Fractional Quantum Hall Phase Transitions and Four-Flux States in Graphene

Graphene and its multilayers have attracted considerable interest because their fourfold spin and valley degeneracy enables a rich variety of broken-symmetry states arising from electron-electron interactions, and raises the prospect of controlled phase transitions among them. Here we report local electronic compressibility measurements of ultraclean suspended graphene that reveal a multitude of fractional quantum Hall states surrounding filling factors $\nu =-1/2$ and $-1/4$. Several of these states exhibit phase transitions that indicate abrupt changes in the underlying order, and we observe many additional oscillations in compressibility as $\nu$ approaches $-1/2$, suggesting further changes in spin and/or valley polarization. We use a simple model based on crossing Landau levels of composite fermions with different internal degrees of freedom to explain many qualitative features of the experimental data. Our results add to the diverse array of many-body states observed in graphene and demonstrate substantial control over their order parameters.

Figure 1

(a) Schematic of the measurement setup. (b) Inverse compressibility $d\mu /dn$ as a function of filling factor $\nu$ and magnetic field $B$. (c) Finer measurement around $\nu =-4/7$. Panels (b) and (c) have identical color scales. (d) and (e) $d\mu /dn$ as a function of $B$ and composite fermion Landau level (CF LL) index $p$. Panels (d) and (e) have identical color scales. Principal FQH states are marked by black lines and are labeled. The dashed lines in panel (b) mark where higher-denominator FQH states in the standard CF sequence would be expected to occur.

Figure 2

(a) Schematic of CF LL energies $E_p^{\pm }$ divided by and plotted against $B^{1/2}$. Crossings (black circle) between spin-up and -down CF LLs (colored arrows) correspond to phase transitions. (b) and (c) Numerical simulations of $d\mu /dn$ as a function of $B$ and $p$ assuming either minimal charge inhomogeneity (b), or more realistic density fluctuations (c). Black ovals correspond to the black circle in panel (a). Both panels use the same color scale.

Figure 3

$d\mu /dn$ (a) and chemical potential $\mu$ relative to its value at $\nu =-1/2$ (b) as a function of $\nu$ at $B=11.9\mathrm{T}$. (c) and (d) Steps in chemical potential $\Delta {\mu }_{\nu }$ [green labels in panel (b)] of FQH states as a function of $B$ at measured multiples of $\nu =1/3$ and $1/5$ (c), and $\nu =1/7$ and $1/9$ (d).

Figure 4

(a)–(c) $d\mu /dn$ as a function of $\nu$ and $B$. Prominent four-flux FQH states are labeled. (d) $\Delta {\mu }_{\nu }$ of the ${}^4\mathrm{CF}$ states as a function of $B$.