Competing Fractional Quantum Hall and Electron Solid Phases in Graphene

We report experimental observation of the reentrant integer quantum Hall effect in graphene, appearing in the $N=2$ Landau level. Similar to high-mobility $\mathrm{GaAs}/\mathrm{AlGaAs}$ heterostructures, the effect is due to a competition between incompressible fractional quantum Hall states, and electron solid phases. The tunability of graphene allows us to measure the $B\text{−}T$ phase diagram of the electron solid phase. The hierarchy of reentrant states suggests spin and valley degrees of freedom play a role in determining the ground state energy. We find that the melting temperature scales with magnetic field, and construct a phase diagram of the electron liquid-solid transition.

Figure 1

RIQHE in the $N=2\mathrm{L}\mathrm{L}$ of graphene. (a) Schematics of the gate defined Hall bar device used in this experiment. (b) Longitudinal (left axis) and Hall (right axis) conductivity as a function of the filling factor measured at $B=23\mathrm{T}$ and $T=0.3\mathrm{K}$. Greyed regions highlight the four RIQHE and dashed vertical lines mark the four FQH states. Additionally, dashed horizontal lines mark the nearest integer value where the RIQHE is expected to be quantized.

Figure 2

Magnetic field dependence of the electron solid phase. (a) Numerical calculation of the energy as a function of filling for Laughlin and electron solid states in the $N=2\mathrm{LL}$ for different $d/l_B$ ratio, where $d$ is the distance to the metallic gates. The Laughlin states are represented by crosses while the one-electron and two-electron bubble phases are represented by solid and dashed lines, respectively. (b) Longitudinal (right) and Hall (left) conductivity as a function of filling factor for selected magnetic fields measured at 0.3 K. Dashed lines mark the presence of FQH states and greyed regions highlight the two RIQHE. Curves are vertically shifted for clarity.

Figure 3

Broken valley symmetry for the RIQHE of graphene. Longitudinal resistance as a function of filling factor and temperature, measured at 23 T, for the lower spin branch of the $N=2\mathrm{LL}$.

Figure 4

Temperature dependence of the electron solid phase. (a) Longitudinal and Hall resistance as a function of filling factor and temperature for the $R6a$ and $R6b$ reentrant states, measured at 27 T. White dashed boxes mark the FQH state $\nu =6+1/5$ and $\nu =6+4/5$. Gray dashed lines indicate boundaries, using parabolic function $T_c(\overline{\nu })=T_c(\overline{{\nu }_c})-\beta (\overline{\nu }-\overline{{\nu }_c}{)}^2$ for each RIQHE, consistent with [25]. (b) line cut at $\nu =6.35$ for $R_{\mathrm{xx}}$ (left axis) and $R_{\mathrm{xy}}$ (right axis) at 27 T. (c) $B\text{−}T$ phase diagram of the $R6a$ state, dashed red line is the linear fit, dashed blue line is the square root fit.