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

Abstract

In this paper we analyze the properties of a dc superconducting quantum interference device (SQUID) when the London penetration depth $\ensuremath{\lambda}$ is larger than the superconducting film thickness $d$. We present equations that govern the static behavior for arbitrary values of $\ensuremath{\Lambda}={\ensuremath{\lambda}}^{2}∕d$ relative to the linear dimensions of the SQUID. The SQUID's critical current ${I}_{c}$ depends upon the effective flux $\ensuremath{\Phi}$, 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 $\ensuremath{\Lambda}$, 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 $\ensuremath{\Lambda}$. 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 $\ensuremath{\Lambda}∕w$ and $a∕w$.