Ab initio quasiparticle band structure of ABA and ABC-stacked graphene trilayers

We obtain the quasiparticle band structure of ABA and ABC-stacked graphene trilayers through ab initio density-functional theory (DFT) and many-body quasiparticle calculations within the $GW$ approximation. To interpret our results, we fit the DFT and $GW$ $\pi$ bands to a low-energy tight-binding model, which is found to reproduce very well the observed features near the $K$ point. The values of the extracted hopping parameters are reported and compared with available theoretical and experimental data. For both stackings, the self-energy corrections lead to a renormalization of the Fermi velocity, an effect also observed in previous calculations on monolayer graphene. They also increase the separation between the higher-energy bands, which is proportional to the nearest-neighbor interlayer hopping parameter ${\gamma }_1$. Both features are brought to closer agreement with experiment through the self-energy corrections. Finally, other effects, such as trigonal warping, electron-hole asymmetry, and energy gaps, are discussed in terms of the associated parameters.

Figure 1

Arrangement of carbon atoms on the ABA (left) and ABC (right) graphene trilayers. Atoms belonging to the A and B sublattices of each layer are represented by yellow and black spheres, respectively. The red dashed lines indicate the tight-binding parameters included in the model used to fit our first-principles results (see text).

Figure 2

Comparison between the LDA (black lines) and $GW$ (red lines) band structures obtained from our calculations for ABA (left) and ABC (right) stacking. The Brillouin-zone path is along the M-K-$\Gamma$ directions from left to right, and it is centered in K. The Fermi level is set to zero in all cases, both in LDA and $GW$.

Figure 3

Comparison between the quasiparticle bands (black lines) and the corresponding adjusted TB bands (red lines) for ABA (left) and ABC (right) stacking. The path is the same as in Fig. 2, and the Fermi level is set to zero in both cases. Fitted parameters are shown in Table 2.