Correlated Insulating States in Twisted Double Bilayer Graphene

We present a combined experimental and theoretical study of twisted double bilayer graphene with twist angles between 1° and 1.35°. Consistent with moiré band structure calculations, we observe insulators at integer moiré band fillings one and three, but not two. An applied transverse electric field separates the first moiré conduction band from neighboring bands, and favors the appearance of correlated insulators at $1/4$, $1/2$, and $3/4$ band filling. Insulating states at $1/4$ and $3/4$ band filling emerge only in a parallel magnetic field ($B_{||}$), whereas the resistivity at half band filling is weakly dependent on $B_{||}$. Our findings suggest that correlated insulators are favored when a moiré flat band is spectrally isolated, and are consistent with a mean-field picture in which insulating states are established by breaking both spin and valley symmetries at $1/4$ and $3/4$ band filling and valley polarization alone at $1/2$ band filling.

Figure 1

(a) Schematic of a moiré pattern formed by two Bernal stacked graphene bilayers (red and blue) twisted relative to one another. The dashed lines indicate the moiré lattice. (b) The moiré Brillouin zone of TDBG (black) near the corners of the two bilayer graphene Brillouin zones (red and blue). (c) TDBG band structure calculated for $\theta =1.01°$. The inset shows an enlarged view of the lowest energy bands near the $\gamma$ point. Gaps at charge neutrality and between conduction subbands are labeled. Similar gaps are present between the corresponding valence subbands. (d) Schematic of the Hall bar structure used to probe TDBG samples, with top and bottom graphite gates. The inset shows an optical micrograph of a typical sample, with a $5\mu \mathrm{m}$ scale bar.

Figure 2

(a) Contour plot of ${\rho }_{xx}$ vs $V_{\mathrm{TG}}$ and $V_{\mathrm{BG}}$ measured at 1.5 K in TDBG with $\theta =1.01°$. Resistivity maxima are observed at charge neutrality, $\pm 1n_s$, and $\pm 3n_s$, as well as at $+1/2n_s$. The inset axes indicate the directions of varying $E$ and $n$ values. (b) Calculated band gaps between different moiré conduction (solid) and valence (dashed) band as a function of the on-site energy difference between layers, ${\mathrm{\Delta }}_V$. (c) ${\rho }_{xx}$ vs $n$ measured at different $T$ values, at $V_{\mathrm{TG}}=-0.8\mathrm{V}$. Top inset: Enlarged view of ${\rho }_{xx}$ vs $n_s$ at 1.5 K, which shows several resistivity maxima at fractional band fillings. (d) ${\rho }_{xx}$ vs $T$ at different band fillings marked by arrows in (c). Dashed lines are a guide to the eye. Inset: $d{R}_{xx}/dT$ vs $n$ in units of $n_s$ in the linear region.

Figure 3

(a) ${\rho }_{xx}$ vs $E$ and $n$ measured in a TDBG sample with $\theta =1.33°$, at $T=1.5\mathrm{K}$. The top axis shows $n$ in units of $n_s$. (b) ${\rho }_{xx}$ and ${\rho }_{xy}$ vs $n$ measured at $B=1\mathrm{T}$, and $E\simeq 0.4\mathrm{V}/\mathrm{nm}$ [dashed line in (a)]. The dashed lines indicate ${\rho }_{xx}$ maxima and ${\rho }_{xy}$ sign changes at half or full MB fillings. (c) ${\rho }_{xx}$ vs $E$ and $n/n_s$ measured at $B_{\parallel }=14\mathrm{T}$, and $T=1.5\mathrm{K}$. Insulating states develop at $n_s/4$ and $3n_s/4$. (d) ${\rho }_{xx}$ normalized to the zero field value vs $B_{\parallel }$ at different band fillings, as marked in (c).

Figure 4

(a) ${\rho }_{xx}$ vs $n$ and $B$ measured in a TDBG sample with $\theta =1.10°$ at 0.3 K, which shows the Hofstadter’s butterfly with fans originating from the CNP, $-1n_s$, $-2n_s$, and $\pm 4n_s$, and emergent fans at $-3n_s$, $-3/2n_s$, and $+2n_s$. (b) Summary of the fans observed in (a) data. The right-hand $y$ axis is the magnetic flux per moiré unit cell ($\varphi$) in units of the flux quantum (${\varphi }_0=h/e$). The lower table shows the filling factors for each fan. (c) ${\rho }_{xx}$ vs $n$ measured at $B=8.5\mathrm{T}$. Dashed lines indicate the originating subband and Landau level filling factor for each identifiable minimum. Colors correspond to the fans in (b).