arguments; it will be shown that it is possible to calculate analytically \emph{all} Fourier components of the hopping amplitudes for any kind of commensurate structure with large period; the calculations will be extended beyond the perturbative regime in the interlayer coupling to address the electronic structure and local density of states in the very small angle limit. \\[4pt] [1] J.~M.~B. Lopes dos Santos, \emph{et. al}, Phys. Rev. Lett. \textbf{99}, 256802, (2007).\\[0pt] [2] Z.~Ni, \emph{et. al.}, Phys. Rev. B \textbf{77}, 235403, 2008.\\[0pt] [3] A.~Luica, \emph{et. al}, Phys. Rev. Lett. \textbf{106}, 126802 (2011).\\[0pt] [4] G.~Li, \emph{et. al}, Nature Physics \textbf{6}, 109 (2010).\\[0pt] [5] S. Shallcross, \emph{et. al}, Phys. Rev. B \textbf{81}, 165105 (2010).\\[0pt] [6] G.~de Laissardiere, \emph{et. al}, Nano Letters \textbf{10},804 (2010).\\[0pt] [7] E.~J. Mele, {Phys. Rev. B} \textbf{81}, 161405 (2010).   

Moire Bloch Bands in Twisted Bilayer Graphene

A moir\'{e} superlattice pattern is formed when two copies of a periodic lattice are overlaid with a relative twist. I will address the electronic structure of a twisted two-layer graphene system by generalizing the Dirac equation continuum models that are used to describe single-layer graphene and untwisted bilayers. In the Dirac model electrons in graphene have a pseudospin degree-of-freedom corresponding to the honeycomb sublattice dependence of wavefunction amplitudes. The continuum model of twisted bilayers can be derived systematically [1] by assuming that interlayer tunneling amplitudes are non-nocal with a range that is large compared to the honeycomb lattice constant, and leads to an appealing picture in which the tunneling operator has a position-dependent pseudospin dependence that simply reflects the local registry between the two honeycombs. The continuum model twisted bilayer Hamiltonian is therefore periodic, with the periodicity of the moir\'{e} pattern, and insensitive to the incommensurability of the microscopic Hamiltonian. I will discuss the properties of the Bloch bands of this periodic Hamiltonian, which become highly non-trivial at small twist angles. In particular the Dirac velocity crosses zero several times as the twist angle is reduced and vanishes at a discrete set of magic angles. I will also briefly discuss the Hofstadter butterfly spectral patterns [2] created by incommensurability between the moir\'{e} pattern and magnetic lengths when a twisted bilayer is placed in an external magnetic field, and the electronic structure [3] of a single graphene layer that is twisted with respect to a boron nitride substrate.\\[4pt] [1] R. Bistritzer and A.H. MacDonald, PNAS {\bf 108}, 12233 (2011). \hfill \newline [2] R. Bistritzer and A.H. MacDonald, Phys. Rev. B {\bf 84}, 035440 (2011). \hfill \newline [3] A. Raoux and A.H. MacDonald, arXiv:1112.nnnn.   

Anomalies in the magneto-optical conductivity of twisted multilayer epitaxial graphene

The nature of the electronic coupling between carbon layers in twisted multilayer graphene is an intriguing and hot-debated issue. We measured the Faraday rotation and optical absorption spectra of twisted graphene multilayers grown on the carbon face of SiC in the far-infrared range in magnetic fields up to 7 Tesla [1,2]. Multiple spectral components are identified, which include a quasi-classical cyclotron resonance, originating from the highly doped graphene layer closest to SiC, transitions between low-index Landau levels (LLs), which stem from quasineutral outer layers and a broad optical absorption background, which provenance is less obvious. Electron- and hole-type LL transitions are optically distinguished and shown to coexist. The variation of the Fermi velocity is about 10 percent across the layers. Our central observation is an anomalously small optical spectral weight of the individual LL transitions, which is likely caused by the unusual electronic coupling between randomly stacked graphene layers. \\[4pt] [1] I. Crassee \emph{et al.} Nature Phys. \textbf{7}, 48 (2011). \newline\noindent[2] I. Crassee \emph{et al.} Phys Rev B \textbf{84}, 035103 (2011).   

Quantum Hall Effect, Screening and Layer-Polarized Insulating States in Twisted Bilayer Graphene

We present a study of electronic transport in dual-gated twisted bilayer graphene. Despite the sub-nanometer proximity between the layers, we identify independent contributions to the magnetoresistance from the graphene Landau level spectrum of each layer. We demonstrate that the filling factor of each layer can be independently controlled via the dual gates, which we use to induce Landau level crossings between the layers. By analyzing the gate dependence of the Landau level crossings, we characterize the finite inter-layer screening and extract the capacitance between the atomically-spaced layers. At zero filling factor, we observe magnetic and displacement field dependent insulating states, which indicate the presence of counter-propagating edge states with inter-layer coupling.   

Formation of superlattice structures by interlayer bonding in twisted bilayer graphene

We present a computational analysis of carbon nanostructure formation from twisted bilayer graphene, upon creation of interlayer covalent C-C bonds. The analysis is based on a combination of first-principles density functional theory calculations and classical molecular-dynamics simulations. We demonstrate that the resulting configurations constitute a novel class of stable structures and that their features are determined by the relative angle of rotation between the two graphene planes of the bilayer. For small angles of rotation (near 0 degrees), interlayer covalent bonding generates superlattices of diamond-like nanocrystals embedded within the graphene layers; for rotation angles near 30 degrees, superlattices of caged fullerene-like configurations are generated. We calculate the electronic band structure of these superlattices and show that their band gap can be controlled through selective hydrogenation and creation of interlayer bonds. We also show that the linear dispersion around the K point in the first Brillouin zone (Dirac cones), characteristic of single-layer and non-bonded twisted bilayer graphene, is preserved for some of these structures in spite of the introduction of sp3 bonds due to hydrogenation and interlayer C-C bonding.   

Phonon mediated conductance in misoriented graphene bilayers

Electrical transport in a misoriented graphene bilayer is facilitated by umklapp processes whose strength is known to decrease rapidly with the number of atoms in the commensurate cell. As the misorientation angle is reduced, the number of atoms increases, and the umklapp conductance in an ideal (infinitely large and defect free) bilayer becomes negligible. We show that at room temperature coupling to the out-of-plane phonon vibrations leads in a conductance several orders of magnitude larger than that produced by pure electronic umklapp. The most relevant phonons originate from the flexural modes of the monolayer, vibrating out-of-phase in the bilayer, with energies around 80 cm$^{-1}$ near the $\Gamma $-point. These phonon modes disperse nearly quadratically away from the center of the Brillouin zone. As the misorientation angle is reduced, the relevant phonon wavevector connecting the two Fermi surfaces in monolayers is reduced as well, which results in larger phonon mediated conductance. This is the opposite behavior to that expected from the umklapp conductance. We will present calculations of phonon mediated conductance as a function of misorientation angle, doping, temperature, and applied bias, for a tight-binding electron-phonon Hamiltonian.   

Probing interlayer interactions in twisted bilayer graphene with Raman spectroscopy

Chemical vapor deposition (CVD) growth or artificial layer-by-layer assembly of graphene typically produces multi-layer regions in which the layers are twisted with respect to each other, but the electronic and optical properties of this new material are still under investigation. In particular, little is known about how the twist angle affects the Raman signature of this material. We use a combination of dark-field transmission electron microscopy (DF-TEM) and widefield Raman imaging (WRI) to study the Raman signature of bilayer CVD graphene regions with known twist angles. We find that the intensities of the G and 2D peaks vary predictably with twist angle. In particular, we observe a strong G band enhancement at a specific twist angle that depends on our excitation energy. To explain this behavior, we model the electronic band structure of twisted bilayer graphene; the interaction between layers creates new saddle point van Hove singularities, and these energy states can enable a fully resonant G band pathway for a specific angle and excitation energy. This G band resonance feature is a very sensitive probe of twist angle and interlayer interactions.   

Strong Rotational Angle Dependence of Raman Spectroscopy in Rotated Double-Layer Graphene

We perform a complementary Raman spectroscopy and transmission electron microscopy (TEM) study, as well as electronic-structure and Raman calculations, on suspended rotated double-layer graphene. Graphene Raman spectra show a strong dependence on the rotational angles between two stacked layers. For low-angle mis-orientations ($<\sim $ 10 degrees), double-layer graphene exhibits Raman signature closer to AB-stacked bilayer graphene. Double-layers with high rotational angles ($>\sim $ 15 degrees), on the other hand, display Raman spectra similar to monolayer graphene. Rotational angle dependent modifications of the electronic band structure in double-layer graphene can explain this trend and a G peak enhancement at certain middle angles. The computed electronic band structures and key features of the graphene Raman peaks including the blue shift, width and intensity of the 2D peaks agree well with experimental data.   

Effect of stacking on the transport properties of few-layer graphene: NEGF-DFT investigation

We study the effect of the stacking order on the electronic transport properties of few-layer graphene (FLG) by performing ab initio calculations based on the formalism that combines non-equilibrium green's functions and density functional theory. The stacking of the layers - Bernal (ABA), Rhombohedral (ABC) or even a combination between them - have to be considered as an extra degree of freedom and consequently the FLG properties are highly sensitive to the stacking configuration. In particular, the band structures and the transport properties of FLG present a behavior markedly distinct from that for single layer. We consider FLG from trilayer to dodecalayer in both stacks. We show that for FLG, in the Rhombohedral stacking, a simple counting of bands cannot be used to predict the amount of conducting channels. We also show that for FLG, in the Rhombohedral stacking, the outermost layers dominate the contribution to the transmittance whereas for the Bernal stacking the innermost layers also present a significant contribution to the transmittance. Moreover, we investigate the effect of bias voltage and the temperature in the transport properties of FLG. We acknowledge the INCT/CNPq, CAPES and FAPESP for the financial support.   

First principles study of trilayers of graphene-BN-graphene

The stability, electronic structure and electronic transport properties of graphene-BN-graphene (C-BN-C) trilayers are studied in the framework of density functional theory. Different stacking formats, i.e., AAA, ABA and ABC stackings are considered. The ABA stacking is found to be most energetically favorable, followed by ABC and AAA stackings. The interlayer spacing of trilayers are close to those of corresponding C-BN bilayers, while the intralayer bond length can be regarded as the weighted mean of constituent layers. All considered configurations are found to be metallic, independent of stacking formats. When an external electric field is applied perpendicularly, electronic band structures undergo stacking-dependent variations. While both AAA and ABA stackings show good tunability of energy gap, ABC stacking shows less flexibility of gap tuning. We will also present the results of the electronic transport properties which are modeled by sandwiching trilayers between gold contact electrodes.