Commensuration and Interlayer Coherence in Twisted Bilayer Graphene

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 preempts 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.Comment: 4 pages RevTeX, 3 jpg figure

Electronic structure of turbostratic graphene

We explore the rotational degree of freedom between graphene layers via the simple prototype of the graphene twist bilayer, i.e., two layers rotated by some angle $\theta$. It is shown that, due to the weak interaction between graphene layers, many features of this system can be understood by interference conditions between the quantum states of the two layers, mathematically expressed as Diophantine problems. Based on this general analysis we demonstrate that while the Dirac cones from each layer are always effectively degenerate, the Fermi velocity $v_F$ of the Dirac cones decreases as $\theta\to 0^\circ$; the form we derive for $v_F(\theta)$ agrees with that found via a continuum approximation in Phys. Rev. Lett., 99:256802, 2007. From tight binding calculations for structures with $1.47^\circ \le \theta < 30^\circ$ we find agreement with this formula for $\theta \gtrsim 5^\circ$. In contrast, for $\theta \lesssim 5^\circ$ this formula breaks down and the Dirac bands become strongly warped as the limit $\theta \to 0$ is approached. For an ideal system of twisted layers the limit as $\theta\to0^\circ$ is singular as for $\theta > 0$ the Dirac point is fourfold degenerate, while at $\theta=0$ one has the twofold degeneracy of the $AB$ stacked bilayer. Interestingly, in this limit the electronic properties are in an essential way determined \emph{globally}, in contrast to the 'nearsightedness' [W. Kohn. Phys. Rev. Lett., 76:3168, 1996.] of electronic structure generally found in condensed matter.Comment: Article as to be published in Phys. Rev B. Main changes: K-point mapping tables fixed, several changes to presentation

Liquid-crystal patterns of rectangular particles in a square nanocavity

Using density-functional theory in the restricted-orientation approximation, we analyse the liquid-crystal patterns and phase behaviour of a fluid of hard rectangular particles confined in a two-dimensional square nanocavity of side length $H$ composed of hard inner walls. Patterning in the cavity is governed by surface-induced order, capillary and frustration effects, and depends on the relative values of particle aspect ratio $\kappa\equiv L/\sigma$, with $L$ the length and $\sigma$ the width of the rectangles ($L\ge\sigma$), and cavity size $H$. Ordering may be very different from bulk ($H\to\infty$) behaviour when $H$ is a few times the particle length $L$ (nanocavity). Bulk and confinement properties are obtained for the cases $\kappa=1$, 3 and 6. In the confined fluid surface-induced frustration leads to four-fold symmetry breaking in all phases (which become two-fold symmetric). Since no director distorsion can arise in our model by construction, frustration in the director orientation is relaxed by the creation of domain walls (where the director changes by $90^{\circ}$); this configuration is necessary to stabilise periodic phases. For $\kappa=1$ the crystal becomes stable with commensuration transitions taking place as $H$ is varied. In the case $\kappa=3$ the commensuration transitions involve columnar phases with different number of columns. Finally, in the case $\kappa=6$, the high-density region of the phase diagram is dominated by commensuration transitions between smectic structures; at lower densities there is a symmetry-breaking isotropic $\to$ nematic transition exhibiting non-monotonic behaviour with cavity size.Comment: 31 pages, 15 figure

Quantum interference at the twist boundary in graphene

We explore the consequences of a rotation between graphene layers for the electronic spectrum. We derive the commensuration condition in real space and show that the interlayer electronic coupling is governed by an equivalent commensuration in reciprocal space. The larger the commensuration cell, the weaker the interlayer coupling, with exact decoupling for incommensurate rotations and in the θ → 0 limit. Furthermore, from first-principles calculations we determine that even for the smallest possible commensuration cell the decoupling is effectively perfect, and thus graphene layers will be seen to decouple for all rotation angles