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

  • Scientific Reports 7, Article number: 8011 (2017)
  • doi:10.1038/s41598-017-08710-7
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Abstract

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.

Introduction

Measuring noise provides an uncompromising test of microscopic and nanoscopic superconducting sensors1,2,3,4,5, such as superconducting quantum interference devices (SQUIDs), for ultra-sensitive detection of weak and local magnetic signals. A hybrid superconducting magnetometer6, 7 based on the proximity effect8 (superconducting quantum interference proximity transistor, SQUIPT9) has demonstrated in experiments high responsivity to magnetic flux9,10,11,12 and theoretically13 the noise is predicted to be very low, comparable to or below 50 nΦ0/Hz1/2 obtained with state-of-the-art nanoSQUIDs4, 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 amplifier14,15,16,17 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 part18, probed via the tunnel junction. The figure of merit of a SQUIPT magnetometer is the noise-equivalent flux (NEF) or flux sensitivity19, which has been considered theoretically in Ref. Giazotto2011. In the earliest experimental realization9, the NEF was limited by the preamplifier contribution to the noise, and estimated to be ~20μ Φ 0 / Hz 1/2 . In a subsequent optimized device with a shorter proximity junction, 500 nΦ0/Hz1/2 has been obtained at 240 mK in a low-frequency (sub-kHz) cross-correlation measurement, still limited by the room-temperature amplifier noise12. Recently, 260 nΦ0/Hz1/2 at 1 K was reported for a fully superconducting device20. 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 junction13.

Figure 1
Figure 1

Typical sample and the noise measurement setup. (a) False color scanning electron micrograph of the SQUIPT device, together with a zoomed-in view of the Cu island (green) embedded in the superconducting Al loop (brown). The Al tunnel probe (blue) contacts the middle of the proximity SNS junction. (b) Schematic view of the DC and noise measurement system in the dilution refrigerator. Fluctuations of current through the sample are converted to voltage noise at the HEMT amplifier input by a resonant circuit formed mainly by the inductors L and L l on the sample holder, and the distributed cable capacitance C coax that connects the sample to the cryogenic amplifier residing directly in the helium bath.

Besides hindering sensor operation, the shot noise21, 22 in the electrical current of a biased conductor provides information on quantum transport in mesoscopic structures beyond the average current23. 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 junctions27, 28 and recently for QPCs as well29. 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 MHz14,15,16. 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 amplifiers14, and its adaptations have been employed to study the noise of QPCs30, 31 and QDs32, 33, including demonstration of the quantum of thermal conductance for heat flow in a single electronic channel34. 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 μΦ0/Hz1/2 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 3He/4He 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 amplifier17 (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 VSD 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 Ll. This line is low-pass filtered by the resistance Rl = 330 Ω and capacitance Cl = 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 Cc. At frequencies of the order of the resonance at f 0 =1/(2π L C coax )4.2MHz, formed by the inductance L = ( L 1 + L l 1 ) 1 16.5μH (due to the coils L = Ll = 33 μH on the sample holder) and the capacitance Ccoax ≈ 92 pF (mainly due to distributed cable capacitance between the sample holder and the amplifier), the capacitors Cc = Cl 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 RLCcoax circuit, to be present in the observed voltage noise spectral density. In Fig. 1(b), the phenomenological resistor R50kΩ denotes the parasitic losses in the circuit, mainly the inductors L and Ll. It accounts for the losses in the circuit when the differential resistance of the sample R S (V,Φ)=dV/dIR. 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 215 samples14, and a desired number of spectra are averaged together to improve the signal-to-noise ratio.

DC transport measurements

Figure 2(a) displays the experimental current–voltage (IV) characteristics of the device recorded at T = 60 mK at two different magnetic fields, Φ = 0 and Φ = 0.5 Φ0, which correspond to maximum and minimum minigap opened in the normal metal DoS35, 36, respectively. At large biases | V | 0.5mV the resistance of the tunnel junction approaches the asymptotic normal-state value RT ≈ 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 Φ0, is observed due to the small Al loop inductance compared to that of the SNS weak link11, 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 tunneling37, 38, assuming the junction to be embedded in an effective RC environment.

Figure 2
Figure 2

DC transport measurements. (a) IV characteristics at T = 60 mK, measured at two values of magnetic flux Φ = 0 (red solid line) and Φ = 0.5 Φ0 (blue solid line), respectively. (b) Flux modulation of the IV curve (solid lines) around zero bias voltage at four values of magnetic flux between Φ = 0 and Φ = 0.5 Φ0, together with the theoretical model at each flux (dotted lines). (c) Current modulation I(Φ) at various fixed bias voltages VΔ/e, and (d), in the sub-gap region close to zero bias voltage. Here the bias voltages are indicated in microvolts. (e) Measured flux-to-voltage curves V(Φ) at several values of the bias current through the device. (f) Current responsivity ∂I/∂Φ and (g), voltage responsivity ∂V/∂Φ as functions of the magnetic flux at the optimum bias points, V = 0.249 mV and I = 4.2 nA, respectively.

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 | I / Φ | max 8nA/ Φ 0 and | V / Φ | max 0.2mV/ Φ 0 are reached at V ≈ 249 μV and I ≈ 4.2 nA, respectively. The transfer functions close to these optimum bias values are plotted in Fig. 2(f) and (g), whereas the corresponding I(Φ) and V(Φ) characteristics are shown in bold in Fig. 2(c) and (e). Note that the maxima at Φ = n Φ0 (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 I S =2e | I | 22. The blue dots in Fig. 3(a) show examples of measured spectral densities of voltage noise, referred to the HEMT amplifier input. They were recorded at the base temperature with fixed magnetic flux Φ ≈ 0 through the interferometer loop, at the few indicated values of bias voltage V across the device. The solid lines result from nonlinear least squares fitting to14

S V (f)= S V A + R eff 2 S I 1 + ( f 2 f 0 2 ) 2 / ( f Δ f ) 2 ,
(1)

showcasing how the white current noise SI is filtered by the bandpass response of the RLC circuit, centered around f0 (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, R eff = ( R S 1 + R 1 ) 1 is the effective resistance in the circuit, and S I = S I S + S I R + S I A denotes the total current noise, composed of the current fluctuations of the sample ( S I S ), equilibrium noise of the parasitic resistance R ( S I R =4 k B T/R), and a background term ( S I A ), attributed to the amplifier current noise (see also Figs 8 and 9 and related discussion in the Methods Section). The current noise of the sample S I S = S I shot + ( I / Φ ) 2 S Φ can be further separated into shot noise S I shot in the quasiparticle tunneling current, and external flux noise SΦ mediated by the transfer function ∂I/∂Φ. Given the responsivity of the present sample, in our setup with f0 in the MHz regime we expect the second term to be negligible. In the experiment, we have investigated the dependence of SV and hence SI on V, Φ, and T.

Figure 3
Figure 3

Noise measurements. (a) Power spectral density of the measured voltage noise at Φ = 0 for the indicated values of the bias voltage V (blue dots), plotted on a semilogarithmic scale. The solid lines are fits to Eq. 1. (b) Bias dependence of the peak height R eff 2 S I , extracted from fits to Eq. 1, for a few equally spaced values of magnetic flux between Φ = 0 (red solid line) and Φ = 0.5 Φ0 (blue solid line). (c) Total current noise S I vs. the DC bias, measured at T = 4.2 K with the junction in the normal state, and at T = 60 mK in the superconducting state, together with the theoretical predictions (see text for details). The dashed horizontal line indicates the background noise level, independent of V and Φ.

We make the fits to Eq. 1 using S V A , f0, the peak height P 0 = R eff 2 S I , and the peak width Δf=2π L f 0 2 / R eff as adjustable parameters. Here, Δf gives directly the full width at half maximum (FWHM) of the peak in SV in the limit f 0 Δf. Of the four parameters, the background level S V A 3× 10 18 V 2 /Hz due to the amplifier voltage noise, and the resonance frequency f0 ≈ 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 P0. The different curves correspond to a few equally spaced flux values between Φ = 0 and Φ = 0.5 Φ0, 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 P0 in Fig. 3(b) resembles that of RS(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 Reff is a parallel combination of RS and the constant parasitic resistance R.

Figure 3(c) shows the bias dependence of the total current noise SI extracted in the above manner from noise spectra similar to those in panel (a). The two curves correspond to measurements at bath temperature T = 4.2 K with the SQUIPT fully in the normal state (top), and at the base temperature T = 60 mK (bottom) at a constant magnetic flux close to Φ = 0. At T = 4.2 K, the measured noise is well explained by assuming S I S =(2eV/ R T )coth(eV/2 k B T), shown by the pink solid line, see, e.g., ref. 22 and references therein. We use the high bias shot noise, i.e., the linear asymptotic increase of SI with V, to calibrate the total gain of the setup, by requiring that the slope of SI vs. V equals 2e|I|. The value is in reasonable agreement with the nominal amplifier gains and expected losses in the circuit. At T = 60 mK we also find the noise to be dominated by the shot noise of the SQUIPT tunnel junction: Despite the nonlinear IV at VΔ/e, with increasing V the noise increases as S I S 2e | I ( Φ ) | . Here at T = 60 mK, for the theoretical model for simplicity we use the noise of an NIS tunnel junction, approximately valid for a SQUIPT at magnetic flux Φ = 0.5 Φ0 in which case the minigap in the normal metal vanishes35, 36 (cf. subsection “Quasiparticle current fluctuations in a hybrid tunnel junction” under Methods). As evident in Fig. 3(c), at base temperature the expected S I R 0.7× 10 29 A 2 /Hz is much smaller than the background term S I A 2.3× 10 27 A 2 /Hz (48 fA ) 2 /Hz. The origin of the large background current noise requires further study in future work: it is approximately an order of magnitude larger than the amplifier current noise S I A (13 fA ) 2 /Hz found in Ref. Arakawa2013. It is notable that both the peak width Δf and height P0 reflect strongly the bias- and flux-dependent changes in Reff and hence RS(V, Φ), cf. Figure 3(b). On the other hand, as illustrated by Fig. 3(c), the noise SI P0Δf2 calculated from these parameters follows SI |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 Φ0, 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 I S 2e | I | . In particular this is well satisfied at Φ = 0.5 Φ0. 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 Reff when the peak is at its narrowest (lowest height). It originates from the residual interfering peaks in the background noise of SV, present for example at f ≈ 4.02 MHz in Fig. 3(a).

Figure 4
Figure 4

Bias dependence of the current noise. IV characteristics of the SQUIPT (red and blue dots) compared to the measured current noise S I S (red and blue solid lines) vs. bias voltage, at the two extreme flux values, Φ = 0 and Φ = 0.5 Φ0.

Flux noise characterization

Two basic figures of merit of the SQUIPT are the transfer function ∂I/∂Φ and the noise-equivalent flux NEF= S I S 1/2 / | I / Φ | (flux sensitivity)19. To characterize the flux sensitivity of the device, we measure the flux dependence of I(Φ) and S I S (Φ) simultaneously in the same setup at several bias voltages V around the onset of the quasiparticle current. Examples of the resulting periodic modulations of I and S I S are shown in Fig. 5(a) and (b), respectively, demonstrating good qualitative agreement with S I S | I ( Φ ) | . Figure 5(c) further plots the flux-to-current transfer function ∂I/∂Φ, again obtained by numerical differentiation of I(Φ). With S I S and ∂I/∂Φ at hand, we obtain the NEF curves shown in Fig. 5(d) for two bias values: V = 0.24 mV resulting in the lowest NEF (green), and V = 0.29 mV giving the highest ∂I/∂Φ. The dotted lines use S I S obtained by direct fitting of the measured SV spectra as discussed above. On the other hand, considering the uncertainties in the fitting procedure, the solid lines assume full shot noise S I S =2e | I ( Φ ) | with I(Φ) taken from the DC measurement. A reasonable fit to the SV spectra is obtained also under this assumption, resulting only in slight changes in the fitted values of Reff.

Figure 5
Figure 5

Characterization of flux noise performance of the voltage-biased SQUIPT device. (a) DC current and (b), current noise as a function of magnetic flux at several values of bias voltage, measured simultaneously in the same setup at 60 mK. (c) Responsivity ∂I/∂Φ at several biases. (d) Flux sensitivity at the 2nd and 3rd lowest bias voltages V = 0.24 mV and 0.29 mV in panels (a –c). The dotted lines use S I S obtained by direct fitting of the measured S V spectra, whereas the solid lines assume full shot noise S I S =2e | I ( Φ ) | with I(Φ) from the DC measurement. Each color in panels (a)–(d) corresponds to a specific bias voltage.

We achieve the minimum value of NEF ≈ 4 μΦ0/Hz1/2 (green solid/dotted line) at the optimum working point V = 0.24 mV, Φ ≈ 0.4 Φ0. For comparison, low frequency (f ~ 100 Hz) flux noise NEF ≈ 0.5 μΦ0/Hz1/2 has been reported for a current-biased, optimized Al-SQUIPT in a room-temperature cross correlation setup12. Likewise, improved flux sensitivity figures down to ≈50 nΦ0/Hz1/2 have been recently reported for nanoSQUIDs1,2,3,4,5. 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 responsivity11,12,13. For example, fabricating the interferometer loop from a larger-gap superconductor39 and making a shorter normal metal wire11, 12, the transfer function can be enhanced by a few orders up to μA/Φ012 under voltage bias, and flux noise in the nΦ0/Hz1/2 range has been predicted9, 13. Similarly, for devices which reach the maximum responsivity in the supercurrent branch39, we expect minimum values of the flux noise in the range of 50 nΦ0/Hz1/2. Even higher-bandwidth readout of SQUIPT detectors, closely related to fast NIS tunnel junction thermometers40,41,42, 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 junctions43 or nanobridge weak links44, with flux noise down close to 20 nΦ0/Hz1/2 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 μΦ0/Hz1/2, 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 CF4 (for Ge) and O2 (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.

Cryogenic HEMT amplifier

The design of our cryogenic amplifier follows directly the one introduced in Ref. Arakawa2013. The device consists of passive components including surface mount metal-film resistors and laminated ceramic capacitors, and two Avago ATF-34143 high-electron-mobility transistors as the only active elements. The PCB board is placed in a brass shield box with the outer size 34 mm × 34 mm [see Fig. 6(a)]. In order to reduce 1/f noise of the amplifier, we prepared the double-HEMT amplifier using two transistors in parallel17.

Figure 6
Figure 6

Cryogenic HEMT amplifier. (a) Inside view of the cryoamplifier with a pair of ATF-34143 HEMT transistors. (b) Source-drain current I SD and (d) gain as a function of the supply voltage V SD at two different temperatures T = 4.2 K (blue solid line) and T = 300 K (red solid line) at 3 MHz. By decreasing the temperature, I SD decreases while the gain increases. (c) Frequency dependence of the gain at V SD = 2 V at two different temperatures.

The typical source-drain current ISD [Fig. 6(b)] and gain [Fig. 6(d)] as a function of the supply voltage VSD are plotted at two different temperatures T = 4.2 K (blue solid lines) and T = 300 K (red solid lines) at 3 MHz. As observed in Ref. Arakawa2013, at low temperatures ISD reduces while the gain increases. Furthermore, the gain varies only weakly in the saturation region V SD 1.2V. For the noise measurement setup, we prepared the LC resonance circuit with the resonant frequency to be close to 4 MHz. Figure 6(c) shows the frequency dependence of the gain at VSD = 2 V, remaining approximately constant (~3.4 at T = 4.2 K) at the frequencies of interest.

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 Cc and Cl behave as shorts at frequencies close to f0, whereas ZR denotes the impedance of the parallel RLC circuit, defined via

1/ Z R (ω)=1/R+1/(iω L )+iω C coax ,
(2)

with ω = 2πf. The cryogenic amplifier probes the voltage Vin, 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 ~ f0. δVA and δIA 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 δIR, with spectral density S I R . For the total sample noise we write δIS = δIshot + (∂I/∂Φ)δΦ, corresponding to S I S = S I shot + ( I / Φ ) 2 S Φ .

Figure 7
Figure 7

Circuit model for evaluating the voltage fluctuations referred to the amplifier input.

It is now straightforward to write down Kirchhoff’s laws for the circuit. Considering the voltage fluctuation ΔVin(ω) at the amplifier input, we have

Δ V in (ω)= Z eff (ω)[δ I S +δ I R +δ I A ]+δ V A .
(3)

Here Zeff(ω) is the parallel impedance of the sample and the RLC circuit

1/ Z eff (ω)=1/ R S +1/ Z R (ω).
(4)

Equation 3 now directly yields the spectral density of the total voltage fluctuations at the amplifier input as

S V (ω)= Z eff (ω) Z eff (ω)[ S I S + S I R + S I A ]+ S V A .
(5)

Noting the definition of the effective resistance R eff = ( R S 1 + R 1 ) 1 and Eq. 2, we see that Eq. 5 can be explicitly rewritten to obtain Eq. 1 in the main text.

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 I shot 2e | I | still holds down to relatively low sub-gap bias voltages V. In the SNS proximity junction, the density of states nN(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 nS(ε), biased by voltage V. Starting from a generic tunnel Hamiltonian, the current noise for a SQUIPT with tunnel resistance RT can be written as

S I shot ( ω , V , φ ) = 2 R T 1 d ε n N ( ε , φ ) n S ( ε e V ) { f ( ε e V )[1 f ( ε + ω ) ] + [1 f ( ε e V ) ] f ( ε ω )} .
(6)

Here we assume a narrow probe electrode and neglect the dependence of nN(ε, ϕ) on the position along the SNS junction10, 13. In Eq. 6, f(ε) = 1/[exp(ε/kBTe) + 1] denotes the Fermi-Dirac (quasi-) equilibrium distribution function where Te and kB are electron temperature and Boltzmann constant, respectively. We further assume the low frequency limit ω k B T e ,eV,Δ, yielding

S I shot (V,φ)=2 R T 1 dε n N (ε,φ) n S (εeV){f(εeV)[1f(ε)]+[1f(εeV)]f(ε)}.
(7)

For simplicity, let us consider the SQUIPT device at magnetic flux Φ = 0.5 Φ0, in which case we approximate nN = 1 and obtain

S I shot (V)=2 R T 1 dε n S (ε){f(εeV)[1f(ε)]+[1f(εeV)]f(ε)}.
(8)

Figure 8 displays the IV characteristics of such a NIS tunnel junction together with the current noise from Eq. 8 calculated at Te = 0.05 K, assuming the superconducting Al gap Δ = 200 μeV. For an NIN junction we can further set nN = nS = 1, resulting in

S I shot (V)=2 R T 1 dε{f(εeV)[1f(ε)]+[1f(εeV)]f(ε)}.
(9)
Figure 8
Figure 8

Current noise of a NIS junction. (a) Normalized IV characteristics of a NIS tunnel junction (blue dashed line) together with the current noise S I shot (red solid lines) vs. bias voltage calculated at T = 0.05 K, with the superconducting Al gap Δ = 200 μeV. Here, the dimensionless parameter γ is the ratio between NIS junction asymptotic resistance at high bias voltage and the sub-gap resistance, used in the modeling of a smeared BCS density of states n S (E)= | Re [ ( E / Δ + i γ ) / ( E / Δ + i γ ) 2 1 ] | . (b) I and S I shot as in (a) but plotted on a semilogarithmic scale.

This can be directly integrated to yield S I shot =(2eV/ R T )coth(eV/2 k B T e ). Two basic cases are then immediately obtained from this expression, namely (i) for e | V | k B T e the equilibrium thermal noise 4kBTe/RT, and (ii) full shot noise 2e|I| in the limit e | V | k B T e .

Temperature dependence of the total current noise

Besides the measurements at T = 4.2 K and T = 60 mK discussed in the main text, we have probed the total current noise S I = S I S + S I R + S I A at Φ ≈ 0.5 Φ0 in a range of bath temperatures below 500 mK. Figure 9(a) plots the bias dependence of SI for various bath temperatures T between 69 mK and 430 mK. At each bias we observe a slight increase in SI 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 I R =4 k B T/R. This is supported by Fig. 9(b) where we plot the values of SI at V = 0. At zero bias, the sample noise S I S =4 k B T/ R S is negligible compared to S I R due to R S R at sub-gap voltages. The measured temperature dependence of the zero-bias noise agrees with S I = S I R + S I A , i.e., a linear increase with slope 4kB/R, on top of a background set by S I A .

Figure 9
Figure 9

Temperature dependence of the total current noise. (a), Bias dependence of the total noise S I at various values of the bath temperature at Φ ≈ 0.5 Φ0. (b), Temperature dependence of S I at V = 0 and Φ ≈ 0.5 Φ0. As expected for thermal noise of R eff (≈R at V = 0), the signal increases approximately linearly with slope 4k B/R (solid blue line) with increasing T on top of a constant background S I A 2.3× 10 27 A 2 /Hz.

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Acknowledgements

We acknowledge Micronova Nanofabrication Centre of Aalto University for providing the processing facilities. We thank O.-P. Saira and M. Meschke for helpful discussions. The work has been supported by the Academy of Finland Center of Excellence program (project number 284594). J. T. P. acknowledges support from Academy of Finland (Contract No. 275167).

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  1. Low Temperature Laboratory, Department of Applied Physics, Aalto University School of Science, P.O. Box 13500, FI-00076, Aalto, Finland

    • R. N. Jabdaraghi
    • , D. S. Golubev
    • , J. P. Pekola
    •  & J. T. Peltonen

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Contributions

R.N.J. fabricated the device. R.N.J. and J.T.P. performed the experiments, analyzed the data, and wrote the manuscript. D.S.G. and J.P.P. provided theory support. J.P.P. discussed at all stages of the measurement with R.N.J. and J.T.P. All authors discussed the results and their implications, and contributed to editing the manuscript.

Competing Interests

The authors declare that they have no competing interests.

Corresponding author

Correspondence to R. N. Jabdaraghi.

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