Little-Parks oscillations with half-quantum fluxoid features in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ microrings

In a microring of a superconductor with a spin-triplet equal-spin pairing state, a fluxoid, a combined object of magnetic flux and circulating supercurrent, can penetrate as half-integer multiples of the flux quantum. A candidate material to investigate such half-quantum fluxoids is ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. We fabricated ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ microrings using single crystals and measured their resistance behavior under magnetic fields controlled with a three-axis vector magnet. Proper Little-Parks oscillations in the magnetovoltage as a function of an axially applied field, associated with fluxoid quantization, are clearly observed using bulk single-crystalline superconductors. We then performed magnetovoltage measurements with additional in-plane magnetic fields. By carefully analyzing both the voltages $V_{+}$ ($V_{-}$) measured at positive (negative) current, we find that, above an in-plane threshold field of about 10 mT, the magnetovoltage maxima convert to minima. We interpret this behavior as the peak splitting expected for the half-quantum fluxoid states.

Figure 1

(a) Schematic profile of the free energies for IQF states and a HQF state. A HQF state may become energetically favorable under in-plane magnetic field; it is realized above a threshold in-plane field value. (b) Expected change of the magnetoresistance oscillations. Peak splittings with in-plane field are expected when HQF states are realized.

Figure 2

(a) False-colored scanning ion microscope (SIM) image of sample A (yy075). The blue and yellow regions are the ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ crystal and the silver paint, respectively. (b) Magnified SIM image of sample A. (c) Scanning electron microscope (SEM) image of sample B (yy150). The thickness of samples A and B are 1.3 and 2.0 $\mu \mathrm{m}$, respectively. Resistance of (d) sample A and (e) sample B as functions of temperature. Both rings show superconducting transition above 1.5 K and several other transition steps. The dashed lines indicate the temperatures at which the magnetoresistance and magnetovoltage shown in Fig. 3 are measured.

Figure 3

Magnetoresistance $R\left(H_z\right)$ of (a) sample A and (b) sample B without in-plane fields. Both oscillatory periods and amplitudes agree with those of simulations for the Little-Parks oscillations. (c) Comparison of resistance and voltage as functions of $H_z$ for up-sweeps at 0.3 mT/min under constant in-plane fields $H_y$. At ${\mu }_0{H}_y=8\mathrm{mT}$, the difference in the peak positions in $V_{+}\left(H_z\right)$ and $V_{-}\left(H_z\right)$ results in apparent resistance-peak splitting because the resistance is evaluated as $\left[V_{+}\left(H_z\right)-V_{-}\left(H_z\right)\right]/2I$. For ${\mu }_0{H}_y=20\mathrm{mT}$, however, dips at ${\mu }_0{H}_z=\pm 1.3\mathrm{mT}$ are clearly observed even in the raw voltage $V_{+}$. Hence the HQF dips in the resistance is not an artifact originating from averaging. Note that for $V_{-}$, its absolute values $|V_{-}|$ are plotted with vertical offsets. (d) Effects of in-plane magnetic field $H_y$ on the magnetovoltage $V_{+}\left(H_z\right)$ of sample A, including the data shown in (c). Magnetovoltage peaks split above ${\mu }_0{H}_y=12\mathrm{mT}$ as indicated with arrows, and the width of the splitting becomes wider with increasing $H_y$, as expected for HQF states. Measurements are repeated twice in each condition to demonstrate good reproducibility. Each set of curves has a 0.1-$\mu \mathrm{V}$ offset for clarity. The dashed curves are a guide to the eye.