Using quantitative magneto-optical imaging to reveal why the ac susceptibility of superconducting films is history independent

Measurements of the temperature-dependent ac magnetic susceptibility of superconducting films reveal reversible responses, i.e., irrespective of the magnetic and thermal history of the sample. This experimental fact is observed even in the presence of stochastic and certainly irreversible magnetic flux avalanches, which, in principle, should randomly affect the results. In this work, we explain such a paradoxical result by exploiting the spatial resolution of magneto-optical imaging. To achieve this, we successfully compare standard frequency-independent first-harmonic ac magnetic susceptibility results for a superconducting thin film with those obtained by ac-emulating magneto-optical imaging (acMOI). To demonstrate the possibilities of the acMOI technique, we further explore the experimental data. A quantitative analysis provides information regarding flux avalanches, reveals the presence of a vortex-antivortex annihilation zone in the region in which a smooth flux front interacts with preestablished avalanches, and demonstrates that the major impact on the flux distribution within the superconductor happens during the first ac cycle. Our results establish acMOI as a reliable approach for studying frequency-independent ac field effects in superconducting thin films while capturing local aspects of flux dynamics, otherwise inaccessible via global magnetometry techniques.

Figure 1

Temperature-dependent ac susceptibility of $a$-MoSi film under $H_{\text{rem}}$ obtained using the MPMS. (a) Data acquired as the temperature is decreased from the normal state using a probe magnetic field of $f=1$ Hz and amplitude varying from ${\mu }_{0}h=0.01$ to 0.35 mT. Purple arrow indicates the onset of flux avalanches, characterizing the paramagnetic reentrance region. (b) Data acquired both as the temperature is decreased from the normal state ($T\downarrow$) and increased from the Meissner state ($T\uparrow$) with $f=1$ Hz and ${\mu }_{0}h=0.01$ mT (smooth regime) and ${\mu }_{0}h=0.25$ mT and 0.38 mT (avalanche regime).

Figure 2

ac-emulated magneto-optical imaging (acMOI). A dc magnetic field is progressively applied in stairlike discrete steps. The dc field intensity is varied from zero to $h_{\text{dc}}^{\text{max}}$, then from $h_{\text{dc}}^{\text{max}}$ to $-h_{\text{dc}}^{\text{max}}$, and finally from $-h_{\text{dc}}^{\text{max}}$ to zero. Successively repeating this field cycle emulates an applied low-frequency ac magnetic field. After each field step, a MO image of the sample is recorded, as exemplified in the detail of the first field cycle. Additional parameters may be controlled, as schematically represented by an increase followed by a decrease in the temperature.

Figure 3

Example of typical $M_{\text{MOI}}\left(h_{\text{dc}}\right)$ loops obtained for measurements as the temperature is increased ($T\uparrow$) at (a) 4.5 K and (b) 6.5 K for ${\mu }_0{h}_{\text{dc}}^{\text{max}}=0.24$ mT or $h_{\text{dc}}^{\text{max}}\approx 191$ A/m. Comparison between the temperature-dependent ac susceptibility of the $a$-MoSi film under $H_{\text{rem}}$ obtained using the MPMS, and (c) a quantitative acMOI analysis and (d) a semiquantitative acMOI analysis. In both panels, the MPMS measurements are carried out at $f=0.05$ Hz for better correspondence with the slower MO measurements. MO images are taken as the temperature is increased ($T\uparrow$) and decreased ($T\downarrow$), with ac-emulated field amplitudes of 0.10 and 0.24 mT, thus in the smooth and avalanche regimes, respectively.

Figure 4

Comparison between direct MO images and differential MO images of the $a$-MoSi film taken at (a) 2.9 K and (b) 6.5 K as the temperature is increased ($T\uparrow$) after ZFC to the base temperature and as it is decreased ($T\downarrow$) from above $T_c$. Data are acquired using an ac-emulating applied field with an amplitude corresponding to 0.10 mT, thus in the smooth regime. The contrast in each image was individually adjusted for optimal visualization of the flux penetration. The lower left corner detail indicates the point of the field cycle at which each image was captured.

Figure 5

Comparison between direct MO images and differential MO images of the $a$-MoSi film taken at (a) 2.9 K and (b) 5.5 K as the temperature is increased ($T\uparrow$) after ZFC to the base temperature and as it is decreased ($T\downarrow$) from above $T_c$. Data are acquired using an ac-emulating applied field with an amplitude corresponding to 0.24 mT, thus in the avalanche regime. The contrast in each image was individually adjusted for optimal visualization of the flux penetration. The lower left corner indicates the point of the field cycle at which each image was captured. Black arrows indicate regions of further flux penetration as discussed in Sec. 6.

Figure 6

Top row: demonstration of the process used to obtain quantitative data on single avalanches. See the main text for a detailed explanation. Main panel: the total magnetic flux of each individual avalanche triggered in the $a$-MoSi film as the temperature is increased ($T\uparrow$) after ZFC to the base temperature and decreased ($T\downarrow$) from above $T_c$. Results are plotted against the area of the respective avalanches. Solid horizontal lines represent the net $\mathrm{\Delta }{\mathrm{\Phi }}_{\text{aval}}$ obtained by summing the flux of all avalanches triggered at the same temperature as $T$ is increased (red) or as $T$ is decreased (blue). Dashed lines are guides to the eye.

Figure 7

MOI of the $a$-MoSi film at 5 K as the temperature is increased after ZFC. The images were captured during ac-emulating field cycles with ${\mu }_0{h}_{\text{dc}}^{\text{max}}=0.24$ mT, and they demonstrate how an incoming flux front interacts with previously established avalanches. Panels (a) and (b) show the same MO images with different color scales to highlight different features of the flux penetration dynamics. Dashed rectangles highlight regions in which it is possible to observe the vortex-antivortex annihilation zone.

Figure 8

MOI of the $a$-MoSi film at (a) 6 K as $T$ is increased after ZFC and (b) 5 K as $T$ is decreased from above $T_c$. The images were captured after ac-emulating field cycles with ${\mu }_0{h}_{\text{dc}}^{\text{max}}=0.24$ mT were completed, i.e., under ${\mu }_0{h}_{\text{dc}}=0$ mT. The first row shows direct measurements, whereas the second row shows differential results. The differential images were obtained by subtracting the images after a field cycle from those obtained after the previous field cycle, i.e., the one in the previous column on the first row.

Figure 9

Temperature-dependent ac susceptibility under $H_{\text{rem}}$ of the $a$-MoSi film obtained using the MPMS. The probe field amplitude is kept constant at ${\mu }_{0}h=0.1$ mT while $f$ is varied between 0.05 and 1000 Hz. Inset shows an amplification of the graph region highlighted by the dashed rectangle in the main panel.

Figure 10

ac susceptibility of $a$-MoSi film under $H_{\text{rem}}$ as a function of $h$ obtained using the MPMS. The different measurement runs reflect results obtained by averaging different number of field cycles. All measurements were carried out at $f=1000$ Hz and $T=2$ K.