High operating temperature in V-based superconducting quantum interference proximity transistors

Here we report the fabrication and characterization of fully superconducting quantum interference proximity transistors (SQUIPTs) based on the implementation of vanadium (V) in the superconducting loop. At low temperature, the devices show high flux-to-voltage (up to 0.52 mV/Φ0) and flux-to-current (above 12 nA/Φ0) transfer functions, with the best estimated flux sensitivity ~ 2.6 μΦ0/(Hz)1/2 reached under fixed voltage bias, where Φ0 is the flux quantum. The interferometers operate up to T bath  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\simeq $$\end{document}≃ 2 K, with an improvement of 70% of the maximal operating temperature with respect to early SQUIPTs design. The main features of the V-based SQUIPT are described within a simplified theoretical model. Our results open the way to the realization of SQUIPTs that take advantage of the use of higher-gap superconductors for ultra-sensitive nanoscale applications that operate at temperatures well above 1 K.

Within the Mattis-Bardeen theory 1 , the kinetic inductance L kin of a superconducting strip with length l, width w and thickness t is given by L kin =hR/π∆, where ∆ is the superconducting order parameter and R = ρl/wt is the normal state resistance of the superconducting strip (here ρ is the normal state resistivity). This expression can be used to estimate the kinetic inductance of the superconducting loop of the SQUIPT when the ring consist of a single superconductor.
For a comparison, we consider a sinusoidal current-phase relation for the superconducting weak-link, which in the short junction limit is valid when the temperature is not too small compared to the critical temperature 2 . Under this assumption, the minimal kinetic inductance at zero phase bias φ = 0 reads L NW kin ≈hR NW /π∆, where R = ρ NW l NW /w NW t NW is the normal state resistance of the weak link.
The ratio between the kinetic inductance of the wire and the ring is where the superscripts NW, R refer to the nanowire and ring, respectively. The dimensions of the Al nanowire (device A) are l NW = 150 nm, t NW = 20 nm and w NW = 60 nm. We assume ρ NW 5 µΩ cm, which is the typical resistivity for 25 nm Al layer at 4.2 K evaporated in past experiments, consistently with the values reported in the literature 3-6 . If we consider a ring made of Al with dimensions l NW = 6 µm, t NW = 50 nm and w R = 1 µm and same resistivity (although in general the resistivity drops by increasing the thickness of the layer) we obtain L NW kin /L R kin ∼ 1.05. The resistivity of the vanadium may vary quite strongly depending on evaporation conditions. Considering Figure S1. V-Al bilayer scheme. The strip has total thickness t tot = t V + t Al , where t V and t Al are the thicknesses of vanadium and aluminium, respectively. literature values 7-11 , we estimate the V layer resistivity ρ V to range approximately from the same resistivity of the Al 5 µΩ cm to a value 5 times larger 25 µΩ cm. As a consequence, this would produce a potentially large deviation from the ideal condition L NW kin /L R kin 1. When the superconducting ring is made of a bilayer, the situation is more involved (as we detail in the next subsection): In first approximation it is possible to model the total kinetic inductance of the bilayer as the parallel of the kinetic inductance of the two layers. This simple calculation shows how the inclusion of the Al underlayer provides a suitable geometry for the good phase biasing of the device, independently of the specific properties of the vanadium layer.

S2 Bilayer modeling
The spectral properties of the V-Al bilayer in the dirty limit can be modeled within the Usadel formalism 12 . The problem formulation is similar to the one given by Fominov and Feigel'man for the properties of a thin NS bilayer 13 . In the numerical computation we model the bilayer as a superconducting strip with total thickness t = 100 nm and we assume a ratio 1:1 (t Al = t V = 50 nm) between the two layers, accordingly to the experimental realization ( Fig. 1). We assume a clean interface between the two layers. S1/S5  A parameter relevant for the properties for the bilayer is where ρ X and D X are the normal state resistances and the diffusion constants of the materials X =Al,V (Einstein relation D −1 X = ρ X e 2 ν X is assumed and ν X is the density of states at the Fermi level). The coupling constants in the two superconducting layers λ X = − ln(∆ X /2E X c ) depend in the weak coupling limit on the cutoff energy E X The density of states at the Fermi level ν X = N X (E F )d X /M X are taken from the literature. Here N X is the density of states at the Fermi level for atom (N Al (E F ) = 0.208 eV −1 , N V (E F ) = 1.31 eV −1 ) 14 , d X is the mass density (d Al = 2.7 g/cm 3 , d V = 6.0 g/cm 3 ) 15 and M X is the atomic mass (M Al = 26.98 u , M V = 50.94 u) 16 . The Debye temperature is assumed to be the same for both materials θ Al D = θ V D = 400 K. For the Al layer we choose a critical temperature equal to the bulk value T Al C = 1.2 K, corresponding to a zero temperature order parameter ∆ Al = 178 µeV, and typical resistivity ρ Al = 5 µΩ cm and Dynes parameter Γ Al /∆ Al = 10 −4 obtained through electron beam evaporation.
As already stated before, the properties of the vanadium deposited through electron beam evaporation are extremely sensitive to the evaporation conditions. In accordance with the discussion in the previous section, we consider ρ V = 5 µΩ cm and ρ V = 25 µΩ cm as minimal and maximal resistivity in the numerical computation. Similarly apply to the critical temperature of the vanadium, which can be significantly smaller than the bulk value 17 , depending on the evaporation rate. In our numerical computation we set T V c = 3.5 K, which is reasonable due to the low evaporation rate and previous realizations 18 . Finally we consider two cases for the Dynes parameter of vanadium: a very ideal situation Γ V /∆ V = 10 −4 and an extremely leaking layer Γ V /∆ V = 0.6. The latter seems to describe better the results of our experiment as we show in Fig.2, where the DOS at the bottom of the Al layer is compared to the effective BCS DOS used in the main text. In particular the resistivity of the Vanadium plays a role in the determination of the energy gap of the bilayer, but does not affect significantly the subgap density of states. In particular, the large subgap conductance observe in the experiment must be associated to an high effective Dynes parameter in the V layer even in this model. Notably, the results compare quite well with the effective BCS model used in the main text.
In this model, the kinetic inductance of the bilayer is evaluated as L R kin =h/2eI S (φ ), where the supercurrent dispersion I S (φ ) is computed starting from the solution of the Usadel equation. An approximate expression for ultrathin layers 13 can be obtained in the Cooper limit 19 , where the superconducting energy gap is homogeneous along the bilayerṪhe kinetic inductance of the ring is therefore given by the parallel of the kinetic inductances of the two layers: In Tab. S1, we see how the approximate expressions for the kinetic inductance compare to the values obtained through the rigorous calculation. Generally, the approximation underestimates the kinetic inductance somewhat, and does not take nonzero Dynes parameters into account. for the numerical computation parameters chosen above.

S3 Impact of the finite width of the probe
In the main text is stated that, in order to simplify the calculation, we disregard the finite width of the probe. First, we show that the high subgap conductance observed in the differential conductance curves is not originated by the finite extension of the probe. In Fig. S3, panel a) we compare the effective DOS used in the main text with the DOS obtained after averaging over the probe width < N >= 1 w x 0 +w/2 where we set an ideal Dynes parameter Γ R /∆ R = 10 −3 . The subgap conductance in the latter case is too small to explain the experimental results. Then we quantify the relative deviation between the simplified expression N(E, Φ, x 0 ) and the integrated expression < N > through the figure of merit In Fig.S3 panel b) we plot this function for different values of Φ = 0 (there is no deviation at Φ = 0). We see that the maximum relative deviation is always smaller or equal than 1%.

S4 Theoretical flux dependence
For completeness, in this section we discuss the theoretical flux dependence obtained from the theoretical model adopted throughout the main text. The plots corresponding to the panel of the Fig. 3 of the main text are displayed in Fig. S4 We note S3/S5 that the comparison with the experimental data is certainly less satisfactory compared to the differential conductance curves. In particular the predicted oscillation is larger than the observed (especially at larger voltages/currents) and the curves are quite smoother around 0.5 Φ 0 + nΦ 0 . Notably, despite these deviations, the maximum current-to-flux and voltage-to-flux transfer functions are close to the one observed in the experiment. This plot explain why a large deviation in the temperature evolution of the swing is observed in the theoretical curves in Fig. 5 of the main text, whereas the temperature evolution of the maximum transfer function works better.