Quantitative Transport Measurements of Fractional Quantum Hall Energy Gaps in Edgeless Graphene Devices

Owing to their wide tunability, multiple internal degrees of freedom, and low disorder, graphene heterostructures are emerging as a promising experimental platform for fractional quantum Hall (FQH) studies. Here, we report FQH thermal activation gap measurements in dual graphite-gated monolayer graphene devices fabricated in an edgeless Corbino geometry. In devices with substrate-induced sublattice splitting, we find a tunable crossover between single- and multicomponent FQH states in the zero energy Landau level. Activation gaps in the single-component regime show excellent agreement with numerical calculations using a single broadening parameter $\mathrm{\Gamma }\approx 7.2\mathrm{K}$. In the first excited Landau level, in contrast, FQH gaps are strongly influenced by Landau level mixing, and we observe an unexpected valley-ordered state at integer filling $\nu =-4$.

Figure 1

Edgeless graphene devices. (a) Steps for fabricating internal contacts. (i) Dry transfer produces a hBN/graphite/hBN/graphene/hBN/graphite heterostructure. (ii) A hole is etched in the top hBN and first graphite layer. (iii) The stack is flipped upside down, exposing the second graphite layer. (iv) The exposed graphite is etched and (v) another hBN flake deposited. (vi) A hole is etched through the entire stack to expose the graphene for (vii) edge contacting [5]. (b) Optical micrograph of device A. Scale bar is $10\mu \mathrm{m}$. (c) Conductance of device B at low magnetic fields. The insulating state persisting through $B=0$ at charge neutrality is associated with broken $AB$ sublattice symmetry [23, 26]. (d) Thermally activated transport at charge neutrality in device A. The measured activation gap ${\mathrm{\Delta }}_{AB}\approx 114\mathrm{K}$.

Figure 2

Fractional quantum Hall gaps in the zero energy Landau level. (a)–(c) Magnetotransport data from device A at (a) $B_{\perp }=B_T=18\mathrm{T}$ (b) 14 T, and (c) 3 T. (d) LL energy diagram in gapped monolayer graphene. The shaded region around the $|A\downarrow ⟩\mathrm{LL}$ indicates energy intervals of $\pm 0.05E_C$; single component physics for $\nu \in (-1,0)$ is expected only when no other LLs are within this range. At low $B_{\perp }$, Coulomb interactions mix this level with the $|A\uparrow ⟩\mathrm{LL}$, while at high $B_{\perp }$ growing antiferromagnetic anisotropy leads to another LL crossing and mixing with the $|B\downarrow ⟩\mathrm{LL}$ [18]. (e) $B_T$ dependence of $n/3$ activation gaps for $B_{\perp }=4\mathrm{T}$. The gaps at $-2/3$ and $-4/3$ grow with $B_T$ below $B_T^{*}\approx 6.4\mathrm{T}$ before saturating. Black lines indicate expected slope for spin-flip excitations involving $s=1$ and $s=2$ flipped spins [29]. (f) $\mathrm{\Lambda }$ level energy diagram for $\nu =-4/3$. For $B_T<B_T^{*}$; the polarized $-4/3$ state has low energy excitations consisting of spin-reversed particle-hole pairs, while for $B_T>B_T^{*}$ state admits only spinless particle-hole excitations. (g) FQH activation gaps in device A at $B_{\perp }/B_T=10\mathrm{T}/14\mathrm{T}$. Dashed lines are numerical results for a single-component system [30] with a constant phenomenological broadening $\mathrm{\Gamma }=7.2\mathrm{K}$ subtracted.

Figure 3

Fractional and integer quantum Hall gaps in the first excited LL. (a) Conductance of device A measured at $B=10\mathrm{T}$. (b) Measured FQH activation gaps. Dashed lines are linear fits to the function $\mathrm{\Delta }=e{B}_{\mathrm{eff}}/m_{\mathrm{cyc}}-\mathrm{\Gamma }$ (defined in the main text) for each FQH series labeled by the numerals (I)–(VIII). (c) Composite fermion cyclotron mass $m_{\mathrm{cyc}}$ and (d) broadening $\mathrm{\Gamma }$ extracted from \the linear fits for the different FQH series I–VIII in the first excited LL.

Figure 4

Valley-ordered (VO) phase at $\nu =-4$ in device B (for device A, see Fig. S4 [25]). (a) Conductance in the first excited Landau level at $B_{\perp }=B_T$. Two insulating states are visible, at low and high $B_{\perp }$, noted schematically in (c). (b) Conductance in the first excited LL in tilted field, with $B_{\perp }=0.67B_T$. The low-$B_{\perp }$ state is suppressed. (d) Level crossing model for the phase transition. A VO state driven by a valley splitting ${\mathrm{\Delta }}_V$ (which may have either single-particle or many-body origin) is suppressed by the Zeeman splitting $E_Z$, which favors a spin-polarized (SP) state.