Noise of a superconducting magnetic flux sensor based on a proximity Josephson junction

We demonstrate simultaneous measurements of DC transport properties and flux noise of a hybrid superconducting magnetometer based on the proximity effect (superconducting quantum interference proximity transistor, SQUIPT). The noise is probed by a cryogenic amplifier operating in the frequency range of a few MHz. In our non-optimized device, we achieve minimum flux noise ~4 μΦ0/Hz1/2, set by the shot noise of the probe tunnel junction. The flux noise performance can be improved by further optimization of the SQUIPT parameters, primarily minimization of the proximity junction length and cross section. Furthermore, the experiment demonstrates that the setup can be used to investigate shot noise in other nonlinear devices with high impedance. This technique opens the opportunity to measure sensitive magnetometers including SQUIPT devices with very low dissipation.

Measuring noise provides an uncompromising test of microscopic and nanoscopic superconducting sensors [1][2][3][4][5] , such as superconducting quantum interference devices (SQUIDs), for ultra-sensitive detection of weak and local magnetic signals. A hybrid superconducting magnetometer 6, 7 based on the proximity effect 8 (superconducting quantum interference proximity transistor, SQUIPT 9 ) has demonstrated in experiments high responsivity to magnetic flux [9][10][11][12] and theoretically 13 the noise is predicted to be very low, comparable to or below 50 nΦ 0 /Hz 1/2 obtained with state-of-the-art nanoSQUIDs 4,5 . Yet the intrinsic limits to flux noise performance of such a device have not been experimentally investigated in detail up to now. Here, we present a measurement of flux noise of a SQUIPT using a cryogenic amplifier [14][15][16][17] operating in the frequency range of a few MHz.
A SQUIPT interferometer consists of a superconducting loop interrupted by a short normal-metal wire in direct metal-to-metal contact while an additional superconducting probe electrode is tunnel-coupled to the normal region, cf. Fig. 1(a). Its operation relies on the phase dependence of the density of states (DoS) in the normal part 18 , probed via the tunnel junction. The figure of merit of a SQUIPT magnetometer is the noise-equivalent flux (NEF) or flux sensitivity 19 , which has been considered theoretically in Ref. Giazotto2011. In the earliest experimental realization 9 , the NEF was limited by the preamplifier contribution to the noise, and estimated to be µΦ20 /Hz 0 1/2 . In a subsequent optimized device with a shorter proximity junction, 500 nΦ 0 /Hz 1/2 has been obtained at 240 mK in a low-frequency (sub-kHz) cross-correlation measurement, still limited by the room-temperature amplifier noise 12 . Recently, 260 nΦ 0 /Hz 1/2 at 1 K was reported for a fully superconducting device 20 . However, the challenging task has remained to observe directly the non-bandwidth-limited intrinsic flux noise performance of the hybrid superconducting magnetometer devices, predicted to be determined by shot noise in the current through the probe tunnel junction 13 .
Besides hindering sensor operation, the shot noise 21,22 in the electrical current of a biased conductor provides information on quantum transport in mesoscopic structures beyond the average current 23 . It has been measured in various systems, including quantum point contacts (QPCs) 24,25 and quantum dots (QDs) 26 , and found to provide an accurate means of primary thermometry for metallic tunnel junctions 27,28 and recently for QPCs as well 29 . A successful technique for measuring the shot noise of high-impedance semiconducting samples relies on a cryogenic amplifier based on a high electron mobility transistor (HEMT) and an RLC tank circuit with resonance frequency of a few MHz [14][15][16] . Such an approach avoids the ubiquitous amplifier 1/f noise and signal loss due to low pass filter formed by cable capacitance and the high sample inductance. For enhanced sensitivity, the method extends straightforwardly to cross-correlation of signals from two amplifiers 14  conductance for heat flow in a single electronic channel 34 . In this work we use the technique to characterize the flux noise of a hybrid superconducting tunnel junction magnetometer. Here we present simultaneous measurements of the DC transport properties and current noise of a SQUIPT interferometer, and use them to infer a NEF ≈ 4 μΦ 0 /Hz 1/2 in the non-optimized structure with a proximity SNS junction of length l ≈ 245 nm. We show that the low-temperature readout of RLC filtered shot noise can be applied to the study of nonlinear devices once changes in the differential resistance are taken into account, cf. gate-tunable semiconducting devices where the resistance depends only weakly on the bias voltage.

Results
Noise measurement setup. A typical SQUIPT based on a superconducting aluminium loop placed into a perpendicular magnetic field is presented in Fig. 1(a), with an enlarged view of the weak link region depicted in the top inset. It is fabricated using conventional methods of electron beam lithography and metal deposition through a suspended mask (see further fabrication details in the Methods Section). The noise measurement setup is installed in a 3 He/ 4 He dilution refrigerator with base temperature close to 60 mK as shown in Fig. 1(b). As the main elements, our home made double-HEMT cryogenic amplifier 17 (see Fig. 6 in the Methods Section for amplifier characterization at room temperature and 4.2 K) and the inductors of the LC resonant circuit are placed in the liquid helium bath and on the sample holder at base temperature, respectively. The voltage source V SD is used to bias the amplifier. A bias voltage V is applied to the SQUIPT tunnel probe electrode, and the average current I is measured with a room-temperature current amplifier through the line with inductor L l . This line is low-pass filtered by the resistance R l = 330 Ω and capacitance C l = 22 nF. An identical filter is included in the biasing line of the tunnel probe but omitted in Fig. 1(b) for clarity.
Simultaneously with measurement of the average current I, current noise through the SQUIPT is probed by the HEMT amplifier via the capacitor C c . At frequencies of the order of the resonance at π (due to the coils L = L l = 33 μH on the sample holder) and the capacitance C coax ≈ 92 pF (mainly due to distributed cable capacitance between the sample holder and the amplifier), the capacitors C c = C l can be considered as electrical shorts. Importantly, this results in a robust peak signature of the white shot noise of the sample, filtered by the characteristic band-pass response of the RL′C coax circuit, to be present in the observed voltage noise spectral density. In Fig. 1(b), the phenomenological resistor  Ω R k 50 denotes the parasitic losses in the circuit, mainly the inductors L and L l . It accounts for the losses in the circuit when the differential resistance of the sample . The signal is further amplified by another stage (SRS SR445A) at room temperature, and low pass (LP) filtered by a commercial 5 MHz filter to avoid aliasing. The amplified voltage signal is finally captured by a 16-bit digitizer running continuously at 50 MSamples/s, converted into spectral density of voltage noise by windowing and Fast Fourier Transform of blocks with typically 2 15 samples 14 , and a desired number of spectra are averaged together to improve the signal-to-noise ratio. DC transport measurements. Figure 2  maximum and minimum minigap opened in the normal metal DoS 35,36 , respectively. At large biases  . V 0 5mV the resistance of the tunnel junction approaches the asymptotic normal-state value R T ≈ 60 kΩ. Figure 2(b) further shows an enlarged view of the flux dependence of the sub-gap current. Full phase modulation, i.e., complete suppression of the supercurrent at Φ = 0.5 Φ 0 , is observed due to the small Al loop inductance compared to that of the SNS weak link 11,12 . The shape of the supercurrent peaks shows good agreement with a theoretical calculation (dotted lines) based on the P(E) theory of incoherent Cooper pair tunneling 37,38 , assuming the junction to be embedded in an effective RC environment.
We next characterize the flux responsivity of the SQUIPT device by measuring current I(Φ) and voltage V(Φ) modulations at different values of bias voltage or current applied to the tunnel probe. Figure 2(c) and (e) illustrate some of such current and voltage modulations in the bias range from 0.246 mV to 0.369 mV and 0.14 nA to 7 nA, respectively. Furthermore, Fig. 2(d) shows the measured current modulation at several sub-gap bias voltages. With the I(Φ) and V(Φ) characteristics at hand, we obtain the flux-to-voltage transfer function ∂V/∂Φ and flux-to-current transfer function ∂I/∂Φ by numerical differentiation. The maximum absolute values  Fig. 2(c) and (e). Note that the maxima at Φ = n Φ 0 (with integer n) in the V(Φ) characteristics in panel (e) cross over to minima at higher bias currents, with analogous behavior evident in the current modulations in panel (b). This reflects the influence of the phase-dependent density of states on the IV characteristics at bias voltages slightly above the sum of the probe lead Al gap Δ and the proximity-induced minigap in the Cu wire.

Shot noise measurements.
We now turn to a description of the SQUIPT noise measurements. For a tunnel junction-based device such as the SQUIPT, we expect the spectral density of the current shot noise to follow = S e I 2 I S

22
. The blue dots in Fig. 3(a) show examples of measured spectral densities of voltage noise, referred to showcasing how the white current noise S I is filtered by the bandpass response of the RLC circuit, centered around f 0 (refer to Fig. 7 and subsequent discussion in the Methods Section for details). Above, S V A is the input voltage noise of the amplifier, can be further separated into shot noise S I shot in the quasiparticle tunneling current, and external flux noise S Φ mediated by the transfer function ∂I/∂Φ. Given the responsivity of the present sample, in our setup with f 0 in the MHz regime we expect the second term to be negligible. In the experiment, we have investigated the dependence of S V and hence S I on V, Φ, and T.
We make the fits to Eq. 1 using S V A , f 0 , the peak height = P R S 0 e ff 2 I , and the peak width π ∆ = ′ f Lf R 2 / 0 2 eff as adjustable parameters. Here, Δf gives directly the full width at half maximum (FWHM) of the peak in S V in the limit Of the four parameters, the background level ≈ × − S 3 10 V /Hz V 18 2 A due to the amplifier voltage noise, and the resonance frequency f 0 ≈ 4.18 MHz can be kept fixed, whereas the peak height and width depend systematically on V, Φ, and T. With the fitting procedure established, Fig. 3(b) demonstrates typical bias dependence of the extracted values of the peak height P 0 . The different curves correspond to a few equally spaced flux values between Φ = 0 and Φ = 0.5 Φ 0 , whereas the vertical arrows indicate the bias voltages at Φ = 0 for the spectra displayed in panel (a). It is noteworthy that the bias dependence of P 0 in Fig. 3(b) resembles that of R S (V, Φ), i.e., the differential resistance of the sample. It arises due to the factor R eff 2 in the definition of the peak height, and the fact that R eff is a parallel combination of R S and the constant parasitic resistance R.   Arakawa2013. It is notable that both the peak width Δf and height P 0 reflect strongly the bias-and flux-dependent changes in R eff and hence R S (V, Φ), cf. Figure 3(b). On the other hand, as illustrated by Fig. 3(c), the noise S I ∝ P 0 Δf 2 calculated from these parameters follows S I ∝ |I(Φ)|, highlighting the contribution of the shot noise of the tunnel junction.
In Fig. 4 we show the bias dependence of the noise in more detail at the two extreme flux values Φ = 0 and Φ = 0.5 Φ 0 , noting that the shot noise directly reflects changes in the average current I(V, Φ). For an explicit comparison with the average current, we plot the corresponding IV characteristics in the same panel, showing that indeed ≈ S e I 2 I S . In particular this is well satisfied at Φ = 0.5 Φ 0 . In the S I S curve at Φ = 0, we attribute the apparent excess noise around zero bias (at the gap edge) to an uncertainty in the fitting to extract the exact value of R eff when the peak is at its narrowest (lowest height). It originates from the residual interfering peaks in the background noise of S V , present for example at f ≈ 4.02 MHz in Fig. 3(a).  A reasonable fit to the S V spectra is obtained also under this assumption, resulting only in slight changes in the fitted values of R eff .
We achieve the minimum value of NEF ≈ 4 μΦ 0 /Hz 1/2 (green solid/dotted line) at the optimum working point V = 0.24 mV, Φ ≈ 0.4 Φ 0 . For comparison, low frequency (f ~ 100 Hz) flux noise NEF ≈ 0.5 μΦ 0 /Hz 1/2 has been reported for a current-biased, optimized Al-SQUIPT in a room-temperature cross correlation setup 12 . Likewise, improved flux sensitivity figures down to ≈50 nΦ 0 /Hz 1/2 have been recently reported for nanoSQUIDs [1][2][3][4][5] . Significant improvements to our initial demonstration of the MHz-range SQUIPT noise performance are  expected to result from optimizing the geometry of the device and the consequently enhanced responsivity [11][12][13] . For example, fabricating the interferometer loop from a larger-gap superconductor 39 and making a shorter normal metal wire 11,12 , the transfer function can be enhanced by a few orders up to μA/Φ 0 12 under voltage bias, and flux noise in the nΦ 0 /Hz 1/2 range has been predicted 9, 13 . Similarly, for devices which reach the maximum responsivity in the supercurrent branch 39 , we expect minimum values of the flux noise in the range of 50 nΦ 0 / Hz 1/2 . Even higher-bandwidth readout of SQUIPT detectors, closely related to fast NIS tunnel junction thermometers [40][41][42] , is possible by embedding the device in a lumped element or coplanar waveguide resonator with resonance frequency in the range of several hundred MHz or several GHz, respectively. This is similar to work on quantum-limited dispersive SQUID magnetometry with conventional Al tunnel junctions 43 or nanobridge weak links 44 , with flux noise down close to 20 nΦ 0 /Hz 1/2 and bandwidth of the order of 10 MHz.

Discussion
In summary, we have investigated the flux noise performance of a SQUIPT interferometer based on shot noise measurements with a cryogenic amplifier at frequencies of the order of a few MHz. This represents the first noise study of such a hybrid interferometer not limited by the low-bandwidth room-temperature readout. The setup is capable of resolving the shot noise of a current I ~ 100 pA in a typical probe junction in an averaging time of the order of 30 s. In future work, the performance can be further improved by employing a lower-noise room-temperature amplifier, and by using the cross-correlation of signals from two low-temperature amplifiers to reject the uncorrelated background S V A while reliably picking out the signal due to S I S . In the present device we reach shot-noise-limited flux sensitivity of the order of μΦ 0 /Hz 1/2 , which can be significantly improved upon optimizing the dimensions of the SNS weak link and the readout tunnel probe.

Methods
Fabrication details. The sample is fabricated using electron beam lithography (EBL) and electron beam evaporation of the Al and Cu thin films. A single lithography step relying on a Ge based hard mask is used to define patterns for multi-angle shadow evaporation of the NIS tunnel probe and the proximity SNS weak link in a single vacuum cycle. The starting point is an oxidized Si substrate onto which we first spin coat a 900 nm thick layer of P(MMA-MAA) copolymer. Subsequently, a 22 nm thick film of Ge is deposited by electron beam evaporation, followed by spin coating a 50 nm thick polymethyl methacrylate (PMMA) layer. The EBL step is followed by first developing the chip in 1:3 solution of methyl isobutyl ketone (MIBK) and isopropanol (IPA) for 30 s, rinsing in IPA and drying. Reactive ion etching (RIE) with CF 4 (for Ge) and O 2 (for the copolymer layer) is then used to create a suspended mask with proper undercut profile for shadow evaporation. The metals are deposited by electron-beam evaporation: first, 25 nm of Al is deposited and oxidized in-situ for 1 min with pure oxygen pressure of 1 millibar to form the tunnel barrier of the normal metal-insulator-superconductor (NIS) probe.
Next, approximately 15 nm of copper is evaporated to complete the NIS junction and to form the normal metal part of the SNS proximity weak link. Immediately after this, the superconducting Al loop with 120 nm thickness is deposited to form clean contacts to the copper island, which completes the structure. Figure 1(a) shows an SEM image of a resulting SQUIPT device, illustrating the thick Al loop interrupted by the short Cu wire, as well as the thin Al tunnel probe electrode in the middle. Model for evaluating the spectrum of voltage noise. Figure 7 shows a simplified circuit model of the setup in Fig. 1(b), including relevant noise sources for calculating the total voltage noise probed by the HEMT amplifier at its input. Here, it is assumed that the capacitors C c and C l behave as shorts at frequencies close to f 0 , whereas Z R denotes the impedance of the parallel RLC circuit, defined via

Cryogenic
R c oax with ω = 2πf. The cryogenic amplifier probes the voltage V in , applied to the gate of its HEMT transistor. The amplifier input voltage and current noise spectral densities are denoted by S V A and S I A , respectively. In the following they will be assumed to be white at the frequencies of interest f ~ f 0 . δV A and δI A represent the corresponding voltage and current noise sources. In Fig. 7, the input impedance of the amplifier is assumed to be high. The equilibrium current fluctuations in the RLC circuit (i.e., the resistance R) are denoted by δI R , with spectral density S I R .
For the total sample noise we write δI S = δI shot + (∂I/∂Φ)δΦ, corresponding to = + ∂ ∂Φ Φ S S I S ( / ) It is now straightforward to write down Kirchhoff 's laws for the circuit. Considering the voltage fluctuation ΔV in (ω) at the amplifier input, we have Here Z eff (ω) is the parallel impedance of the sample and the RLC circuit Quasiparticle current fluctuations in a hybrid tunnel junction. Here we show that despite the non-constant densities of states in both electrodes of the SQUIPT tunnel junction and the nonlinear IV characteristic, the simple approximation ≈ S e I 2 I shot still holds down to relatively low sub-gap bias voltages V. In the SNS proximity junction, the density of states n N (ε, φ) in the proximized normal metal depends on the phase difference φ between the S electrodes. This phase-and hence flux-dependent DoS is probed by a tunnel junction with a pure superconducting counterelectrode with the BCS DoS n S (ε), biased by voltage V. Starting from a generic tunnel Hamiltonian, the current noise for a SQUIPT with tunnel resistance R T can be written as Here we assume a narrow probe electrode and neglect the dependence of n N (ε, φ) on the position along the SNS junction 10,13 . In Eq. 6, f(ε) = 1/[exp(ε/k B T e ) + 1] denotes the Fermi-Dirac (quasi-) equilibrium distribution function where T e and k B are electron temperature and Boltzmann constant, respectively. We further assume the low frequency limit ω ∆  k T eV , , A at Φ ≈ 0.5 Φ 0 in a range of bath temperatures below 500 mK. Figure 9(a) plots the bias dependence of S I for various bath temperatures T between 69 mK and 430 mK. At each bias we observe a slight increase in S I towards higher T, whereas the V-dependence is always dominated by the shot noise of the SQUIPT probe junction. The background term S I A is expected to be independent of the sample holder temperature, and S I S has only weak temperature dependence at bias voltages around the gap edge. Hence, most of the T-dependence in Fig. 9(a) should be determined by = S k T R 4 / I B R . This is supported by Fig. 9