Pressure-induced reentrant Dirac semimetallic phases in twisted bilayer graphene

External hydrostatic pressure ($P$) is another controllable and clean knob to tune the energy position of van Hove singularities in twisted bilayer graphene (TBLG), besides the interlayer rotation angle ($\theta$). Based on total energy density functional theory calculations, we report here the ground-state properties of high-angle TBLG with two families due to their even and odd sublattice-exchange parity, hereafter parity for simplicity, under vertical hydrostatic $P$. Even (odd) parity refers to rotate one layer with respect to the other in a bilayer graphene with stacking AA (AB). We observed a Dirac semimetal (0–20 GPa) semiconductor (20–70 GPa) phase transition at $\theta =21.8^{\circ }$ and $13.4^{\circ }$; however, a reentrant Dirac semimetallic phase is observed for $\theta =9.4^{\circ }$ and $P\ge 70$ GPa whenever TBLG has even parity. Meanwhile, TBLG systems with odd parity and different $\theta$ remain metallic but with an enhanced trigonal warping for $P>50$ GPa. Indeed, the semiconductor phase is only present for high $\theta$. The phase transition in TBLG with even parity and the metallic character of TBLG with odd parity are due to band inversion effect assisted by a band-to-band repulsion mechanism shown by unfolded bands. This work shows the relevance of external $P$ and parity in TBLG electronic structure and their possible implications in experiments.

Figure 1

Top view of TBLG with $\theta =21.8^{\circ }$ and (a) even parity and (b) odd parity. The black line encloses the periodic supercell. Red hexagon depicts the $C_6$ symmetry centered in a hollow position in (a) while the red triangle is for top position in (b) having a $C_3$ symmetry. Blue and orange spheres represent carbon atoms at the bottom and top graphene monolayer, respectively. (c) Relative energy between optimized TBLGs with even parity and odd parity as a function of $P$ and/or $\varepsilon$. The TBLG systems with even parity were taken as reference with zero energy. The inset shows the relationship between $P$ and $\varepsilon$ for different bulk TBLGs system in comparison with the calculated data for bulk graphite in this work (bulk G: this work) and with data from Ref. [26] (bulk G: Ref. [26]). The relative energy is in meV per formula unit (f.u.) of graphene.

Figure 2

(a), (b) Show atomic displacement fields of carbon atoms in the bottom layer with different even (E) or odd (O) parity, and high $P$. Empty gray circles represent the initial atomic positions and arrowheads shows the direction and the size of atomic displacement $\mathrm{\Delta }$. (c) Modulus of maximum $\mathrm{\Delta }$ as a function of $P$ for different values of $\theta$ and parity.

Figure 3

(a,) (d) Show the logarithm scale density of states (DOS) as a function of energy, pressure, and compression for TBLG with $\theta =21.8^{\circ }$ and different parity, even (E) and odd (O). (b), (c) Show the band structure and their corresponding DOS of TBLG, respectively, with even parity at four different pressures. (e), (f) Plots represent the same as (b) and (c) but for odd parity. Dirac point (DP), first (VHS1), second (VHS2), and third (VHS3) van Hove singularities and band gap (GAP) can be identified in (a) and (d). The Fermi level is set at 0 eV.

Figure 4

The same as in Fig. 3 but now for $\theta =13.2^{\circ }$ with different parity, even (E) and odd (O).

Figure 5

The same as in Fig. 3 but now for $\theta =9.4^{\circ }$ with different parity, even (E) and odd (O). Additionally, pseudogaps (pGAPs) can be identified. Green circles show Dirac cones.

Figure 6

Contour plots of eigenvalue states as a function of $k_x$ and $k_y$ in momentum space for $\theta =9.4^{\circ }$ with different parity, even (E) and odd (O), and $P=70$ GPa for the valence band maximum (VBM) and conduction band minimum (CBM). The first Brillouin zone edges are marked by the black hexagon while irreducible Brillouin zone is delimited by red triangle with high-symmetry points. $k_x$ and $k_y$ are in ${\AA }^{-1}$.

Figure 7

Effective band structures calculated through the unfolding approach for $\theta =9.4^{\circ }$ with (a) even parity and (a) odd parity at different pressures. The unfolding was performed onto the $K-M k$ path of the graphene primitive cell in blue, where TBLG supercell states are projected onto both graphene primitive cells. The top (bottom) graphene layer is highlighted with blue (orange) lines, where the width of the lines is proportional to spectral weight, $W(k,I)$. The line widths are scaled ×1, ×1, ×1.5, ×1.5, and ×2.5 for $P=5$, 14, 26, 44, and 70 GPa, respectively, with respect to the line width at 0 GPa. The line width in red insets with yellow background are scaled $\times 4$ as well. The Fermi energy is set to 0 eV.

Figure 8

$P$ vs atomic volume data for bulk graphite employing different equations of states (EOSs) for several basis-set sizes and exchange-correlation functionals, as local density approximation (LDA), generalized gradient approximation (GGA), and van der Waals (vdW).