First-principles studies of electric field effects on the electronic structure of trilayer graphene

A gate electric field is a powerful way to manipulate the physical properties of nanojunctions made of two-dimensional crystals. To simulate field effects on the electronic structure of trilayer graphene, we used density functional theory in combination with the effective screening medium method, which enables us to understand the field-dependent layer-layer interactions and the fundamental physics underlying band gap variations and the resulting band modifications. Two different graphene stacking orders, Bernal (or ABC) and rhombohedral (or ABA), were considered. In addition to confirming the experimentally observed band gap opening in ABC-stacked and the band overlap in ABA-stacked trilayer systems, our results reveal rich physics in these fascinating systems, where layer-layer couplings are present but some characteristics features of single-layer graphene are partially preserved. For ABC stacking, the electric-field-induced band gap size can be tuned by charge doping, while for ABA band the tunable quantity is the band overlap. Our calculations show that the electronic structures of the two stacking orders respond very differently to charge doping. We find that in the ABA stacking hole doping can reopen a band gap in the band-overlapping region, a phenomenon distinctly different from electron doping. The physical origins of the observed behaviors were fully analyzed, and we conclude that the dual-gate configuration greatly enhances the tunability of the trilayer systems.

Figure 1

Schematics of the (a) ABC and (b) ABA stacking orders of trilayer graphene and the (c) dual- and (d) single-gate configurations used in simulations. The first Brillouin zone of trilayer graphene is shown in the insert in (a). In (c) and (d) the net charge density on the trilayer graphene is denoted by $\sigma$, and the electric field between graphene and gate electrodes is denoted by $E$. Vacuum (permittivity ${\epsilon }_0)$ is used as the medium between graphene and gate electrodes in our simulations.

Figure 2

The band structure of ABC-stacked trilayer graphene near the Fermi energy under electric fields (a) $E=0$ and (b) ${\epsilon }_{0}E=0.023\mathrm{C}/{\mathrm{m}}^2$ with the trilayer graphene kept neutrally charged. The energy gap at the $K$ point is denoted as $E_K$, and the smallest band gaps along the $\mathrm{\Gamma }\text{−}K$ and $\mathrm{\Gamma }\text{−}M$ paths are denoted as $E_{g1}$ and $E_{g2}$. In (c) the energy gap at electric field of ${\epsilon }_{0}E=0.023\mathrm{C}/{\mathrm{m}}^2$ is plotted on a $k$-point mesh centered at the $K$ point. The dependence on the electric field is plotted in (d).

Figure 3

The band structure (left in each panel) and the density of states (right) of ABC-stacked trilayer graphene are shown for different electric fields and charge dopings. The electric field is ${\epsilon }_{0}E=0.006\mathrm{C}/{\mathrm{m}}^2$ for (a)–(c) and ${\epsilon }_{0}E=0.030\mathrm{C}/{\mathrm{m}}^2$ for (d)–(f). The net charge density on the trilayer graphene is $-7.6\times {10}^{12}{\mathrm{cm}}^{-2}$ for (a) and (d), zero for (b) and (e), and $+7.6\times {10}^{12}{\mathrm{cm}}^{-2}$ for (c) and (f). The band structure is plotted along the same path as in Figs. 2 and 2. The layer-projected density of states on the central graphene layer is denoted by solid (black) lines, and on the other two graphene layers by dashed (red) and dotted (blue) lines.

Figure 4

The band gap of ABC-stacked trilayer graphene as a function of the net charge density. The electric field strength is ${\epsilon }_{0}E=0.030\mathrm{C}/{\mathrm{m}}^2$ (squares) and ${\epsilon }_{0}E=0.006\mathrm{C}/{\mathrm{m}}^2$ (circles). The changes in the potential energy felt by electrons along the direction of the electric field are plotted in the insets, where the positions of the top and bottom graphene layer are denoted by vertical dashed lines.

Figure 5

The band structure of ABA-stacked trilayer graphene under electric fields of (a) 0 and (b) $0.023\mathrm{C}/{\mathrm{m}}^2$, respectively. In (c) the energy gap at electric field of ${\epsilon }_{0}E=0.023\mathrm{C}/{\mathrm{m}}^2$ is plotted on a $k$-point mesh centered at the $K$-point. The widths of the band overlaps induced by electric fields marked as $\mathrm{\Delta }k$ and $\mathrm{\Delta }E$ in (b) are plotted in (d) as a function of the electric field.

Figure 6

The band structure (left in each panel) and the density of states (right in each panel) of ABA-stacked trilayer graphene at different electric fields and charge dopings. The electric field is $E=0$ for (a)–(c) and ${\epsilon }_{0}E=0.015\mathrm{C}/{\mathrm{m}}^2$ for (d)–(f). The net charge density on the trilayer graphene is $-9.5\times {10}^{12}{\mathrm{cm}}^{-2}$ for (a) and (d), 0 for (b) and (e), and $+9.5\times {10}^{12}{\mathrm{cm}}^{-2}$ for (c) and (f). The band structure is plotted along the same path as in Figs. 5 and 5. The layer-projected density of states on the central graphene layer is denoted by solid (black) lines and on the other two graphene layers by dashed (red) and dotted (blue) lines. The total density of states is denoted by dashed black lines.

Figure 7

The energy gap between the two peaks in the total DOS of ABA-stacked trilayer graphene as a function of the net charge density shown for electric field strength $E=0$ (squares and black solid line) and ${\epsilon }_{0}E=0.015\mathrm{C}/{\mathrm{m}}^2$ (circles and red dashed line). Insets show the changes in the potential energy felt by electrons along the direction of the electric field (black solid lines), where the positions of the graphene layers are denoted by vertical dashed lines.

Figure 8

The density of electrons and holes imposed by gate voltages on (a) ABC-stacked and (b) ABA-stacked trilayer graphene. The net charge density is $-9.5\times {10}^{12}/{\mathrm{cm}}^2$ for electron doping and $+9.5\times {10}^{12}/{\mathrm{cm}}^2$ for hole doping. The densities along the $z$ axis after integrating over the $x\text{−}y$ plane are shown in (a1) and (b1). The isosurfaces of electron density are shown in (a2) and (a3) and hole density in (b2) and (b3); the contour value is set as $\pm 0.05\times {10}^{-3}/{\AA }^3$.

Figure 9

(a), (b) Lattice constants, (c) the $C_{33}$ component of elastic constant, and (d) the interlayer binding energy calculated using variations of van der Waals density functionals listed in Table 1. The range of experimental results are denoted by the two black dashed lines, except (a) for which there is no spread. References for the experimental results are: (a) Ref. [29], (b) Refs. [29, 40], (c) Refs. [41, 42, 43, 44], and (d) Refs. [43, 45, 46].