Electronic structure of turbostratic graphene

By S. Shallcross, Sangeeta Sharma, E. Kandelaki, O. Pankratov. 2009

356 citations recorded by the source corpus.

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 $\ensuremath{\theta}$. 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 $\ensuremath{\theta}\ensuremath{\rightarrow}0\ifmmode^\circ\else\textdegree\fi{}$; the form we derive for ${v}_{F}(\ensuremath{\theta})$ agrees with that found via a continuum approximation in [J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Phys. Rev. Lett. 99, 256802 (2007)]. From tight-binding calculations for structures with $1.47\ifmmode^\circ\else\textdegree\fi{}\ensuremath{\le}\ensuremath{\theta}l30\ifmmode^\circ\else\textdegree\fi{}$ we find agreement with this formula for $\ensuremath{\theta}\ensuremath{\gtrsim}5\ifmmode^\circ\else\textdegree\fi{}$. In contrast, for $\ensuremath{\theta}\ensuremath{\lesssim}5\ifmmode^\circ\else\textdegree\fi{}$ this formula breaks down and the Dirac bands become strongly warped as the limit $\ensuremath{\theta}\ensuremath{\rightarrow}0$ is approached. For an ideal system of twisted layers the limit as $\ensuremath{\theta}\ensuremath{\rightarrow}0\ifmmode^\circ\else\textdegree\fi{}$ is singular as for $\ensuremath{\theta}g0$ the Dirac point is fourfold degenerate, while at $\ensuremath{\theta}=0$ one has the twofold degeneracy of the $AB$ stacked bilayer. Interestingly, in this limit the electronic properties are in an essential way determined globally, in contrast to the ``nearsightedness'' [W. Kohn, Phys. Rev. Lett. 76, 3168 (1996)] of electronic structure generally found in condensed matter.

Subfields: Atomic and Molecular Physics, and Optics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/197bbb534b12fb88d956af30ce1943d70820abb8

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

Unconventional superconductivity in magic-angle graphene superlattices (2018, background): “One important parameter in the commensurate TBG structures is r, which can be intuitively understood as the number of ‘apparent’ moiré pattern wavelengths it takes to fully recover the lattice periodicity.(19,55) The simplest commensurate structures with r = 1 are called ‘minimal’ structures.”
Electronic properties of graphene-based bilayer systems (2015, background): “[154, 368, 369, 378] characterized a superstructure by two co-prime integers p and q, which can be connected to our notation (m0, r) as follows: p = r/2, q = m0 + r/2 for even r and p = r, q = 2m0 + r for odd r.”
Band structure of twisted bilayer graphene: Emergent symmetries, commensurate approximants, and Wannier obstructions (2018, background): “The content of this section has all been discussed in existing literature [34, 39, 40]; here, we simply review these results in an attempt to clarify some potential confusions which arose in light of the recent interest of TBG.”
Origin of Magic Angles in Twisted Bilayer Graphene. (2018, background): “Early studies [35, 36, 38, 39] set w0 = w1 which we denote as the Bistritzer-MacDonald continuum model (BM-CM).”
Colloquium: Graphene spectroscopy (2014, background): “Calculations predict a strong dependence of the velocity on the twist angle (Lopes dos Santos, Peres, and Castro Neto, 2007, 2012; Shallcross et al., 2010; Trambly de Laissardiére, Mayou, and Magaud, 2010; Bistritzer and MacDonald, 2011).”
Interlayer interaction in general incommensurate atomic layers (2015, background): “When the rotation angle is small, in particular, the system exhibits a moiré interference pattern of which period can be much greater than the atomic scale, and such a long-period modulation is known to strongly influence the low-energy electronic motion [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].”