Phase shifter based on an ultrathin superconducting bilayer with a through-hole for a superconducting device

Applying a radian phase shift other than 2 π is a key issue for superconducting circuits, such as flux qubits. The magnetic flux is useful when generating a phase shift. However, a quantized magnetic flux accompanying a trapped vortex in a superconductor does not possess a phase shifter function. The magnetic flux generated by an external field generates noise. In this study, we propose a phase bias system that does not require an external field during operation. We confirm the phase shift of a direct current superconducting interference device (SQUID) placed on an ultrathin superconducting Nb bilayer with a through-hole by cooling it to a temperature below the superconducting transition temperature with an external field. Although the cause of the phase shift in our system is unclear, we believe that it may be caused by a fractional quantum in the bilayer. When the SQUID is replaced by a qubit, the phase shift can be applied to a phase bias.

Phase difference solitons between components (i-solitons) in 3 multicomponent superconductors, which are not found in conventional superconductors, have been discussed and have been shown to increase the possibility of fractional flux quantum generation [10][11][12][13].
Hence, we developed a new ultrathin bilayer system to investigate these new topological objects found in multicomponent superconductors. In 2018, we confirmed the fractional quantization of magnetic flux in the ultrathin bilayer system with the use of a scanning magnetometer [14].
The scanning magnetometer provides a precise measurement of the magnetic flux distribution [15]. However, it requires the presence of a person throughout the measurement process, and it is difficult to measure the magnetic field dependence at low temperatures automatically. To investigate the basic properties of an ultrathin bilayer in detail, we placed a direct current superconducting interference device (DC-SQUID) on the ultrathin bilayer with the use of a device process.
Given that the structures of qubits and SQUIDs are similar, we believe that our system will facilitate the function of reinforcing/remodeling conventional flux qubits [16].
The direct combination of a SQUID and target to produce a magnetic flux has been attempted in other multicomponent superconductors to attain a fractional vortex [18]. However, with the exception of one of our previous publications [16], to our knowledge, there is no other report on the identification of the fractional vortex. We explore herein the basic properties of this device with the use of a bilayer with a through-hole. 4 Fig. 1(a) shows that the system used in this study consists of an external copper coil and a chip with the developed ultrathin bilayer disk. The coil wire had a diameter of 0.85 mm, and the coil had an internal diameter of 7.4 mm. The space between the coil and chip was ~4.5 mm. This coil generated the field, which was oriented perpendicular to the surface of the chip. Fig. 1(b) shows the schematics of the chip, comprising a Nb coil, lines connecting the terminals, and two types of SQUID arrays; one array type is used for calibration and the other for detecting fractional flux quantum trapped in the bilayer. In the right-hand-side row of the chip, the bilayer disk was present underneath a series array of 100 SQUIDs, whereas in the left row, the bilayer disk was absent. The length of the SQUID arrays was 1.2 mm, and the separation between the centers of the two SQUID arrays was 24 μm. These arrays were surrounded by a rectangular Nb coil, with an inner width and length of 1 mm and 5 mm, respectively, and a Nb linewidth of 200 μm. Figs. 1 (c)-(e) show the SQUID and bilayer in detail. The disk with a diameter of 10 μm consisted of two 20-nm-thick Nb layers. We inserted a 5-nm-thick Al layer between the two Nb layers, and the top surface of this Al layer was oxidized. The disk consisted of three layers. The special feature of our device is a central through-hole with a diameter of 2 μm in the bilayer, which was designed with a Josephson current density of 300 A cm 2 5 between the two Nb layers. The bilayer was covered with a 100-μm-thick silicon oxide layer, and the DC-SQUIDs (with a designed critical current of 76 μA) were placed on it. The DC-SQUID has two square junctions of dimensions 3 μm × 3 μm, and the critical current was designed to be 38 μA in one junction. The series array of 100 SQUIDs improved the signal-to-noise ratio because of the averaged results of the SQUIDs. We quantified the transport properties of the SQUIDs by using a typical instrument [16,19,20]. We inferred that the total flux that passed through the SQUIDs was characterized by the I C values of the SQUIDs. In the I-V characteristic, 6 we determined the positive currents measured at 90% (I + C (90%) and 10% (I + C (10%) of the maximum voltage, as well as the negative currents measured at 90% (I -C (90%)) and 10% (I -C (10%)) of the minimum voltage.

Experiments
When all the SQUIDs were in the same I C state (high/low state), I ± C (10%) ≈ I ± C (90%). In all the measurements, the SQUID bias currents were first increased from 0 to 100 μA, then decreased to −100 μA, and finally increased back to 0 μA. The current flowing in the upward direction in Fig. 1(d) is the current in the positive direction.
The flux quantum was trapped as the temperature decreased and while the temperature passed through the superconducting transition point in the presence of an external field (this phenomenon is called "field cooling" (FC)), hereafter referred to as Φ FC . The device temperature was first increased to 12-14 K, which was higher than the critical temperature of superconductivity for the bilayer (i.e., T bilayer C = 7.73 K); subsequently, the temperature was reduced following the application of a magnetic field. The magnetic field was then removed at temperatures in the range of 6-6.5 K. After cooling below 5.5 K, we re-applied the field to measure the external field dependency of I C at low temperatures. The field applied at low temperatures is designated as Φ ex . The downward direction of the magnetic field is determined to be the positive direction.
The applied magnetic field is positive when the current flow appears clockwise by looking at the chip from above.
Φ FC denotes the magnetic field applied at high temperatures (above 6.5 7 K) when FC is generated by the external copper coil, whereas Φ ex denotes the magnetic field applied at low temperatures (below 5.5 K) after FC is generated by the Nb coil. The reasons behind the use of the two types of coils are heating and resistance. The external coil was made of copper wire, which was heated when the current was applied. When the external coil is used for a long period, it becomes difficult to maintain the coil at a constant temperature, and thus, becomes unsuitable for generating Φ ex . In contrast, the Nb coil has a large resistance above T Nb C = 9.23 K, and it is difficult to apply current to it at temperatures > T Nb C .
Thus, the Nb coil is unsuitable for generating Φ FC and operates at higher temperatures (12-14 K).
We conducted two types of calibrations for the external field. For one type of calibration, the external field was represented by the total flux inside the SQUIDs without the bilayer when the external field was applied. Subsequently, we calibrated the magnetic field with the left SQUID. The flux unit was represented by Φ 0 in this case. For the other type of calibration, the external field was represented by the total flux inside the SQUIDs with the bilayer when the external field was applied. The flux unit was represented by Φ 0 in this case. The appropriateness of the unit depends on the situation, as has been discussed later.
We investigated two types of I C changes in the SQUIDs, which were measured at temperatures of 4-5 K. In the Φ FC -I C characteristic 8 measurements, the FC process was performed each time before the I-V characteristics were measured; the Φ FC value changed at every measurement. In these Φ FC -I C measurements, we maintained Φ ex = 0.
In the Φ ex -I C characteristic measurements, the FC process was performed only once, i.e., the I-V characteristics were measured by varying Φ ex , without increasing the device temperature again.

Results and Discussion
The I-V characteristics (Figs. 2(a) and 2(b)) depict the high and low I C states. Typically, the SQUID voltage jumped to ~280 mV, which corresponded to a value which was equal to 100 times the gap of the superconducting Nb [21,22]. This means that all SQUIDs were in the "running state," which we were able to confirm.  Φ 0 is used as the calibration value in Fig. 3. The reason behind the use of Φ 0 as the calibration value in Fig. 3 is that the superconductors cancel a part of the applied field at temperatures below T C . In our system, T bilayer C (critical temperature of the bilayer) is 7.73 K [23,24], and Φ ex is always applied below T bilayer C . Φ 0 has the following relationship with Φ 0 : Similarly, changes in the period have been observed in the Sr 2 RuO 4 -Ru eutectic microplates with a micro-DC-SQUID [18]. 10 The I C of the SQUID varies at Φ 0 intervals with the modulation of the external magnetic fields. The value of I C is usually maximized at Φ ex = 0.
The shift of the location where I C is maximized indicates the presence of a magnetic flux other than the external field Φ ex . In Fig. 3, this shift is equal to 0.45 ± 0.01Φ 0 .
Furthermore, we investigated the value of Φ FC at which the phase shift occurred (Fig. 4). When the phase shift occurs, the SQUIDs should be in a low I C state with Φ ex = 0. As is evident from Fig. 4  The fractional flux is generated in the ultrathin bilayer; the fractional value is determined as the ratio of the thickness of the upper to the lower layer [14,25]. Fractional flux quanta were confirmed in an ultrathinbilayer system with a hole present only in the upper layer [14,16]. We believe that the same phenomenon may occur in an ultrathin bilayer system with a through-hole. If a flux quantum is generated in the given by the extra loop following the attachment of a SQUID [30].
Further, the proposed system does not require new materials, such as magnetic materials, other than Nb [31,32].
We confirmed that a research study has been published on a ring 13 composed of a thick bilayer [33]. However, while the geometric shape of this bilayer is similar to the geometry of our ultrathin bilayer, fundamental content differences exist. In the case of the conventional bilayer, flux propagates through layers. In our ultrathin bilayer, flux goes through a pinhole in the center of the bilayer. The physical principles, properties, dynamics, basic formulas, underlying phenomena, and the role of the magnetic flux are completely different in these two cases. The "fractional phase shift" in the thick bilayer ring was also investigated [34], wherein the phase shift was induced by an externally injected current. Similar to the phase shift induced by the additional ring [30], the extra space was needed, and this system could potentially feed the noise induced by the external current lead. Using the flux trapped in the ultrathin bilayer system we can avoid these drawbacks.

Conclusion
It is worth noting that this system can be utilized as a phase shifter. We intend to investigate this phase shifting mechanism for use in a qubit. In addition, this system is useful for studying fractional flux quanta. If it can be proven that this phase shift is due to fractional flux quanta, and it can be used to assess whether fractional flux quanta are generated in multilayer systems, and multicomponent superconductors would thus become considerably simpler.