Energy Gaps and Layer Polarization of Integer and Fractional Quantum Hall States in Bilayer Graphene

Owing to the spin, valley, and orbital symmetries, the lowest Landau level in bilayer graphene exhibits multicomponent quantum Hall ferromagnetism. Using transport spectroscopy, we investigate the energy gaps of integer and fractional quantum Hall (QH) states in bilayer graphene with controlled layer polarization. The state at filling factor $\nu =1$ has two distinct phases: a layer polarized state that has a larger energy gap and is stabilized by high electric field, and a hitherto unobserved interlayer coherent state with a smaller gap that is stabilized by large magnetic field. In contrast, the $\nu =2/3$ quantum Hall state and a feature at $\nu =1/2$ are only resolved at finite electric field and large magnetic field. These results underscore the importance of controlling layer polarization in understanding the competing symmetries in the unusual QH system of BLG.

Figure 1

(a) $dG/dn(B,n)$ between $B=0$ and 4 T. Arrows and numbers indicate filling factors. (b) $G(n)$ traces at several $B$.

Figure 2

(a)–(c). Data from device 1 at $B=29\mathrm{T}$: $G(E_{\perp },\nu )$, $G(\nu )$ line traces at zero (blue) and finite $E_{\perp }$ (red), and $G(E_{\perp })$ line trace along $\nu =1$ (green). (d)–(f). Similar data set from device 2 with higher mobility at $B=28\mathrm{T}$.

Figure 3

(a) $G(V,E_{\perp })$ of device 2 at $B=20$. (b) $G(V)$ line traces at $E_{\perp }=0$ and $E_{\perp }=-20\mathrm{mV}/\mathrm{nm}$ at 23 T. (c) Measured LL gap $\mathrm{\Delta }(B)$ at $E_{\perp }=0$ (blue) and $E_{\perp }=-20$ and $-40\mathrm{mV}/\mathrm{nm}$ (red), respectively. (d) Schematics of electronic configurations of the different $\nu =1$ phases. T (B): top (bottom) layer; S (AS): their symmetric (antisymmetric) combination; 0 and 1 are the orbital indices; solid (dotted) lines represent occupied (empty) levels.

Figure 4

(a) $G(B,\nu )$ of device 1 at $E_{\perp }=35\mathrm{mV}/\mathrm{nm}$. (b) $G(B,\nu )$ of device 2 with back gate only. (c)–(d) Solid lines: $G(n)$ and $G(\nu )$ at $E_{\perp }=35\mathrm{mV}/\mathrm{nm}$ and $B=21$, 24, 27, 31 T, respectively. Dotted lines: data of device 1 at $E_{\perp }=0$ and $B=29\mathrm{T}$. (e) $G(V,E_{\perp })$ and $G(E_{\perp })$ at $V=0$ line trace of device 2 at $B=21\mathrm{T}$ and $\nu =2/3$. (g) Line traces of $G(V)$ at $E_{\perp }=17$ (blue) and $E_{\perp }=25\mathrm{mV}/\mathrm{nm}$ (brown). (h) LL gap of $\nu =2/3$ state versus $B$ field. Symbols: data. Blue dotted line and brown solid lines are fits using linear $B$ and $\sqrt{B}$ dependences, respectively.