Response of thin-film SQUIDs to applied fields and vortex fields: Linear SQUIDs

In this paper we analyze the properties of a dc superconducting quantum interference device (SQUID) when the London penetration depth $\lambda$ is larger than the superconducting film thickness $d$. We present equations that govern the static behavior for arbitrary values of $\Lambda ={\lambda }^2∕d$ relative to the linear dimensions of the SQUID. The SQUID’s critical current $I_c$ depends upon the effective flux $\Phi$, the magnetic flux through a contour surrounding the central hole plus a term proportional to the line integral of the current density around this contour. While it is well known that the SQUID inductance depends upon $\Lambda$, we show here that the focusing of magnetic flux from applied fields and vortex-generated fields into the central hole of the SQUID also depends upon $\Lambda$. We apply this formalism to the simplest case of a linear SQUID of width $2w$, consisting of a coplanar pair of long superconducting strips of separation $2a$, connected by two small Josephson junctions to a superconducting current-input lead at one end and by a superconducting lead at the other end. The central region of this SQUID shares many properties with a superconducting coplanar stripline. We calculate magnetic-field and current-density profiles, the inductance (including both geometric and kinetic inductances), magnetic moments, and the effective area as a function of $\Lambda ∕w$ and $a∕w$.

Figure 1

$I$ enters the main body of the SQUID from the counterelectrode below through two Josephson junctions (small black squares, labeled $a$ and $b$) and divides into the currents $I_1=I∕2-I_{\mathrm{d}}$ and $I_2=I∕2+I_{\mathrm{d}}$ as shown.

Figure 2

$I_{\mathrm{c}}∕2I_0$ vs ${\Phi }_{\mathrm{f}}∕{\varphi }_0$, calculated from Eqs. (17, 18), for $\pi L{I}_0∕{\varphi }_0=0$, 1, 2, 3, 4, and 5 (bottom to top).

Figure 3

Sketch of central portion of the long SQUID considered in Sec. 3.

Figure 4

Profiles of the sheet current $J_I\left(x\right)$, Eq. (49), stream function $G_I\left(x\right)$, Eq. (50), and magnetic induction $B_I\left(x\right)$ for the equal-current case ($B_{\mathrm{a}}=0$, $I_1=I_2=I∕2>0$). Shown are the examples $a∕w=0.3$ with $\Lambda ∕w=0$ (solid lines with dots), 0.03 (dot-dashed lines), 0.1 (dashed lines), 0.3 (dotted lines), and 1 (solid lines). Here $B∕{\mu }_0$ and $J$ are in units $I_2∕w$ and $G$ in units $I_2$.

Figure 5

Magnetic-field lines in the equal-current case for $a∕w=0.1$ and $\Lambda ∕w=0$, 0.01, 0.1, and 50 (or ∞).

Figure 6

Profiles of $J_{\mathrm{d}}\left(x\right)$, Eq. (54), $G_{\mathrm{d}}\left(x\right)$, Eq. (57), and magnetic induction $B_{\mathrm{d}}\left(x\right)$ for the circulating-current case ($B_a=0$, $-I_1=I_2>0$). Shown are the examples $a∕w=0.3$ with $\Lambda ∕w=0$ (solid lines with dots), 0.03 (dot-dashed lines), 0.1 (dashed lines), 0.3 (dotted lines), and 1 (solid lines). $B∕{\mu }_0$ and $J$ are in units $I_2∕w$ and $G$ in units $I_2$.

Figure 7

Magnetic-field lines in the circulating-current case for $a∕w=0.1$ and $\Lambda ∕w=0$, 0.01, 0.1, and 50 (or ∞).

Figure 8

(a) Solid curves show the inductance $L=L_{\mathrm{m}}+L_{\mathrm{k}}$ [Eq. (55)] vs $a∕w$ calculated for $\Lambda ∕w=0$, 0.03, 0.1, 0.3, and 1. See the text for descriptions of analytic approximations shown by the open circles, open squares, dots, and dashes. (b) Solid curves show the geometric inductance $L_{\mathrm{m}}$ [Eq. (58)] vs $a∕w$ for $\Lambda ∕w=0$, 0.1, and 1 and the kinetic inductance $L_{\mathrm{k}}$ [Eq. (59)] vs $a∕w$ for $\Lambda ∕w=0.03$, 0.1, 0.3, and 1. See the text for descriptions of analytic approximations shown by the dotted and dashed curves.

Figure 9

(a) Solid curves show the inductance $L=L_{\mathrm{m}}+L_{\mathrm{k}}$ [Eq. (55)] vs $\Lambda ∕w$ for $a∕w=0.01$, 0.1, 0.4, 0.8, 0.95, and 0.99. See the text for descriptions of analytic approximations shown by the dotted and dashed curves. (b) Solid curves show both the geometric inductance $L_{\mathrm{m}}$ [Eq. (58)] and the kinetic inductance $L_{\mathrm{k}}$ [Eq. (59)] vs $\Lambda ∕w$ for $a∕w=0.01$, 0.1, 0.4, 0.8, 0.95, and 0.99. See the text for descriptions of analytic approximations shown by the dotted, dashed, and dot-dashed curves.

Figure 10

The magnetic moment $m_{\mathrm{d}}$ for the circulating-current case ($B_{\mathrm{a}}=0$, $-I_1=I_2=I_{\mathrm{d}}>0$) plotted versus $a∕w$ for various values of $\Lambda ∕w$ in units $2wl{I}_{\mathrm{d}}$. These curves coincide with those in Fig. 13 below.

Figure 11

Profiles $J_{\mathrm{f}}\left(x\right)$, Eq. (62), $G_{\mathrm{f}}\left(x\right)$, Eq. (64), and magnetic induction $B_{\mathrm{f}}\left(x\right)$ for the flux-focusing case ($B_{\mathrm{a}}>0$, $I_1=I_2=0$). Shown are the examples $a∕w=0.3$ with $\Lambda ∕w=0$ (solid lines with dots), 0.03 (dot-dashed lines), 0.1 (dashed lines), 0.3 (dotted lines), and 1 (solid lines). $B$ and ${\mu }_{0}J$ are in units $B_{\mathrm{a}}$, and $G$ in units $w{B}_{\mathrm{a}}∕{\mu }_0$.

Figure 12

Magnetic-field lines in the flux-focusing case for $a∕w=0.1$ and $\Lambda ∕w=0$, 0.01, 0.1, and 0.5.

Figure 13

The effective area $A_{\mathrm{eff}}$, Eq. (63), plotted versus the gap half-width $a∕w$ for several values of $\Lambda ∕w=0$, 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, and 10. The lower right curves show the same data shifted and stretched along $a∕w$. The dots show the exact result (A19) in the limit $\Lambda ∕w\to 0$. For $\Lambda ∕a⩾10$ one has $A_{\mathrm{eff}}∕2wl\approx (1+a∕w)∕2$, Appendix . These curves coincide with Fig. 10 since $A_{\mathrm{eff}}=m_{\mathrm{d}}∕I_{\mathrm{d}}$.

Figure 14

The effective area $A_{\mathrm{eff}}$, Eq. (63), plotted versus $\Lambda ∕w$ (range $7\times {10}^{-6}$ to 14) for several values of $a∕w=0$, 0.001, 0.003, 0.01, 0.03, 0.1, and 0.2. Same data as in Fig. 13.

Figure 15

The minimum of the magnetic induction in the flux-focusing case, $B_{\mathrm{f}}\left(0\right)=B_{\mathrm{f}}(x=0)$, referred to the applied field $B_{\mathrm{a}}$ and plotted versus the half gap width $a∕w$. Top: The ratio $B_{\mathrm{f}}(x=0)∕B_{\mathrm{a}}$, tending to unity for $a∕w\to 1$ and for $\Lambda ∕w⪢1$, and diverging for $a∕w\to 0$ when $\Lambda =0$. Bottom: The same ratio multiplied by $a∕w$ to avoid this divergence and fit all data into one plot. The lower right plot depicts the small-gap data two times enlarged along the ordinate, and shifted and five times stretched along the abscissa. The circles show the approximation $B_f\left(0\right)∕B_{\mathrm{a}}\approx (w∕a)∕\ln (4w∕a)$ good for $a∕w⩽0.3$ (Ref. 46).

Figure 16

The minimum field $B_{\mathrm{f}}\left(0\right)=B_{\mathrm{f}}(x=0)$ for the flux-focusing case as in Fig. 15 but plotted versus $\Lambda ∕w$ (range $7\times {10}^{-6}$ to 14) for several values of $a∕w=0$, 0.001, 0.003, 0.01, 0.03, 0.1, and 0.2.

Figure 17

The magnetic moment $m_{\mathrm{f}}$ for the flux-focusing case plotted versus $a∕w$ for various values of $\Lambda ∕w$ in units $w^{2}l{B}_{\mathrm{a}}∕{\mu }_0$.

Figure 18

Profiles of the sheet-current density $J\left(x\right)$ [second term in Eq. (62)] and magnetic induction $B\left(x\right)$ generated when a perpendicular magnetic induction $B_{\mathrm{a}}$ is applied in the zero-fluxoid case when $\Phi =0$, $I_1=-I_2>0$, and $I=0$. Shown are the examples $a∕w=0.3$ with $\Lambda ∕w=0$ (solid lines with dots), 0.03 (dot-dashed lines), 0.1 (dashed lines), 0.3 (dotted lines), and 1 (solid lines). $B$ and ${\mu }_{0}J$ are in units $B_{\mathrm{a}}$.

Figure 19

Magnetic-field lines in the zero-fluxoid case $\Phi =0$, $B_{\mathrm{a}}>0$, and $I=0$ for $a∕w=0.1$ and $\Lambda ∕w=0$, 0.01, 0.1, and 0.5.

Figure 20

The magnetic moment $m$ for the zero-fluxoid case $\Phi =0$, $B_{\mathrm{a}}>0$, and $I=0$ plotted versus $a∕w$ for various values of $\Lambda ∕w$ in units $w^{2}l{B}_{\mathrm{a}}∕{\mu }_0$. At $a=\Lambda =0$ one has $-m=\pi w^{2}l{B}_{\mathrm{a}}∕{\mu }_0$.