Anomalous Sequence of Quantum Hall Liquids Revealing a Tunable Lifshitz Transition in Bilayer Graphene

Bilayer graphene is a unique system where both the Fermi energy and the low-energy electron dispersion can be tuned. This is brought about by an interplay between trigonal warping and the band gap opened by a transverse electric field. Here, we drive the Lifshitz transition in bilayer graphene to experimentally controllable carrier densities by applying a large transverse electric field to a h-BN-encapsulated bilayer graphene structure. We perform magnetotransport measurements and investigate the different degeneracies in the Landau level spectrum. At low magnetic fields, the observation of filling factors $-3$ and $-6$ quantum Hall states reflects the existence of three maxima at the top of the valence-band dispersion. At high magnetic fields, all integer quantum Hall states are observed, indicating that deeper in the valence band the constant energy contours are singly connected. The fact that we observe ferromagnetic quantum Hall states at odd-integer filling factors testifies to the high quality of our sample. This enables us to identify several phase transitions between correlated quantum Hall states at intermediate magnetic fields, in agreement with the calculated evolution of the Landau level spectrum. The observed evolution of the degeneracies, therefore, reveals the presence of a Lifshitz transition in our system.

Figure 1

Bilayer graphene band structure. (a) The whole Brillouin zone up to $3\mathrm{eV}$ from the neutrality point, including low-energy bands (blue) and split bands (yellow). (b) Valley $K$ dispersion near the valence-band top in gapped BLG. In valleys $K$ and $K^{\prime }$, the dispersion is inverted as a result of time-reversal symmetry. Insets show characteristic cross sections of $\epsilon (p)$ discussed in the main text.

Figure 2

Characterization of the device. (a) Schematics of the device: a bilayer graphene flake is transferred onto the $B{N}_b$ (bottom) flake and contacted. The flake is then covered by another h-BN layer ($B{N}_t$, top), on top of which a metal top gate is patterned. (b) Optical microscope image of the device. The Ohmic contacts buried under $B{N}_t$ appear dark yellow, while the top gate is highlighted in red. (c) Conductance map of the device ($B=0\mathrm{T}$, $T=1.6\mathrm{K}$). The displacement field axis and the density axis of the dual-gated area are shown, as well as the four regions of different polarities. At high displacement field, the conductance through the device is suppressed (a gap opens). (d) Conductance map measured at 6 T. (e) Normalized transconductance map at 6 T: in the unipolar cases when $\nu <{\nu }_{\text{lead}}$, all the broken-symmetry states of the area under the top gate are visible (parallel lines running right or left of the displacement field axis). The red dashed line indicates the cut along which the left Landau fan in Fig. 3 is measured.

Figure 3

Magnetotransport through dual-gated bilayer graphene. (a) Conductance cuts taken along the white dotted lines in (b) (right panel), where $n_{\text{gate}}$ is varied with $V_{\mathrm{TG}}$. A contact resistance $R_C=850\mathrm{\Omega }$ has been subtracted. At high magnetic field (black curves), all the broken-symmetry states are visible. At lower magnetic fields (red curves), the first conductance plateaus to appear correspond to filling factors $-3$ and $-6$. (b) Normalized transconductance LL spectra measured at $V_{\mathrm{BG}}=-50\mathrm{V}$ ($D=-0.95\mathrm{V}\mathrm{\cdot }{\mathrm{nm}}^{-1}$ at $n_{\text{gate}}=0$), along the red dashed line shown in Fig. 2, and $V_{\mathrm{BG}}=-61\mathrm{V}$ ($D=-1.14\mathrm{V}\mathrm{\cdot }{\mathrm{nm}}^{-1}$ at $n_{\text{gate}}=0$). Corresponding filling factors are labeled. We observe a crossing between filling factors $-4$ and $-5$. (c) Calculated LL spectra corresponding to gap sizes $u=70$ and $82\mathrm{meV}$. Blue (black) lines refer to valley $K$ ($K^{\prime }$).