SQUID noise thermometers with lithographically defined metal sensing elements

We present SQUID noise thermometers for sub-kelvin thermometry that employ lithographically defined metal thin-film resistors as the temperature sensors. The resistors with values of about 1 milliohm are lithographically fabricated using Au, PdAu and superconducting Nb wiring. In order to ensure constant resistance values in the whole temperature range of operation, the resistive structures are designed to avoid superconducting proximity effects. The thin-film resistor circuits can be easily combined with different types of multi-purpose dc SQUID current sensors to adapt noise temperatures, uncertainties and measurement speeds as well as robustness against magnetic fields in different measurement setups.


Introduction
The sub-kelvin temperature range has become increasingly relevant for a variety of applications, most notable superconducting quantum technology.Today, easy-to-use refrigerators are manufactured and deployed on an industrial scale and enable straightforward access to millikelvin (mK) temperatures.Furthermore, dilution refrigerators equipped with nuclear demagnetization refrigerators that are already commercially available, will open up the microkelvin (μK) range for both fundamental and applied research.Thermometry in the mK and μK range that is in accordance with the International System of Units (SI) is the basis to determine with sufficient precision and traceability the temperature at which such refrigerators operate.
Johnson noise thermometry using SQUID sensors [1][2][3] has been developed to measure thermodynamic temperatures T predominantly below about 4 K and demonstrated to be applicable down to below 100 μK [4].There are four major variants of SQUID-based Johnson noise thermometers (JNTs), namely the resistive SQUID noise thermometer (RSQUID-JNT) [5], the quantum roulette noise thermometer (QRNT) [6], the current sensing noise thermometer (CSNT) [7] and the magnetic field fluctuation thermometer (MFFT) [8,9].The RSQUID-JNT, CSNT and the MFFT have been included in the most recent mise en pratique for the definition of the kelvin in the SI [10].While the RSQUID-JNT and a primary version of the MFFT (pMFFT [11]) have been developed for metrology, CSNTs and MFFTs are being used as practical thermometers, and a MFFT system is commercially available [12].
The temperature sensing element in a CSNT is an electrical resistor R that is galvanically coupled to the input coil of a SQUID current sensor as shown in figure 1.This way, the thermal current noise of the resistor at temperature T is measured as a thermal noise by the SQUID sensor.The power spectral density of this thermal noise is directly proportional to T.
In the MFFT, the temperature sensing element is a bulk metale.g.copper or silverpart placed at the temperature T. It has a comparably large volume (≈cm 3 ) and an electrical resistivity ρ.Brownian motion of electric charge carriers inside the sensing element gives rise to fluctuating magnetic fields at its surface that are sensed by a SQUID sensor as a thermal magnetic flux noise that is also directly proportional to T. The commercial Magnicon MFFT-1 uses a specifically designed gradiometric SQUID sensor with integrated flux antennas [13], and the SQUID sensor chip is placed directly onto the MFFT-1 copper sensing element.To minimize thermal gradients in both a CSNT and a MFFT configuration, the sensing elements should be as well as possible thermally coupled to the refrigerator stage the temperature of which is to be measured and parasitic thermal loading be avoided.Also, for both variants the electrical resistance or resistivity, respectively, of their sensing elements need to be highly constant in the targeted temperature range of operation.Otherwise, deviations of the proportionality of the thermal flux noise measured by the SQUID sensor and the temperature to be inferred from it occur.A CSNT configuration offers more flexibility to combine or vary resistors and multipurpose SQUID current sensors.Arguably, the need for galvanic contacts to the sensing element can be considered a disadvantage of the CSNT compared to the MFFT as the contacts can present a technical complication or even be potential source of device failure.Contrary, the 'lumped-element' circuit formed by the CSNT resistor R and the SQUID current sensor input coil with inductance L, as shown is figure 1, is simple to analyse.The RL-circuit has a first-order low-pass frequency response with the characteristic frequency The development of both primary and practical CSNTs for sub-kelvin temperatures has been significantly advanced over the last about 20 years by the Quantum Fluids and Solids Group of Royal Holloway University of London [7,14].Their CSNTs use sensor resistors with values from a few hundreds of μΩ to > 1 Ω that have been formatted into appropriate dimensions by means of cutting high-purity copper foils and platinum tungsten wires with thicknesses in the range of about 10 to 50 μm.Electrical contacts are made by soldering superconducting wires to the resistors.The resistor elements are on one side thermally and electrically grounded using a clamp and screw to a sensor resistor assembly which is mounted to the refrigerator stage.
In [3] a sensor resistor with a value of 5 μΩ is realised using a bulk piece of high-purity silver onto which niobium is locally deposited to form wire bonding contact pads of well-defined shape and size.The resistance value is defined by the resistivity of the silver in combination with the distance and dimensions of the niobium contact pads.The massive sensor element can be easily thermally anchored directly to the refrigerator stage, very similar to a MFFT sensing element.
We have developed CSNTs that employ lithographically defined metal thin-film resistors as the sensing elements.The configuration and fabrication of the thin-film resistors is detailed in section 2. Demonstration measurements of the new CSNT configuration down to 10 mK in direct comparison to the commercial Magnicon MFFT-1 are presented in section 3.

Lithographically defined Au-PdAu thin-film resistors
In the present work, resistors are realised as integrated planar thin-film structures by means of microfabrication and lithographic methods.Thin-film deposition and patterning allows highly reproducible and effective waferscale fabrication of such chip resistors.Furthermore, well-defined integrated resistor networks can be prepared.We have chosen a bilayer of gold (Au) and palladium gold (Pd 52 wt% Au 48 wt% -PdAu) to form the resistor.Both materials are compatible to common thin-film deposition techniques.Their high resistance to corrosion ensures long-term stability of their electrical properties.Nb is used to form superconducting leads.Lengths and distances of the Nb leads define the resistor structure and its value.The reason to use an Au-PdAu bilayer as the resistor material, instead of Au or PdAu, is discussed in detail below.shown that forms additional wiring (strip-line and co-planar).Nb-II does not contact the Au layer nor the PdAu layer, but only Nb-I.The Nb-1 leads are arranged such that 4 resistor segments of same dimensions, each denoted R in the electrical circuit scheme, are formed.The value of R is determined by the sheet resistance of the Au-PdAu bilayer as well as the length of the Nb-I leads and the parameter d Nb-Idistance of the Nb-I leads on the bilayer region.It can be seen, that at both contact lead pairs, 1-2 and 3-4, a serial-parallel connection of the 4 resistor segments is formed.The resistance of the serial-parallel connection equals R. Each of the contact lead pairs 1-2 or 3-4 can be used to connect the resistor to the SQUID input coil.The contact lead pair not connected to the input coil is available to apply a known test current to the resistor segments.When doing so, a well-defined power is dissipated in the resistor.The configuration is chosen to enable in situ power loading measurements on the sensor resistor without the applied current, and associated noise currents, flowing into the SQUID input coil.As shown in figure 2, the Au thin-film structure extends over a much larger area than the resistor region.The Au layer banks to the sides of the Au-PdAu bilayer region significantly increase the total volume of the Au layer and its contact area to the substrate.This way they function as large-area 'cooling fins' [15] attached to the resistor structure in the bilayer region.Additionally, they act as contact pads for wire bonding connections for thermal and electrical grounding of the resistor to a refrigerator stage (see section 3).The thin-film resistors were fabricated as follows.First, the Au layer was thermally evaporated from high-purity (99.99%)Au onto a thermally oxidized 3′′ silicon wafer in high vacuum and with a deposition rate of 0.2 nm s −1 .A thin, ≈5 nm, Ti layer was used as an adhesion layer.PdAu was then deposited locally onto the Au layer, as shown in figure 2. This was done by physical vapor deposition (PVD) from a PdAu target following an Ar plasma etch to clean the Au surface.Nb was deposited by PVD, as well.Patterning of all layers was performed by standard photolithography and lift-off microstructuring.
The thickness of the Au layer, as deposited and including the Ti adhesion layer, was measured at different positions across the wafer to be 100 K 103 nm.Its sheet resistance, measured at 300 K and in liquid helium at 4.2 K was 0.36 Ω and 0.096 Ω, respectively.These values correspond to Au layer resistivities of ρ Au,300K = 3.65 μΩ cm −2 and ρ Au,4.2K = 0.97 μΩ cm −2 .The sheet resistance of the Au-PdAu bilayer was measured, as well, and values of 0.38 Ω at 300 K and 0.145 Ω at 4.2 K were found.The resistivity of our PdAu deposited on thermally oxidized Si is typically ≈350 μΩ cm −2 , and its residual resistance ratio Therefore, unsurprisingly, the Au sheet resistance determines the sheet resistance and RRR of the Au-PdAu bilayer of which the resistor element is formed.Note, the somewhat lower sheet resistance values of the bare Au layer and its higher RRR compared the Au-PdAu bilayer.This indicates that the thickness of the Au underneath the PdAu has slightly reduced, or its resistivity increased due to the PdAu deposition including the predeposition Ar plasma etch.
We have opted to form the resistor elements from the Au-PdAu bilayer, and not from Au, to avoid superconducting proximity effects that could cause variations in the resistance values in the CSNT temperature range of operation.Thin-film systems that combine normal-conductors and superconductors are prone to such proximity effects.The so-called longitudinal proximity effect (LoPE) has been studied both experimentally and theoretically in the context of superconducting transition edge sensors (TESs) [16,17].Here, the material properties and dimensions of superconducting leads significantly affect the electrical transport properties of the TES elements they connect to.In [18] a TES has been demonstrated that directly exploits the LoPE in a thin-film structure composed of Au as the TES element and Nb leads.A superconducting transition is induced in the intrinsically normal-conducting Au by the Nb leads, and the resistance of the Au region strongly becomes temperature dependent.Whether or not LoPE-induced superconducting transitions occur in a given structure, as well and their transition temperatures, is affected by the material compositions, the thicknesses and distances of the superconducting leads and normal-conducting layer material as well as the electron mean free path in the normal conductor [16].We have performed resistance versus temperature measurements on test structures made from our thermally evaporated Au onto which Nb was PVD-deposited and patterned.Curves of test samples are exemplarily shown in figure 3 in the temperature range 0.4 K to 10 K.The samples were fabricated of the same wafer, only the distance d Nb between the Nb structures was varied.As shown in the inset, the resistance measurement contacts were made to the Au layer of the samples.The sharp drop in resistance at about 9 K is caused by the Nb structures becoming superconducting and short-circuiting the underlaying Au regions.However, the sample resistances were observed to decrease further at lower temperatures indicating LoPEinduced superconducting transitions of the Au regions in between the Nb structures.As shown, variations in d Nb between 2 μm and 5 μm caused pronounced differences in transition temperatures and widths.
The results on our Nb-Au test structures show that if one were to use this material combination for the resistor elements, large Nb lead distances, likely in the range of tens of micrometers, would be required to obtain constant values of R. In the Au-PdAu bilayer, the function of the PdAu, with its significantly higher resistivity than Au, is to suppress LoPE-induced changes of R within the temperature range of CSNT operation, while the Au-PdAu bilayer sheet resistance is still determined by the Au resistivity.This way, much smaller Nb lead distances compared to Nb-Au can be chosen and, hence, more compact practical resistor elements be obtained.Note, that resistor elements could also be realized in a configuration in which the Nb-I leads are first formed on the substrate and then coved by PdAu onto which the Au is deposited.This 'inverted' configuration compared to the one shown in figure 1 would have the advantage that the Nb-I thickness could be reduced.The Au in such a PdAu-Au bilayer, however, would have to cross over the step edges of the patterned PdAu to connect to the Au side bank cooling fins.Here, discontinuities in the Au layer could occur.The signals of both MFFTs and CSNT are typically analyzed in the frequency domain.Individual time series of the SQUID output signals are sampled and the thermal noise spectra S Φ ( f ) are obtained by fast Fourier transform (FFT) calculations.Individual spectra of N Avg time series at a given temperature are typically rootmean-square averaged with equal weighting to obtain thermal noise spectra from which the temperature is extracted.The commercial MFFT-1 is normally used in relative primary mode [19].That means, the thermal noise spectrum of a given specimen is obtained, with low uncertainty, at one known and traceable reference temperature T ref to perform a calibration.There are different methods to analyse the thermal noise spectra to extract sought temperatures [20].These methods are not specific to either MFFTs or CSNTs, but equally applicable to both.For a chosen analysis method, a frequency range Δf = {f low , K , f up } of typically equally spaced frequencies needs to be selected from the noise spectra.An estimated value T of the sought temperature is then calculated from the measured thermal noise within Δf at this temperature.It is meaningful to choose Δf so that the average noise temperature T Noise of a given MFFT or CSNT is below a limit suitable for a targeted uncertainty for the lowest temperature to be measured.The parameter T Noise ( f ) is defined as the temperature at which the thermal noise of the sensing element equals the noise of the SQUID sensor without the sensing element attached, i.e., the readout noise [7].Hence, a meaningful comparison of different SQUID noise thermometers can be based on their noise temperatures.Figure 4 depicts the noise temperatures of the commercial Magnicon MFFT-1 and of CSNTs that use a sensing element with R = 1 mΩ and state-of-the-art commercially available dc SQUID current sensors [21,22].The noise temperatures are frequency dependent because both the thermal noise and the intrinsic SQUID noiseincluding low frequency excess noise [23] vary differently with frequency f.The noise temperature and its frequency dependence of a MFFT-1 depends besides many other parameters on the geometry and the resistivity of its Cu sensing element and is unique to every thermometer specimen.The range of T Noise shown in figure 4 in grey represents typical values for MFFT-1 thermometers.The graphs for the CSNT noise temperature ranges represent combinations of 1 mΩ sensing resistors with SQUID current sensors that have input coils of inductances of 70 nH and 155 nH.Noise levels of state-of-the-art dc SQUID current sensors have been used.The upper limits of the CSNT noise temperature ranges assume input-referred current noise levels of single SQUID current sensors, the lower limits the overall noise levels achievable with two-stage SQUID sensor cascades.
The comparison in figure 4 shows that the considered CSNT configurations can achieve up to a factor of 5 lower minimum T Noise ( f ).Also, the frequency range for which the noise temperatures of the CSNTs are below those of the MFFT-1 extend from <0.2 kHz to above 10 kHz.This means that a larger measurement frequency range Δf can be chosen for the CSNTs.Lower noise temperatures and larger measurement frequency ranges directly translate into smaller relative uncertainties u rel (T) when using the considered CSNT configurations compared to the MFFT-1, assuming equal measurement frequencies spacings and measurement time [19].In the following a CSNT configuration using an Au-PdAu sensor resistor with R ≈ 1.25 mΩ is discussed.Figure 5 shows the thermometer setup.The resistor chip is located on a solid body made from conventional oxygen-free copper (OF-Cu).Next to the resistor chip the SQUID sensor chip is placed.Both chips are affixed onto the OF-Cu body using epoxy resin suitable for low temperature use.Conventional Au bonding wires are used to connect the Au layer banks to the OF-Cu body for heat sinking.The sensor resistor and the SQUID input coil are connected by short chip-to-chip wire bond connections using conventional Al bond wire that becomes superconducting below about 1.2 K. Electrical connections of the readout electronics to the SQUID sensor are made via a printed circuit board on the copper body and a socket connector.The copper body fits into a cylindrical superconducting magnetic shield made of solid niobium to diminish parasitic electromagnetic signal coupling to the SQUID sensor.The dimensions of the copper body, socket connector and niobium magnetic shield are identical to the corresponding components of MFFT-1 thermometers.Similarly, the threaded stud on the copper body is used to mount the CSNT onto the cryostat stage the temperature of which is to be measured.
We have performed measurements below 1 K to directly compare a CSNT prototype and a MFFT-1 thermometer.The CSNT used the aforementioned sensor resistor with R ≈ 1.25 mΩ.The current sensor was a single SQUID, fabricated in niobium technology, with an integrated input coil of nominal inductance per design of L = 150 nH.Both the CSNT and the MFFT-1 were operated in a flux-locked loop (FLL) [24].Both thermometers were placed, in proximity, on a separate plate that is directly connected to the mixing chamber plate of a commercial 3 He/ 4 He dilution refrigerator.That plate was also equipped with a calibrated superconductive reference device (SRD) [25].The SRD provided 6 reference temperature points T 2000 traceable to the Provisional Low Temperature Scale of 2000 (PLTS-2000) [26], namely T SRD = T 2000 = 207.69mK, 160.58 mK, 99.22 mK, 65.72 mK, 31.68 mK, and 15.13 mK.The setup allowed a precise and highly stable temperature regulation to the reference temperature points.The relative uncertainties of the reference temperatures range between 0.05% and 0.4%.Note, that the minimum temperature of the dilution refrigerator was about 10 mK.
In figure 6 the thermal noise spectra of the CSNT and the MFFT-1 acquired at a temperature of T 2000 = 160.58mK are shown.The two spectra were obtained with N Avg = 107 999 and a resolution bandwidth of 1 Hz.The CSNT's thermal noise spectrum corresponds to a first-order low-pass frequency response with the characteristic frequency f c,CSNT = 1.25 mΩ/(2π × 158.8 nH) = 1252.79Hz.The MFFT-1 thermal noise spectrum is similar to, but not exactly first-order low-pass-like.We define its characteristic frequency f c,MFFT-1 as the 3 dB frequency at which the MFFT-1 thermal noise power has dropped with increasing frequency to about half (−3 dB ≈ 0.5012) the value found in the limit f → 0 Hz.For the MFFT-1 specimen used here f c,MFFT-1 = 597.04Hz.For both JNTs, f c is determined from a two-parameter fit.In case of the MFFT-1, two other parameters describing the slope of the thermal noise spectrum [9] are kept constant.
As discussed above, for both the CSNT and the MFFT-1 the frequency response of the thermal noise spectra needs to be constant in the targeted temperature range of operation.As shown in [27] this requirement is fulfilled for the MFFT-1.In light of potential LoPE-induced changes of the value R of the sensor element used in the CSNT, we examined the constancy of the thermal noise spectrum shape over the temperature range 1 K to 10 mK for both JNTs based on their characteristic frequencies f c .CSNT and MFFT-1 thermal noise spectra were acquired at the 6 reference temperature points, at a refrigerator temperature of about 20 mK as well as at the lowest refrigerator temperature of about 10 mK.The values of f c,CSNT and f c,MFFT-1 were obtained from each of these spectra.Figure 7 shows these characteristic frequencies normalized to the two values, f c,CSNT,160.58mK and f c,MFFT-1,160.58mK, respectively, observed at T ref = T 2000 = 160.58mK.The values of f c,CSNT are reproducible to within 0.2% for all reference temperatures as well as for the lowest temperature of 10 mK, showing that the CSNT's thermal noise spectrum shape is preserved down to the lowest measurement temperature.This indicates that changes in R, LoPE-induced or otherwise, of the Au-PdAu sensor resistor element used do not exceed a level of 0.1% in the temperature range 1 K to 10 mK, which otherwise would be detrimental to the CSNT operation.
Principally, when considering f c,CSNT as the indicator that the noise spectrum shape remains unchanged with temperature, a variation in R could be obscured by a corresponding change in L. However, in the temperate range below 1 K the inductance of the niobium thin-film input coil that dominates L can be expected to remain highly stable.Assuming a critical temperature of the niobium thin film input coil of >8 K the relative variation with temperature below 1 K towards 10 mK of the London penetration depth λ Nb ( T ) can be estimated to be <0.01%.Consequently, a variation in L that would 'mask' a change in R can be ruled out.
The parallel CSNT and MFFT-1 operation allowed their direct comparison against the PLTS-2000 as measured with the SRD.In figure 8    Concluding this section, we note that compared to the MFFT-1 the prototype CSNT should be more robust with regard to magnetic background fields in operation.We measured the magnetic field sensitivities of the MFFT-1 SQUID sensor type with integrated gradiometric flux antennas as well as of the SQUID current sensor type used in the measurements presented above.The two sensor types were exposed to a homogenous magnetic flux density B, oriented normal to the planes of the sensor chips, and flux changes in the sensors upon varying B were measured.Magnetic field sensitivities B/Φ were found to be ≈3 μT/Φ 0 for the MFFT-1 SQUID sensor type compared to ≈16 μT/Φ 0 for the SQUID current sensor type used in the prototype CSNT.Given that the two thermometer modules use identical Nb magnetic shields, a correspondingly smaller parasitic sensitivity to background magnetic fields of the CSNT can be expected.

Summary
We have developed CSNT sensing elements for the sub-kelvin range in the form of integrated metal thin-film circuits.The resistor circuits are based on Au-AuPd bilayers with superconducting Nb wiring.They are fabricated on Si chips and are meant to be combined with multi-purpose SQUID current sensors, most easily by chip-to-chip superconducting bond wires.Resistance values in the range of 1 mΩ have been realized.In combination with commercially available low-noise dc SQUID current sensors, CSNT noise temperatures below 10 μK over practical measurement frequency ranges of hundreds of Hz to a few kHz can be obtained.A prototype CSNT was investigated in direct comparison with the commercial MFFT-1 thermometer.Constancy of the resistance of the Au-AuPd bilayer-based sensing element was found in the temperature range of 1 K to 10 mK.Furthermore, the temperature values obtained with the prototype CSNT were found to be consistent to within 1% with those of the MFFT-1 as well as with the PLTS-2000.We expect our CSNT also to be functional at temperatures below 10 mK and even below the range covered by the PLTS-2000, 1 K to 0.902 mK.However, meaningful measurements would then inevitably require taking into account the configuration specific T Noise within the frequency range for the CSNT temperature estimation.
As shown here, CSNT sensing elements in the form of integrated thin film circuits can be purely resistive.However, combinations of resistor and compact inductor elements are technically feasible, as well.Resistorinductor combinations as sensing elements would have thermal current noise spectra with higher-order lowpass frequency responses.This could be an approach to favourably modify the frequency characteristics of noise temperatures compared to conventional CSNTs with purely resistive sensing elements.
The CSNT prototype setup presented here is a stand-alone thermometer meant to be used on platforms of sub-kelvin refrigerators.Lithographically defined metal thin-film resistors as noise thermometry sensors are also particularly suited to be integrated on components, for instance, bolometric or calorimetric detectors for radiation or particles that are operated in such refrigerators [28].This way, direct measurements of the component temperature are possible in a precise and inherently dissipationless manner and can provide valuable insight in the operation of such components.
Figure 2 schematically depicts the layer and wiring configuration.As shown in figure 2(a), a large-area Au thin-film structure with lateral dimensions ≈mm is locally covered by a PdAu thin-film.Superconducting leads are formed from a first niobium (Nb-I) layer and are located on the Au-PdAu bilayer.Figure 2(b) details the wiring layout.A second niobium (Nb-II) layer is

Figure 1 .
Figure 1.Scheme of a current sensing noise thermometer (CSNT).The resistor R at temperature T is directly connected to the input coil of a SQUID current sensor.The thermal current noise of the resistor is represented as the current source I N .It flows into the SQUID sensor input coil with inductance L and generates thermal magnetic flux noise in the SQUID via the mutual inductance M. The power spectral density of the noise current source I N is S IN = 4 k T/R, with k-Boltzmann constant.The resistor and the input coil form a RL-circuit with a first-order low-pass frequency response S I ∝ 1/[1+ ( f/f c , CSNT ) 2 ] with the characteristic frequency f c, CSNT = R/2π L.Not shown is the separate feedback coil typically used to operate the SQUID current sensor in a flux-locked loop.

Figure 2 .
Figure 2. Schematic layer and wiring configuration of lithographically defined metal thin-film resistors.(a) Top view and crosssection along dotted line of normal-conducting Au and Au-PdAu layers and superconducting Nb-I wiring with d Nb-Idistance of the Nb-I leads on the Au-PdAu bilayer.(b) Detailed wiring layout of region indicated in (a) by the dotted rectangle and electrical circuit scheme.

Figure 3
Figure 3 Resistance versus temperature curves of Au/Nb test structures with varied distance d Nb between pattererned Nb (grey) on Au (yellow).The inset shows the cross-section schematic of the test samples with the measurement probes connecting to the Au layer.

3 .
Dimensioning and operation of a CSNT with Au-PdAu thin-film resistors in the temperature range 1 K to 10 mK

Figure 4 .
Figure 4. Ranges of typical noise temperatures for commercial MFFT-1 thermometers, shown in grey, and of CSNTs with sensor resistors R = 1 mΩ combined with dc SQUID current sensors that have input coils of inductance 70 nH (green) and 155 nH (blue).Noise levels of state-of-the-art dc SQUID current sensors have been assumed for both the MFFT and the CSNT.

Figure 5 .
Figure 5. Schematic drawing and photograph of a CSNT setup.Thin-film resistor and SQUID sensor chips are glued onto the Cu body.Brass socket connector, wiring to the PCB and niobium magnetic shield are shown in the photograph, only.
the CSNT and MFFT-1 temperatures T JNT are plotted relative to T 2000 with T 2000 .Here, T JNT denote the estimates of the two JNTs obtained based on the improved non-parametric evaluation [20], N avg,ref is the number of FFT averages at reference temperature, and S JNT,th are the measured

Figure 6 .
Figure 6.CSNT and MFFT-1 thermal noise spectra at T 2000 = 160.58mK measured with N Avg = 107 999 and a resolution bandwidth of 1 Hz for both the CSNT and MFFT-1.The left axis quantifies the magnetic flux noise power spectral density in each SQUID sensor, and the right axis is the input-referred current noise power spectrum density of the CSNT.The values R = 1.25 mΩ and L = 158.8nH are obtained from the CSNT thermal noise frequency response and the known temperature T 2000 .For the MFFT, a second-order lowpass filter becomes effective just below its cut-off frequency of 10 kHz.

Figure 7 .
Figure 7. Characteristic frequencies of the CSNT and MFFT-1 thermal noise spectra normalized to their values observed at T ref = T 2000 = 160.58mK.The graph also includes the f c values obtained at about 20 mK as well as at the lowest refrigerator temperature of about 10 mK.For both JNTs, the error bars shown include the statistical uncertainty from a two-parameter fit.

Figure 8 .
Figure 8.Comparison of relative primary temperature determinations obtained from the CSNT and the MFFT-1 at the 6 reference temperatures traceable to the PLTS-2000.Both thermometers were calibrated at T ref = T 2000 = 160.58mK.The error bars shown include uncertainty components from all quantities appearing in equation (1).