Electrostatic fate of $N$-layer moiré graphene

Twisted $N$-layer graphene (TNG) moiré structures have recently been shown to exhibit robust superconductivity similar to twisted bilayer graphene (TBG). In particular for $N=4$ and $N=5$, the phase diagram features a superconducting pocket that extends beyond the nominal full filling of the flat band. These observations are seemingly at odds with the canonical understanding of the low-energy theory of TNG, wherein the TNG Hamiltonian consists of one flat-band sector and accompanying dispersive bands. Using a self-consistent Hartree-Fock treatment, we explain how the phenomenology of TNG can be understood through an interplay of in-plane Hartree and inhomogeneous layer potentials, which cause a reshuffling of electronic bands. We extend our understanding beyond the case of $N=5$ realized in experiment so far. We describe how the Hartree and layer potentials control the phase diagram for devices with $N>5$ and tend to preclude exchange-driven correlated phenomena in this limit. To circumvent these electrostatic constraints, we propose a flat-band paradigm that could be realized in large-$N$ devices by taking advantage of two nearly flat sectors acting together to enhance the importance of exchange effects.

Figure 1

(a) Device schematics. We consider $N$-layer graphene with alternating twist angles in a double-gated setup. Here $\theta$ is the physical twist angle. (b) Schematics of layer charge distribution [see Eq. (8)] for $N=11$, showing the three sectors with lowest effective twist angle, $k=1$: red, $k=2$: orange, $k=3$: green. In experiment to date, $k=1$ is the flattest, “magic” sector. (c) Single-particle band structure for $N=11$. (d) Band filling of the magic sector at different total gate densities. $25\times {10}^{12}{\mathrm{cm}}^{-2}$ is the threshold of dielectric breakdown in current hBN-based samples [29].

Figure 2

(a) Experimental data of the Hall density vs total filling showing the the cascade transitions (arrows) for $N=3,4,5$, see Ref. [22] for details on the samples and measurements. (b) Corresponding experimental data for $T_C$ domes for $N=3,4,5$. (c) Colormap of ${\nu }_{\text{total}}$ needed to reach filling ${\nu }_{\text{magic}}=3$ of the magic sector in the ${\varepsilon }_{\parallel }\text{−}{\varepsilon }_{\perp }$ plane. (d) Interacting structures and densities of states for TPG at ${\nu }_{\text{magic}}\approx 4$, including in-plane Hartree and Fock (HFX) terms from Eq. (3). Shown are the $k=1$ magic sector (red) and $k=2$ nonmagic TBG-like sector (blue). Dashed red lines denote the location of the Fermi level. (e) Same as (d) but with layer Hartree potentials and Fock (XFL). (f) Same as (d) but including all terms, that is, HFL. (g) Total magic-sector filling (top) and flavor-resolved magic filling (bottom), showing the cascade with in-plane Hartree and Fock (HFX) for $N=3,4,5$. (h) Same as (g), but with out-plane Hartree and Fock (XFL) (i) Same as (g), but including all the terms (HFL).

Figure 3

(a) ${\nu }_{\text{total}}$ as a function of layer number $N$ at ${\nu }_{\text{magic}}=4$ choosing $k=1$ as the magic sector at ${\varepsilon }_{\parallel }=10, {\varepsilon }_{\perp }=6, {\varepsilon }_{\text{strain}}=0\%$. (b) Same as (a) for $k_{\text{magic}}=2$. (c) Bandwidth at ${\nu }_{\text{magic}}=4$ for the choice of $k_{\text{magic}}=1$ (red) and $k_{\text{magic}}=2$ (blue). Dashed curves are for finite strain ${\varepsilon }_{\text{strain}}=0.2\%$. (d) Effective interaction parameter $r_s$ at ${\nu }_{\text{magic}}=4$ for $k_{\text{magic}}=1$ (red) and $k_{\text{magic}}=2$ (blue). Dashed lines are at finite strain ${\varepsilon }_{\text{strain}}=0.2\%$.

Figure 4

Dependence of $\overline{\langle u_{\mathbf{k}+\mathbf{G},\alpha }|u_{\mathbf{k}},\alpha \rangle }$, a quantity that controls the in-plane Hartree correction, on twist angle for $N=2$.

Figure 5

Same as Figs. 2–2 in the main text but without imposed strain.

Figure 6

${\nu }_{\text{magic}}$ as a function of ${\nu }_{\text{total}}$ at different ${\varepsilon }_{\perp }$ and ${\varepsilon }_{\parallel }$ for $N=5$ at strain ${\varepsilon }_{\text{strain}}=0.2\%$.

Figure 7

Strain dependence of charge in the magic sector depending on the gate charge increases from left to right. Here we take ${\varepsilon }_{\perp }=6, {\varepsilon }_{\parallel }=10$.

Figure 8

Dependence of the magic-sector filling on interaction strength and the number of layers with gate charge $n$ increasing from left to right. Here we work at zero strain ${\varepsilon }_{\text{strain}}=0.0\%$.

Figure 9

Interaction strength dependence of total charge needed to fill the magic sector completely. (Left) ${\varepsilon }_{\text{strain}}=0\%$. (Right) ${\varepsilon }_{\text{strain}}=0.2\%$.

Figure 10

(a) Dependence of the interaction suppression factor on momentum (in units of $G=|{\mathbf{G}}_{\mathbf{\text{moiré}}}|$) for $N=5,10$ and $k=1,2$. (b) Dependence of the $r_s$ plot in Fig. 3 in the main text on the choice of $\lambda$ characterizing the typical momentum for Coulomb interactions.

Figure 11

${\varepsilon }_{\parallel }$ and ${\varepsilon }_{\perp }$ dependence of the $r_s$ plot from the main text.

Figure 12

Dependence of the $r_s$ on working at ${\nu }_{\text{magic}}=3.6$ or ${\nu }_{\text{magic}}=4$ and of using the bandwidth (BW) or standard deviation $\sigma$ as a measure of the width of the active magic bands.

Figure 13

Physical twist angle for choosing $k_{\text{magic}}=1$ (red) or $k_{\text{magic}}=2$ (orange) as a function of the layer number $N$.