*General introduction.* Single-valuedness of superconducting order parameter has profound consequences. One of them is that multiply-connected SC structures have discreet fluxoid states. This allows superconducting rings to support metastable dissipationless currents, which is one of the defining phenomenological properties of superconducting state of matter (Leggett 2006). Transitions between fluxoid states are realized by phase slips, when the phase winding changes by multiple of \(2 \pi\). This topological protection gives fluxoid states robustness to small perturbations, and makes them attractive as principal degrees of freedom for superconducting devices, memory etc. For this reasons superconducting structures with multiply-connected geometry became progenitors of many superconducting devices of great practical importance such as SQUIDs, superconducting qubits etc. Fluxoid states in more complicated multiply-connected devices might find future applications in electronics (flux-tronics). In superconducting films, all said above translates from fluxoid states to vortices, which are topological defects of superconducting order parameter in continuous case, and, thus, are of fundamental interest. Vortex core, being a singularity, hosts quasiparticles, and if combined with topological insulator might host Majorana mode. Finally, in some cases, in superconductors, \(h/2e\) periodicity is predicted to be replaced by \(h/4e\) or \(h/e\) (half-quantum vortices in SRO, s-wave rings with \(R<\xi\), maybe some other examples??). Testing those hypotheses experimentally requires careful observation of vortices/fluxoid states. To sum it up, the ability to set, read and control fluxoid states in superconducting structures is one of great interest for both fundamental physics and applications.

*Overview of the progress in area and summary of the results.* MFM technique is one of the oldest scanning probe techniques and the first one for magnetic imaging. Concieved as a technique to characterize magnetic memory, it became popular for imaging of magnetic domain walls and vortices in superconductors. Its main advantage is simplicity and high spatial resolution. Shortcomings include difficult quantitative interpretation, coupling to magnetic field gradient rather than field itself, and exposing sample to magnetic field of the tip, which could possibly perturb the state of sensitive samples. It is worth mentioning a few of active measurements, in which the magnetic field of the tip was crucial. These include measurements of penetration depth and pinning strength for vortices by dragging them (reference is needed).

In this paper we demonstrate how MFM can be used to map and control fluxoid or vortex states of superconducting structures. This is entirely based on magnetomechanical interaction between MFM tip and dynamics of single vortex (or phase slip in 1D superconductor). Technique exploits inhomogeneous magnetic field created by MFM tip. It enables to address the vortex states which are inaccessible by application of global homogenous magnetic field. This mode of MFM has strong similarities with single electron electric force microscopy e-EFM (Woodside 2002, Zhu 2005, Stomp 2005, Dâna 2005, Azuma 2006, Zhu 2008, Bennett 2010, Cockins 2010), in which cantilever couples to the motion of single electron on/off quantum dot. By analogy, we propose to call it \(\Phi_0\)-MFM. Apart from mapping and controlling fluxoid states we show that it enables to measure phase slip rate. It is done by taking advantage of stochastic resonance interaction between dynamics of cantilever and thermally activated phase slips. First, we will demonstrate the principles of technique on example of superconducting ring. We discuss model of stochastic resonance interaction of thermally activated phase slips and the motion of cantilever. Measured phase slip rates are compared to Langer-Ambegaokar theory. After that, we apply it to a ring with crossbar, which has richer fluxoid states and their dynamics. It is demonstrated how \(\Phi_0\)-MFM allows to map and characterize distinct phase-slip processes.

The schematics of the setup is shown on Fig.\ref{SetupDiagram}. MFM cantilever is made by gluing a SmCo magnetic particle onto a silicon cantilever and giving it desired shape with FIB, fig.\ref{SetupDiagram}. We use ulta-soft silicon cantilever with spring constant about \(1.8 \times 10^{-4}\) N/m (\(2.5\times 10^{-4}\) N/m when measured in tester, must be checked) and resonant frequency \(\sim 7675\) Hz. We use home-build MFM operated in a frequency detection mode(Albrecht 1991). Laser interferometer is used to detect the motion of the cantilever. Cantilever is self-oscillated at its resonant frequency. Feedback loop with automatic gain control(AGC) is used to mountain the desired oscillation amplitude and monitor changes in damping. The shift of the resonant frequency is measured by phase-lock-loop(PLL). The resonant frequency shift and increase of damping allows to measure the amplitude of external forces that are in-phase and out-of-phase with the motion of cantilever, accordingly. \[f_{in} \approx \frac{2 \Delta f}{f_0} \cdot k x_0,\qquad f_{out} = - \frac{2(\Delta \gamma/2 \pi)}{f_0}\cdot k x_0,\quad\text{where}\quad\gamma_0=\omega_0/2 Q\]

Cantilever in our setup is oriented vertically, and the magnetic moment of the SmCo paticle is normal to the surface of the sample. The horizontal oscillations of the cantilever produce ac magnetic field on the surface of the sample with spatially varying amplitude. The amplitude is zero right under the tip and have the opposite signs in front and behind the tip. (Comment: Maybe, it’s worth including plots of ac modulation amplitude).

To demonstrate the idea of the technique let us consider a superconducting ring. For each position of the ring, MFM tip will apply flux trough the ring \(\Phi (r_{tip})\) which is a function of the position of the tip. As the tip moves, the fluxoid state that minimizes the energy of the ring in field can change. In points where two fluxoid states with the lowest energies become degenerate, small oscillations of the tip can drive the ring so that the current in the ring repeatedly switches. This is possible when the energy barrier for phase slip is low enough to allow thermally activated PS. The switching of supercurrent which is synchronized (at least statistically) with the motion of cantilever generates force acting on MFM tip, which is stronger than other forces that come from interaction of SC ring and MFM tip. The averaged over oscillations of the cantilever force in general has an amplitude and phase offset, that depends on the properties of the dynamics of driven phase slips. By measuring resonant frequency shift and change of damping we can find both in-phase and out-of-phase component of this force. The points of scan that correspond to switching of fluxoid states form concentric circles around the center of the ring. For a thin SC ring, for which magnetic screening can be neglected, the fluxoid transition lines correspond to points where half integer number of flux quanta is applied through the ring \(\Phi_{tip}=(n+1/2)\Phi_0\), where n is integer. This fact can be used to recover the distribution of the magnetic field on the surface created by MFM tip. The simplicity of the procedure and the fact that it requires only a thin superconducting ring makes it potentially useful for calibrating MFM tips. After calibration, the tips can be used as a source of local magnetic field in combination with transport or other measurements.

Two fluxoid states with winding numbers N=n and N=n+1 cross around \(h=n+1/2\), where \(h=\Phi/\Phi_0\) is flux through the ring in units of flux quanta. For applied flux such that \(|E_{n+1}(h)-E_n(h)|<k_B T\) and \( |E_{barrier}-E_{n,n+1}|\lesssim k_BT\) both fluxoid states can be occupied, and thermally activated phase slip events between them happen stochastically. The dynamics of the probability \(P_n\) of the system to stay in state \(n\) is governed by the relaxation rate \(\nu_r=\tau_n^{-1}+\tau_{n+1}^{-1}\), where \( \tau_{n, n+1}\) -average time in state \(n, n+1\) between phase slip events: \[dP_n/dt=- \nu_r \cdot P_n +\tau_{n+1}^{-1}\] At the point of energy level crossing \(\tau_n=\tau_{n+1}\) and phase slip rate \(\nu_{PS}=2 \cdot \nu_r\). Small flux modulation due to MFM tip oscillations \(h(t) =h_0+\delta h \sin(\omega t)\) produce small periodic changes in \(E_n(h)\) and \(E_n(h)\). When frequency of the cantilever \(\omega\) becomes comparable to the relaxation rate of fluxoid dynamics \(\nu_r\) the average state of the system becomes statistically synchronized with the oscillations of the cantilever. This can be described by a time varying part of the probability to find ring in state \(n\): \[\begin{aligned} &P_n(t)=P_n(h_0) +\frac{ \nu_r}{\sqrt{\nu_r^2+\omega^2} } \cdot \frac{dP^{eq}}{dh}\cdot \delta h \cdot \sin(\omega t+\theta) \\ &\theta=\arctan\left(-\frac{\nu_r}{\omega}\right)\\ &P^{eq}(h)=\frac{1}{1+\exp(-(E_{n+1}(h)-E_n(h))/k_B T)}\end{aligned}\] where \(dP^{eq}(h)\) - equilibrium probability of being in state \(n\). Depending on fluxoid state the current in the ring exerts a force on cantilever \(F=\alpha I_n\), where \(\alpha \) is magnetic coupling between current in the ring and MFM tip. When \(\xi< 2R\) the cubic term in flux dependance of current is weak (of the order of \(\sim 0.25 (\xi/R)^2\)) and we can consider the current difference between two fluxoid states to be independent on flux \(I_{n+1}(h)-I_{n}(h)\approx \Delta I\).

As a result of statistically synchronized switching of supercurrent stochastic force acting on cantilever have the following time-average amplitudes of in-phase and out-of phase components: \[f_{in} =- \frac{ \nu_r^2}{\nu_r^2+\omega^2} \cdot \frac{dP^{eq}}{dh}\cdot \alpha \cdot\Delta I\cdot \delta h\] \[f_{out} = \frac{\omega \cdot \nu_r}{\nu_r^2+\omega^2} \cdot \frac{dP^{eq}}{dh} \cdot \alpha \cdot\Delta I\cdot \delta h\] Let us notice that the ratio of in-phase and out-of-phase component of the force gives the relaxation rate in units of cantilever resonant angular frequency: \[\frac{ \nu_r}{\omega}=\frac{f_{in}}{f_{out}}=\frac{2 \pi \Delta f}{\Delta \gamma}\] this allows to measure \(\nu_r\) even without knowing absolute calibration for coupling of currents to cantilever.

All samples were made from aluminium by lift-off process. 45 nm Al layer was deposited on top of \(5\,\text{nm}\) Ti layer. Sample 1 has a shape of the ring with radius \(R=1.40\,\mu m\), wall width \(w=212 \pm 15\, \text{nm}\) (Fig. \ref{3um_SEM}). Superconducting transition temperature is \(T_c=1.163\) K (Fig. \ref{3um_Tc}).

MFM images of the frequency shift for Sample 1 at several different tip-sample separations taken at \(T=1.1425\) K shown on figure \ref{3um_heights}. Dark circles correspond to transitions between fluxoid states. The frequency shift fades in the region of horizontal diameter of the ring, because the modulation of flux becomes very small for corresponding positions of the tip. By accepting the furthest from the center visible transition line as one between states N=0 and N=1, we can enumerate all of them. In order to study temperature dependance of the signal that corresponds to transition between fluxoid states with N=3 and N=4, we took a series of short line-scans ( marked by red circle on figure \ref{3um_transition}). Small scans across the transition at several different temperatures are shown on Fig. \ref{3um_peaks}. The temperature dependance of the amplitudes of in-phase and out-of-phase forces was measured for the transition between fluxoid states with N=3 and N=4 (Fig. \ref{3um_SR}). The out-of-phase force reaches its maximum at T=1.1387 K (t=0.98), 24 mK below the transition. The ratio of in-phase and out-of-phase signals \(\Delta f/\Delta \gamma\) can be calculated with reasonable error between 1.1372 K and 1.1445 K in 7.3 mK range. Corresponding increase of \(\nu_r\) is from \(0.2\omega_0\) to \(255 \omega_0\) or from 8.7 kHz to 12.3 MHz, which gives dynamic range of 32 dB.

In order to model the temperature dependance of the phase slip rate we use Langer-Ambegaokar-McCumber-Halperin theory (Langer 1967, McCumber 1970). It was proposed by Langer et al. (1967) that the rate of thermally activated phase slips in 1D superconducting wires is \(\Gamma (T)= \Omega (T) \exp (-\Delta F (T)/k_B T)\), where \(\Delta F\) is energy barrier for phase slips. In case of 1D wire: \[\Delta F_{wire}= \frac{8\sqrt 2}{3} \xi wd \cdot \frac{H_c^2 }{8\pi}\]

In case \(2 \pi R\gg \xi\) this result for \(\Delta F\) can be used for a 1D ring and it becomes: \[\Delta F= \frac{8\sqrt 2}{3} \xi wd \cdot \frac{H_c^2 }{8\pi}-2 \pi R wd \frac{H_c^2}{8 \pi} \cdot 2\frac{\xi^2}{R^2} (h-n)^2\] \[\Delta F= \xi wd \cdot \frac{H_c^2 }{8\pi} \left( \frac{8\sqrt 2}{3} - \frac{\pi \xi}{R}\right),\quad\text{ for $h=1/2$}\] For Sample 1 at \(T=1.14\) K we can estimate \(\xi=768\) nm, so \(\xi/2\pi R=0.088\). For the case \(2 \pi R \approx \xi\), the expression for energy barrier was refined for the case of 1D ring by Zhang et al. (1997). It was later suggested by McCumber et al. (1970) that \(\Omega = (L/\xi)(\Delta F/k_BT)^{0.5}/\tau\), where \(\tau=\pi \hbar / 8 k_B (T_c-T)\).

From the suppression of superconducting transition by external magnetic field, we estimate coherence length to be \(\xi(0) =108 \pm 8 \, \text{nm}\) (Fig. \ref{3um_xi}) (here uncertainty comes mo

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