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SQUIDs (superconducting quantum interference devices) are sensitive detectors of magnetic field [1, 2]. When equipped with an inductively coupled coil, SQUIDs may be used as sen-sitive current amplifiers. In such setups, noise level below 1 pA (

√Hz)−1 can be achieved [2]. Furthermore, when imped-

ance matching devices are used, such as transformers [3–11] or cryogenic current comparators [12–19], noise levels in the fA (

√Hz)−1 range are accessible. Due to their exquisite

sensitivity, such SQUID-based current sensors are often used for Johnson noise thermometry [20–26] or as null detectors in high precision bridge measurements in quantum metrology [12–19].

For certain SQUID applications, the source resistance to be measured, also referred to as the sample, is exposed to magn-etic fields. Such an environment is known to be extremely harsh for several reasons. Inductive pick-up due to changing magnetic fields and also due to mechanical vibrations of the

sample placed in a magnetic field induces spurious electrical signals interfering with the circuit operation. Flux motion in the filaments of superconducting solenoids causes addi-tional noise [2]. Furthermore, any uncontrolled rf signals will directly couple to an exposed sample and will interfere with the operation of the SQUID. Under such circumstances flux-locked operation of the SQUID is often not possible [2]. These unwanted effects can be mitigated using low-pass fil-ters placed between the sample and the SQUID. Such SQUID circuits with filters were used to measure samples exposed to magnetic fields up to 0.1 T [27], 0.5 T [28], and 3.2 T [29].

A dc-SQUID interfaced with a coreless transformer used for fA (

√Hz)−1 level current noise measurements of kΩ level

resistors was recently described in [26]. This current ampli-fier is stable when there are no externally applied magnetic fields. However, when the source resistance is replaced with a two-dimensional electron gas exposed to an external magn-etic field, we found that the flux-locked operation of the SQUID could no longer be achieved at magnetic fields as low as 20 mT. Here we describe a low pass filter connected to

Measurement Science and Technology

Stabilizing a SQUID current amplifier in high magnetic fields

V Shingla1 , E Kleinbaum1,4, L N Pfeiffer2, K W West2 and G A Csáthy1,3

1 Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, United States of America2 Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, United States of America3 Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, United States of America

E-mail: gcsathy@purdue.edu

Received 11 June 2018Accepted for publication 21 August 2018Published 14 September 2018

AbstractSQUID-based galvanometers are unstable when connected to sources exposed to strong magnetic fields. We present a SQUID-based current amplifier with a 17 fA (

√Hz)−1 current

noise, which is unconditionally stable with a source exposed to magnetic fields up to 10 T. Stability is achieved by inserting a four-pole low-pass filter between the source and the current amplifier. With this circuit, we demonstrate Johnson noise measurements below 1 K for source impedances in the kΩ range subjected to strong magnetic fields.

Keywords: dc SQUID, low noise current amplifier, two-dimensional electron gas

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V Shingla et al

Stabilizing a SQUID current amplifier in high magnetic fields

Printed in the UK

105903

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© 2018 IOP Publishing Ltd

29

Meas. Sci. Technol.

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10.1088/1361-6501/aadbe1

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Measurement Science and Technology

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4 Present address: Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, United States of America

2018

1361-6501

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https://doi.org/10.1088/1361-6501/aadbe1Meas. Sci. Technol. 29 (2018) 105903 (5pp)

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the primary of the transformer of this SQUID-based current amplifier which allows a stable flux-locked operation of the SQUID in large magnetic fields. This filter limits the band-width of the circuit to about 100 Hz, but it is able to maintain a low amplifier noise of 17 fA (

√Hz)−1. With such a bandwidth

limiting filter, we were able to considerably extend the sta-bility range the SQUID-based current amplifier to magnetic fields up to 10 T, the largest field available in our instrument.

1. Low pass filter

We use the current-sensing circuit described in [26], which consists of a commercial dc-SQUID [30] connected to an impedance matching superconducting transformer of effec-tive turn ratio 212. We found that when a R = 3.25 kΩ source resistor is connected to this current amplifier, the overall gain is RT = Vout/Ipr = 2.25 GΩ, the circuit bandwidth is 600 Hz, and the amplifier noise is 2.3 fA (

√Hz)−1. Here Vout is

the output voltage of the SQUID electronics and Ipr the pri-mary current of the transformer. In this circuit, the impedance matching transformer was mounted on the mixing chamber of a dilution refrigerator, located in the cancellation region of a 10 T superconducting solenoid [26].

For the present work, we replaced the source resistor with a two-dimensional electron gas mounted onto the copper tail of our dilution refrigerator and which is exposed to an external magnetic field. The electron gas is mounted to be perpendicular to the applied magnetic field. Under such conditions, the sample impedance is close to the clas-sical Hall resistance [31, 32]. In a magnetic field of 4 T, the

impedance of our sample of electron density 3.7 × 1011/cm2 is 6.75 kΩ. As noted earlier, even the application of a modest magnetic field prevents the stable operation of the SQUID in the flux-locked mode. To illustrate this, in figure 1 we plot the output voltage of the SQUID electronics as a function of time when the two-dimensional electron gas is exposed to 4 T. We close the fluxed-locked mode of the SQUID near t = 0, but the output of the SQUID electronics runs away at t = 0.85 min. We then reset the SQUID electronics output to a vanishing voltage at t = 1.2 min. However, the output of the electronics runs away again at t = 1.65 min. Figure 1 captures another unsuccessful reset at t = 1.85 min. Longer wait times of a few minutes result in overload to 10 V of the electronics (not shown). We thus find that the circuit cannot be stabilized and hence is unusable at 4 T. We note that in [26] the transformer was magnetically shielded and it was mounted onto the mixing chamber of our wet dilution refrigerator. In contrast, in the present work the transformer was mounted onto the 1K pot and the SQUID sensor onto the still plate of the refrigerator.

In order to stabilize our current amplifier, following ideas from [27–29], we used a low pass filter. Our filter is placed at the input of the impedance matching transformer. However, in contrast to past designs [27–29], we employed a four-pole filter. The advantage of such a filter is an enhanced attenua-tion above the cut-off frequency as compared to single pole designs. In order to avoid introducing any additional Johnson noise, we opted for not using any resistors as filter elements.

The circuit digram of our four-pole low-pass filter, together with further details of the SQUID current amplifier, are shown in figure 2. The resistor on the left end is the source resistor to be measured, of the order of a few kΩ. The filter consists of two capacitors, one inductor, and it takes advantage of the effec-tive inductance of the transformer primary Leff = L2 = 0.87 H [30]. Values of the circuit elements are C1 = 0.5 μF, C2 = 5 μF and L1 = 8.3 H.

The SPICE simulation of the filter response for RS = 8.8 kΩ is shown in figure 3. Our sample impedance reaches such a value near 5 T. The filter gain is defined as the ratio of the current injected into the source resistor RS and the current in the transformer primary. The filter response has a peak near 100 Hz, after which it abruptly decreases. The response, however, below about 20 Hz is nearly flat. Figure 3 also displays the measured filter response. For this measurement we made use of the bias network formed by the resistors R3 = 100 Ω and R4 = 1 MΩ. Measurement and simulation are in good agreement. Furthermore, the measured filter attenuation past 200 Hz of 75 dB/decade is in reasonable agreement with 80 dB/decade, the expected attenuation of an ideal four-pole filter.

An essential part of our filter design is the inductor L1 = 8.3 H. It is wound on a high permeability toroidal core [33]. We used 570 turns of copper wire AWG 43. A photo of

Figure 1. The output voltage Vout of the SQUID electronics as a function of time when the two-dimensional electron gas is exposed to an external magnetic field of 4 T. The flux-locked loop is closed at 0.02 min. The electron gas is held at 0.25 K and is connected to the primary of the transformer using twisted copper wires shielded by a CuNi capillary tube.

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the inductor is shown in figure 4. For protection, the inductor was wrapped in teflon tape and attached with Elmer’s Rubber Cement to a Pb shield. The inductance at room temperature is 18 H, which decreases to about half at low temperatures. Similar changes in the inductance were reported in other cores of high permeability [34].

Once the filter was installed, the SQUID current ampli-fier became unconditionally stable to the maximum magn-etic field available of 10 T and sample temperatures below 1 K. We attribute stability to the steep attenuation of the four-pole filter.

2. Noise measurements

To characterize our circuit, we measured its input referred noise. The input referred noise iin was obtained from the voltage noise measured at the output of the SQUID electronics which was divided by the total transimpedance gain of the circuit of 2.25 GΩ. Noise spectra are displayed in figure 5. For these measurements, the input bias voltage vin was dis-connected and the wire was left floating. The spectral depend-ence of the current noise is consistent with the expected filter frequency response as it displays a peak near 100 Hz. In this region a large numer of spikes are seen due to the vibrations of our refrigerator generated from the various pumps. Spectra, however, are clean below 15 Hz.

In order to understand the noise generated in our circuit, in figure 6 we plot the dependence of the power spectral density of the current noise i2in measured at 13 Hz on the temperature of the source. We observe a linear temperature dependence with an offset along the vertical scale. Such a behavior can be understood considering two sources for the noise: the temper-ature-dependent Johnson noise of the electron gas iR [35] and a temperature-independent amplifier noise iamp. These noise contributions obey the i2in = i2R + i2amp relation. The Johnson noise, plotted as a continuous line in figure 6, is consistent with expected values for a 17 kΩ resistor, the impedance of the electron gas at 10 T. Our analysis yields an amplifier noise iamp = 17 fA (

√Hz)−1. We conclude that the stability of the

circuit in strong magnetic field was obtained at the expense of an increased noise level in comparison to 2.3 fA (

√Hz)−1, the

measured value at zero field in the absence of the four-pole filter. The excess noise contribution is attributed to dissipation in the magnetic core of the toroidal inductor L1 [34].

Figure 2. The circuit diagram of the four-pole filter connected to the transformer primary and other details of our circuit. The mounting places of various circuit elements within the dilution refrigerator are indicated by color of the background. RS is the sample to be measured.

Figure 3. Simulated and measured filter gain for RS = 8.8 kΩ. For clarity, the measured filter gain is vertically shifted.

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3. Conclusions

We found that a sensitive SQUID current amplifier becomes unstable when connected to a two-dimensional electron gas exposed to strong magnetic fields. However, a four-pole low-pass filter inserted between the electron gas and the current amplifier resulted in unconditional stability in magnetic fields up to 10 T. In comparison to other SQUID-based circuits, we therefore achieved an improved circuit stability at considerably larger values of the magnetic field. This improvement came, however, at the expense of a lim-ited bandwidth and a noticeable increase in the noise of the current amplifier.

Acknowledgments

Measurements at Purdue were supported by the NSF grant DMR 1505866, while sample growth efforts at Princeton University were funded by the Gordon and Betty Moore Foundation through the EPiQS initiative Grant GBMF4420, and by the National Science Foundation MRSEC Grant DMR-1420541.

ORCID iDs

V Shingla https://orcid.org/0000-0002-9497-1857

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