Dendritic flux penetration in Pb films with a periodic array of antidots

We explore the flux-jump regime in type-II Pb thin films with a periodic array of antidots by means of magneto-optical measurements. A direct visualization of the magnetic flux distribution allows us to identify a rich morphology of flux penetration patterns. We determine the phase boundary $H^{*}\left(T\right)$ between dendritic penetration at low temperatures and a smooth flux invasion at high temperatures and fields. For the whole range of fields and temperatures studied, guided vortex motion along the principal axes of the square pinning array is clearly observed. In particular, the branching process of the dendrite expansion is fully governed by the underlying pinning topology. A comparative study between macroscopic techniques and direct local visualization sheds light onto the puzzling $T$- and $H$-independent magnetic response observed at low temperatures and fields. Finally, we find that the distribution of avalanche sizes at low temperatures can be described by a power law with exponent $\tau \sim 0.9\left(1\right)$.

Figure 1

Schematic drawing (not to scale) of a Pb thin film with lateral dimensions $w_1$ and $w_2$ and thickness $t$. The black squares represent the array of antidots with lateral dimension $b$ and lattice parameter $d$.

Figure 2

MO images of the Pb sample AD75 with antidots showing different types of flux penetration: (a) fingerlike dendritic penetration at ${\mu }_{0}H=1.2\mathrm{mT}$ and $T=4.5\mathrm{K}$ and (b) ${\mu }_{0}H=1.2\mathrm{mT}$ and $T=5.5\mathrm{K}$, (c) treelike dendritic outburst coexisting with smooth flux penetration at ${\mu }_{0}H=1.5\mathrm{mT}$ and $T=6\mathrm{K}$, and (d) smooth profile at ${\mu }_{0}H=1.5\mathrm{mT}$ and $T=6.5\mathrm{K}$. A saw-tooth-like magnetic wall domain artifact from the magneto-optical garnet is observed in the bottom part of the images. (e) and (f) MO images of a Pb plain film at $T=2.5\mathrm{K}$ for ${\mu }_{0}H=0.4\mathrm{mT}$ and ${\mu }_{0}H=1.2\mathrm{mT}$, respectively. The scale bar in each figure corresponds to 0.5 mm.

Figure 3

Phase boundary lines, $H^{*}\left(T\right)$, separating dendritic from smooth penetration for different samples. Open symbols correspond to ac-susceptibility measurements (Ref. 13), while filled symbols are values obtained by MO imaging. (a) Results for samples with antidots AD65 and AD75. (b) Comparison between a plain film (PF15) and a sample with antidots (AD15) (see Table ).

Figure 4

(a) Three-dimensional plot of magnetic field near the edge of the sample (AD75) with antidots at $T=4\mathrm{K}$ and ${\mu }_{0}H=1.8\mathrm{mT}$. (b) Profiles of magnetic field along a dendrite indicated by $A\text{−}{A}^{\prime }$ in (a) for different applied fields. (c) Sequence of magnetic field profiles along the line $B\text{−}{B}^{\prime }$ in (a) (transverse to the dendrites). For clarity the curves are displaced by 1.5 mT along the vertical axis. The difference in external field between each consecutive profile is 0.2 mT.

Figure 5

Avalanches as a function of applied field for sample AD75. (a) Total magnetic flux, ${\Phi }_T$, (b) area of the sample covered by avalanches, $A_T$, and (c) average size of avalanches, $⟨\Phi ⟩$, at each field step as a function of $H$ for different temperatures.

Figure 6

Log-log plot of the distribution of avalanche sizes for sample AD75 at different temperatures. The size of an avalanche is given by the moved flux $\Phi$. The straight solid lines are linear fits, yielding a power-law exponent $\tau =0.9\left(1\right)$.