Kinetic Inductance of Few-Layer NbSe$_2$ in the Two-Dimensional Limit

TL;DR

This work investigates the kinetic inductance of atomically thin NbSe$_2$, a two-dimensional van der Waals superconductor, using $\lambda/4$ coplanar waveguide resonators with $h$BN-encapsulated NbSe$_2$. The NbSe$_2$ film contributes a large, thickness-dependent kinetic inductance that scales roughly as $L_{k,\mathrm{sq}} \propto 1/d$, reaching $\approx 1.2$ nH per square in the monolayer, and exhibits a thickness-driven crossover from clean to dirty limit behavior. A unified model combining clean and dirty-limit contributions, with $L_{k,\mathrm{sq}} = L_{k,\mathrm{sq,clean}} + L_{k,\mathrm{sq,dirty}}$ and $L_{k,\mathrm{sq,dirty}} \propto R_s/T_c$, captures the data and clarifies the role of surface scattering and multi-band effects. The measured Kerr nonlinearity is modest ($K/2\pi$ from $-0.008$ to $-14.7$ Hz/photon), making NbSe$_2$ a promising linear high-inductance material for superconducting quantum devices and detectors, while the fabrication approach is extensible to other two-dimensional superconductors.

Abstract

Van der Waals (vdW) superconductors remain superconducting down to the monolayer limit, enabling the exploration of emergent physical phenomena and functionality driven by reduced dimensionality. Here, we report the characterization of the kinetic inductance of atomically thin NbSe$_2$, a two-dimensional van der Waals superconductor, using superconducting coplanar waveguides and microwave measurement techniques familiar to circuit quantum electrodynamics (cQED). The kinetic inductance scales inversely with the number of NbSe$_2$ layers, reaching 1.2 nH/$\Box$ in the monolayer limit. Furthermore, the measured kinetic inductance exhibits a thickness-dependent crossover from clean- to dirty-limit behavior, with enhanced dirty-limit contributions emerging in the ultra-thin regime. These effects are likely driven by increased surface scattering, multi-band superconductivity, and geometric confinement. Additionally, the self-Kerr nonlinearity of the NbSe$_2$ films ranges from $K/2π$ = -0.008 to -14.7 Hz/photon, indicating its strong potential in applications requiring compact, nearly linear, high-inductance superconducting quantum devices and detectors. The fabrication and characterization techniques demonstrated here are extensible to the investigation of other two-dimensional superconductors.

Figures (12)

• Figure 1: Superconducting resonators and terminations to characterize the kinetic inductance of NbSe2.a, Crystal structure of monolayer 2H-NbSe2, top-down and side views. b, Measurement circuit schematic. CPW resonators are capacitively coupled to a common transmission line and terminated either directly to ground via aluminum (control) or through a hBN-NbSe2-hBN heterostructure (experiment). The resonant frequency $f_\mathrm{r}$ of the aluminum $\lambda$/4 resonator is determined by the effective inductance and capacitance of the CPW. Termination by NbSe2 introduces additional kinetic inductance, shifting the resonant frequency to a new value $f_\mathrm{r'}$. (Zoom-in of inductor) hBN-NbSe2-hBN heterostructure contacted by Al electrodes with both RF and DC input signals applied across the sample and terminated at ground. c, Simulated resonator spectra illustrating the effect of increased inductance on the resonant frequency. $L_\mathrm{k0}$ denotes the inductance of the aluminum-terminated resonator, while $L_\mathrm{k}$ represents the additional inductance introduced by the NbSe2 sample.
• Figure 2: Measurement configuration and device fabrication for the kinetic inductance measurement of NbSe2.a, Optical micrograph of a 5 $\times$ 5mm^2 chip containing CPW resonators, a shared feedline, DC bias lines, on-chip LC filters, and a ground plane, all patterned from 250nm-thick aluminum on a high-resistivity silicon substrate. Only one (of four) DC bias lines is used in this experiment. b - c, Optical micrographs of the NbSe2-terminated and aluminum-terminated $\lambda$/4 CPW resonators, respectively. d - g, Illustration of the edge-contact fabrication process. The NbSe2 (purple) flake is fully encapsulated by hBN (blue) and patterned via reactive-ion etching (RIE) to expose its edges. After in-situ argon ion milling, superconducting edge contacts are formed by angled aluminum evaporation with substrate rotation.
• Figure 3: Microwave measurements of NbSe2-terminated $\lambda$/4-resonator.a, Transmission coefficient ($|S_{21}|$) of a NbSe2-terminated $\lambda$/4-resonator measured at 10mK (Device ID D6). Inset: Complex I–Q plane showing the measured response (blue) and Lorentzian fit (red). b, Transmission coefficient ($|S_{21}|$) of NbSe2-terminated $\lambda$/4-resonator resonator as a function of input microwave power ($P_{rf}$). The resonant frequency ($f_\mathrm{r,Al-NbSe_2}$) shifts to the lower frequency with increasing microwave power. The onset of the frequency shift---defined as exceeding one standard deviation from the low-power baseline ($f_\mathrm{r,Al-NbSe_2} =$ 4.785GHz)---is observed at an input power of -100dBm (indicated by the green arrow). At -82dBm, the resonant peak bifurcates (blue arrow), indicating the onset of a bistable regime. c, Linear fit of the resonant frequency shift ($\Delta f = f_\mathrm{r,Al-NbSe_2}(n_\mathrm{r}= 1) - f_\mathrm{r,Al-NbSe_2}(n_\mathrm{r}$)) as a function of resonator photon number ($n_\mathrm{r}$) in the NbSe2-terminated $\lambda$/4-resonator. d, Optical image of the NbSe2-terminated $\lambda$/4-resonator with a DC bias line connected at the microwave input port. e, DC bias dependence of the NbSe2-terminated $\lambda$/4-resonator measured at a fixed microwave power of $P_{rf} =$ -120dBm. The resonant frequency shifts to the lower frequency with increasing bias current. The onset of the frequency shift—defined as exceeding one standard deviation from the baseline ($f_\mathrm{r,Al-NbSe_2} =$ 4.785GHz)—is observed at a bias current of $1\mu A$ (maroon arrow). f, Extracted resonant frequency shift ($\Delta f = f_\mathrm{r,Al-NbSe_2}(I_\mathrm{DC}= 0) - f_\mathrm{r,Al-NbSe_2}(I_\mathrm{DC})$) and kinetic inductance shift ($\Delta L_\mathrm{k,sq} = L_\mathrm{k,sq}(I_\mathrm{DC}= 0) - L_\mathrm{k,sq}(I_\mathrm{DC})$) of the NbSe2-terminated resonator as a function of DC bias current. Measured data (circle-green and sky-blue) and theoretical fit (line-green) are shown.
• Figure 4: Thickness dependence of kinetic inductance in NbSe2.a, Sheet kinetic inductance ($L_\mathrm{k,sq}$) from microwave measurements (circle-red) of nine NbSe2 devices with varying film thicknesses. The blue dashed line shows a $1/d$ scaling fit. b, Comparison of measured $L_\mathrm{k,sq}$ in NbSe2 with values reported for other high-impedance superconducting thin films bretz2022highcoumou2012microwavesamkharadze2016highshearrow2018atomicfrasca2023nbnwinkel2020implementation. c Sheet kinetic inductance ($L_\mathrm{k,sq}$) from microwave measurements is plotted as a function of $R_\mathrm{s}/T_c$, where $R_\mathrm{s}$ and $T_\mathrm{c}$ values are obtained from both our transport measurements (red circles) and previous studies (green and black circles)khestanova2018unusualcao2015quality. The blue dashed line is a linear fit with a slope of 16.3 and a y-intercept of 80.0pH, marked as the clean-limit contribution to $L_\mathrm{k,sq}$. d, The magnified view of the lower region for clarity.
• Figure S1: Al and NbSe2 interface characterization and DC characteristics of NbSe2.a, Optical micrograph of a pseudo-4-probe DC device where hBN flakes fully encapsulate NbSe2 flake, the current is applied between two terminals, and voltage is measured across those two terminals without taking into account the line resistance of the measurement setup. b, Resistance(T) for Al-NbSe2-Al device showing the superconducting transition of NbSe2 (d = 6 nm) at 5.4 K and the superconducting transition of Al at 1.3 K. Inset represents the 4-pt measurement setup for measuring the contact resistances of the Al/NbSe2 interfaces. c, Total Resistance of the device below Al transition, which has dropped to zero within the noise floor of the measurement. d, Temperature dependence of the resistance for devices with NbSe2 thicknesses of 4 nm, 6 nm, and 8 nm. To extract the $T_\mathrm{c}$, we use the mean-field definition of $T_\mathrm{c}$, which corresponds to half of the normal state resistance. e, Differential resistance (dV/dI) measurement for the d = 8 nm thick NbSe2 of 4-point DC measurement. The plot shows the critical current $400\mu A$ for that device. f, Current-voltage (I-V) relation on the same device.