Electronic properties of asymmetrically doped twisted graphene bilayers

Rotated graphene bilayers form an exotic class of nanomaterials with fascinating electronic properties governed by the rotation angle $\theta$. For large rotation angles, the electron eigenstates are restricted to one layer and the bilayer behaves like two decoupled graphene layers. At intermediate angles, Dirac cones are preserved but with a lower velocity and van Hove singularities are induced at energies where the two Dirac cones intersect. At very small angles, eigenstates become localized in peculiar moiré zones. We analyze here the effect of an asymmetric doping for a series of commensurate rotated bilayers on the basis of tight-binding calculations of their band dispersions, density of states, participation ratio, and diffusive properties. While a small doping level preserves the $\theta$ dependence of the rotated bilayer electronic structure, larger doping induces a further reduction of the band velocity in the same way as a further reduction of the rotation angle.

Figure 1

Dirac cones and total DOS of a doped rotated bilayer, schematic diagram.

Figure 2

Band dispersions for doped bilayers with $\mathrm{\Delta }\epsilon =0.2$ eV. (a) (5,9) bilayer ($\theta =18.{73}^{\circ }$), (b) (6,7) bilayer ($\theta =5.{08}^{\circ }$), and (c) (12,13) bilayer ($\theta =2.{65}^{\circ }$). Lines (dashed line) are TB calculations for bilayers (monolayer).

Figure 3

Band dispersions for doped bilayers: (a) (6,7) bilayer ($\theta =5.{08}^{\circ }$), (b) (12,13) bilayer ($\theta =2.{65}^{\circ }$). The onsite energy difference $\mathrm{\Delta }\epsilon$ between $p_z$ orbitals in the two layers is (solid line) $\mathrm{\Delta }\epsilon =0$, (circle) $\mathrm{\Delta }\epsilon =0.2$ eV, (star) $\mathrm{\Delta }\epsilon =0.4$ eV, (square) $\mathrm{\Delta }\epsilon =0.6$ eV. Arrows show the energy difference $\mathrm{\Delta }{E}_D$ between bands at K (Table 2). $E\left(\vec{k}\right)$ has been computed for more then 50 $\vec{k}$ values in the $k$ scale shown in these figures.

Figure 4

$\mathrm{\Delta }{E}_D$ versus $\mathrm{\Delta }\epsilon$ in rotated doped bilayers. [Inset: maximum value $\mathrm{\Delta }{E}_{Dm}$ of $\mathrm{\Delta }{E}_D$ versus the $\theta$ angle. Points line is guide for the eyes. The dashed lines show $\mathrm{\Delta }{E}_{\mathrm{vHs}}$ given by Eq. (7), i.e., here $\mathrm{\Delta }{E}_{\mathrm{vHs}}\simeq 0.15\theta \left(\text{deg}\right)-0.24$.]

Figure 5

Velocity at Dirac point (slope of the band along $KM$ at $K$ and Dirac energy) versus $\mathrm{\Delta }\epsilon$ in rotated doped bilayers. The vertical dashed lines show the values $\mathrm{\Delta }{\epsilon }_m$ for which $\mathrm{\Delta }{E}_D\left(\mathrm{\Delta }\epsilon \right)$ is maximum.

Figure 6

Band dispersions for (25,26) bilayer ($\theta =1.{30}^{\circ }$): (a) undoped bilayer ($\mathrm{\Delta }\epsilon =0$), (b) doped bilayer $\mathrm{\Delta }\epsilon =0.2$ eV. (c) Zoom around energy $E_{D1}$: (dashed line) $\mathrm{\Delta }\epsilon =0$, (c.1) $\mathrm{\Delta }\epsilon =0.05$ eV, (c.2) $\mathrm{\Delta }\epsilon =0.1$ eV, (c.3) $\mathrm{\Delta }\epsilon =0.2$ eV, (c.4) $\mathrm{\Delta }\epsilon =0.3$ eV, (c.5) $\mathrm{\Delta }\epsilon =0.4$ eV. The arrows show the energy difference $\mathrm{\Delta }{E}_D$.

Figure 7

Average velocity $V_x$ along the $x$ axis ($V=\sqrt{2}{V}_x$) versus energy $E$. (a) (Dashed line) graphene and undoped ($n,m$) bilayers ($\mathrm{\Delta }\epsilon =0$). (b) (6,7) bilayer and (c) (25,26) bilayer: (solid line) undoped ($\mathrm{\Delta }\epsilon =0$), (circle) $\mathrm{\Delta }\epsilon =0.2$ eV, (star) $\mathrm{\Delta }\epsilon =0.4$ eV, and (square) $\mathrm{\Delta }\epsilon =0.6$ eV.

Figure 8

Total density of states (DOS) in (a) (5,9), (b) (6,7), (c) (12,13), and (d) (25,26) bilayers. The onsite energy difference $\mathrm{\Delta }\epsilon$ between $p_z$ orbitals in the two layers is (solid line) $\mathrm{\Delta }\epsilon =0$, (circle) $\mathrm{\Delta }\epsilon =0.2$ eV, (star) $\mathrm{\Delta }\epsilon =0.4$ eV, (square) $\mathrm{\Delta }\epsilon =0.6$ eV. The DOS curves have been shifted vertically for clarity (the origin of the DOS for each curve is indicated by the horizontal line on the lower left corner). (e) Energy difference between the energies $\mathrm{\Delta }{E}_{\mathrm{vHs}}$ of vHs, $\mathrm{\Delta }{E}_{\mathrm{vHs}}=E_{+}-E_{-}$, versus rotation angle for different doping values. The fine solid line is guide for the eye. Dashed line shows $\mathrm{\Delta }{E}_{\mathrm{vHs}}$ given by Eq. (7), i.e., here $\mathrm{\Delta }{E}_{\mathrm{vHs}}\left(\mathrm{eV}\right)\simeq 0.15\theta \left(\mathrm{deg}\right)-0.24$.

Figure 9

Density of states (DOS) calculated by recursion in doped (a) (6,7), (b) (25,26) bilayers with $\mathrm{\Delta }\epsilon =0.4$ eV: (solid line) total DOS, (triangle up) average DOS in layer 1 (undoped layer), (triangle down) average DOS in layer 2 (doped layer). (a) The total DOS in (6,7) bilayer, calculated by diagonalization in reciprocal lattice, is also shown with fine line with empty circle. It is very close with total DOS calculated by recursion method.

Figure 10

TB (a) average participation ratio and (b) average layer participation ratio in ($n,m$) undoped bilayers ($\mathrm{\Delta }\epsilon =0$).

Figure 11

TB average participation ratio $\langle p\rangle$ and average layer participation ratio $\langle p\rangle$* in (a) (5,6), (b) (6,7), (c) (12,13), and (d) (25,26) bilayers. The onsite energy difference $\mathrm{\Delta }\epsilon$ between $p_z$ orbitals in the two layers is (black line) $\mathrm{\Delta }\epsilon =0$, (circle) $\mathrm{\Delta }\epsilon =0.2$ eV, (star) $\mathrm{\Delta }\epsilon =0.4$ eV, (square) $\mathrm{\Delta }\epsilon =0.6$ eV.

Figure 12

Diffusivity $D$ versus scattering time $\tau$ in graphene (5,9), (6,7), and (25,26) bilayers. For (25,26) bilayers, the Boltzmann (B) term and non-Boltzmann (NB) term are shown. $\mathrm{\Delta }\epsilon =0$ for (a) energy $E_{\mathrm{F}}$ close to the Dirac energy ($E_D=0$), (b) $E_{\mathrm{F}}=0.1$ eV. (c) $\mathrm{\Delta }\epsilon =0.2$ and $E_{\mathrm{F}}=-0.1$ eV, (d) $\mathrm{\Delta }\epsilon =0.6$ and $E_{\mathrm{F}}=-0.3$ eV.