Intrinsic Paramagnetic Meissner Effect Due to $s$-Wave Odd-Frequency Superconductivity

In 1933, Meissner and Ochsenfeld reported the expulsion of magnetic flux—the diamagnetic Meissner effect—from the interior of superconducting lead. This discovery was crucial in formulating the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity. In exotic superconducting systems BCS theory does not strictly apply. A classical example is a superconductor-magnet hybrid system where magnetic ordering breaks time-reversal symmetry of the superconducting condensate and results in the stabilization of an odd-frequency superconducting state. It has been predicted that under appropriate conditions, odd-frequency superconductivity should manifest in the Meissner state as fluctuations in the sign of the magnetic susceptibility, meaning that the superconductivity can either repel (diamagnetic) or attract (paramagnetic) external magnetic flux. Here, we report local probe measurements of faint magnetic fields in a $\mathrm{Au}/\mathrm{Ho}/\mathrm{Nb}$ trilayer system using low-energy muons, where antiferromagnetic Ho (4.5 nm) breaks time-reversal symmetry of the proximity-induced pair correlations in Au. From depth-resolved measurements below the superconducting transition of Nb, we observe a local enhancement of the magnetic field in Au that exceeds the externally applied field, thus proving the existence of an intrinsic paramagnetic Meissner effect arising from an odd-frequency superconducting state.

Popular Summary

Since its discovery in 1911, superconductivity has remained one of the most intriguing phase transitions in the field of condensed-matter physics. Superconductivity, which occurs in certain materials cooled below a critical temperature, involves the disappearance of electrical resistance and the expulsion of external magnetic flux. In 1933, Meissner and Ochsenfeld first reported the expulsion of magnetic flux from the interior of superconducting lead. Their discovery, now known as the Meissner effect, enables the levitation of magnetic objects (e.g., Maglev trains in Japan). For certain unconventional forms of superconductivity, an inverse paramagnetic Meissner effect has been predicted in which superconductivity attracts external flux at a superconductor interface with certain forms of magnetism. In such a system, levitation would not be possible; instead, external flux would be amplified. Here, we probe superconductivity in a thin film of gold coupled to an antiferromagnet-superconductor system using low-energy muons.

Our experimental setup consists of an Au/Ho/Nb trilayer in which each layer has a thickness measured in nanometers. We apply an external field, and we conduct measurements below the superconducting transition of Nb (8.5 K). We find that muon particles implanted with a specific energy in gold experience a change in their precession frequency, which is directly related to the magnitude of the local magnetic flux they experience. Because of the extreme sensitivity of this magnetometry technique to magnetism—smaller than 0.1 G—we detect an enhancement of the local magnetic field in gold that exceeds the externally applied field in the superconducting regime. This result provides a direct observation of the paramagnetic Meissner effect and demonstrates that conventional Meissner screening is not a universal property of superconductivity.

We expect that our findings will motivate future experiments for harnessing magnetic energy that is generated.

Figure 1

Simulated muon stopping distributions in $\mathrm{Au}/\mathrm{Ho}/\mathrm{Nb}$. (a) Experimental $\mathrm{LE}\text{−}\mu \mathrm{SR}$ setup in transverse-field configuration and normalized muon stopping profiles $p(z,E)$ in $\mathrm{Au}/\mathrm{Ho}/\mathrm{Nb}$ simulated for a few representative implantation energies $E$. (b) Experimental asymmetry data determined from the counting events of the positron detectors for muons implanted in $\mathrm{Au}/\mathrm{Ho}/\mathrm{Nb}$ with $E=4.5\mathrm{keV}$ at 3 K (blue dots) and single-energy asymmetry fit (blue curve). (c) Fourier transform of the asymmetry in (b), which represents the magnetic-field distribution.

Figure 2

Average local magnetic field in $\mathrm{Au}/\mathrm{Ho}/\mathrm{Nb}$ as a function of the muon implantation energy and mean stopping distance. (a) ${\overline{B}}_{\text{loc}}$ values from single-energy asymmetry fits versus implantation energy (bottom $x$ axis) and mean stopping distance (top $x$ axis) in the normal state (magenta circles 10 K) and superconducting state (blue circles 5 K), and in (b) identical data showing the inverse Meissner state in Au. The continuous lines are a guide to the eye. (c),(d) Remeasured data after warming and cooling, but the superconducting state is now measured at 3 K.

Figure 3

Magnetization and screening current profile from the $\mathrm{Ho}/\mathrm{Nb}$ interface in $\mathrm{Au}/\mathrm{Ho}/\mathrm{Nb}$. Local magnetic field $B_{\text{loc}}(z)$ determined as global-energy fit of the $\mathrm{LE}\text{−}\mu \mathrm{SR}$ measurement data (red curve, left $y$ axis) and theoretical model (blue curve, left $y$ axis); calculated dimensionless screening current density $J_x(z)$ flowing parallel to the $x$ axis in Fig. 1 inside the plane of the thin film heterostructure (gray curve, right $y$ axis). Dashed lines show that the position of the maximum in $B_{\text{loc}}(z)$ coincides with that of the null in $J_x(z)$.