$\pi$-SQUIDs based on Josephson contacts between high-$T_c$ and low-$T_c$ superconductors

For phase-sensitive studies on the pairing-symmetry in a given superconductor, its combination with a well-characterized isotropic superconductor into a superconducting quantum interference device (SQUID) provides a very fruitful method. With a well-controlled orientation and position of the hybrid Josephson junctions involved, the angular dependence of both the phase and the magnitude of the order parameter wave function can be determined. In view of this, and motivated by recent proposals for quantum-logic devices, we have investigated low-inductance dc SQUIDs with different geometries. These SQUIDs are based on ramp-type $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7∕\mathrm{Au}∕\mathrm{Nb}$ junctions oriented toward the $⟨100⟩\text{−}\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ crystal axes. For SQUIDs with perpendicularly oriented junctions, maxima in the critical current are observed at applied fluxes of $\pm \frac{1}{2}{\Phi }_0$, confirming the predominant $d_{x^2\text{−}{y}^2}$ pairing symmetry in $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$. The measured critical current-modulations for these $\pi$-SQUIDs correspond well to simulations. No indications are found for any time-reversal symmetry breaking, for instance resulting from imaginary symmetry admixtures.

Figure 1

Potential surface $U({\phi }_1,{\phi }_2)$ of a SQUID (a), and of a $\pi$-SQUID (b) with identical parameters given by: $L=5.2$ $p\mathrm{H}$, $I_{\mathrm{c}}=100\mu \mathrm{A}(=\frac{1}{2}{I}_{\mathrm{c},\mathrm{SQUID}})$, ${\alpha }_{\mathrm{J}}=0.2$, and ${\alpha }_L=0$, and in the absence of an externally applied magnetic field or bias current. The energy is plotted on a logarithmic scale with an offset of $+0.01E_{\mathrm{J}}$ for the SQUID and $-1.28E_{\mathrm{J}}$ for the $\pi$-SQUID to stress the minima (black). (c) 3D perspective view of the potential surface of the $\pi$-SQUID in (b), here with a bias current of $I_{\text{bias}}=200\mu \mathrm{A}$. A possible trajectory of the particle with mass $M$ is schematically indicated.

Figure 2

Spontaneous flux in $\pi$-SQUIDs for varying inductance $L{I}_{\mathrm{c}}$, and ratio of the junction critical currents $\gamma$: (a) Behavior of the junction phase-differences, and (b) the corresponding self-generated magnetic flux. The simulation is normalized to the smallest of the junction critical currents $I_{\mathrm{c}1}$: $I_{\mathrm{c}2}=\gamma I_{\mathrm{c}1}=\gamma I_{\mathrm{c}}$ with $\gamma ⩾1$. For $\gamma \to \infty$ the $\pi$-SQUID-ring contains effectively only one junction. In the inset of (b) the minimum inductance $L{I}_{\mathrm{c}}$ is presented for varying $\gamma$ above which flux is generated, and the dependence for which the maximum current is circulating in the $\pi$-SQUID-ring $({\phi }_1=\pi ∕2)$.

Figure 3

$S$ states in SQUIDs (a, c and e) and $\pi$-SQUIDs (b, d and f) for different values of $\lambda =(2\pi ∕\Phi )$ $L{I}_{\mathrm{c}}$: the total flux in the SQUID-ring as a function of the externally applied flux. In (a) and (b) are the rf SQUIDs presented, in (c) and (d) the symmetric dc SQUIDs, and in (e) and (f) the asymmetric dc SQUIDs with $\gamma =I_{\mathrm{c}2}∕I_{\mathrm{c}1},\gamma >1$, the degree of asymmetry of the junction critical currents. Curves with negative slope are unstable. The arrows illustrate switching between two $S$-states $n\leftrightarrow n+1$ as the externally applied flux is varied.

Figure 4

Simulation of dc $\pi$-SQUIDs with finite size junctions. The inductance is $L=16$ $p\mathrm{H}$ and the $I_{\mathrm{c}0,\mathrm{SQUID}}=100\mu \mathrm{A}$. The applied magnetic flux induces three SQUID oscillations within the first zero of the junction Fraunhofer dependence, i.e., $k_i=3$ (A) symmetric $\pi$-SQUID, (B) symmetric SQUID with an order parameter induced phase-shift of $0.75\pi$, (C) asymmetric $\pi$-SQUID with $L_1=2L_2$, and (D) asymmetric $\pi$-SQUID with $I_{\mathrm{c}1}=2I_{\mathrm{c}2}$. The offsets for the latter three all equal $\Delta \Phi =1∕8{\Phi }_0$. Only (B) is not point symmetric in the origin. Asymmetry in the junction critical currents (D) yields a reduced modulation depth indicated with the vertical arrows.

Figure 5

Design of low inductance $\pi$-SQUIDs. (a) Planar design I, (b) planar design II, and (c) multi-layer design.

Figure 6

Critical current as a function of the applied magnetic field at $T=4.2\mathrm{K}$ of $\pi$-SQUID $\Pi \mathrm{10}$ with the planar I design (top), and “standard” SQUID C10 (bottom). The insets present an enlargement of the $I_{\mathrm{c},\mathrm{SQUID}}\text{−}B$ dependence near zero field. Both SQUIDs are in the low inductance limit. The magnetic field is applied perpendicular to the substrate plane and the characteristics are recorded with a voltage criterion of $2.5\mu \mathrm{V}$.

Figure 7

Critical current as a function of the applied magnetic field at $T=4.2\mathrm{K}$ of $\pi$-SQUID $\Pi 3$ with the planar II design (top), and “standard” SQUID C3 (middle). The insets present schematically the corresponding configuration and an enlargement of the $I_{\mathrm{c},\mathrm{SQUID}}\text{−}B$ dependence near zero field. Both SQUIDs are in the low inductance limit. The magnetic field is applied perpendicular to the substrate plane and the characteristics are recorded with a voltage criterion of $2.5\mu \mathrm{V}$. Simulation (bottom) with method of Langrangian multipliers of the $I_{\mathrm{c},\mathrm{SQUID}}-{\Phi }_{\mathrm{a}}$ dependence of dc $\pi$-SQUID $\Pi 3$ using an order parameter induced phase-shift of $\pi$. The employed parameters are $L_1=2.4p\mathrm{H}$, $L_2=2.0p\mathrm{H}$, $I_{\mathrm{c}1}=50.8\mu \mathrm{A}$, $I_{\mathrm{c}2}=43.2\mu \mathrm{A}$, and $k_1=3.7$, $k_2=4.7$. The shift of the peak positions from $\frac{1}{2}{\Phi }_0$ is $\Delta \Phi =0.015{\Phi }_0$, resulting in a difference in peak position for positive and negative bias-current of $0.03{\Phi }_0$.

Figure 8

$I\text{−}V$ characteristics at $T=4.2\mathrm{K}$ of the SQUIDs and $\pi$-SQUIDs corresponding to Figs. 7, 8: C10, C3, and $\Pi \mathrm{10}$ and $\Pi 3$. The SQUIDs are characterized at zero field and at $\frac{1}{2}{\Phi }_0$. Flux induced or $d$-wave induced $LC$-resonance structures are observed for these SQUIDs, indicated with arrows.

Figure 9

Critical current as a function of the applied magnetic field at $T=4.2\mathrm{K}$ of $\pi$-SQUID $\Pi m$ (top) and SQUID Cm (bottom), both prepared with the multi-layer design. The SQUIDs are in the low inductance limit. The magnetic field is applied parallel to the substrate plane and the characteristics are recorded with a voltage criterion of $5\mu \mathrm{V}$. The $I\text{−}V$ characteristics in the inset are measured at zero field and for the maximum critical current at $\sim \frac{1}{2}{\Phi }_0$. Resonance structures are observed below the gap voltage at $1.2\mathrm{mT}$, probably due to Fiske resonances in the individual junctions.