Theory of the Raman spectrum of rotated double-layer graphene

We study theoretically the Raman spectrum of the rotated double-layer graphene, consisting of two graphene layers rotated with respect to each other by an arbitrary angle $\theta$. We find a relatively simple dependence of the Raman $G$ peak intensity on the angle $\theta$. On the other hand, the Raman $2D$ peak position, intensity, and width show a much more complicated dependence on the angle $\theta$. We account for all of these effects, including dependence on the incoming photon energy, in good agreement with the experimental data. We find that it is sufficient to include the interaction between the graphene layers on the electronic degrees of freedom (resulting in the occurrence of Van Hove singularities in the density of states). We assume that the phonon degrees of freedom are unaffected by the interaction between the layers. Furthermore, we decompose the Raman $2D$ peak into two components having very different linewidths; these widths are almost independent of the angle $\theta$. The change in the intensity and the peak position of one of these two components gives insight into the dependence of the overall Raman $2D$ peak features as a function of the angle $\theta$. Furthermore, we study the influence of the coherence on the Raman signal, and we separately study the influence of the interaction between the layers on the electron wave functions and energies. Additionally, we show regions in the phonon spectrum giving rise to the Raman $2D$ peak signal. This work provides an insight into the interplay between the mechanical degree of freedom (angle $\theta$) and the electronic degrees of freedom (singularities in the density of states) in rotated double-layer graphene. Additionally, this work provides a way to establish experimentally the value of the rotation angle $\theta$ using Raman spectroscopy measurement. This procedure becomes even more robust if one repeats the Raman spectroscopy measurement with a different incoming photon energy.

Figure 1

A few examples of the rotated double-layer graphene for the special values of angle $\theta$ for which the structure is superperiodic. Carbon atoms from the fixed graphene layer are shown with the red dots. Blue dots indicate the carbon atoms in the layer rotated by the angle $\theta$. The unit cell (supercell) of the rotated double-layer graphene is indicated with black lines. Integers $p$ and $q$ for each case are indicated on top of each panel. Panel (a) corresponds to the double-layer graphene in which layers are not misaligned with respect to each other, $\theta =0^{\circ }$ (AA stacking). On the other hand, panel (i) corresponds to the case in which the angle $\theta$ is maximal, $\theta ={60}^{\circ }$ (AB stacking). Panels (b) through (h) cover range of angles from $0^{\circ }$ to ${30}^{\circ }$: specifically, they are $9.{43}^{\circ }$ (b), $13.{17}^{\circ }$ (c), $15.{18}^{\circ }$ (d), $16.{43}^{\circ }$ (e), $17.{90}^{\circ }$ (f), $21.{79}^{\circ }$ (g), and $27.{80}^{\circ }$ (h).

Figure 2

Unfolding of the rotated double-layer Brillouin zone (thick black line) onto two single-layer Brillouin zones (red and blue). Panel (a) corresponds to $p=1$, $q=3$, $\theta =21.{79}^{\circ }$, $N=7$ [corresponding real-space cell is shown in panel Fig. 1g], while panel (b) corresponds to $p=3$, $q=7$, $\theta =27.{80}^{\circ }$, $N=13$ [corresponding real-space cell is shown in Fig. 1h]. Arrows indicate $\mathit{G}$ vectors from the set $\mathcal{G}$ (see main text). Determination of the set of vectors $\mathcal{G}$ is relatively easy for these choices of $p$ and $q$, while it becomes more involved for some other choices since some elements of $\mathcal{G}$ must point outside of the single-layer Brillouin zone.

Figure 3

Feynman diagrams included in the first-order Raman calculation for the Raman $G$ peak. Time is increasing from the left to the right, photons are indicated with wavy line, while the phonon is shown with the dashed line. Electrons and holes are drawn with arrows in the opposite direction with respect to time. Explicit expression for the diagram in panel (a) is given in Eq. (15).

Figure 4

Feynman diagrams included in the second-order Raman calculation for the Raman $G$ peak. Conventions are as in Fig. 3. Explicit expression for the diagram in panel (a) is given in Eq. (16).

Figure 5

Density of states in our supercell tight-binding model near the Fermi level (at the zero energy) for the rotated double-layer graphene at varying angles $\theta$ from $6.0^{\circ }$ to $17.9^{\circ }$. Lighter gray lines correspond to larger values of $\theta$. For each $\theta$ we find two large Van Hove singularities next to each other, with similar energy. Steplike singularity arises from the energy maximum or minimum (as a function of momentum), while the logarithmic divergence arises from the energy saddle point. As angle $\theta$ is increased, these singularities move further away from the Fermi level (compare lighter and darker gray lines in the figure). Thick black line is showing the density of states of a single-layer graphene, multiplied by two, so that it can be compared more easily to the rotated double-layer graphene case.

Figure 6

Calculated Raman $G$ peak intensity as a function of angle $\theta$ for two incoming photon energies [1.96 eV in black and 2.41 eV in red (gray)]. The range of angle $\theta$ shown is from $0^{\circ }$ to ${30}^{\circ }$. For range ${30}^{\circ }<\theta <{60}^{\circ }$, we find almost the same Raman $G$ peak intensity for $\theta ={30}^{\circ }+\Delta$ as for the $\theta ={30}^{\circ }-\Delta$ case. Intensity is measured relative to a single-layer graphene. We find $\sim$70 fold enhancement in the Raman $G$ peak intensity for 1.96-eV incoming photon energy near the critical angle ${10}^{\circ }$. This enhancement shifts to the higher angles $\theta$ for higher incoming photon energy (2.41 eV), consistent with shift in the Van Hove singularity. At angles away from both sides of the critical angle, we find Raman $G$ peak enhancement close to 2 (as would be expected in the limit of no interaction between the layers). Comparison with experimental data (in good agreement with our calculation) is shown in Ref. 13.

Figure 7

Dependence of the Raman $G$ peak enhancement (relative to a single layer) at the critical angle on the electron and the hole lifetime $\gamma$. The incoming photon energy in this calculation equals 1.96 eV. Fitted functional dependence of the Raman $G$ peak enhancement ($G_{\mathrm{enh}}$) is indicated with a dotted line, and equals $G_{\mathrm{enh}}=2.58\left({\mathrm{eV}}^2\right){(\gamma /2)}^{-2}$.

Figure 8

Raman intensity for the $G$ peak of a single-layer graphene computed in four different ways. Horizontal axis shows the difference in the electronic energies $\Delta E$ appearing in the energy denominators of the Feynman diagrams as in Eq. (15). The vertical axis shows the Raman $G$ peak intensity if the sum (13) is performed only using electron-hole pairs separated in energy up to $\Delta E$. Dotted lines show the results when the sum in Eq. (13) is performed incoherently over electron and hole states. Both dotted lines are downscaled 300 times in intensity. Solid lines show the results when the sum is performed coherently. Blue (gray) lines show results when the sum is performed only over two Feynman diagrams shown in Figs. 3a and 3b, while black lines shows results when the sum is performed over all 12 Feynman diagrams. Comparing the solid black line to the other three lines, we see the influence of the coherence in the electronic sum in Eq. (13), the influence of all 12 Feynman diagrams from Fig. 3, and the influence of performing the sum up to energies larger than the incoming photon energy ${\omega }_{\mathrm{in}}$ (in this calculation, ${\omega }_{\mathrm{in}}=$ 1.96 eV).

Figure 9

Supercell tight-binding model calculated position, intensity, and width of the Raman $2D$ peak. Black line indicates results for the incoming photon energy of 1.96 eV, while the red (gray) line shows results for the incoming photon energy of 2.41 eV. Horizontal axis gives angle $\theta$ of the rotated double-layer graphene. The range of angle $\theta$ shown is from $0^{\circ }$ to ${30}^{\circ }$. For range ${30}^{\circ }<\theta <{60}^{\circ }$, we find almost the same Raman $G$ peak intensity for $\theta ={30}^{\circ }+\Delta$ as for the $\theta ={30}^{\circ }-\Delta$ case. To be consistent with Ref. 13 and other experimental work, the fit was performed to the Lorentzian function. Similar results (especially for the position and the intensity) are obtained by a fit to the Gaussian function (see black line in Fig. 14). Intensity is defined as the area under the peak (not peak height). Width is defined as the full width at half of the peak maximum (FWHM). Peak intensity and peak position are defined relative to a single-layer graphene. See main text for more details. Comparison with experimental data (in good agreement with our calculation) is shown in Ref. 13.

Figure 10

Position, intensity, and width of the Raman $2D$ peak using a more approximate method (continuum model). Conventions are the same as in Fig. 9, but the range of vertical scales is not the same as in Fig. 9.

Figure 11

Regions of the phonon Brillouin zone contributing the most to the Raman $2D$ peak, with characteristic triangular regions around the Brillouin zone $K$ point. Region that contributes the most to the Raman $2D$ peak is shown in red color. Region of the Brillouin zone with intermediate intensity is shown in yellow, and that of zero intensity in blue. Color scale for each panel is scaled individually to the largest intensity for that panel since otherwise the overall intensity of the panels for small angles would be too small (see Fig. 9 showing decrease of the Raman $2D$ peak intensity at small angles $\theta$). The Brillouin zone of the bottom (solid line) and the top (dashed line) graphene layers is indicated. Contributions to the $2D$ peak of only one layer (bottom) are shown for simplicity, and we only show the region of the Brillouin zone close to the $K$ point (approximately the same region is indicated with dashed line in Fig. 12). The angle $\theta$ is increasing going from panel (a) to panel (h) and it equals 4.41${}^{\circ }$ (a), 7.93${}^{\circ }$ (b), 8.61${}^{\circ }$ (c), 9.43${}^{\circ }$ (d), 10.42${}^{\circ }$ (e), 11.64${}^{\circ }$ (f), 13.17${}^{\circ }$ (g), and 27.80${}^{\circ }$ (h). Calculation is performed with the incoming photon energy of 1.96 eV. Large transfer of weight is seen close to the critical angle in panel (e) when the Brillouin zone $K$ point of the top layer is overlapping with the triangular region in the phonon Brillouin zone.

Figure 12

Panel (a) shows the sketch of the overlapping Dirac cones of the two graphene sheets (shown in red and blue). Dirac cones are centered at the Brillouin zone edge points $K$ of each graphene layer. Black arrows indicate the overlap region in which the interaction between the graphene layers introduces a hybridization gap in electron and hole states. Panel (b) shows the sketch of the isoenergy curves (for both layers) in the electron Brillouin zone separated in the energy by the amount equal to the incoming photon energy. The angle $\theta$ is close to the critical angle. Brillouin zone of each layer is indicated with red and blue hexagons. Overlap region is indicated with the black arrow, as in panel (a). Panel (c) shows the sketch of the nesting vectors in the phonon Brillouin zone connecting two Dirac cones in the electron Brillouin zone (corresponding to the same graphene layer). By construction, phonon nesting vectors have opposite trigonal warping to that of the electrons and are twice as far away from the Brillouin zone edge point $K$. Approximately the same region of the phonon Brillouin zone as in Fig. 11 is indicated with a dashed line.

Figure 13

Comparison of a single Gaussian fit (thin red line) and a two-Gaussian fit (thin yellow line) of the calculated Raman $2D$ profile (thick black line) for the single-layer graphene (left panel) and the rotated double-layer graphene with $\theta =6.4^{\circ }$ (right panel). Two Gaussian components of the two-Gaussian fit are shown in green (broad component) and blue (narrow component). See Fig. 14 for dependence of broad and narrow components on angle $\theta$.

Figure 14

Fit of the calculated Raman $2D$ peak to a single Gaussian (black line) and to two Gaussians (broad Gaussian component is in green and narrow in blue). Peak position and intensity for all three lines are given relative to the single Gaussian fit of the $2D$ Raman peak in the single-layer graphene. Other conventions are as in the Fig. 9. Narrow Gaussian component for some values of angle $\theta$ has negligible intensity, which makes the fitting procedure ill conditioned. For that range of angles, the position and width of the narrow component are drawn with a straight dotted line. Calculation is performed for a single incoming photon energy 1.96 eV.

Figure 15

Calculated position, intensity, and width of the Raman $2D$ peak for the incoming photon energy of 1.96 eV. Dotted lines show results of the calculation in which the influence of the electron hopping terms between two graphene layers affects only electron eigenenergies (red) or only electron wave functions (blue). See main text for more details. Other conventions are the same as in Fig. 9.

Figure 16

Calculated Raman $2D$ profiles [$I_2({\omega }_{\mathrm{out}}=\omega )$ from Eq. (14)] for the rotated double-layer graphene (thin black lines), the single-layer graphene multiplied by two (thick red line), the AB stacked double-layer graphene (blue), and the AA stacked double-layer graphene (green). The rotated double-layer graphene spectra are shifted in the vertical direction, proportionally to the angle $\theta$, for clarity. Raman $2D$ profile of the single-layer graphene (red) is shifted vertically proportional to $\theta ={30}^{\circ }$.