Energetics of the complex phase diagram of a tunable bilayer graphene probed by quantum capacitance

Bilayer graphene provides a unique platform to explore the rich physics in quantum Hall effect. The unusual combination of spin, valley, and orbital degeneracy leads to interesting symmetry-broken states with electric and magnetic field. Conventional transport measurements, like resistance measurements, have been performed to probe the different ordered states in bilayer graphene. However, not much work has been done to directly map the energetics of those states in bilayer graphene. Here, we have carried out the magnetocapacitance measurements with electric and magnetic field in a hexagonal boron nitride encapsulated dual-gated bilayer graphene device. At zero magnetic field, using the quantum capacitance technique we measure the gap around the charge neutrality point as a function of perpendicular electric field and the obtained value of the gap matches well with the theory. In the presence of a perpendicular magnetic field, we observe Landau-level crossing in our magnetocapacitance measurements with electric field. The gap closing and reopening of the lowest Landau level with electric and magnetic field shows the transition from one ordered state to another. Furthermore, we observe the collapsing of the Landau levels near the band edge at higher electric field ($\overline{D}>0.5$ V/nm), which was predicted theoretically. The complete energetics of the Landau levels of bilayer graphene with electric and magnetic field in our experiment paves the way to unravel the nature of ground states of the system.

Figure 1

(a) Optical image of the device. Scale bar 5 $\mu \text{m}$. (b) Schematic of the device architecture and measurement scheme. (c) Color plot of measured total capacitance $\left(C_t\right)$ as a function of backgate voltage ($V_{bg}$) and topgate voltage ($V_{tg}$) at $T\sim$ 240 mK. Black solid line shows the $\overline{D}$ axis, and white solid line shows the $n$ axis. (d) Cut lines showing $C_t$ as a function of topgate voltage $V_{tg}$ for several values of backgate voltage ($V_{bg}$).

Figure 2

(a) Color plot of total capacitance ($C_t$) as a function of electric field ($\overline{D}$) and Fermi energy $E_F$. (b) Extracted quantum capacitance ($C_q$) with $E_F$ for different value of $\overline{D}$. (c) Blue scattered plots shows the extracted band gap as a function of electric field $\overline{D}$. Red dashed line shows the calculated band gap with $\overline{D}$ (Refs. [16, 45]).

Figure 3

(a) Color plot of the measured total capacitance ($C_t$) for $\overline{D}=0$ as a function of magnetic field ($B$). The data was recorded synchronously by sweeping $V_{tg}$ ($V_{tg}$) keeping $\overline{D}=0$. (b) Extracted quantum capacitance ($C_q$) as a function of $E_F$ for different values of magnetic field for $\overline{D}=0$. The solid lines are the single-particle LL energy spectrum of BLG as discussed in the main text. (c) Color plot of the measured total capacitance ($C_t$) as a function of ($V_{tg}$, $V_{bg}$) for $B=10\text{T}$. (d) Extracted quantum capacitance ($C_q$) as a function of Fermi energy $E_F$ and electric field ($\overline{D}$) at $B=10\text{T}$. (e) Color plot of $C_q$ as a function of $\overline{D}$, and $B$ for $E_F=0$. (f) $C_q$ as a function of $E_F$, for small value of $\overline{D}$ [zoomed-in region of Fig. 3 labeled in white dashed box]. Different insulating phase has been labeled by I and II.

Figure 4

(a) Critical electric field (${\overline{D}}_c$) as a function of $B$. (b) The LL energies for $\nu =0$ state as a function of $B$ for $\overline{D}=0$.

Figure 5

(a) LL energies as a function of electric field ($\overline{D}$) for $B=6$ T. (b) Extracted quantum capacitance ($C_q$) as a function of $E_F$ for different values of magnetic field at $\overline{D}=0.8\text{V/nm}$. (c) Evolution of the $\nu =2$ gap as a function of electric field for $B=10\text{T}$. (d) Theoretical LL energies as a function of interlayer bias ($U$) for $B=6$ T.