Contactless cavity sensing of superfluid stiffness in atomically thin 4Hb-TaS$_2$

TL;DR

The paper introduces a contactless cavity-based approach to measure the microwave conductivity and thereby the superfluid phase stiffness in atomically thin superconductors, using on-chip Nb resonators to probe 2D materials without galvanic contacts. Applied to a 4Hb-TaS2 few-layer flake, the authors extract the complex sheet conductance via a circuit-model mapping of resonance parameters, enabling a detailed view of the gap structure. They find a nodeless superconducting gap with $2\Delta_0/(k_B T_c)=3.9(1)$, a small condensed spectral weight $\eta\approx0.19$, and a phase stiffness $T_\theta \approx 1.1\times10^3$ K at low temperature, implying pairing-dominated Tc and arguing against surface nodal superconductivity on the 1H-TaS$_2$ termination. This contactless technique offers a general, fabrication-friendly route to study microwave conductivity and gap structure in diverse 2D superconductors, with implications for understanding unconventional superconductivity in atomically thin systems.$

Abstract

The exceptional tunability of two-dimensional van der Waals materials offers unique opportunities for exploring novel superconducting phases. However, in such systems, the measurement of superfluid phase stiffness, a fundamental property of a superconductor, is challenging because of the mesoscopic sample size. Here, we introduce a contact-free technique for probing the electrodynamic response, and thereby the phase stiffness, of atomically thin superconductors using on-chip superconducting microwave resonators. We demonstrate this technique on 4Hb-TaS$_2$, a van der Waals superconductor whose gap structure under broken mirror symmetry is under debate. In our cleanest few-layer device, we observe a superconducting critical temperature comparable to that of the bulk. The temperature evolution of the phase stiffness features nodeless behavior in the presence of broken mirror symmetry, inconsistent with the scenario of nodal surface superconductivity. With minimal fabrication requirements, our technique enables microwave measurements across wide ranges of two-dimensional superconductors.

Figures (4)

• Figure 1: Device structure and microwave measurements. (a) Schematic of the chip design, showing two Nb $\lambda/4$ CPW resonators (sample and reference) of different lengths, coupled to a central transmission line in a hanger geometry. (b) Optical micrograph of the region outlined by the dashed box in (a). The vdW sample is placed away from the voltage node and bridges the signal trace and ground plane (inset). The dashed line marks the 4Hb-TaS$_2$ flake, while the solid green and blue lines denote the edges of the top and bottom hBN flakes, respectively. (c) Cross-sectional schematic of the sample region. A 4-layer 4Hb-TaS$_2$ flake is capacitively coupled to the resonator through a thin (2.2 nm) hBN dielectric. A thick ($>200~\text{nm}$) top hBN layer serves both as encapsulation and mechanical support. (d) Amplitude of the microwave transmission coefficient, $|S_{21}|$, versus frequency and temperature near the reference resonator's fundamental mode. The dashed line shows the resonance frequency $f_0$ extracted from fits to the complex $S_{21}$ for each temperature. (e) Internal linewidth of the resonance, $\kappa \coloneq f_0/2Q_i$, as a function of temperature. (f, g), same as (d, e) for the sample resonator. $T_\text{c}$ and $T_{\text{c,Al}}$ mark the superconducting critical temperatures of the sample and the Al wirebonds connecting the CPW ground planes (not shown in (b)), respectively.
• Figure 2: Separating sample response from background. (a) Cavity perturbation fit to the temperature evolutions of $f_0$ (blue) and $\kappa$ (orange) in the reference resonator, where the surface impedance of Nb is modeled using Mattis-Bardeen (MB) theory. Data at $T<T_{c,\text{Al}}$ are excluded from the fitting because the Al wirebonds are not included in the model. (b) $f_0$ of the sample resonator plotted against that of the reference at the same temperature. The ratio $g_{\text{S}}/g_{\text{R}}$ is extracted using a linear fit to the data above $T_\text{c}$. (c) Temperature evolution of the modified $f_0'$ after background subtraction. (d) Same as (c) for $\kappa'$. Dashed line indicates the upper bound of $\kappa_{\text{bkg,S}}$ from background losses unrelated to 4Hb-TaS$_2$. $T_\text{c}$ indicates the superconducting transition temperature.
• Figure 3: Circuit model mapping between $(f_0',\kappa')$ and $\sigma$. (a) Schematic of the unmeandered sample resonator. The Nb and sample flake are shown in gray and red, respectively. The structure is discretized into $n+1$ segments, each with their own cross-sectional geometry. $d_n$ denotes the length of segment $n$. Although the segments illustrated here are sparse (for clarity), the actual segmentation used is dense enough for our calculation to converge. (b) Circuit model of the structure in (a). $R_m$, $L_m$, $G_m$, and $C_m$ denote the resistance, inductance, admittance, and capacitance per unit length for segment $m$, respectively. $Z_{\text{load}}$ is a load resistor that accounts for $\kappa_{\text{bkg,S}}$. $Z_{\text{in},n}$ represents the input impedance at the end of segment $n$. (c) Cross-sectional schematic using segment 2 as an example. $G_2$ is determined by the sum of impedance between points A and B, B and C, and C and D ($Z_{\text{AB}}$, $Z_{\text{BC}}$, and $Z_{\text{CD}}$). Inset: cross-sectional circuit model for the overlap region between sample and Nb, where $Z_{\text{AB}}$ equals the input impedance of the finite-length transmission line. (d) Bottom: $|Z_{\text{in},n}|$ calculated for a given set of $\sigma$, $d_n$, and $Z_\text{load}$, plotted on a log scale as a function of frequency. Data are fit to the input impedance of an effective resonator circuit (top) with parameters $R'$, $L'$, and $C'$, which allows the extraction of $f_0'$ and $\kappa'$. The fitting range is adaptively set to approximately $40\kappa'$ centered at $f_0'$.
• Figure 4: Temperature evolution of sheet conductance $\sigma$ at 6 GHz for a 4-layer 4Hb-TaS$_2$ flake. Error bars represent uncertainties from unmeasured background dissipation in the sample resonator; those smaller than the sizes of the circle and square markers are omitted. The imaginary part, $\sigma_2$ (blue squares), is fit using the Mattis-Bardeen theory in the dirty limit with a BCS-like gap evolution (Eqs. \ref{['eq:MB2']} and \ref{['eq:gap']}). The fit is shown in solid blue along with all fitting parameters extracted.