Mitigation of Magnetic Flux Trapping in Superconducting Electronics Using Moats

Abstract

Magnetic flux (vortex) trapping remains a major obstacle to very large scale integration in superconducting electronics. Moats -- etched regions in circuit layers placed in ground planes and around critical circuitry -- offer a simple passive approach to sequester flux. Here, we systematically examine the effectiveness of moat arrays in superconducting niobium films as a function of geometry (size, shape, and density) and background magnetic field. By measuring the vortex expulsion field, we estimate the flux saturation number and flux trapping temperature for a range of geometries. We find that many moat designs effectively sequester flux in magnetically shielded environments (< 1 $μ$T), with high-aspect-ratio rectangular "slit" moats providing the strongest mitigation at minimal area cost. However, our measurements show that moats alone do not eliminate flux trapping in non-ideal films, as vortices can preferentially pin at material defects. These results provide design guidance for flux mitigation in superconducting integrated circuits and highlight the need for combined optimization of circuit geometries and materials.

Figures (8)

• Figure 1: (a) Layout of test chip used to characterize flux trapping in square moat (antidot) arrays. The spacing between moats varies across rows (0.5--80 $\upmu$m), while the moat side length varies across columns (1--80 $\upmu$m). The highlighted region shows the array with moat side length $a = 4~\upmu$m and spacing $s = 10~\upmu$m. (b) Magnetic images of the $a = 4~\upmu$m, $s = 10~\upmu$m array at $B_r < B_{\mathrm{exp}}^{\mathrm{meas}}$ (left) and $B_r > B_{\mathrm{exp}}^{\mathrm{meas}}$ (right). Vortices that first appear below $B_{\mathrm{exp}}^{\mathrm{meas}}$ are circled in white (left). Across temperature cycles and background fields, vortices consistently nucleate at these locations.
• Figure 2: (a)--(b) Example images of a moat array with $a_x=a_y=4~\upmu$m, $s_x=6~\upmu$m, and $s_y=14~\upmu$m. (c)--(d) Example images of a moat array with $a_x=36~\upmu$m, $a_y=1~\upmu$m, $s_x=4~\upmu$m, and $s_y=13~\upmu$m. In each image, the first few moats in the upper left are marked (white squares in (a)--(b), white stripes in (c)--(d)), and the first vortices to appear in the film are circled in white. These vortices reproducibly emerge at the same locations across temperature cycles, indicating nucleation and/or pinning at defect sites.
• Figure 3: Example expulsion-field measurements for (a) square moat arrays ($a_x=a_y$, $s_x=s_y$), (b) square arrays with anisotropic spacing ($a_x=a_y$, $s_x\neq s_y$), and (c) rectangular arrays with anisotropic spacing ($a_x\neq a_y$, $s_x\neq s_y$). For each array, the vortex areal density $n_v(B_r)$ is obtained by imaging the vortex distribution at each $B_r$, and $B_{\mathrm{exp}}^{\mathrm{meas}}$ is extracted from the onset of a linear regime in $n_v(B_r)$ by extrapolating to $n_v=0$ (with slope $m$ varying slightly between arrays but remaining $\sim \Phi_0^{-1}$). We find $B_{\mathrm{exp}}^{\mathrm{meas}}$ depends most strongly on moat spacing, with a smaller but nontrivial dependence on moat size and shape. In all cases, $n_v \neq 0$ at $B_{\mathrm{exp}}^{\mathrm{meas}}$, as vortices appearing below this field nucleate and/or pin reproducibly at defect sites in the film.
• Figure 4: Measured expulsion field vs. $\langle N_{\Phi_0}\rangle \Phi_0n_\mathrm{moat}$, assuming $\langle N_{\Phi_0}\rangle=1$ (Eq. \ref{['eq:Bexp_basic']}). The purple, black, and orange lines indicate the expected behavior for $\langle N_{\Phi_0}\rangle =1$, 3, and 5, respectively. Square arrays with $s \geq 20~\upmu$m and arrays with anisotropic spacing agree well with $\langle N_{\Phi_0}\rangle=1$. For $s < 20~\upmu$m and highly asymmetric moats ($a_x/a_y \geq 30$), the data instead fall within $1 \leq \langle N_{\Phi_0}\rangle\leq 5$, indicating multiple flux quanta per moat before vortices enter the film. For example, at comparable moat densities, $a_x = 36$$\upmu$m, $a_y=1$$\upmu$m slits trap $\sim5\times$ more flux than $a = 4$$\upmu$m squares.
• Figure 5: Measured expulsion field of the rectangular moat arrays vs. the expected expulsion field assuming the scaling laws described in Eq. \ref{['eq:Bexp_slits_2d']} and $\beta=1$, which approximate the expulsion field based on the moat spacing. For reference, the purple line illustrates the expected behavior for $\beta=1$. The data shows good agreement for $\beta\approx1$ for the square moats with anisotropic spacing, but deviates from this scaling for the slit moats, indicating a non-trivial dependence on moat size and aspect ratio. For simplicity, we have included the measured $a_x=4$$\upmu$m, $a_y=3$$\upmu$m, $s_x=14$$\upmu$m, and $s_y=16$$\upmu$m moats in the square rather than slit dataset, as its aspect ratio is much closer to that of squares ($a_x/a_y=$ 1) than slits ($a_x/a_y \geq 30$).