Landau level splitting in rotationally faulted multilayer graphene

We show that valley degeneracy in rotationally faulted multilayer graphene may be broken in the presence of a magnetic field and interlayer commensurations. This happens due to a simultaneous breaking of both time-reversal and inversion symmetries leading to a splitting of Landau levels linear in the field. Our theoretical work is motivated by an experiment [Y. J. Song et al., Nature (London) 467, 185 (2010)] on epitaxially grown multilayer graphene where such linear splitting of Landau levels was observed at moderate fields. We consider both bilayer and trilayer configurations and, although a linear splitting occurs in both cases, we show that the latter produces a splitting that is in quantitative agreement with the experiment.

Figure 1

The two trilayer configurations discussed in the text. Upper panel: schematic representation; lower panel: atomic positions for top (blue, small), middle (red, medium) and bottom (green, large) layers. The angles between the top and middle layers are exaggerated for clarity.

Figure 2

Energy splitting ${\Delta }_n$ of LL${}_n$ for $n=1$ (blue, long-dashed), $n=2$ (red, medium-dashed), and $n=3$ (red, short-dashed) as a function of $B$ in trilayer configuration A for ${\gamma }_1=15\text{meV}$ and $V=40\mathrm{meV}$. In black the experimentally determined $B$ dependence of LL${}_1$: ${\Delta }_1=1\mathrm{meV}B/\mathrm{T}$.

Figure 3

${\Delta }_n/B$ as a function of $V_1$ and $V_2$. Inset: ${\Delta }_n$ as a function of $B$ at $V_1=143\mathrm{meV}$ and $V_2=170\mathrm{meV}$ for calculations that are (i) perturbative in ${\mathcal{V}}_{12}$ with $\zeta =1$ [Eq. (7); solid (green)]; (ii) nonperturbative in ${\mathcal{V}}_{12}$ with $\zeta =1$ [dashed (blue)]; and (iii) nonperturbative in ${\mathcal{V}}_{12}$ with $\zeta =1.2$ [dotted (red)].