Negative electronic compressibility in charge islands in twisted bilayer graphene

We report on the observation of negative electronic compressibility in twisted bilayer graphene for Fermi energies close to insulating states. To observe this negative compressibility, we take advantage of naturally occurring twist-angle domains that emerge during the fabrication of the samples, leading to the formation of charge islands. We accurately measure their capacitance using Coulomb oscillations, from which we infer the compressibility of the electron gas. Notably, we not only observe the negative electronic compressibility near correlated insulating states at integer filling, but also prominently near the band insulating state at full filling, located at the edges of both the flat and remote bands. Furthermore, the individual twist-angle domains yield a well-defined carrier density, enabling us to quantify the strength of electronic interactions and verify the theoretical prediction that the inverse negative capacitance contribution is proportional to the average distance between the charge carriers. A detailed analysis of our findings suggests that Wigner crystallization is the most likely explanation for the observed negative electronic compressibility.

Figure 1

(a) Schematic cross section of a tBLG device, including an illustration of the moiré lattice forming between the two graphene layers (bottom left) and the resulting stacking order (bottom right). Here, $\theta$ denotes the twist angle between the layers and $\lambda$ the moiré wavelength. (b) Illustration of the constricted Hall-bar device D1, with the width of the constrictions labeled. (c) Optical microscopy image of device D1. (d) Four-point resistance as a function of gate voltage $V_g$ and bulk filling factor $\nu$ at two temperatures ($T$), highlighting the fragile superconducting region (SC). (e) Four-point resistance as a function of gate voltage $V_g$ and temperature of the 750-nm-wide constriction, measured with the contacts highlighted by arrows in panel (c). Band insulators (BIs), correlated insulators (CIs), and a fragile superconducting dome (SC) are visible.

Figure 2

Two-point conductance $G$ as a function of gate voltage $V_g$ over the entire gate voltage range. Zoom-ins close to the band insulating states reveal high-frequency conductance oscillations.

Figure 3

(a) Exemplary two-point conductance ($G$) trace close to the band insulators or constriction C1. (b) Conductance as a function of gate voltage and bias voltage $V_b$ in a small gate voltage range.

Figure 4

Magnetic field dependence of the oscillations. (a) Gradient of the voltage drop (measured in the four-point configuration) as a function of gate voltage and magnetic field. (b) Power spectral density of the oscillations in panel (a) as a function of magnetic field.

Figure 5

Schematic illustration of twist-angle variations and resulting band structure variations over the sample geometry. (a) Illustration of the formation of twist-angle domains over the sample geometry, where the local variation of the band structure is drawn on the right-hand side. (b) The Fermi level increases, and in this example the areas with a small twist angle will become insulating since the Fermi level is in their band gap, while other areas remain conducting. (c) Further increasing the Fermi level results in the formation of confinements, where the energy levels are quantized.

Figure 6

Observation of negative compressibility and determining the interaction strength $S$. (a) Power spectrum $P_{\omega }$ vs capacitance $C$ and gate voltage $V_g$ on constriction C1. The conductance trace (right vertical axis) is included for easy comparison. (b) Zoom-in of $P_{\omega }$ near the band insulators at $\nu =-4$. Equation (5) is fit to two frequency components. The inset is scaled vertically by a factor 5 to highlight component C1.2. (c) Histogram of all the values of $S$ determined from fitting and determining the mean of $S$.

Figure 7

Maxima in the power spectrum used to fit Eq. (5), plotted with $1/C$ on the vertical axes and the average distance between carriers $1/\sqrt{n}=a$ on the horizontal axes such that Eq. (5) becomes a straight line. The fits to each frequency component are made with a fixed value of $S=0.84$. The frequency components are divided into panels such that (a) shows components found near $\nu =\pm 4$, on the side of the flat band (see the simplified band structure in the inset), (b) shows components found near $\nu =\pm 4$, on the side of the remote band, and (c) shows frequency components which are found near fractional fillings of the superlattice.

Figure 8

Empirical cumulative distribution function (CDF) of the size of the confinements $\sqrt{A}$ compared to the sizes of twist-angle domains estimated from the results of Uri et al. [23]. Insert shows a schematic of the confinements and the “leads” along the twist-angle boundaries where transport is thought to occur.

Figure 9

(a) Phase of two prominent frequency components extracted from the fast Fourier transform in Fig. 4. (b) Quantum oscillation frequency of the magnetoresistance oscillations (see Supplemental Material S1 [31]), compared to the quantum oscillations in the phase of the Coulomb oscillations, as a function of gate voltage. (c) Zoom-in of the quantum oscillation frequency in the case of the Coulomb oscillation.

Figure 10

Band structure of $1.{018}^{\circ }$ tBLG in the presence of long-ranged electron-electron interactions. Hartree corrections render the band structure of tBLG filling dependent, which leads to a pinning of the van Hove singularity to the Fermi energy $E_{\mathrm{F}}$ and additional band flattening at the tip of the remote valence and conduction bands.