Acoustoelectric Probing of Fractal Energy Spectra in Graphene/hBN Moiré Superlattices

Abstract

Moiré superlattices with long-range periodicity exhibit Hofstadter energy spectra under accessible magnetic fields, enabling the exploration of emergent quantum phenomena through a hierarchy of fractal states. However, higher-order features, located at elevated energies with narrow bandwidths, typically require high carrier densities and remain difficult to resolve using conventional electrical transport due to limited sensitivity and strong background conductivity. Here, we utilize acoustoelectric (AE) transport to probe high-order fractal states and the Hofstadter spectrum in graphene/hBN moiré superlattices. Surface acoustic waves on a ferroelectric LiNbO$_3$ substrate generate an AE voltage proportional to the derivative of electrical conductivity, significantly enhancing sensitivity to weak spectral features. Combined with substrate-induced high electron doping, this technique resolves fractal Brown-Zak oscillations up to the fifth-order and provides the first AE observation of the Hofstadter butterfly, revealing high-order fractal magnetic Bloch states and symmetry-broken Landau levels over a wide carrier density range. Our results establish AE transport as a powerful derivative-sensitive probe for emergent fractal quantum states in moiré-engineered 2D systems.

Figures (4)

• Figure 1: Integration of Gr/hBN moiré superlattice with SAWs on a LiNbO3 substrate. (a) Schematic of the device with IDTs and AE measurement configuration. (b) Cross-sectional schematic of the device with the graphite top gate. (c) Frequency dependence of AE voltage $V^{\mathrm{AE}}_{xx}$ (top) and the IDT reflection spectrum (bottom), with a peak at $f_\mathrm{SAW} = \qty{314}{MHz}$. (d) Carrier density $n$ at $V_\mathrm{G} = \qty{3}{V}$ and primary Dirac-point position $V_\mathrm{PDP}$ versus temperature. (e) Comparison of $R_{xx}$ and $V^{\mathrm{AE}}_{xx}$ versus $V_\mathrm{G}$ at 80K. Vertical dashed lines denote the primary Dirac and secondary Dirac points.
• Figure 2: Quantum oscillations in Gr/hBN moiré superlattice measured by AE transport. (a) $V^{\mathrm{AE}}_{xx}$ versus magnetic field $B$ at 80K and 2K at fixed carrier density $n = 5.4 n_{0} = \qty{2.46e12}{cm^{-2}}$. (b) Oscillatory component $\Delta V^{\mathrm{AE}}_{xx}$ and its field derivative $- \partial V_{xx}^\mathrm{AE}/\partial B$ plotted as functions of $\mathit{\Phi}_{0}/\mathit{\Phi}$, showing clear BZ oscillations. (c) Positions of maxima and minima in $\Delta V^{\mathrm{AE}}_{xx}$, plotted in $1/B$ versus $q$ for the primary BZ oscillations at $\mathit{\Phi}/\mathit{\Phi}_{0} = 1/q$ (80K). (d) Positions of maxima and minima in $V^{\mathrm{AE}}_{xx}$, plotted in $1/B$ versus LL filling factor $\nu$ from SdH oscillations at 2K.
• Figure 3: High-order fractal BZ oscillations in Gr/hBN moiré superlattice resolved via electrical and AE transport. (a-b) $\sigma_{xx}$ and $V^{\mathrm{AE}}_{xx}$ versus normalized carrier density $n/n_{0}$ and $B$ at 80K. Horizontal dashed lines mark magnetic Bloch states at flux values $\mathit{\Phi}/\mathit{\Phi}_{0} = p/q$. (c) Vertical line cuts of (a) and (b) at $n = 15.7 n_{0} = \qty{7.15e12}{cm^{-2}}$, showing $\sigma_{xx}$ and $V^{\mathrm{AE}}_{xx}$ versus $B$. (d) Schematic of magnetic and moiré unit cells at $\mathit{\Phi}/\mathit{\Phi}_{0} = p/q$. (e) Second derivative of $\sigma_{xx}$ and first derivative of $V^{\mathrm{AE}}_{xx}$ versus $\mathit{\Phi}_{0}/\mathit{\Phi}$ at $n = 15.7 n_{0}$. Vertical dashed lines mark high-order fractal BZ oscillations, resolved in AE transport up to the fifth order.
• Figure 4: Electrical and AE transport observation of Hofstadter butterfly in Gr/hBN moiré superlattice. (a-b) $\sigma_{xx}$ and $V^{\mathrm{AE}}_{xx}$ versus $n/n_{0}$ and $B$ at 3.5K. (c-d) Zoomed-in views of (a) and (b) highlighting linear Landau-fan trajectories (sloped =dashed lines), fractal states at commensurate fields (horizontal dashed lines), and symmetry-broken Landau fans (purple dashed lines). (e) Schematic of the symmetry-broken LLs in graphene. Red areas represent =fourfold-degenerate LLs, and blue areas indicate partially symmetry-broken LLs. (f) High-resolution AE map of $V_{xx}^\mathrm{AE}\left( n/n_{0}, B \right)$ within the dashed rectangle region in (d), resolving intersections of primary and secondary Landau fans at commensurate flux values $\mathit{\Phi}/\mathit{\Phi}_{0} = 1/3$, $2/7$, $1/4$ and $1/5$ (marked as stars). RF power: 0dBm in (b) and (d), 3dBm in (f).