Tunable Luttinger liquid and correlated insulating states in one-dimensional moiré superlattices

TL;DR

This work demonstrates a tunable one-dimensional Luttinger liquid in directly grown CNT/hBN moiré superlattices, where gate control over miniband filling enables access to commensurate states at $f=1/4$ and $f=1/2$ and reveals correlated insulating behavior. Transport exhibits Luttinger-liquid scaling with conductance $G(T) \propto T^{\alpha}$ and differential conductance $dI/dV \propto V^{\alpha}$, with $\alpha$ approaching 1 at commensurate fillings and the Luttinger parameter $g$ being strongly suppressed, indicating extreme correlation. The combination of 1D moiré physics and gate-tunability provides a versatile platform for exploring LL physics under periodic potentials and potential transitions to exotic phases (e.g., Luther-Emery liquids) in coupled chains.

Abstract

Two-dimensional moiré superlattices have been extensively studied, and a variety of correlated phenomena have been observed. However, their lower-dimensional counterpart, one-dimensional (1D) moiré superlattices, remain largely unexplored. Electrons in 1D are generally described by Luttinger liquid theory, with universal scaling relations depending only on the Luttinger parameter g. In particular, at half-filling, Umklapp scattering plays a crucial role, as it can significantly change the conductance-temperature scaling relation and lead to Mott insulators. However, this prediction has never been observed since doping an empty band to half-filling was extremely difficult. Here, we show that the marriage of moiré superlattices and 1D electrons makes it possible to study the Luttinger liquid in an exceptionally wide filling region simply by electrical gating. We perform transport measurements on 1D moiré superlattices of carbon nanotubes on hexagonal boron nitride (hBN) substrates, and observe correlated insulating states at 1/4 and 1/2 fillings of the superlattice mini-band, where Umklapp scattering becomes dominant. We also observe a T-linear conductance at these commensurate fillings over a range of temperatures. Strikingly, the T-linear conductance leads to a strongly suppressed Luttinger parameter, suggesting a state of extreme correlation.

Figures (4)

• Figure 1: 1D moiré superlattices of CNTs on hBN. (a) Schematic of a CNT/hBN heterostructure. (b) Top view of the CNT/hBN heterostructure, showing a clear 1D moiré pattern. (c) Band structure of a (10,0) CNT on hBN substrate. New bandgaps are opened by the moiré potential, generating narrow minibands. (d) An as-grown CNT on hBN substrate. All bending angles are 120°, implying a perfect alignment with the hBN substrate. Scale bar: 1 $\mu$m. (e) Dependence of the CNT/hBN interlayer stacking energy (black curve) and the moiré period (red) on the twist angle. At zero twist angle the stacking energy is minimal and the moiré period is maximal $\sim$ 14 nm. (f) A zoom-in AFM topography image exhibits a well-defined periodic pattern along the CNT. Arrows indicate each local maximum in height along the CNT. Scale bar: 15 nm. (g) Experimental height profile (blue) extracted from the AFM image in (f), showing a period of $\sim$ 14 nm and a corrugation of $\sim$ 0.04 nm. The orange line is the MD simulated height profile.
• Figure 2: Emergence of correlated insulating states at 1/2 and 1/4 fillings. (a) Schematic of a CNT device with 1D moiré pattern. (b) Color plot of differential conductance as a function of $V_g$ and $V_b$ at 3 K. Insulating states can be observed at the charge neutrality point (CNP), the 1/4 filling, the 1/2 filling and full filling point of the hole mini-band. Insulating states at 1/4 and 1/2 filling are attributed to correlated insulators. (c) Gate-dependent differential conductance of the CNT device at 3 K. Equally spaced resistance peaks are induced by Coulomb blockade. The separation between these peaks $\Delta V_{\mathrm{g}}\approx\;57\;mV$ corresponds to the gate voltage required to charge the device with one electron/hole. (d) A zoom-in color plot of differential conductance as a function of $V_g$ and $V_b$. Well-defined Coulomb diamonds can be observed. (e) Color plot of conductance against carrier density and temperature.
• Figure 3: Luttinger - liquid behavior of the 1D CNT superlattice at different fillings. (a) The conductance at 1/2 filling as a function of temperature T, showing a clear power law behavior with a component $\alpha\;{\sim}\;0.98$. (b) dI⁄dV measured at 1/2 filling as a function of electrical bias at a few representative temperatures. A power function fitting (dash line) yields the same power-law exponent. (c) The same data as in (b), but plotted as a scaled conductance $(dI/dV)/T^\alpha$ versus a scaled excitation $eV⁄(k_B T)$, where all data collapse to a single curve, as predicted by the Luttinger-liquid theory. (d) Temperature dependence of the conductance at three representative fillings (1/2, 1/4 and an arbitrary filling). (e) Scaled conductance versus scaled excitation at the three representative fillings, which contains all the data we measured in various temperatures ranging from 3 K to 200 K. All of them fit the universal scaling.
• Figure 4: Filling - dependent Luttinger parameter and phase diagram of the 1D CNT moiré superlattice. (a) Extracted power-law exponent $\alpha$ (black line) and Luttinger parameter g (red and purple lines in different filling regions) as a function of the filling factor. The Luttinger parameter g drops to zero at 1/4 and 1/2 fillings, indicating possible breakdown of the Luttinger liquid. (b) Schematic of electron - electron Umklapp scattering in one - dimensional moiré superlattice at half filling. (c) A schematic phase diagram for the 1D CNT superlattice, which behaves as a tunable Luttinger liquid. The correlated insulators appear at 1/4 and 1/2 filling. Gray areas near zero filling and full filling refer to band insulators (BI).