Coupled Superconducting Phase and Ferromagnetic Order Parameter Dynamics
Via a direct coupling between the magnetic order parameter and the singlet Josephson supercurrent, we detect spin-wave resonances, and their dispersion, in ferromagnetic Josephson junctions in which the usual insulating or metallic barrier is replaced with a weak ferromagnet. The coupling arises within the Fraunhofer interferential description of the Josephson effect, because the magnetic layer acts as a time dependent phase plate. A spin-wave resonance at a frequency implies a dissipation that is reflected as a depression in the current-voltage curve of the Josephson junction when . We have thereby performed a resonance experiment on only Ni atoms.
The coupled dynamics of the electromagnetic field and a Josephson junction has a number of manifestations and is very well understood Josephson (); shapiro (); fiske (); likharev (). When the usual insulating or metallic barrier is replaced with a weak ferromagnet there is a coupling to another field, namely the spontaneous magnetisation of the ferromagnet. Spin-waves are elementary spin excitations which can be viewed as both spatial and time dependent variations of the magnetisation. In a ferromagnet the lowest energy excitation, the Ferromagnetic Resonance (FMR), corresponds to the uniform precession of the magnetisation around an externally applied magnetic field at the frequency . This mode can be resonantly excited by an alternative (ac) magnetic field that couples directly to the magnetisation, as described by the Landau-Lifshitz equations Kittel (). The Josephson phase difference between the two superconductors has its own dynamics. A bias voltage causes to become time-dependent so that , where and is arbitrary. Corresponding to the ac Josephson effect Josephson (), for our junctions, to a good approximation, the resulting ac Josephson current density is , where is the critical current density.
In analogy with the A-phase 3Heref () of He, coupled magnetic and phase oscillations should exist in ferromagnetic superconductors with triplet pairing, but have never been observed. We show here that a similar coupling for singlet superconductors can be realised in a Josephson junction with a ferromagnetic barrier. The dynamical coupling stems from the spatial interference of the Aharonov-Bohm phase caused by , resulting in the spatial dependence of . The ac Josephson current produces an oscillating magnetic field and when the Josephson frequency matches the spin wave frequency, , this resonantly excites . Due to the nonlinearity of the Josephson effect, there is a rectification of current across the junction, resulting in a dip in the average dc component of at voltage . The principal result reported here is the observation of these coupled dynamics.
Magnetised Josephson junctions Kontos () require weak ferromagnetic materials and nanosized junction area to keep the overall magnetic flux in the junction smaller than the flux quantum . An electron microscope image of a typical ferromagnetic junction used in this study is shown in Fig. 1(a), while Fig. 1(b) is a schematic representation of the different layers. The superconducting electrodes comprise 50 nm of Nb ( K), while the barrier is 20 nm of PdNi ( K). The current-voltage (IV) characteristics are measured using current bias and are reported, as function of the applied in-plane field, in the right insert of Fig. . The IV characteristics are not hysteretic, and overall they correspond closely to those expected for a junction with a conductive barrier stewart (); mccumber (). The junction normal resistance is , and the Josephson coupling is V, as expected for ferromagnetic junctions of this thickness Kontos (), yielding the critical supercurrent of A. The hostile nature of even a weak ferromagnetic environment for singlet Cooper pairs is illustrated by a similar junction with 70 nm of nonmagnetic Pd which, despite the almost four times larger thickness, has a larger critical current A.
For a square junction of side , the total supercurrent is given by the integral barone ()
where the last term is the Aharonov–Bohm phase rowell (), involving the vector potential . We use a gauge where , the direction being perpendicular to the junction surface [see Fig. 1]. Therefore , where reflects time dependent fields and . Here is the component of the static magnetisation , the applied field is in the direction, and are the actual and magnetic thickness of barrier and the London penetration depth. Equations (1) and (2) are used to describe both the statics and the dynamics of our junctions.
In the absence of a bias , we are dealing with static fields, and the Equations (1) and (2) lead to the Fraunhofer pattern rowell (). The magnetisation of the barrier has the same effect as inserting a wedge shaped phase plate in front of the slit, it displaces the diffraction pattern. Experimentally, the diffraction pattern is shifted to the right for increasing (positive ), and the left for decreasing (negative ), fields. This illustrates the linear nature of the coupling to . In Fig. 2, the dotted curves are a fit using Eqs. (1) and (2), along with the magnetisation data measured on a trilayer with the same cross section as the junction [see the left insert of Fig. 2]. The periodicity and the asymptotic behaviour of the measured diffraction pattern attest to the high quality of our junctions. They confirm the close-to-uniform current distribution and single-harmonic current-phase relation, while the reproduction of the shift with the two sweep directions, using experimental magnetostatic data, confirms the validity of our description.
The dynamical coupling reflects a similar phase contribution due to , but which now has both a temporal and a spatial dependence, the equivalent of a phase plate in the optical analog with a similarly dependent refractive index . The dynamics of the magnetisation (timescale of ns) is much slower than the diffusion time through the ferromagnetic layer ( ps). The Josephson coupling is thus adiabatic with respect to the magnetisation dynamics. This assumption is implicit in Eqs. (1) and (2). The signal is seen for , implying Eq. (1) can be linearized. The dc magnetic signal then corresponds to barone ()
where the bar denotes a time average. Substituting for and using , following both a time and space integration by parts, the dc signal reflecting the magnetic resonance is
with , , where is the dynamic susceptibility. This has an appealing interpretation in terms of magnetic losses. Here, as illustrated by Fig. 1(b), is the magnetic field which circulates inside the junction by virtue of the ac Josephson current. The junction lateral size is smaller than both and the skin depth for the frequencies involved. The displacement current is therefore negligible and all that is needed is to integrate Ampère’s law in order to determine . More details of these calculations are given elsewhere barnes (). The current due to is
where is the flux due to the radio frequency field and and ; , reflect the geometrical structure of the coupling. As the equilibrium magnetisation is along the axis, the magnetic resonance signal is contained in and , the Fourier transforms of the imaginary part of the susceptibility. Therefore, the total dc current within the Resistively Shunted Josephson junction (RSJ) model stewart (); mccumber () is
where and are the junction resistances for dc and frequency . A simple physical argument can account for the three terms in Eq. (6). The average power dissipated in the junction is and so the first term, , corresponds to the Ohmic loss at dc, while is the similar loss at . The key third term represents a self-inductance , stemming from the ferromagnet, in parallel with the junction and modeled as an LCR oscillator [see Fig 1(c)], where reflects the magnetic damping. At the magnetic resonance frequency, energy is absorbed by the ferromagnet, causing the oscillator to be lossy. This actually reduces the effective junction resistance, leading to a dip in . In this manner, the Josephson junction rectifies the self-induced magnetic resonance.
This coupling to the magnetic system is evident in the measured dynamical resistance curves reported in Fig. 3. We measure the dynamical resistance rather than the IV characteristics to improve amplitude resolution. The mode labeled FMR (Ferromagnetic Resonance) is seen only for ferromagnetic junctions. There is good agreement between the experiment, solid curves, and theory, dotted curves. The magnetic resonance mode observed in our experiments reflects the properties of a thin film of the ferromagnet PdNi. Magnetisation curves , measured directly for a large area trilayer with the same cross section as the junction, are shown in the insert of Fig. 2. They indicate that is perpendicular to the junction plane, a conclusion reinforced by earlier anomalous Hall effect measurements on similar thin films kontos2001 (). The FMR mode, shown in Fig. 3, occurs at V. This is unambiguously identified as such, since the frequency agrees, without fitting parameters, with the Kittel formula Kittel1 ()
for the in-plane magnetic field dependence of the uniform FMR mode when the anisotropy field is perpendicular to the plane. The anisotropy field G and the magnetisation at saturation G are both determined directly from the static magnetisation data, and , where is the Bohr magneton. For comparison, the ferromagnetic resonance of a macroscopic Nb/PdNi/Nb trilayer has been measured in a conventional GHz cavity spectrometer at K with field applied parallel to the substrate. The cavity FMR, shown in the bottom insert of Fig. 3, occurs at G, again exactly as predicted by Eq. (7). Displayed in the top insert of Fig. 3 is the comparison of the resonant mode in the Josephson junction (solid square) and in the macroscopic trilayer (open square). The dotted curve shows the frequency of the FMR mode calculated from the Kittel formula [Eq. (7)]. The spectra presented in the main part of Fig. 3 contain an extrinsic broadening caused by a lock-in modulation voltage of V. For the ac modulation voltage of 0.5 V, the junction resonance width saturates at 0.5 V, which corresponds to the conventional resonance width (150 G). The signal amplitude corresponds to a resonant susceptibility of approximately 10, consistent with the FMR mode measured in a microwave cavity and reported in the bottom insert of Fig. 3.
In order to demonstrate that the magnetic system is coupled to the super- but not to the normal current, we have performed Shapiro step shapiro () measurements, reported as the dynamical resistance in Fig. 4(a). The junction is irradiated with microwaves of frequency GHz at 35 mK. The Shapiro steps arise from the mixing of the microwave signal with the ac Josephson effect and are smaller replicas of the zero-voltage current step displaced from zero voltage by , where , and is an integer. We do not observe half integer Shapiro steps, indicating negligible higher harmonics in the current-phase relation. However, as expected within the RSJ model, the ferromagnetic resonance can be exited at voltage likharev (). The sub-harmonic for is visible in the spectrum in Fig. 3. As shown in Fig. 4(a), it is reproduced as a side-band to each regular step when . Experimentally, we do not have available a high enough frequency to separate similar side-bands for the main FMR mode at .
Finally, the field dependence of the resonance at has been studied in more detail in the second derivative, [Fig. 4(b)], where the minima correspond to . Measurements were limited in field due to the rapid decrease of the critical current above 800 G.
In the insert of Fig. 4(b), we show as a function of the applied magnetic field. The error bars are due to the drift of the amplifier. The solid curve is Eq. (7), without fitting parameters, with the spatial dependence of FMR taken into account. The spatial dependence of the spin-waves leads to an additional term to Eq. (7) given by , where is the spin wave momentum and , where meV is the PdNi exchange energy and nm the lattice constant. Since the width of the junction is only about 500 nm, this leads to a small but finite correction to the uniform FMR energy which is larger than the direct effect of the applied dc field. Illustrated in this manner is the direct determination of spin-wave dispersion using the present technique.
In conclusion, we have demonstrated the dynamical coupling of the superconducting phase with the spin waves in a ferromagnet and measured their dispersion. We have performed a photon free FMR experiment on about 10 Ni atoms, which would be infeasible with standard FMR techniques, and have illustrated a new methodology for the study of spin dynamics. There are direct and implied applications to spintronics and nanomagnetism spinbook ().
We thank J. Gabelli, B. Reulet, D. Feinberg, R. Melin, Z. Radović, I. Martin and M. Houzet for stimulating discussions. M.A. is indebted to H. Bouchiat for an illuminating conversation and to A. Thiaville and H. Hurdequint for many tutorials about spin dynamics. This work was in part supported by CREST of JST, and EPSRC(UK).
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