Critical role of device geometry for the phase diagram of twisted bilayer graphene

The effective interaction between electrons in two-dimensional materials can be modified by their environment, enabling control of electronic correlations and phases. Here, we study the dependence of electronic correlations in twisted bilayer graphene (tBLG) on the separation to the metallic gate(s) in two device configurations. Using an atomistic tight-binding model, we determine the Hubbard parameters of the flat bands as a function of gate separation, taking into account the screening from the metallic gate(s), the dielectric spacer layers, and the tBLG itself. We determine the critical gate separation at which the Hubbard parameters become smaller than the critical value required for a transition from a correlated insulator state to a (semi)metallic phase. We show how this critical gate separation depends on twist angle, doping, and the device configuration. These calculations may help rationalize the reported differences between recent measurements of tBLG's phase diagram and suggest that correlated insulator states can be screened out in devices with thin dielectric layers.

Figure 1

Twisted bilayer graphene devices with a single gate (left) and two gates (right). The separation between the tBLG and the gate(s) is denoted by $\xi$. For the double-gated device, the separation is set to be equal on either side.

Figure 2

(a) On-site Hubbard parameter $V_{00}$ as a function of distance $\xi$ between tBLG and the metallic gate(s) for a twist angle of ${1.12}^{\circ }$ for the single-gate and double-gate device (see Fig. 1 for device geometries). (b) Long-range corrected on-site Hubbard parameter $U^{*}$ as a function of $\xi$ for a twist angle of ${1.12}^{\circ }$ for the single-gate and double-gate device. The horizontal dotted-dashed lines denote critical values of $U^{*}$ obtained by multiplying the critical $U^{*}/t$ values from atomistic RPA calculations of tBLG [27] with the hopping parameter for neighboring moiré Wannier functions. Dotted lines through the data points correspond to fits that are used to extract critical gate separations in Fig. 3. The solid vertical line denotes the length of the moiré unit cell.

Figure 3

Critical value of gate distance as a function of twist angle for (a) two additional electrons per moiré unit cell ($-2e$), (b) charge neutrality ($0e$), and (c) two additional holes per moiré unit cell ($2e$) for a device with a single gate. The grey regions indicate correlated insulator states [either ferromagnetic insulators (FMIs) or antiferromagnetic insulators (AFMIs)], while the white regions denotes either metallic (M) or semimetallic (SM) phases. Note that we employ the critical $U^{*}/t$ values obtained from atomistic RPA calculations of tBLG [27].

Figure 4

Hubbard parameters as a function of the separation $\xi$ of the metallic gate(s) from the tBLG for several twist angles (indicated in the legend). The panels in the left-hand (right-hand) column are for the single-gate (double-gate) configuration. (a),(b) On-site Hubbard parameters $V_{00}$; (c),(d) nearest-neighbor Hubbard parameters $V_{01}$; (e),(f) long-ranged corrected on-site Hubbard parameters $U^{*}$. The vertical lines correspond to the moiré unit cell length for each twist angle.

Figure 5

Ratio of on-site Hubbard parameter $[{\tilde{V}}_{00}=V_{00}\left(\xi \right)/V_{00}\left(\infty \right)]$ to the nearest-neighbor Hubbard parameter $[{\tilde{V}}_{01}=V_{01}\left(\xi \right)/V_{01}\left(\infty \right)]$, both of which are rescaled to their Coulomb limit value, as a function of gate separation for the studied twist angles (same color and symbol coding as Fig. 4). (a) Single-gate device configuration; (b) double-gate device configuration.

Figure 6

Intralobe (closed symbols) and interlobe (open symbols) contributions to the on-site Hubbard parameter $V_{00}$ (a),(b) and nearest-neighbor Hubbard parameter $V_{01}$ (c),(d) as a function of gate separation $\xi$ for the studied twist angles (same color and symbol coding for the different twist angles as Fig. 4). The panels in the left-hand (right-hand) column are for the single-gate (double-gate) configuration. The vertical lines correspond to the moiré unit cell length for each twist angle.

Figure 7

Critical value of gate separation ${\xi }_{\mathrm{c}}$ as a function of twist angle from the magic angle for two additional electrons per moiré unit cell ($-2e$, left panel), charge neutrality ($0e$, middle panel), and two additional holes per moiré unit cell ($2e$, right panel), for a device in the double-gate configuration at a temperature of $T\approx 0.3\text{K}$. In the grey regions, FMI denotes a ferromagnetic insulator and AFMI denotes an antiferromagnetic insulator, while in the white regions, M denotes metal and SM denotes semimetal.

Figure 8

Critical value of gate separation as a function of twist angle from the magic angle for two additional electrons per moiré unit cell ($-2e$, left panel), charge neutrality ($0e$, middle panel), and two additional holes per moiré unit cell ($2e$, right panel), for a device in the single-gate configuration at a temperature of $T\approx 5\text{K}$. In the grey regions tBLG is an antiferromagnetic insulator (AFMI), while in the white regions, M denotes metal and SM denotes semimetal.

Figure 9

Critical value of gate separation as a function of twist angle from the magic angle for two additional electrons per moiré unit cell ($-2e$, left panel), charge neutrality ($0e$, middle panel), and two additional holes per moiré unit cell ($2e$, right panel), for a device in the double-gate configuration at a temperature of $T\approx 5\text{K}$. In the grey regions tBLG is an antiferromagnetic insulator (AFMI), while in the white regions, M denotes metal and SM denotes semimetal.