Pressure-induced metal-insulator transition in twisted bilayer graphene

Recent experiments on twisted bilayer graphene (TBLG) have observed insulating states for two and three unit charges per moiré supercell, whereas the quarter-filling state (QFS) remained metallic. Subsequent experiments show that under hydrostatic pressure the QFS turns insulating for a certain window of pressure. In fact, the resistivity of the 1/2-filling and 3/4-filling states are also enhanced in the same pressure-window. Using pressure-dependent band structure calculations we compute the ratio of the potential to the kinetic energy, $r_s$. We find a window of pressure for which $r_s$ crosses the threshold for a triangular Wigner crystal, thereby corroborating our previous work that the insulating states in TBLG are driven by Wigner physics, A key prediction of this work is that the window for the onset of the hierarchy of Wigner states that obtains at commensurate fillings conforms to a dome shape under pressure. We also predict the optimal condition for Wigner crystallization to be around $1.5\mathrm{GPa}$. Consequently, TBLG provides a new platform for the exploration of Wigner physics and its relationship with superconductivity.

Figure 1

Interlayer tunneling for different pressures as a function of distance from the site of rotation. With increasing pressure, the interlayer distance decreases causing the tunneling strength to increase. Inset: The dominant contribution to $t\left(\mathit{r}\right)$ comes from $V_{pp\sigma }$ which leads to a simpler expression in Eq. (6).

Figure 2

Pressure-dependent band dispersion of TBLG for $\theta =1.{25}^{\circ }$. The parameters used in obtaining these are listed in the Table 1. With increasing pressure, the low energy bands become flatter; however, beyond $\sim 1.45\pm 0.1$ GPa, the bandwidth increases subsequently. The reason behind such an optimal behavior can be understood from Eq. (12).

Figure 3

With increasing external pressure, the interlayer distance decreases by ${\delta }_d\%$ (blue curve or right axis) which is described in Eq. (7). Reduced separation enhances interlayer tunneling, $w\approx t_{\perp }/3$ (green curve or top axis), as can be seen from Eq. (11). This causes an increase (red curve or left axis) in the effective magic angle, ${\theta }_{\text{magic}}^{\text{eff}}$, where the band become the flattest; see Eq. (12). The dots correspond to the reported values of pressure where the measurements of [24] were performed.

Figure 4

For the device D2 of [24], we compute $r_s$ (red dots along with computational error associated with coarse graining of the $k$ space) of the quarter-filling state. The blue curve provides a guide for the eye. Its dome-like behavior can explain a similar feature seen in the conductance of the quarter-filling state in the experiment of [24]. For the pressure window of $\sim 1–3$ GPa the system enters the Wigner crystallization regime, $r_s\gtrsim 37$. Similar behavior is obtained for $n_e=n_s/2$ and $3n_s/4$.

Figure 5

(a) The chemical potential dependence (smoothly fitted) of the carrier concentration for various pressures, as obtained in Eq. (16). (b) $r_s$ as a function of pressure obtained for $\nu =2,3$ states. Similar to the $\nu =1$ state discussed in the main text they also exhibit dome-like behavior.