Electronic structure and optical properties of twisted multilayer graphene

We study the electronic and optical properties of twisted trilayer and tetralayer graphene structures using a combination of tight-binding and density-functional theory methods. Band structures, densities of states, and optical absorption spectra are calculated for a variety of layer stackings and twisting angles. Systematic trends of all properties are obtained and compared to the more well known case of twisted bilayer graphene. For trilayer and tetralayer structures, we find, respectively, two and three well-defined absorption peaks in the infrared/visible range that shift with twisting angle, in contrast to the single peak observed in bilayer graphene. In addition, systems containing Bernal-stacked layers present an extra peak in the infrared which is related to transitions between parabolic bands and does not shift with twisting angle. The observed trends may be used to identify the twisting angle and the number of layers in multilayer graphene samples. In particular, magic angles are predicted for the trilayer and tetralayer structures.

Figure 1

Rotation between two hexagonal Brillouin zones by a generic angle $\theta$ (top) and the schematic representation of the band structure superposition for electronic dispersions with two monolayers [t(1+1)LG, bottom left] or two bilayers [t(2+2)LG, bottom right]. Bands from up (u) and down (d) layers are indicated by red and blue, respectively. Optical transition energies are also displayed schematically.

Figure 2

Band structures and DOS for $\theta =13.2^{\circ }[(N,M)=(2,3)]$ for all the cases studied. Black, red, and blue arrows represent the lowest-energy allowed optical transitions between log- and discontinuitylike VHSs. These optical transitions are indicated by circles in Fig. 5. Horizontal dashed lines guide the eyes to these singularities in the DOS.

Figure 3

(a)–(d) Spectral weight for $k$ points inside the first BZ corresponding to the lowest-energy optical transition peak for the t(1+1)LG, t(2+1)LG, t(3+1)LG, and t(2+2)LG structures, respectively, with the same twist angle of $13.2^{\circ }$. Lighter tones of blue represent higher intensities.

Figure 4

Density of states as a function of twist angle for all the cases studied. Black and blue circles mark the log-type VHS closest to the Fermi level for valence and conduction band states, respectively. Twist angles and $(N,M)$ values are indicated for each case.

Figure 5

Optical absorption spectra as a function of twist angle for all the cases studied. Black, red, and blue circles indicate some of the optical transitions (discussed in the text). Twist angles and $(N,M)$ values are indicated for each case.

Figure 6

Comparison of the optical spectra of various $\text{t}\left(n+m\right)\text{LG}$ structures for $13.2^{\circ }$, using (a) tight-binding and (b) DFT methods. Spectra are normalized by monolayer absorption and displaced vertically for clarity. Here t(1+1+1)LG indicates $A\theta A$, and $\text{t}\left(1+1+1^{\prime }\right)\text{LG}$ indicates $A\theta B$ structures.

Figure 7

Angular dependence of the first optical absorption peaks for t(1+1)LG (solid black line), t(2+2)LG (solid red line), and t(2+1)LG (solid blue line) and the second transition of t(2+1)LG (dashed blue line), fitted from our tight-binding results. From the intercepts at the horizontal axes, magic angles for the various structures are predicted. The solid symbols represent DFT calculations of the absorption peaks, and the open symbols are experimental results [31]. Note that we cannot describe the second transition of t(2+1)LG for very small angles since there is no anticrossing between the monolayer and second bilayer bands close the $K$ point.

Figure 8

Tight-binding energy difference between the lowest-energy optical transition for t(1+1)LG and the other twisted multilayer structures $\left[\mathrm{t}\right(n+m\left)\mathrm{LG}\right]$ studied in this work. The lines are spline fits to guide the eye. The shaded region between the t(1+1+1)LG and $\text{t}\left(1+1+1^{\prime }\right)\text{LG}$ curves represents a general $A{\theta }_1{\theta }_2$ stacking.