Circuit QED detection of induced two-fold anisotropic pairing in a hybrid superconductor-ferromagnet bilayer

Abstract

Hybrid systems represent one of the frontiers in the study of unconventional superconductivity and are a promising platform to realize topological superconducting states. Owing to their mesoscopic dimensions, these materials are challenging to probe using many conventional measurement techniques, and require new experimental probes to successfully characterize. In this work, we develop a probe that enables us to measure the superfluid density of micron-size superconductors using microwave techniques drawn from circuit quantum electrodynamics (cQED). We apply this technique to a paradigmatic hybrid system, the superconductor/ferromagnet bilayer, and find that the proximity-induced superfluid density is two-fold anisotropic within the plane of the sample and exhibits power law temperature-scaling which is indicative of a nodal superconducting state. These experimental results are consistent with the theoretically predicted signatures of induced triplet pairing with a nodal $p$-wave order parameter. Moreover, we unexpectedly observe drastic modifications to the microwave response at frequencies near the ferromagnetic resonance, suggesting a coupling between the spin dynamics and induced superconducting order in the ferromagnetic layer. Our results offer new insights into the unconventional superconducting states induced in superconductor/ferromagnet heterostructures and simultaneously establish a new avenue for the study of fragile unconventional superconductivity in low-dimensional materials such as van der Waals heterostructures.

Figures (8)

• Figure 1: Device geometry and ferromagnetic resonance. a. False-colored scanning electron micrograph of the superconductor/ferromagnet (S/F) bilayer composed of a 30 nm permalloy film deposited directly on top of a 25 nm thick Nb film, as shown in the cross-section in panel b. The bilayer is integrated into a quarter-wavelength coplanar resonator patterned into the Nb film, shown in an optical micrograph in panel b. The resonator is capacitively coupled to a transmission line and is perforated with artificial flux-pinning holes (inset of panel a.) to improve the resonator performance in magnetic fields. c. Top: At microwave frequencies, the bilayer response can be treated as a circuit of two parallel inductors, corresponding to the kinetic inductances associated with the bulk Nb superfluid density ($L_{\text{Nb}} \sim 1/n_s^{\text{Nb}}$) and the induced superfluid density in the bilayer. Bottom: As a result of their direct contact, the Nb is able to proximity induce superconductivity in the Py strip, leading to the formation of a mini-gap $\Delta_{\text{Py}}$ in the majority spin band. d. Transmission $S_{21}$ across the circuit as a function of in-plane magnetic field $\mu_0 H_\parallel$ oriented along the length of the Py stripe. When the resonator frequency is tuned to the ferromagnetic resonance frequency of the Kittel magnons in the Py, an anti-crossing is observed in the resonator spectrum, where the cavity photons hybridize with the FMR mode to form cavity magnon-polaritons. Note that given the low field $\mu_0 H_\parallel \approx 7$ mT associated with the FMR at this frequency, the magnon-photon hybridization leads to substantial damping of the photon mode even at zero field. The black lines is an overlay of the modelled spectral function of coupled harmonic oscillators (see Supplemental Information), which allow us to extract an effective coupling strength $g/2\pi = 120$ MHz between the resonator photons and Py magnons. e. Transmission spectrum at the third harmonic of the resonator. Anti-crossings associated with magnon-polariton modes are again observed, now at a higher field $\mu_0 H_\parallel \approx 120$ mT, at which the FMR crosses the third harmonic frequency $\approx 11$ GHz. Fitting the transmission spectrum (black lines) yields a similar coupling $g/2\pi = 100$ MHz to that observed at the first harmonic. The broad shoulder on the left-hand side of both sweeps is due to hysteretic effects related to trapped flux in the superconducting resonator.
• Figure 2: Anisotropic temperature dependence of inductance. a. Shift in resonance frequency $\delta f/f_0 = [f(T,H) - f(55 \; \text{mK},H)]/f(55 \; \text{mK},H)$ in an in-plane field $\mu_0 H_\parallel = 300$ mT oriented along the length of the Py stripe, as illustrated in the inset. The comparatively negligible temperature dependence of the resonance frequency of a bare Nb resonator (without a Py stripe) is shown for comparison. b. Shift in the resonance frequency in an in-plane field $\mu_0 H_\perp = 300$ mT oriented perpendicular to the length of the Py stripe. c. Frequency shift in an in-plane field $\mu_0 H_\parallel = 20$ mT; d. Frequency shift in an in-plane field $\mu_0 H_\perp = 25$ mT. In all plots, the grey line is a fit of the data over the full temperature range to the power-law dependence $\delta f/f_0 = \alpha T^n$, with $\alpha$ and $n$ fitting parameters. e. Extracted temperature-scaling exponent $n$ as a function of the upper cutoff of the temperature range over which the data is fit, for the data in each panel b-d. Irrespective of the details of the fit procedure, the scaling exponents for fields parallel and perpendicular to the stripe are distinct.
• Figure 3: Superfluid density for a disorder nodal $p$-wave state. a. Illustration of the S/F bilayer with the in-plane field directions $H_\parallel$, $H_\perp$ indicated. The cross-section schematically depicts how interfacial spin-orbit coupling can convert isotropic spin-singlet pairs in the Nb layer into spin-triplet $p$-wave pairs in the ferromagnet, which can survive into the ferromagnet over lengths scales on the order of the electronic mean free path. b. Superfluid density $\delta n_{s}\left(T\right)=n_{s}\left(T\right)-n_{s}\left(T=0\right)$ as a function of temperature for $\Delta\tau\approx5\times10^{2}$, $\Delta\tau\approx10^{3}$, $\Delta\tau\approx3\times10^{3}$, $\Delta\tau\approx6\times10^{3}$, where $\Delta$ is the triplet gap. Solid and dashed curves correspond to the response along and transverse to the nodes of the superconducting gap, respectively, where darker colors correspond to higher $\tau$. Lines corresponding to temperature scalings of $T,T^2,$ and $T^3$ are included in purple/pink as guides to the eye. The blue-shaded region indicates the range of parameter space compatible with the experimental results.
• Figure 4: Temperature scaling near the ferromagnetic resonance. a. Schematic illustration of the first and third harmonic modes of the resonator and the evolution of the ferromagnetic resonance (Kittel) mode frequency with in-plane magnetic field. b. Temperature dependence of the first harmonic resonator frequency at fields above the ferromagnetic resonance field, i.e. $\delta H_{\text{FMR,1st}} = H - H_{\text{FMR,1st}} >0$. Progressively steeper upturns in the temperature dependence are observed as the ferromagnetic resonance field is approached. c. Temperature dependence of the third harmonic resonator frequency at fields below the ferromagnetic resonance field, $\delta H_{\text{FMR,3rd}} < 0$. The steepness of the upturns again scales with the proximity to the ferromagnetic resonance field. Dashed lines are guides the eye that mark the approximate temperature at which the upturns onset.
• Figure 5: Experimental setup. a. Schematic wiring diagram for the microwave measurement setup. All lines are coaxial cables, with the materials for each segment indicated in the figure. Grey boxes represent attenuators thermally anchored to each plate of the dilution refrigerator. b. Traces of the microwave transmission $S_{21}$ for bare Nb resonators and hybrid resonators terminated with an S/F bilayer, respectively. The fit used to extract the resonance frequency is shown along with the raw data, along with the quality factor estimated from the fit.