Anne Polyakov ‘12
Physics Department, Stony Brook University
Information in superconductor circuits that are to be used in prospective super fast computers is presented by the value or sign of the stored magnetic flux. As an unavoidable drawback, these unique circuits are highly sensitive to the ambient magnetic field that ideally should be less than approximately 0.2 nT for their correct operation. Sophisticated magnetic shields are currently employed for shielding of the Earth’s magnetic field. However, the very low temperatures that are required for these circuits make it difficult to monitor and therefore compensate the residual magnetic field with proper accuracy. In this work, the magnetometer setup was built, which utilizes an ability of the Superconducting Quantum Interference Filter (SQIF) to be used for absolute (without offset) zero-field detection. The setup is capable of measuring absolute magnetic fields with a record accuracy of 3.5 nT, which is 25 times better than the accuracy of known counterparts. The setup is optimized for real-time monitoring and active compensation of residual magnetic field in the vicinity (less then 1 mm) from the superconductor circuit under test. Using the setup, the residual magnetic field and shielding capabilities of magnetic shields made of µ-metal, Bi-2223 High-Tc superconductor ceramic and lead were measured and compared. It was also shown that the setup can be used for direct measurements of the magnetic field produced by magnetic vortices trapped in a superconductor chip. This feature presents a novel way of monitoring and avoiding the trapping of magnetic fluxes in superconductor chips.
Superconducting analog and digital circuits, for example, Rapid Single Flux Quantum (RSFQ) devices dramatically outperform conventional silicon counterparts in speed and energy dissipation . Unfortunately, they operate at very low (helium) temperature of about 4.2K and are extremely sensitive to ambient magnetic fields. When such devices are cooled down below the critical temperature of niobium, in the presence of even very small magnetic field, magnetic vortices  are formed and can become trapped in the superconductor material of the chip, preventing it from operating correctly. This effect is known as “flux trapping” and is considered the main obstacle to practical applications of highly integrated superconductor circuits . Usually, the influence of an external magnetic field is significantly reduced by placing the superconductor chip inside of a magnetic shield, either a superconducting shield or one made of a metal with high magnetic permeability . Unfortunately, it was found that flux trapping still occurred in the majority of cooling cycles, which indicates that the magnetic field that affected the chip was still high enough to cause flux trappings. To guarantee that fluxes will not be trapped at all, the magnetic field B must be smaller than a critical field Bcrit that can create flux equal to a fundamental magnetic flux quantum Φ0 = 2.05•10-15 Wb, through the typical area of a superconductor chip A ≈ 10 mm2. This gives the following estimation of the critical field:
Bcrit = Φ0/A ≈ 2 • 10-10 T = 0.2 nT (1)
However, even the measurement of such weak fields at low temperatures is difficult. This is because the only magnetic sensor that can be used in this situation is a Superconducting Quantum Interference Device (SQUID) , which is now widely used in sensitive magnetometers, with a typical sensitivity of about 10 fT/Hz1/2 . The most common type is a DC SQUID , which consists of a superconducting loop with two Josephson junctions (designated as JJ in Figure 1A).
A Josephson junction is a weak coupling between two superconducting wires, with a very small area of 1-5 µm2. According to the Josephson effect , such a junction remains superconducting until the current through it is less than the critical current Ic, which is usually chosen in the range 10 – 50 µA. When the SQUID is biased by the current slightly exceeding 2∙Ic (because there are two junctions connected in parallel), quantum interference occurs between the two junctions [7,8]. The voltage drop Vs across the SQUID then becomes a periodic function of the flux Φ through the SQUID loop, with a period of Φ0 and minima corresponding to ± k•Φ0, where k= 0, 1, 2 … (Figure 2A). This dependence is called a “flux-voltage characteristic” or “modulation characteristic” of the SQUID. It makes it possible to use the SQUID as a magnetometer or a device sensitive to applied magnetic fields.
The periodicity itself is in fact a major drawback of the SQUID. Because of it, the SQUID cannot distinguish between input fluxes, which differ by the period Φ0. To be able to measure input fluxes in a higher range, a special feedback electronics has to be used, which keeps the SQUID voltage Vs at the point of maximum slope dVs/dΦ by applying the corresponding feedback flux . This method of SQUID readout is referred to as FLL (Flux Locked Loop) and is widely used in many practical applications including biomagnetism, susceptometors, nondestructive evaluation, geophysics, scanning SQUID microscope, nuclear magnetic resonance and quantum computing [5, 8]. However, the FLL method can only measure relative changes of magnetic field, and cannot provide any information about the field itself, or as it is frequently called, the absolute magnetic field.
On the other hand, the absolute magnetic field can be measured directly by determining a position of the main minimum (at k=0) on the modulation characteristics recorded by applying the test magnetic field. This method, however, will obviously work only when the position of the minimum is within ±Φ0, or else it will be impossible to tell which peak the position B=0 belongs to. The range of measured magnetic fields can be widened by increasing the voltage modulation period, which can be achieved by using a SQUID with a smaller loop area. However, this will cause a corresponding reduction in measurement accuracy, because the voltage minimum will become less sharp and it will be more difficult to determine its position on the B axis.
This tradeoff between measurement accuracy and range is avoided in a circuit containing several SQUIDs with different sensitivities to magnetic field and connected in series (Figure 1B). This kind of circuit is known as the Superconducting Quantum Interference Filter (SQIF) [9, 10, 11]. When loop areas of individual SQUIDs are chosen in a such a way that none are multiples of each other, the result is a non-periodic voltage modulation characteristic with a sharp dip at B=0 (Figure 2C). The absence of periodicity allows for dramatically increasing the range of measured DC magnetic fields. Therefore, the SQIF can be used for direct, non-FLL measurements of absolute magnetic fields with high accuracy and a large magnetic field range.
As an absolute magnetic field sensor, the SQIF has been used mostly to measure relatively high magnetic fields, like the Earth’s magnetic field of about 25 µT . The measurement accuracy in  was reported to be restricted by a residual field of about 0.15 µT from the magnetic shields, and flux trapping of about 0.5 µT in the SQIF itself during cooling cycles. Another source of error was a magnetic field created by the SQIF bias current, which limited the accuracy of the SQIF to 0.1 µT in . Authors have mentioned the possibility to compensate this error by using a so called bias reversal technique that switches the bias current polarity , but did not use it in their work. A SQIF magnetometer in  was designed to minimize the field from the SQIF bias current to well below the measurement accuracy of 0.1 µT, which was yet limited at this value by the low curvature of the voltage peak tip. Besides, they also observed an unexpected shift of about 1 µT due to residual field inside their shields. In many applications of SQIF magnetometers the accuracy is not important at all; rather, magnetometers are optimized for best noise limited sensitivity, which can reach 10 fT/Hz1/2 or better values.
As it can be seen, the accuracy of available SQIF magnetometers is not enough for monitoring the magnetic field in the vicinity of superconductor chips, which can trap flux and become non-operational if the ambient magnetic field is greater than the Bcrit= 0.2 nT given by (1). The goals of this research are: a) to create the SQIF magnetometer setup optimized for best possible accuracy of magnetic field measurement, ideally to be able to measure absolute magnetic fields down to 0.2nT, and b) investigate the effectiveness of shields used to protect superconductor circuits from external magnetic fields.
Methods and Results
As the sensitive element of our magnetometer setup, we used a SQIF sensor designed in our laboratory and fabricated by a third party company. The SQIF sensor contained two identical SQIF circuits, fabricated with multi-layer niobium technology on a silicon substrate with dimensions 2.5 x 2.5 mm (Figure 3). The technology uses niobium for all on-chip wiring and a silicon dioxide as an insulator between layers.
The SQIF itself was built from eleven SQUIDs connected in series. For additional increase of the output voltage level, four such SQIFs were connected in series to form a single 44 SQUID array. The SQUID areas have been selected as a geometric progression with a common ratio equal to 0.92. The SQIF chip also contained two heaters (Figure 3) made of molybdenum films, each with a resistance of about 330 Ω . The heaters were used to perform a “thermal deflux” procedure, or removal of trapped fluxes by heating the chip above the niobium critical temperature of 9.3K for about 4 seconds. After this procedure the heater turned off, allowing the chip to cool down to its original temperature of 4.2 K.
We installed the SQIF sensor on the underside of the fiberglass chip holder board (Figure 4) of a low-temperature test probe used in the lab for testing superconductor circuits. The probe consisted of a stainless steel tube with all wiring situated inside, a connector box on its upper end and a chip holder on its lower end. The probe was cooled down by insertion into liquid helium in a non-magnetic Dewar (model Cryofab-100). To be able to measure the magnetic field at a very close proximity to the RSFQ chip, some of the fiberglass material was removed with a milling machine. This allowed for enough space to install the sensor so that the RSFQ chip could be located parallel, at a distance of 0.5 mm from the SQIF surface.
The chip holder had several 16 mm diameter coils containing 8 turns each, used for imposing a specified magnetic field by applying a calculated DC current to the coils. The main coil was mounted on the surface of the chip holder (XY plane in Figure 4), and it created a magnetic field in the Z direction. The holder also had Helmholtz coils (pairs of coils used to create a field with improved uniformity) installed to create a magnetic field in X and Y directions.
The chip was protected from external magnetic fields by a shield consisting of three nested shielding cans with open tops, oriented in the Y direction. The shield was cooled down to liquid helium temperatures together with the chip holder. Two outer cylinders were made of µ-metal, an alloy with a very high magnetic permeability (µ ≈ 1∙105) that contains 75% nickel, 15% iron, copper and molybdenum. The inmost cylinder in our experiment was chosen to be either a high-Tc superconductor (Bi-2223), low-Tc superconductor (lead) or a µ-metal cylinder. The residual magnetic field inside the shield was created by two major sources: penetration of external magnetic field through the shield and a magnetic field created by the shield itself.
Since the chip surface in our configuration was orthogonal to the Z-axis, the flux trapping was determined only by the Z-component of the magnetic field. This presented us with an interesting opportunity to separate contributions of external magnetic field and a shield-generated field. For this purpose, the shield was mounted on a separate stainless steel tube, which was concentric with the chip holder tube and had a slightly larger diameter. This made it possible to manually perform independent rotation around the Y axis of either the chip holder or the shield. With such an arrangement, when the shield was rotated, the chip experienced variations of the Z-component of the magnetic field produced by the shield only. On the other hand, rotation of the chip holder tube caused both the external field and the shield field to contribute to variations of the Z-component of the field.
To precisely control the rotation angles of the chip holder and the shield, we made a special protractor dial, which was attached to the Dewar flange during the experiment (Figure 5). We also attached individual pointers to both the connector box (which rotated with the chip holder) and the shield tube, so that we were able to set the chip holder and shield angles independently, anywhere between 0 and 360° with an accuracy of about ±3°.
The measurements were conducted using a multi-channel automated system, which is used in the lab for performing measurements of superconductor chips. The system can apply DC currents and measure voltages at up to 64 independent channels. All operations were programmed in XLISP language which makes it possible for users to create their own programs of controlling the measurement process. A nice feature of the system was that all intermediate measurement data and a log file with all commands of the current session were automatically saved and could be retrieved in the future for the purpose of reviewing, or to be used as examples. This was very useful for a quick and easy understanding of the XLISP language and principles behind controlling the measurements.
Optimization of the SQIF setup for measurements of absolute magnetic field
To provide the output voltage Vs (Figure 1), the SQIF has to be biased with a constant current Is that slightly exceeds the critical current. The choice of the bias current is very important as it determines the shape of the flux-voltage characteristic of the SQIF. A typical modulation characteristic for an individual SQUID similar to the one used in our SQIF is shown on Figure 6.
It can be seen that at Is > 30 µA, the shape is almost sinusoidal. Below 30 µA, the sign wave becomes distorted and turns almost rectangular at Is ≈ 26 µA. To figure out what shape is better for the SQIF magnetometer, we have simulated the formation of SQIF modulation characteristic in Matlab as a sum of periodic curves for all eleven SQUIDs, with periods distributed in a geometric progression with a factor of 0.92. To simulate distortions of sinusoidal shape we have approximated a modulation curve by the exponent of the cosine wave in the form:
y = exp(A Cos(x)) (2)
By varying the parameter A we were able to simulate all different shapes of modulation characteristics shown in Figure 6. The result of simulation is shown in Figure 7.
We have found that the general shape of the SQIF modulation characteristics (Figure 7C) did not change much with variations in shape of the SQUID’s modulation curve. The SQIF still exhibited the single main minimum at zero magnetic field, and the amplitude of other unwanted minima never exceeded 1/3 of the main minimum at all possible shapes shown in Figure 6. However, the sharpness of the main peak varied significantly with the shape of the SQUID’s modulation curve (Figure 6). Therefore, for better accuracy of determination of peak position we must choose a bias current of the SQIF that provides the sharpest peak shape, which is necessary for improved accuracy of absolute magnetic field measurements.
For initial evaluation of operation of our SQIF, we recorded its modulation characteristics at different bias currents in a wide range of magnetic fields from -15 µT to +15 µT (Figure 8). As it was expected, the SQIF demonstrated a single dip around zero magnetic field, which had the sharpest shape at the bias current Is ≈ 35 µA. For more accurate determination of the best bias current, we repeated the same recording with a smaller step and in a narrower range of magnetic fields from -0.3 µT to 0.3 µT (Figure 9). We have used here both positive and negative bias currents to evaluate operation in bias reversal mode. It can be seen that the position of the voltage peak strongly depends on the bias current of the SQIF and is located symmetrically around B ≈ 0 for positive and negative current values.
This means that the position of the peak on the B axis is mainly determined by the parasitic magnetic field created by the bias current Is of the SQIF. It can be also noticed that this dependence (dashed line in Figure 9) is very non linear, and not proportional to Is as one may expect. This behavior is most probably explained by the non-uniformity of the magnetic field created by the SQIF bias current. In this case, the change in Is can create different changes in magnetic field for different loops of the SQIF, which will cause shape distortions leading to an additional shift of a peak position. In any case, since this effect is intrinsically symmetric with respect to the sign of the bias current, then it can be easily eliminated by taking a middle point between the positive and negative peaks as a measure of the actual magnetic field.
Using data on Figure 9, we chose Is = 37 µA as the best bias current, because it provided the sharpest minimum and thus ensured the best accuracy of determination of the peak position on the B axis. For further improvement of measurement accuracy, the position of a peak was determined by taking measurements at about 20 points located around the minimum on the B axis, with a very fine step of about 0.4 nT. To average the measurement noise and to avoid inaccuracy due to a finite measurement step, the data was then fitted with a 5th order polynomial and a polynomial minimum was calculated (Figure 10). This method made it possible to maximize the precision of determination of the position of voltage minimum.
The whole algorithm was then programmed in XLISP language as a magnetic field measurement function that automatically measured positions of both positive and negative minima and calculated their mean value to compensate for the contribution of the SQIF bias currents. The function returned the value of the actual magnetic field, meaning that the SQIF could now operate as a real magnetometer of absolute magnetic field. Each single measurement of such a magnetometer involved about 120 voltage measurements at different currents through the main magnetic coil, which in all took about 15 s. In the future, the time of this measurement can be noticeably reduced by using a faster voltmeter and special electronic hardware to find the positions of voltage minima.
We checked the stability of our SQIF magnetometer by monitoring a background magnetic field for a long time (about 200 measurements) and calculating the standard deviation of measurements which was found to be about 13 pT. We then repeated the same stability test but in a more harsh condition, by executing a thermo recycling after each unit measurement. Surprisingly, the standard deviation was found to be of about the same order, 16 pT, which means that our SQIF sensor did not trap any magnetic fluxes during the cooling down process.
To demonstrate the magnetometer sensitivity, we have recorded a response of the SQIF magnetometer to a 0.1 nT step-like magnetic field (Figure 11). The magnetic field step was generated by applying an additional current to the same main coil which was used for recording modulation characteristic of the SQIF.
Measurement of magnetic field with SQIF magnetometer
Using the SQIF magnetometer setup we were able to measure the absolute magnetic field inside of our shields and to compare the effectiveness of magnetic shielding provided by different materials. For all experiments, we used the same set of two outmost shielding cylinders made of µ-metal. The third, inmost cylinder was chosen between µ-metal, High-Tc or Low-Tc superconductor. First, we investigated the shielding configuration with a µ-metal inner shield. All three µ-metal shields had been demagnetized before experimentation on a special setup, which subjected the shield to an oscillating 7 Hz sine-wave magnetic field with gradually decreasing amplitude. The initial amplitude was about 10 mT to ensure saturation of the µ-metal material. It was then exponentially reduced with a time constant of about 40 s. The whole demagnetization procedure took about 5 min and was performed inside of a bigger tri-layer µ-metal shield. After demagnetization, the residual magnetic field was checked with the flux-gate magnetometer. Special attention was given to the inner µ-metal shield, which we always tried to demagnetize to less than 3 nT. The dependence of magnetic field on the rotation angles of the µ-metal shield and the chip holder is shown in Figure 12. The data was recorded by setting the angle from 0º to 360º with an increment of 10º and performing a measurement of magnetic field at each position. We must note here again that the graph of the shield rotation (blue curve) represents only the contribution of magnetic field produced by the shield, while the graph of chip holder rotation (green curve) contains contributions of magnetic field from both the shield and an external source (Earth’s magnetic field). It can be seen that both graphs coincide with the measurement accuracy, which means that the contribution of the Earth’s magnetic field is negligible in comparison to the residual magnetization of the shield. From Figure 12 we can estimate the Earth magnetic contribution not to exceed 1 nT, which gives an estimation of the shielding factor to be higher than 2.4∙104. Another interesting observation was an unexpected increase of the residual magnetization of the µ-metal shield from about 2 nT at room temperature to 21 nT at liquid helium temperature. This increase might have been induced by the thermal stress in the shield material and must be taken into consideration when using the µ-metal shield. For better estimation of the shielding factor we have recorded the response of the magnetometer to placing a permanent magnet in the vicinity of the Dewar at a distance of about 40 cm from the SQIF sensor location. The direction of the field was orthogonal to the surface of the SQIF for maximum sensitivity. Prior to placement, the field of this magnet was measured with the flux gate magnetometer and was about 240 µT at a 40 cm distance. The recorded response is shown in Figure 13. It was noticed that after the placing and removing of the magnet, the magnetic field baseline acquired some shift which remained constant at subsequent placements of the magnet in the same polarity. However, when the magnet was placed in the opposite polarity, the baseline was shifted to the opposite direction in a similar way. This shift can be explained by partial re-magnetization of µ-metal shields with the magnet, which affected the residual magnetization acquired by the shield during the cooling down process. The average response from the magnet was about 2 nT, which leads to an estimated shielding factor of about 1.2∙105. Such a high shielding factor is a very important property of the used µ-metal material, which was not originally specified by the manufacturer for application at liquid helium temperatures.
We have also recorded similar data for the inner shield made of a High-Tc superconducting ceramic (Figure 14). Again, the contribution of the Earth’s magnetic field was negligible with respect to the field produced by the shield. The upper and lower parts of the graph were not that symmetric as in the case of the µ-metal shield. This distortion of the graph can be explained by a higher non-uniformity of the field produced by the shield. Because the shield was superconductive, the magnetic field measured by the SQIF was produced by flux trapped in a small area on the shield wall. The value of the magnetic field produced by this frozen flux was around 20 nT, which was about the same as in the case of the µ-metal shield.
The response from the permanent magnet is shown in Figure 15, from which we calculated the shielding factor to be about 1∙106. The shielding factor of a BSCCO 22 12 phase High-Tc superconducting cylinder in  had a maximum of 1∙106, which matches our value very well. This value is significantly higher compared to the µ-metal shield and is still probably limited by the quality of the HTS ceramic shield, which had a small crack in the wall.
Lastly, we measured the Low-Tc shield, which was made of lead. The value of the field produced by this shield was about 7 nT, which is noticeably better than values of the previous two shields. The measured shielding factor was 1.2∙107, which is roughly 10 times better than that of a High-Tc shield. Thus, a lead shield can be considered to provide the best shielding in combination with the two outer µ-metal shields.
Finally, we used our SQIF magnetometer setup to measure magnetic fields in the vicinity of a real superconductor chip. For this purpose we chose the superconductor Quantum Computer chip, the operation of which was expected to be extremely sensitive to the ambient magnetic field. When we rotated the shield (with a High-Tc inner cylinder), we observed jumps in the measured magnetic field baseline between several almost equidistant levels (Figure 16). This indicated that our SQIF has directly measured the magnetic field of fluxes trapped in the superconductor chip. Also, it was noticed that the magnetic field produced by the High-Tc superconductor shield was reduced by a factor of about 7, obviously due to the shielding effect of the ground plane of the superconductor chip.
This unique capability of the SQIF to directly measure the magnetic field of flux trapped in superconductor chips can be extremely valuable for future applications of the SQIF magnetometer setup for monitoring magnetic environments in the vicinity of tested superconductor circuits.
We have built a setup capable of measurement of absolute magnetic fields at liquid helium temperatures with 3.5 nT accuracy that is about 25 times better than in the best known counterparts [9, 12, 13]. The achieved accuracy allowed us to investigate magnetic shielding properties and compare effectiveness of shields made of: µ-metal, High-Tc (Bi-2223) and Low-Tc (lead) superconductors.
We discovered that the residual magnetization of the inner µ-metal shield at helium temperature was about 21 nT, which was much worse than the expected 2 nT value as measured at room temperature. As a result, similar shields (which are now commonly used) could not be recommended for prospective superconductor devices. Lead shield provided much smaller residual field of about 7 nT, which was caused by trapped fluxes and can be significantly reduced in practical applications by thermal defluxing.
We also showed that the tri-layer shielding configuration with lead inner cylinder yields a record shielding factor of 1.2∙107 achieved with a simple cylindrical shield with an open top. The shielding factor of High-Tc and µ-metal shields was much worse, being equal to 1∙106 and 1.2∙105 correspondently.
We have also demonstrated that our setup presents a novel way of direct monitoring of magnetic field trapped in superconductor chips. This is especially valuable for controlling the magnetic environment when testing numerous superconductor circuits, like RSFQ or components of prospective superconductor quantum computers, which can operate only in extremely low ambient magnetic fields.
There are several ways for further improving the accuracy of the setup toward the targeted field of 0.2 nT, below which magnetic fluxes will be not trapped at all. At first, the currently achieved accuracy was totally determined by a constant offset of 3.5 nT, which was most probably produced by a microscopic piece of magnetic material accidentally deposited close to the SQIF circuit. Other possible reasons for this offset include the flux trapping in the niobium layout of the SQIF itself, and also the finite accuracy of the bias reversal operation. These assumptions will be verified when we try the newer SQIF design, which is currently in the production facility. The new design is also expected to generate much less magnetic field by the bias current, which can significantly simplify and speed up the field measurement procedure.
With all the above improvements, we consider the accuracy of 0.2 nT realistically achievable, which will make it possible to minimize the magnetic field below the flux trapping threshold Bcrit and ensure reliable operation of highly integrated superconductor circuits.
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