Two-dimensional arrays of superconducting strips as dc magnetic metamaterials

We theoretically investigate the magnetic response of two-dimensional arrays of superconducting strips, which are regarded as essential structures of dc magnetic metamaterials. We analytically obtain local distributions of the magnetic field for the ideal complete shielding state (i.e., $\Lambda /w\to 0$, where $2w$ is the strip width, $\Lambda ={\lambda }^2/d$ is the Pearl length, $\lambda$ is the London penetration depth, and $d$ is the strip thickness), and derive effective permeability by averaging the local-field distributions. We also perform numerical calculations for a realistic case, taking finite $\Lambda /w>0$ into account. We investigate two types of strip arrays: a rectangular array and a hexagonal array. The resulting effective permeability has large anisotropy that depends on the dimensions and arrangement of the superconducting strips, and the hexagonal array is found to be more advantageous for obtaining large anisotropy than the rectangular array.

Figure 1

Cross section of a two-dimensional (2D) array of superconducting strips in the $xy$ plane. In the $n$th layer at $y=2nb$, the $m$th strip is situated at $|x-2ma|<w$, where $m=0,\pm 1,\pm 2,...,\pm \infty$ and $n=0,\pm 1,\pm 2,...,\pm \infty$.

Figure 2

Magnetic field lines in a rectangular array of superconducting strips for $w/a=0.8$: (a) $b/a=1$, (b) $b/a=0.5$, and (c) $b/a=0.2$. Thick horizontal lines correspond to the cross sections of superconducting strips.

Figure 3

Effective permeability for perpendicular field ${\mu }_{\perp \mathrm{r}}$ as a function of $1-w/a$ for $b/a=5,2,1,0.5,0.2$, and $0.1$. The dashed line is ${\mu }_{\perp \mathrm{r}}/{\mu }_0=1$ for $b/a\to \infty$, and the dot-dashed line is ${\mu }_{\perp \mathrm{r}}/{\mu }_0=1-w/a$ for $b/a\to 0$.

Figure 4

Comparison between the numerical results of the effective permeability ${\mu }_{\perp \mathrm{r}}$ in a rectangular array for $\Lambda /w>0$ (symbols) and the analytical results for $\Lambda /w\to 0$ (lines). (a) Numerical results (dots) for $\Lambda /w=0.01$ and analytical results (lines) for $\Lambda /w\to 0$ of ${\mu }_{\perp \mathrm{r}}/{\mu }_0$ vs $1-w/a$ for $b/a=5,2,1,0.5$, and $0.2$. (b) Numerical results (symbols) for $\Lambda /w=0.1,0.05,0.01$ and analytical results (line) for $\Lambda /w\to 0$ of ${\mu }_{\perp \mathrm{r}}/{\mu }_0$ vs $1-w/a$ for $b/a=0.2$. (c) Numerical results (dots) and analytical results (dashed lines) for $\Lambda /w\to 0$ of ${\mu }_{\perp \mathrm{r}}/{\mu }_0$ vs $\Lambda /w$ for $b/a=0.2$.

Figure 5

Cross section of a hexagonal array of superconducting strips in the $xy$ plane. In the even layer at $y=4nb$, the $m$th strip is at $|x-2ma|<w$, whereas in the odd layer at $y=(4n+2)b$, the $m$th strip is at $|x-(2m+1\left)a\right|<w$, where $m=0,\pm 1,\pm 2,...,\pm \infty$ and $n=0,\pm 1,\pm 2,...,\pm \infty$.

Figure 6

Magnetic field lines in a hexagonal array of superconducting strips for $w/a=0.8$: (a) $b/a=1$, (b) $b/a=0.5$, and (c) $b/a=0.2$. Thick horizontal lines correspond to the cross sections of superconducting strips.

Figure 7

Effective permeability for perpendicular field ${\mu }_{\perp \mathrm{h}}$ as a function of $1-w/a$ for $b/a=5,2,1,0.5,0.2$, and $0.1$. The dashed line is ${\mu }_{\perp \mathrm{h}}/{\mu }_0=1$ for $b/a\to \infty$. The dot-dashed line of ${\mu }_{\perp }/{\mu }_0=1-w/a$ is shown for comparison with Fig. 3.

Figure 8

Comparison between the numerical results of the effective permeability ${\mu }_{\perp \mathrm{h}}$ in a hexagonal array for $\Lambda /w>0$ (symbols) and the analytical results for $\Lambda /w\to 0$ (lines). (a) Numerical results (dots) for $\Lambda /w=0.01$ and analytical results (lines) for $\Lambda /w\to 0$ of ${\mu }_{\perp \mathrm{h}}/{\mu }_0$ vs $1-w/a$ for $b/a=5,2,1,0.5,0.2$, and $0.1$. (b) Numerical results (symbols) for $\Lambda /w=0.1,0.05,0.01$ and analytical results (line) for $\Lambda /w\to 0$ of ${\mu }_{\perp \mathrm{h}}/{\mu }_0$ vs $1-w/a$ for $b/a=0.2$. (c) Numerical results (dots) and analytical results (dashed lines) for $\Lambda /w\to 0$ of ${\mu }_{\perp \mathrm{h}}/{\mu }_0$ vs $\Lambda /w$ for $b/a=0.2$.