## Abstract

Practical temperature measurements in accordance with the international system of units require traceability to the international temperature scales currently in force. Along with the awaited redefinition of the unit of temperature, the kelvin, on the basis of the Boltzmann constant, in future its *mise en pratique* will allow the use of approved methods of primary thermometry for the realization and dissemination of the kelvin. To support this process, we have developed a DC superconducting quantum interference device-based noise thermometer especially designed for measurements of thermodynamic temperature in a broad temperature range from 5 K down to below 1 mK. In this paper, we describe in detail the primary magnetic field fluctuation thermometer and the underlying model applied for the temperature determination. Experimental measurement results are presented for a comparison with the Provisional Low Temperature Scale 2000 between 0.7 K and 16 mK including an uncertainty budget for the measured thermodynamic temperatures. In this set-up, the relative combined standard uncertainty is equal to 0.6%.

## 1. Introduction

Today, any traceable temperature measurement requires a reference to actual international temperature scales, i.e. the International Temperature Scale of 1990 (ITS-90) [1] and the Provisional Low Temperature Scale 2000 (PLTS-2000) [2]. Recent progress made in temperature metrology in general has led to the adoption of the *mise en pratique* of the definition of the kelvin (MeP-K) [3] by the Consultative Committee for Thermometry (CCT) of the International Bureau of Weights and Measures. The MeP-K contains all the information necessary for the practical realization of the kelvin. It is a flexible document, which, in future, will allow the inclusion of high-level methods for primary thermometry. This is a fundamental change in temperature metrology as any traceable thermometry will no longer require the sole usage of the international temperature scales. The methods to be included in the MeP-K should be documented in detail and provided with a complete uncertainty budget approved by the CCT. In addition, at least two different implementations must exist and be compared with each other, as well with already accepted methods. To advance this process and in preparation for the awaited redefinition of the kelvin in terms of the Boltzmann constant, the project ‘Implementing the new kelvin—InK’ [4] was established within the Euramet Metrology Research Programme. The work we report on here is part of the InK project.

In the low-temperature region, reliable thermometry is still a demanding task even though a great variety of thermometers are available for temperature measurements. This is caused by the fact that most of the thermometers exhibit nonlinear characteristics, are prone to external influences and instabilities and may lose thermal contact with the object being measured with decreasing temperature.

In the history of international temperature scales, which started in 1887 when the first international temperature scale was adopted by the Comité International des Poids et Mesures (CIPM), only four different truly low-temperature scales were adopted, namely the 1958 ^{4}He and 1962 ^{3}He vapour-pressure scales [5,6], the 1976 Provisional 0.5 K to 30 K Temperature Scale (EPT-76) [7] and the PLTS-2000 [2]. The ITS-90 is basically identical to the EPT-76 in the temperature range between 0.65 K and 5 K as it is based on the same experimental dataset for the helium vapour pressure but is represented in a different mathematical form.

The PLTS-2000 [2], which was adopted by the CIPM in 2000, is defined by a polynomial relating the melting pressure of ^{3}He to temperature. The definition of the PLTS-2000 ensures high reproducibility as well as high resolution of low-temperature measurements. The PLTS-2000 polynomial is mainly based on three input datasets [8] and consensus values for the four natural features, namely the minimum of the melting curve, the A and B transitions in the liquid ^{3}He and the Néel transition in the solid ^{3}He, which may serve as fixed points of pressure and temperature. The agreement between the input data and the PLTS-2000 is better than 1% down to 20 mK in accordance with the stated uncertainties. However, with decreasing temperature, the differences between the input data increase, reaching 6% at 1 mK, a value that is about six times higher than the individual uncertainties for a coverage factor *k*=1. The reason for this discrepancy is still unresolved and is the cause of the provisional status of the PLTS-2000. For practical thermometry, the realization of the PLTS-2000 is expensive and time-consuming and requires special expertise, which is available only in selected laboratories.

To tackle these challenges as well as in preparation for the implementation of the future MeP-K, we have developed a primary noise thermometer which complies with metrological requirements. The primary magnetic field fluctuation thermometer (pMFFT) is a DC superconducting quantum interference device (SQUID)-based noise thermometer, which measures the thermal magnetic flux noise (TMFN) of a metallic temperature sensor. The thermally activated motion of electric charge carriers in the noise sensor causes fluctuating magnetic fields above its surface, which are sensed by DC SQUID gradiometers as TMFN [9]. Via the Nyquist relation [10], the TMFN is directly related to the thermodynamic temperature.

This paper describes the pMFFT in detail and is organized as follows. The design considerations and the resulting set-up for the pMFFT will be presented in §2. In §3, the theoretical models are described on which the operation of the pMFFT is based, and formulae are derived for the calculation of the temperature measured by the pMFFT. Experimental results for all parameters entering the temperature determination as well as the first comparison measurements of the pMFFT with the PLTS-2000 are presented in §4. The uncertainty budget for the measurement of thermodynamic temperature with the pMFFT is compiled in §5. Finally, a short summary is given in §6.

## 2. Design and theory of operation

The pMFFT consists of four main parts: a metallic noise temperature sensor, a system of two separate gradiometric detection coils read out by two DC SQUID current sensors, a calibration coil and two-channel room temperature electronics to read out the SQUID sensors.

Because the pMFFT has two independent signal channels (=detection coil+SQUID sensor+SQUID electronics+analogue-to-digital converter (ADC)) to detect the TMFN of the temperature sensor the cross-correlation technique is applied for its main operation, yielding the correlation-based temperature *T*_{12}. An independent evaluation of the two channels is possible and yields two temperatures *T*_{1}≡*T*_{11} and *T*_{2}≡*T*_{22} (see (3.11)). All contributions from non-thermal noise sources, which occur uncorrelated in both channels, are suppressed in *T*_{12} but are present in *T*_{1} and *T*_{2}. If non-thermal noise is negligible compared with the thermal noise, which holds for the higher temperatures, then *T*_{1}≅*T*_{2}≅*T*_{12}, provided the correct system parameters are used in (3.11).

In operation, both channels of the pMFFT record time traces of the noise voltages *V* _{i}, *i*=1,2, which are then transformed into the cross-power spectral density *S*_{V,12} and power spectral densities *S*_{V,1}, *S*_{V,2} using the Welch periodogram [11]. Figure 1 shows the schematic diagram of the correlation-based pMFFT including data acquisition and processing to obtain Re[*S*_{V,ij}(*f*)], *ij*=12,11,22, which is proportional to *T*_{ij} apart from possible non-thermal components and a weakly temperature-dependent channel calibration. Here, *f* is the frequency.

To relate the measured voltages *V* _{i}, *i*=1,2, to the flux *Φ*_{D,i} in the detection coils, the signal channels (‘SQUID gradiometers’) are calibrated by means of the calibration coil. During calibration with quasi-DC currents, this coil produces a calculable flux *Φ*_{D,i}=*M*_{i3}*I*_{cal} in the detection coil *i* (*i*=1,2) through the calculable mutual induction *M*_{i3} (§3b).

The second purpose of the calibration coil is to enable an independent *in situ* measurement of the electrical conductivity in the temperature sensor by using AC currents at several frequencies. The conductivity must be known because it enters the temperature calculation.

For the measurement of magnetic flux, the SQUIDs are operated in a closed-loop feedback mode with feedback into the superconducting input circuit (input current-lock, ICL). This feedback scheme is essential for the pMFFT operation as it has two advantages over the common feedback into the SQUID (flux-lock loop, FLL): (i) the flux transfer from the detection coil into the SQUID, which is affected by the total inductance in the input circuit, is independent of frequency and (ii) the interaction between both detection coils and between the detection coils and the temperature sensor is suppressed within the ICL bandwidth. Weak or vanishing interaction between the signal channels is essential for the cross-correlation technique. The ICL feedback scheme is easily implemented using the integrated feedback transformer of PTB SQUID current sensors [12].

### (a) Mechanical construction and sensor material

As the pMFFT was specifically designed for the temperature range of the PLTS-2000 as a compact device, the SQUID sensors are integrated in the metallic body of the temperature sensor and placed close to the detection coils. Figure 2 shows an inside view of the partly assembled pMFFT. The basic shape of the device is a cylinder with a diameter of 18 mm and a length of 100 mm. The whole body is made from high-purity copper of nominal 99.9998% purity (MaTecK GmbH, Jülich, Germany). We have determined a residual resistivity ratio of ≈103. The cylinder has large openings from opposite sides to take up coil chips (39 mm×13 mm) and printed circuit boards (PCBs) from both sides. In addition, there are two (uneven) cut-outs, in which SQUID sensors and bond loops are located in a central position. The remaining material in between forms the actual temperature sensor. This part has a thickness of about 2 mm and a lateral size of 11 mm (13 mm including the raised rim) × 18 mm. More specifically, its sensitive region is only a central part near the cylinder axis, which is virtually defined by the detection coils.

The two Si chips hosting detection coils and a calibration coil are mounted from opposite sides, each at a fixed distance in parallel to the surface of the temperature sensor. In order to allow a defined and minimal gap (≈50 μm or ≈100 μm, depending on the side) between the detection coils and the temperature sensor, the Si chips are mounted face down. Both Si chips are surrounded by PCBs made of FR4. These boards carry the SQUID chips and bond connections to the Si chips. While the PCBs are fixed by screws, small springs (CuSn_{6}) press the Si chips towards the edge guides, always providing a proper alignment. No glue is needed to fix the coil chips. Magnetic shielding of the thermometer's inner volume is provided by an outer Nb tube, which can be easily removed by sliding it to either side.

### (b) Thin-film coil assemblies and superconducting quantum interference device sensors

Both the detection coils and the calibration coil are realized as planar thin-film coils fabricated on 3^{′′} Si wafers using PTB's standard Nb thin-film technology. The photolithography allows a lateral (sub-)micrometre precision. Line width and minimum gaps between the coils are 3 μm.

All coils are designed as series gradiometers and have thus a single degree of freedom. They are realized as coaxial, circular ( simplified calculation) gradiometers with coplanar arrangement ( minimized capacitive coupling) and interleaved turns ( maximized correlation for the TMFN). With radii of approximately 2.0 mm, and approximately 0.7 mm for the outer and the eight inner turns, respectively, the gradiometers use only a fraction of the available chip area ( minimized interaction with the rest of the thermometer body). For comparison, the distance between the coil centre and the side walls of the temperature sensor is 6.5 mm (figure 3*a*).

In order to minimize the magnetic coupling between both signal channels, which is particularly important at frequencies similar to or above the ICL bandwidth, each detection coil consists of two identical gradiometers in series with reversed polarity in one channel. One gradiometer is placed above the temperature sensor and detects the TMFN, whereas the second one is located above the larger void and counterbalances the magnetic coupling of the other gradiometer (figure 3*a*,*b*).

With an estimated total coil inductance of approximately 400 nH, we chose SQUID current sensors with an input inductance of 400 nH (PTB single-stage, double-transformer SQUIDs [12] of size ‘L’) for the readout. Each input circuit is equipped with an additional *RC* damping (*R*=3.9 Ω, *C*=47 nF).

## 3. Theoretical model

In order to enable a comprehensive description of the pMFFT, the actual design of the pMFFT facilitates the use of analytically solvable models, which are precise and efficient but restricted to filamentary conductors and special geometries such as circular loops or infinite slabs. To meet real-world conditions, supplementary models are applied, which provide corrections or bounds for the deviations.

### (a) Thermal noise calculation

The cross-power spectral density (CPSD) of magnetic flux, *S*_{Φ}, inside the detection coils *i*,*j* caused by the TMFN of an infinite conducting slab is calculated according to [13]
3.1with
3.2
3.3and
3.4The detection coils are modelled as filamentary conductors with contours *c*_{1}(*x*,*y*,*z*_{1}), *c*_{2}(*x*,*y*,*z*_{2}), at a height *z*_{1},*z*_{2} above the surface of the conducting slab, which has the electrical conductivity *σ* and a magnetic permeability *μ*=*μ*_{r}*μ*_{0} (figure 4*a*). Here, *μ*_{0} is the magnetic permeability of free space. While *S*_{12} is generally complex its real part, Re(*S*_{12}), contains the desired CPSD. For *c*_{1}=*c*_{2}, the CPSD is equal to the PSDs *S*_{1}≡*S*_{11} or *S*_{2}≡*S*_{22}.

In the temperature range relevant for us, it is sufficient to use the classical limit of the generalized Nyquist relation [14], neglecting quantum corrections. These corrections can be expanded in terms of *β*=*hf*/*k*_{B}*T*, where *h* is the Planck constant, *k*_{B} is the Boltzmann constant and the lowest terms are equal to 1+*β*^{2}/12. For *f*=100 kHz and *T*=1 *mK*, the relative deviation of the power spectral density is less than 2×10^{−6}.

An alternative calculation of the TMFN (in particular but not only useful for finite-element method (FEM) models or to determine the spatial variation of the sensitivity) is based on [15]
3.5On the right-hand side of (3.5), the real part of the complex impedance, Re(*Z*), is replaced by the dissipation power inside the conducting body divided by the square of the exciting AC current *I* flowing through the coil under investigation.

For a rotational symmetric coil parallel to an infinite slab, an analytical calculation of the eddy currents *j*(*r*,*z*) and dissipation power is possible:
3.6where *A*_{III} depends on the (filamentary) loop currents above the surface of the slab. For a single circular loop with radius *a* at a distance *d* above the surface of the slab with thickness *t*,
3.7with the first-order Bessel function *J*_{1} and *k*′=(*k*^{2}+i*ωμσ*)^{1/2}, *ω*=2*πf* [16].

Combining equations (3.5) and (3.6) and confining the integration area to a disc with radius *R*, we obtain
3.8for the detection coil *k* consisting of *N*_{k} loops (turns) in series. The total flux noise from the whole slab is . Although not explicitly given here, equations (3.5), (3.6) and (3.8) can be easily modified to yield the CPSD for two coils.

The influence of the edge effect is estimated by comparing the results for *S*_{Φ,k}( *f*) in the infinite space obtained from the whole slab, , with those originating from a disc with a given radius *R*_{c}, *S*_{Φ,k}( *f*,*R*_{c}). From the numerical evaluation, we find that (*S*_{Φ,k}( *f*,*R*_{c})−*S*_{Φ,k}( *f*))/*S*_{Φ,k}( *f*) is very small on an absolute level (see §4e). As a consequence the effect of the superconducting shield is also very small. Its influence can be estimated with the same value as the upper limit.

Alternatively, two-dimensional FEM software could be used to calculate Re(*Z*) or *P* in the finite problem. The effect of the superconducting shield can then be modelled by a sphere instead of a cylinder to estimate an upper limit within the two-dimensional problem.

Because the applied models solely use filamentary conductors, all effects related to the superconducting nature of the wire conductors are suppressed. It is therefore necessary to take these effects into account by separate corrections in the form of effective coil geometries. The essential effect to be considered is the distortion of the magnetic field by the presence of superconducting structures. In our case, where the penetration depth *λ* is smaller than the thickness of the thin-film coils, the key boundary condition on the surface of the superconductor is that the normal component of the magnetic induction is zero: *B*_{⊥}=0.

To consider this effect, we have adapted the formalism of [17], which treats the flux focusing in thin-film superconducting rings, so that we are able to calculate the proximity effect in multi-turn coils. The resulting corrections Δ*r*_{j,k}, *k*=1,2 lead to effective radii *r*_{j,k} for the filamentary coils, differing from the central contour *r*_{c,j,k} of the conductor (figure 4*b*). Alternatively, numerical inductance calculation software can be used to extract these corrections.

### (b) Mutual inductance calculations

The mutual inductances *M*_{1}≡*M*_{13}, *M*_{2}≡*M*_{23} between detection coils (indices 1,2) and the calibration coil (index 3) depend on the frequency *f* owing to the eddy currents flowing in the temperature sensor between or near the coils.

(a) DC case ( *f*=0): as no eddy currents will be excited, the temperature sensor will not affect *M*_{k}, and the problem reduces to the mere calculation of *M*_{k3} between two filamentary coils (*k* and 3). This can be done using (specialized) formulae derived from the Biot–Savart law or by applying the AC case (b) to . The conductive material nearby can be completely ignored at DC if *μ*_{r}=1.

(b) AC case ( *f*>0): the eddy currents involved in this problem complicate the calculation. An analytical solution is available for coaxial, filamentary circular loops and a parallel infinite slab. Starting from the solution for two single loops, one at (*r*,*z*)=(*a*,*d*) carrying the current *I* and the other one at (*r*,*z*)=(*R*,−*Z*) penetrated by the flux *Φ*, the basic solution is *M*=*Φ*/*I*=2*πR*⋅*A*_{IV}(*R*,−*Z*)/*I*, with
3.9and the first-order Bessel function *J*_{1} and *k*′=(*k*^{2}+i*ωμσ*)^{1/2}, *ω*=2*πf* [16].

Considering that each coil consists of several loops (turns) in series (*i*=1,2,…,*N*_{3} for the calibration coil and *j*=1,2,…,*N*_{k}, *k*=1,2 for the detection coils), we have the following expression:
3.10

The frequency dependence of *M*_{k}(*f*) is evaluated to determine the electrical conductivity *σ*=*σ*_{k} of the slab (see §4c) using the fact that *σ* enters *M*_{k}(*f*) only in the form *fμσ*. We perform a fit (preferably simultaneously to both real and imaginary parts) of the normalized function *M*_{k}(*αf*)/*M*_{k}(0) based on an arbitrary conductivity *σ*_{Ref} to the experimental data . The fit result is *σ*_{k}=*α*_{fit}*σ*_{Ref}.

The influence of the superconducting shield is estimated by comparing the results for *M*_{k} in the infinite space with those obtained in a superconducting sphere, , which has the same radius as the superconducting cylinder. Confining the field into a sphere instead of a cylinder provides an upper limit of the edge effect, which is sufficient for the uncertainty calculation. As the system of detection coils, calibration coil and superconducting sphere is again rotational symmetric, two-dimensional models are adequate to solve this problem.

For the calculation, we numerically solve *j*(*r*,*z*) of the discretized problem on a rectangular grid. Alternatively, any two-dimensional FEM software could be used instead.

### (c) Temperature calculation

The temperature is calculated according to
3.11where *N*_{f} denotes the number of frequency bins considered. The first part of (3.11) is related to the calibration of the signal channels (§4b) and only very weakly depending on temperature, whereas the second part provides the actual temperature information as obtained from the spectral density *S*_{V} (normalized by *S*_{Φ}) and averaged over suitable frequency bins. *T*_{Ref} is an arbitrary reference temperature for calculating the TMFN, which cancels out in (3.11), because *S*_{Φ}∝*T*_{Ref} (3.1). Taking the real part of the CPSD and derived expressions avoids the inclusion of (product) terms, which do not contribute to a correlation carried out in the time domain, and would not normally contribute to a correlation carried out in the frequency domain [18].

## 4. Experiment

### (a) Determination of the geometrical parameters of the thermometer

According to the dependencies *M*_{k}=*M*_{k}(*d*_{3k},*r*_{i,3},*r*_{j,k}), *i*,*j*=1,2,…,9, *k*=1,2, and *S*_{Φ,kl}=*S*_{Φ,kl}(*z*_{k},*z*_{l},*t*,*r*_{i,l},*r*_{j,k}), *kl*=12,11,22, *i*,*j*=1,2,…,9, the involved geometrical parameters as defined in §3, figure 4, must be known.

The coil radii *r*_{c,j,k} are well defined by the photo mask used for lithography, whereas the linewidth *d*_{w,k} (and accordingly the gap between the coplanar lines) depends on the specific wafer processing. The latter quantities are determined by scanning electron microscopy (SEM) imaging using a Raith eLINE system (Raith GmbH, Dortmund, Germany). We have *r*_{c,j,1}=(648.5,666.5,…,774.5,2015.75) μm, *r*_{c,j,2}=(642.5,660.5,…,768.5,1999.00) μm and *r*_{c,j,3}=*r*_{c,j,2} with *u*(*r*_{c,j,k})=0.115 μm. The SEM measurements yield , . The calculated corrections owing to the proximity effect in conductor pairs are equal to Δ*r*_{j,1}=−0.220 μm, Δ*r*_{j,2}=+0.199 μm for the eight inner turns (*j*=1,2,…,8) and Δ*r*_{9,1}=−0.073 μm, Δ*r*_{9,2}=+0.065 μm for the outer turns. In total, we find *u*(*r*_{j,k})≅*u*(*r*_{c,j,k}).

The distance between the coil chips (*d*_{31}=*d*_{32}) and the gaps between the coil chips and the temperature sensor (*d*_{1}=*d*_{2}, *d*_{3}) are measured on the assembled device. Because the chips are mounted face down, the relevant surfaces (carrying the coils) are not accessible to contact measurements. Therefore, we rely on an optical method based on low-coherence interferometry in the near infrared, where silicon is transparent. We use a device equivalent to a Precitec CHRocodile IT 18-3000 (Precitec Optronik GmbH, Neu-Isenburg, Germany) to measure the thickness of the air gap between relevant surfaces. The diameter of the measuring spot can be as small as 13 μm, so that scanning is performed over the whole chip area.

Averaging within a diameter of 5 mm coaxial to the coils we get , and , . The distance between coil chips is measured at the base edge and the top edge of the temperature sensor and averaged to give the value at the coil centre, , . This is combined with *z*_{1} and *d*_{3} to calculate the thickness at the coil centre: , .

The thermal expansivity, which leads to a contraction at low (working) temperatures, is taken into account for all geometrical dimensions *l*=*r*,*d* according to
4.1using the total fractional expansion *c*_{TE} for the temperature range from *T* to 293 K. Below 10 K, where *c*_{TE} varies less than 0.3%, we use the values *c*_{TE,Cu}=337×10^{−5} [19] and *c*_{TE,Si}=233.2×10^{−6} [20].

### (b) Calibration of the superconducting quantum interference device gradiometers

For data acquisition during temperature measurements and calibration, we use a National Instruments PXI-4461 DAQ card (24 bit, 204.8 kSa s^{−1} dynamic signal acquisition and generation) (National Instruments Corporation; see http://www.ni.com).

A DC (or AC) calibration current is impressed into the calibration coil via the calibration resistor *R*_{cal} and additional RC low-pass filtering not shown in figure 1. We have determined its value as *R*_{cal}=998.786 Ω with *u*(*R*_{cal})=0.020 Ω.

When calibrating the signal channel *k* (*k*=1,2), ADC *k* measures the ICL voltage *V* _{cal,k} at the output of the SQUID electronics, whereas the other ADC captures the voltage drop *V* _{Rcal,k} along *R*_{cal}. To eliminate the arbitrary offset voltage of *V* _{k}, we evaluate in (3.11) either voltage differences of a rectangular waveform or amplitudes of a sinusoidal waveform. A typical measurement takes 100 s and uses waveforms with periods of 10 s. Finally, the mutual inductances *M*_{k}(*d*_{3k},*r*_{i,3},*r*_{j,k}) are calculated according to §3b.

### (c) Electrical conductivity of the noise sensor

The measurements necessary to determine the electrical conductivity are very similar to those performed for calibration (§3b) except that sinusoidal waveforms are used, which cover a sufficient frequency range below and above the cut-off frequency. We measure the complex-valued function in the frequency range 0.1 Hz–500 Hz, which is well below the limit owing to the low-pass filtered coil-current detection via *R*_{cal} (flatness of 0.003 dB below 5 kHz).

Fitting these data (simultaneous fit of real and imaginary parts) to the normalized function *M*_{k}(*αf*)/*M*_{k}(0) based on *σ*_{Ref} according to (3.10), we obtain the fit result *σ*_{k}=*α*_{fit}*σ*_{Ref}.

The conductivities determined for both channels between 16 mK and 670 mK were stable around 5.3522×10^{9} (Ω×m)^{−1} within ±0.03%. The typical uncertainty *u*(*σ*_{k}(*T*))/*σ*_{k}(*T*), however, is about 0.1%. The reason for the relatively large uncertainty is a residual dependence of *σ* on the evaluated frequency.

Looking at the temperature calculation in (3.11), we may formally introduce the frequency-dependent temperature:
4.2where *c*_{cal,ij} contains the remaining (frequency-independent) coefficients according to (3.11). In theory, Re[*S*_{V,ij}( *f*,…)] and Re[*S*_{Φ,ij}( *f*,…)] share the same frequency dependence, which is cancelled out in (4.2) and makes *T*_{ij}( *f*) virtually independent of *f*. For our experimental data, however, it turns out that *T*_{ij}(*f*) shows a pronounced variation in the evaluated frequency range (1 Hz–5 kHz), which exceeds the statistical scatter of the data. The most likely reason for this behaviour is a significant spatial inhomogeneity of the conductivity *σ* in the particular temperature sensor used.

To overcome this problem, we have (i) reduced the volume of the temperature sensor effectively sensed by confining the evaluated frequency range and (ii) derived the corresponding mean conductivity from a fit, which minimizes the variance of . In detail, we perform the fit
4.3based on (arbitrary) *σ*_{Ref} and obtain the fit result *σ*_{fit}=*α*_{fit}⋅*σ*_{Ref}. The confined frequency range 420–3600 Hz used for evaluation corresponds to skin depths between 0.32 mm and 0.11 mm.

The fit procedure benefits from the fact that the evaluated frequencies are above the roll-off frequency of the TMFN spectra, which is about 230 Hz. By applying this method, we obtain an average conductivity of 5.8706×10^{9} (Ω×m)^{−1} with a relative uncertainty of 0.6% within the examined temperature range 16–670 mK. For the temperature calculation, we use the individual fit values *σ*_{fit}(*T*) obtained with *T*_{12}( *f*).

### (d) Noise and temperature spectra

An example of the spectra, from which the pMFFT temperatures are determined, is shown in figure 5 for a dataset taken at the lowest temperature measured, i.e. ≈16 mK. In this example, about 420×10^{6} samples were taken in 700 min and processed to a CPSD having 8192 frequency bins within 0.61–5000 Hz after 51 100 averages. The raw spectra contain some interference peaks, which occur mainly but not only at the mains frequency (50 Hz) and harmonics. These peaks can be easily detected and filtered out as the thermal noise spectrum is known to be smooth. This filtering reduces the number of bins only slightly (approx. 2%).

A larger reduction (approx. 38%) results from confining the frequency range to 420–3600 Hz that is used for further evaluation. As described in the previous section, this is necessary in order to obtain correct values for the conductivity *σ*, if *σ* is derived directly from the thermal noise spectrum using the fit procedure (4.3). The fit should result in a flat temperature spectrum *T*_{12}( *f*) within the confined frequency range, and figure 5 shows that this objective is achieved quite well. It is important to note that the shape of *T*_{12}( *f*) beyond the confined frequency range depends solely on the actual (fit-) value of the conductivity, but not on the measured temperature.

### (e) Key tests and verification of the theoretical model

The proper function of the cross-correlation technique was checked at constant temperature (4.2 K) by ‘zero measurements’ without thermal noise from the temperature sensor. In [13], we had used a completely non-metallic environment and a PCB chip holder with the copper layer removed. Now, we repeated this experiment with an MFFT body identical to that shown in figure 2 except for the missing temperature sensor, which was removed. In both experiments, we have compared *Re*[*S*_{Φ,12}(*f*)] with (*S*_{Φ,11}*S*_{Φ,22}/*N*_{avg})^{1/2} after averaging *N*_{avg} windowed periodograms to estimate the CPSD *S*_{12}(*f*). We found a good agreement between both quantities in the examined parameter range (1 Hz–100 kHz, *N*_{avg}≤8×10^{5}) and may conclude that both signal channels are uncorrelated to a level of ≤4×10^{−8}*Φ*_{0} Hz^{−1/2} when compared with the SQUID-referred single-channel white noise of 1.3×10^{−6}*Φ*_{0} Hz^{−1/2}, where *Φ*_{0} is the magnetic flux quantum. This proves the proper function of the cross-correlation technique. We note that the measured CPSD level does not constitute an ultimate limit for the flux noise but is rather an experimentally established upper limit. Moreover, the latter experiment shows that possible thermal noise contributions from the side walls of the MFFT body do not enter into the CPSD.

The model for the TMFN (3.1)–(3.4) assumes an infinite slab, whereas the size of the actual temperature sensor is limited. The corresponding edge effect is estimated using (3.8). As described in §3a, we evaluate the relative deviation of the TMFN by the ratio (*S*_{Φ,k}( *f*,5.5 mm)−*S*_{Φ,k}( *f*,25 mm))/*S*_{Φ,k}(*f*,25 *mm*). We obtain −3.8×10^{−5} at 0.1 Hz, −4.1×10^{−5} at 1 Hz, −9.9×10^{−5} at 10 Hz and −1.1×10^{−6} at 100 Hz. Above 30 Hz, the relative deviation is quickly declining, reaching −2.2×10^{−8} at 1 kHz. We point out that the edge effect in the real set-up involves a redistribution of the eddy currents in the temperature sensor and the whole MFFT body that is not taken into account by the simplified model. However, because the eddy current density is quickly declining with increasing distance from the coil centre, this difference is assumed to be on the same scale as the calculated values within the simplified model. A comparable eddy current redistribution is caused in addition by the superconducting shield around the MFFT body. We take the maximum of the values calculated above as upper bounds for the relative deviation of each effect. Thus, the absolute value of the relative deviation of both edge effects is always less than 0.01% of the TMFN. Considering the frequency range used for temperature evaluation (420–3600 Hz), the upper bound reduces to 1.1×10^{−6} of the TMFN.

The other important edge effect is the influence of the superconducting shield on the mutual inductances *M*_{1} and *M*_{2} between the detection coils and the calibration coil, which are calculated in §3b in the infinite space. An experimental test with a superconducting shield of doubled diameter confirmed that the relative deviation is less than 0.02%. We calculated the edge effect according to §3b in a sphere using a numerical code for the discretized problem. For the shielded problem within a superconducting sphere of radius *r*_{s}, we may introduce the mutual inductance . For our pMFFT set-up, we have to compare with . For the outer detection coil (1), we obtain and . The ratio of yields a relative deviation of −0.002% in temperature.

For the calculation of the TMFN, the detection coils are modelled including on-chip interconnections. If that had been neglected, an error of about 70 μm^{2} per turn (greater than or equal to 1.3 mm^{2}) would have been created. Although this particular feature does not introduce an error, it may set a scale for parasitic areas that are not covered by the model. At low frequencies, the relative deviation of the TMFN is equal to the relative deviation of the coil area. We may assume a parasitic area of 100 μm^{2} per turn, thus resulting in a maximum deviation of ±0.0077%.

### (f) Comparison with the PLTS-2000

First comparison measurements were carried out between the pMFFT and the PLTS-2000. The reference temperatures *T*_{2000} were provided by an MFFT, which was already used in [13] and calibrated according to the PLTS-2000 with a relative combined standard uncertainty of 4×10^{−4}. The relative deviations of the noise temperatures from the PLTS-2000 are shown in figure 6.

The noise spectra, from which the temperature values were derived, were recorded using quite different measuring times of 700 and 40 min, respectively. The high number of samples taken (420 MSa and 480 MSa) was chosen to minimize the influence of the statistical uncertainty components. For a 700 min measurement at 16 mK, the corresponding number of frequency bins used, *N*_{f}, was 5040 and the number of averages of the TMFN spectra, *N*_{avg}, was 51 100. Actually, the limiting factor for the uncertainty comes from the uncertainty of the geometrical parameters (table 1). The error bars shown in figure 6 represent the relative combined standard uncertainties (coverage factor *k*=1).

Between 672 mK and 16 mK, the noise temperatures agree well with the PLTS-2000 within the combined standard uncertainties, demonstrating agreement of the PLTS-2000 with thermodynamic temperature on a 1% level. Nevertheless, our data show a slight tendency to negative values at 15 mK. For comparison, figure 6 also displays the deviations of the background data, on which the PLTS-2000 is based [8], from the PLTS-2000. Our actual data are not sufficient to support or rule out a particular set of the background data in the region below 20 mK. Further measurements are needed at lower temperatures down to 1 mK with reduced uncertainties.

## 5. Uncertainty calculation

The uncertainty for temperature measurements with the pMFFT is estimated according to the *GUM* [21]. As an example, the uncertainty budget for a noise temperature *T*_{pMFFT} of about 16 mK is given in table 1.

Here, *u*(*x*_{i}) denotes the individual uncertainty of the component *x*_{i}, whereas *c*_{i} denotes the corresponding sensitivity coefficient to calculate the uncertainty contribution of *x*_{i} in units of temperature. The dominant contribution to the combined standard uncertainty results from the uncertainty of the distance between the coil chips, as is clearly visible from the data of the last column. We expect to reduce this uncertainty either in the existing set-up by characterizing the flatness of both surfaces of the temperature sensor or in a modified set-up by incorporating the calibration coil and the detection coils into the same layer on a single chip. In contrast to that, the contribution from the dimensions of the thin-film coils is nearly two orders of magnitude lower, owing to the high precision of the photolithography. The different approximations applied in the theoretical models (see §4e) for calculating the mutual inductances, the CPSD of the TMFN and the influence of the superconducting shielding do not significantly contribute to the combined standard uncertainty. Furthermore, the variance of the CPSD, which is increased by non-thermal noise and depends on the number of frequency bins used and the number of averages in the periodogram, can also be kept on a negligible level. Finally, the relative combined standard uncertainty for the noise temperatures is 0.59%. According to the linear characteristic of the pMFFT owing to the Nyquist relation and the practically temperature-independent uncertainty contributions, this relative uncertainty holds for the whole intended operation range from about 5 K down to 1 mK. This expectation is experimentally supported by the fact that measurements using a commercial MFFT (Magnicon GmbH, Hamburg, Germany; see http://www.magnicon.com) in relative primary mode showed agreement with the PLTS-2000 on a 1% level down to 1 mK [22] even though the non-thermal noise level, which increases the uncertainty contribution from the signal-to-noise ratio especially at the lowest temperatures, is in the commercial MFFT more than an order of magnitude higher than that of the pMFFT described here (§4e). At temperatures between 1 K and 5 K, the change of the superconducting properties influencing the operation of the pMFFT can be taken into account [17].

## 6. Conclusion

We have presented a complete description of the pMFFT, a new primary thermometer, which will enable measurements of thermodynamic temperature in the temperature region from 5 K down to 1 mK. The pMFFT is a calculable DC SQUID-based noise thermometer using a cross-correlation set-up. It combines a construction allowing the precise determination of the geometrical dimensions of the thermometer necessary for the temperature calculation with an *in situ* calibration of the two signal channels for the thermal magnetic flux noise. An uncertainty budget was established based on a thorough theoretical and experimental investigation of all parameters contributing to the uncertainty of the temperature measurement. In the current version, the pMFFT allows the measurement of thermodynamic temperature with a relatively combined standard uncertainty of 0.6% (coverage factor *k*=1). We expect to reduce this value further by diminishing considerably the contribution from the geometry parameters. First, preliminary measurements of thermodynamic temperature with the pMFFT down to 16 mK are in good agreement with the PLTS-2000. In future, the pMFFT will contribute to resolve the longstanding discrepancy of the PLTS-2000 background data and enable the direct dissemination of the kelvin in the low-temperature region.

## Authors' contributions

A.K. designed and assembled the thermometer, carried out the pMFFT measurements, analysed the data and drafted the manuscript. J.E. conceived the work, carried out the measurements and data analysis of the PLTS-2000 realization and drafted the manuscript. Both authors gave final approval for publication.

## Competing interests

The authors declare that they have no competing interests.

## Funding

The research reported here was carried out within the EURAMET project ‘InK—Implementing the new kelvin’ (JRP number: SIB01) (http://www.euramet.org/research-innovation/emrp/emrp-calls-and-projects/emrp-call-2011-health-si-broader-scope-new-technologies/, http://projects.npl.co.uk/ink/), and funding is kindly acknowledged from the European Community's Seventh Framework Programme, ERA-NET Plus (grant agreement no. 217257).

## Disclaimer

Identification of commercial equipment in this paper does not imply any recommendation or endorsement by the PTB.

## Acknowledgements

Technical support and assistance during the experiments by D. Heyer, M. Fleischer-Bartsch, M. Regin and C. Aßmann is kindly acknowledged. The authors are also indebted to the PTB's Berlin Institute workshop for fabricating the high-precision main copper body for the pMFFT.

## Footnotes

One contribution of 16 to a Theo Murphy meeting isssue ‘Towards implementing the new kelvin’.

- Accepted January 5, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.