Phase-Slip Avalanches in the Superflow of ${}^4\mathrm{He}$ through Arrays of Nanosize Apertures

In response to recent experiments by the Berkeley group, we construct a model of superflow through an array of nanosize apertures that incorporates two basic ingredients: (1) disorder associated with each aperture having its own random critical velocity, and (2) effective interaperture coupling, mediated through the bulk superfluid. As the disorder becomes weak there is a transition from a regime where phase slips are largely independent to a regime where interactions lead to system-wide avalanches of phase slips. We explore the flow dynamics in both regimes, and make connections to the experiments.

Figure 1

Schematic diagram of the model system. Left: the location of the aperture array on the membrane is indicated by the black region, and the phases of the bulk superfluids far away from the nanoaperture array are labeled ${\Phi }^L$ and ${\Phi }^R$. Center: slice through the membrane, with apertures being represented by breaks in the membrane (white). Right: boundary conditions on hemispherical surfaces near the openings of the $i$th aperture.

Figure 2

Total current through an array of apertures as a function of time, computed in mean-field theory, at various disorder strengths. The Gaussian distributions of critical phase-twists ${\varphi }_{c,i}$ have widths $\sigma$ and means ${\varphi }_c=3\pi$. As the disorder strength is increased, the amplitude of the current oscillations decreases. The sharp drops in the current, which correspond to system-wide avalanches, disappear for $\sigma \gtrsim 0.2$. ($B=0.10\mu \mathrm{m}$; $C=0.19\mu \mathrm{m}$; $J=0.01\mu \mathrm{m}$ corresponding to $65\times 65$ periodic array with $\ell =3\mu \mathrm{m}$, $r_0=15\mathrm{nm}$, and $d=50\mathrm{nm}$.) Inset: comparison between mean-field (solid lines) and exact calculation (dots).

Figure 3

Amplitude of the oscillation of the current $I_{\mathrm{slip}}$ (dashed line), and drop in the current caused by an avalanche (solid line), as functions of the disorder strength for the array parameters used in Fig. 2. A and N-A indicate the SWA and disordered regimes, respectively. Inset: Solid line is the amplitude of the oscillation of the current, as a function of temperature, using the “effective disorder” model described in the text, Eq. (6). Dashed line: ideal amplitude, for the case of perfectly synchronous phase slippage, corresponding to the absence of disorder- and edge-driven inhomogeneity. Dots: experimental values with current rescaled by a factor of 1.5 [1].