Fluctuation-driven superconductivity in nitrogen-doped disordered tungsten–rhenium films

Abstract

We report a systematic study of nitrogen-doped superconducting tungsten–rhenium (WReN) thin films with thicknesses ranging from 5 to 60 nm, focusing on the interplay of disorder, superconducting fluctuations (SFs), quantum fluctuations (QFs), and weak localization (WL). The introduction of nitrogen allows fine-tuning of disorder, making WReN an ideal platform to explore quantum transport phenomena compared to related compounds such as WRe. To characterize both superconducting and normal-state properties, we employ Hall effect, magnetoresistance, and magnetoconductivity (MC) measurements for WReN films. Hall effect measurements provide complementary insight into carrier density, mobility, and disorder in the normal state, while SF, WL, and QF analyses reveal fundamental superconducting parameters, including critical temperature, upper critical field, coherence length and signatures of quantum phase transition. Furthermore, MC analyses were used for extracting relaxation times, which were found to be of the order of picoseconds. These results highlight the key role of disorder in shaping both the quantum transport and superconducting behavior of WReN films, offering a foundation for future studies in quantum materials and superconducting device applications.

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Introduction

In disordered superconductors, phenomena such as superconducting fluctuations (SFs),^1–4 quantum fluctuations (QFs),^5–8 and weak localization (WL)^9,10 remain central topics due to their key role in understanding fundamental superconducting properties and their impact on technological applications. These effects govern important phenomena such as quantum phase transitions (QPTs),^5–8 vortex dynamics,^11–13 and phase slips (PSs).^14–16 A notable example is the quantum Griffiths singularity (QGS),^5–8 a disorder-driven QPT emerging from rare superconducting islands, often associated with superconductor–insulator/metal transition (SIT/SMT) and previously observed in β-tungsten (W) films.⁷

Within the W-family superconductors,^11–17 materials such as sputtered β-W and FIB-fabricated W have been extensively studied for SIT/SMT, vortex phenomena, phase slips, and fluctuation effects. Incorporating SF and QF analyses enables a detailed exploration of these phenomena across different geometries, including thin films, meanders, and three-dimensional nanohelices. Other W-based compounds, such as tungsten silicide (WSi),^18,19 tungsten germanide (WGe),²⁰ and FIB-deposited W,²¹ have demonstrated exceptional performance in superconducting single-photon detectors (SSPDs) and optoelectronic applications.²² These materials exhibit superconducting critical temperatures (T_c) of 3–6 K, significantly higher than that of pure W (T_c ∼ 11 mK (ref. 23), making them suitable for cryogenic operations.

To characterize such superconducting films, temperature-dependent Hall effect measurements provide access to normal-state parameters including carrier density (n_e), Fermi wave vector (k_F), and mean free path (l_e). Together with the incorporation of SFs^1–3 and WL,^9,10 magnetoconductivity analyses under a k_Fl_e > 1 criterion form a comprehensive toolkit for extracting the total phase breaking rate (τ_ϕ) and electron energy relaxation times (τ_s), including electron–electron (τ_e–e), electron–phonon (τ_e–ph), and fluctuation-related (τ_e–fl) contributions.^4,24–28 These quantities significantly influence the performances of superconducting devices. For instance, in the case of SSPDs, a short τ_s enables a fast response and low jitter,^29,30 while a higher τ_e–ph/τ_e–e ratio enhances hotspot size and detection efficiency.^4,31 Quasiparticle relaxation processes are further probed via flux-flow instabilities (FFI).^32–35 Superconductors such as WSi,^4,24,28 NbTiN,²⁵ NbN,²⁶ NbGe,²⁷ MoSi,²⁸ WGe,²⁸ NbRe,³³ and NbReN^34,35 have been extensively investigated in this context. Based on this, we investigated tungsten–rhenium (WRe), a W-family superconductor whose properties can be systematically tuned via nitrogen incorporation.³⁶ A previous study³⁷ showed that the pure WRe and nitrogen-doped WRe (WReN) films deposited on silicon (Si) substrates possess τ_s of the order of picoseconds, comparable to those of the typical SSPD materials. Furthermore, a 25 nm-thick WReN film exhibited higher resistivity (ρ) and larger τ_e–ph/τ_e–e ratios compared to the WRe film of the same thickness, indicating improved potential as a SSPD material.

In the present study, we examine amorphous WReN films down to the ultrathin limit (5–60 nm) deposited on SiO₂/Si substrates via DC magnetron sputtering. We analyze their superconducting parameters, disorder- and magnetic field-driven QPT, and magnetoconductivity based on QFs, SFs and WL effects. For thinner films (5, 10, and 25 nm), we investigated the τ_s and τ_e–ph/τ_e–e ratios incorporating SF and WL contributions, providing insights into their suitability for SSPDs. Temperature-dependent Hall effect measurements complement this analysis by offering information on n_e, k_F, and l_e values. Overall, this work provides new insights into the fundamental properties of WReN and their potential applications in quantum devices and SSPDs.

Experimental methods

The WReN films were sputtered using a DC magnetron sputtering system on SiO₂/Si substrates using a target of W_0.75Re_0.25 stoichiometry (diameter 5 cm and thickness 3 mm) with a purity of 99.99% from Testbourne Ltd. The deposition processes were performed at room temperature under an ultra-high vacuum pressure of the order of 2 × 10⁻⁸ mbar. By precise control over argon (Ar) and nitrogen (N₂) flow, thus on an Ar + N₂ mixture, samples with a concentration of 7.5% N₂ (of total incoming flux) were prepared by using an optimal fixed sputtering power of 150 W. Films of different thicknesses were deposited at a rate of 0.84 Å s⁻¹, monitored using a calibrated quartz crystal microbalance, and subsequently confirmed using a stylus thickness profilometer (Bruker DektakXT). WReN films with thicknesses of 5, 10, 25, 40 and 60 nm were deposited. For simplicity and consistency, these samples are referred to as WN5, WN10, WN25, WN40 and WN60, respectively. A comparative analysis was also performed with the WReN and WRe films deposited on Si substrates with a thickness of 25 nm.³⁷ They are referred to as WN25/Si and W25/Si, respectively.

The X-ray diffraction (XRD) measurements reveal amorphous characteristics of WReN films, consistent with previous studies.^36,37 Scanning electron microscopy (SEM) and energy-dispersive X-ray spectroscopy (EDS) measurements were performed using a ZEISS Sigma 360 instrument equipped with an Oxford Xplore 30 EDS detector. Fig. 1(a) shows a SEM image of a WReN film, highlighting a uniform surface morphology without evidence of phase segregation, together with the selected EDS mapping area (inset). We note the presence of a surface impurity cluster in the SEM image, which has been intentionally used as a reference feature for focusing purposes. The EDS spectrum shown in Fig. 1(b) displays the characteristic peaks of the detected elements, while the table in the inset reports the compositional analysis in terms of weight percentages obtained from different regions of the sample. Fig. 1(c)–(e) show the representative EDS spectra acquired from the mapped region, confirming the presence of a well-homogeneous distribution of W, Re, and N across the film.

Fig. 1 (a) Scanning electron microscopy (SEM) image of a WReN film (60 nm). (b) EDS spectrum with the characteristic elemental peaks of W, Re, and N (main panel) and the corresponding weight percentage table (inset). EDS mapping reveals a homogeneous distribution of elements, W (c), Re (d), and N (e), across the analyzed area.

Superconductivity was studied by performing low-temperature electrical transport measurements in a cryogen-free system (2 K, 7 T) from Cryogenic Ltd. A wire bonder (TPT-wire bonder) was used to obtain four-probe electrical connections in the van der Pauw configuration. The sheet resistance was measured in a longitudinal direction of current to voltage, while Hall resistance measurements were carried out in a vertical direction of current to voltage with an excitation current of 10 μA in both cases using a Keithley 6221 current source operating together with a Keithley 2182A nanovoltmeter.

Hall measurements

Fig. 2(a) shows the Hall resistivity (ρ_xy) measured as a function of the magnetic field (B) in the range of 0–4 T for the WN25 film at various temperatures (in the range of 20–300 K), all above T_c. The Hall coefficient, R_H, is extracted from the linear slope of ρ_xy versus B, and the corresponding n_e is obtained via n_e = |−1/(R_He)|. The temperature dependence of R_H and n_e for WN25 is plotted in Fig. 2(b) using a dual y-axis scale.

Fig. 2 (a) Magnetic field (B) dependence of the Hall resistivity for the WN25 film at different temperatures. (b) Temperature dependence of the Hall resistance, R_H, (left scale) and charge carrier density, n_e, (right scale) for the WN25 sample. (c) Temperature dependence of n_e for all the WReN films. (d) Ioffe–Regel parameter (k_Fl_e) as a function of the temperature for all the WReN films.

The same analysis was performed for all other samples (WN5, WN10, WN40, and WN60). The corresponding dependence of n_e as a function of temperature is shown in Fig. 2(c). From the n_e values, we further estimated key electronic parameters for all WReN films, including k_F = (3π²n_e)^1/3, l_e = ℏk_F/(n_ee²R_snd), and the Fermi velocity v_F = ℏk_F/m. Additionally, the Ioffe–Regel parameter k_Fl_e was calculated and its temperature dependence is presented in Fig. 2(d). Interestingly, all these parameters (k_F, n_e, l_e, and k_Fl_e) are approximately constant throughout the temperature range 20–300 K. Temperature-independent n_e confirms the metallic nature of the films, while constant l_e and the fact that k_Fl_e indicate that the WReN films are disordered, with transport dominated by defects and impurities.³⁸ The estimated Hall parameters at 20 K, along with other superconducting parameters, are summarized in Table 1 and compared with those of the WN25/Si and W25/Si films.

Table 1 Superconducting transition temperature (T_c), superconducting gap (Δ), normal state resistivity (ρ_n), magnetic penetration depth [λ(0)] at 0 K, diffusion constant (D), density of states N(0), charge carrier density (n_e), Fermi wave vector (k_F), mean free path (l_e), and Fermi velocity (v_F) for the WReN films. These values are also compared with those of the WN25/Si and W25/Si films.³⁷ d is the thickness of the samples

Films/parameters	d (nm)	Tc (K)	Δ (meV)	ρn (μΩ cm)	λ(0) (nm)	D × 10^−4 (m^2 s^−1)	N(0) × 10^47 (m^−3 J^−1)	ne × 10^29 (e m^−3)	kF (nm^−1)	le (nm)	vF × 10^6 (m s^−1)	Ref.
a Present work.
WN5	5	4.09	0.61	207	747	0.50	3.75	1.14	15	0.26	1.80	PW^a
WN10	10	5.41	0.81	158	568	0.54	4.55	1.39	16	0.3	1.92	PW^a
WN25	25	5.79	0.86	158	550	0.56	4.43	1.07	14.7	0.35	1.76	PW^a
WN25/Si	25	5.61	0.84	171	581	0.51	4.47	1.19	15	0.31	1.82	37
W25/Si	25	4.78	0.71	127	541	0.77	3.99	4.10	23	0.18	2.76	37
WN40	40	5.77	0.86	180	587	0.56	3.88	1.11	14.9	0.31	1.79	PW^a
WN60	60	5.84	0.87	171	569	0.45	5.02	0.86	13.7	0.38	1.64	PW^a

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Superconducting properties

Fig. 3(a) shows the temperature-dependent sheet resistance, R_s(T), for the WReN films measured without an external magnetic field in the ranges of 3–20 K (main panel) and 3–300 K (inset). We measured the normal state resistance (R_sn) at 20 K, which increases as the film thickness is reduced, due to the limitation in the electron conduction pathways on decreasing film thicknesses. This behavior can be attributed to Anderson localization, which arises due to enhanced disorder in thinner films. This leads to increased electron scattering and is associated with SF and WL effects.^1,3 The BCS mean-field transition temperature (T_c) can be estimated by incorporating the effects of Cooper-pair fluctuations into the resistive transition. These fluctuations are commonly described using the Aslamazov–Larkin (AL)^1,2 and Maki–Thompson (MT)^3,39 models, which are applicable to two-dimensional (2D) or quasi-2D superconductors, and are expressed as:

Fig. 3 (a) Superconducting transition curves for the WReN films in the temperature range of 3–20 K (main panel) and 3–300 K (inset). The red line over the experimental data shows the fitting of eqn (1) based on AL and MT fluctuations. (b) R_s–T transition curves measured at different fixed applied magnetic fields (0–7 T) for the WN25 film. (c) The main panel shows the Arrhenius plots of the presented data in panel (b). The black lines correspond to the Arrhenius equation (AE) of thermally activated flux-flow (TAFF) theory. The inset shows the comparison of activation energies as a function of magnetic field for the 25 nm-thick WReN and WRe films (see the main text for details). (d) Activation energies variation as a function of magnetic field for the WReN films. The black curves represent the activation energy fit to the experimental data (see the main text for details).

where ℏ is the Planck constant, e is the elementary charge, and A and T_c are used as fitting parameters. The R_s(T) curves are well fitted using eqn (1), as shown in Fig. 3(a) by the red curves. We find good agreement between the T_c values extracted within the R_sn/2 criteria and those obtained from the fitting procedures (listed in Table 1). The fitted values are A ∼ 3 for all films. For the thicker films (WN40 and WN60), we did not attempt to fit the data using the 2D formula because their higher dimensionality makes such an analysis inappropriate. For WN25, T_c is slightly higher than that for WN25/Si. From the inset data in Fig. 3(a), the residual resistance ratio RRR = R^{300 K}_sn/R^{20 K}_sn is below unity for all samples, indicating a nearly non-metallic character. In addition, normal-state resistivities (ρ_n) at 20 K are also listed in Table 1. Notably, ρ_n for WN25 lies between the values reported for WN25/Si and W25/Si.^36,37 These ρ_n values were further used to estimate the zero-temperature penetration depth, ,^15,40 and the values are listed in Table 1 for all WReN films.

The resistive transitions, R_s(T), in the presence of applied perpendicular B up to 7 T is shown in Fig. 3(b) for WN25, with its Arrhenius-type representation, ln R_s vs. 1/T, shown in the main panel of Fig. 3(c). Increasing B shifts superconducting transitions to lower temperatures and broadens them, indicative of thermally activated vortex dynamics.^41–43 Fitting R_s(T,B) to the Arrhenius relationship R_s(T,B) = R_o(B)exp(−U_o(B)/k_BT) yields activation energies U_o(B), and the fitted black lines crossed at a single-meeting point, T_M ∼ 5.88 K, close to the T_c of WN25. The extracted U_o(B) values are slightly higher than for WN25/Si but lower than for W25/Si³⁷ (inset of Fig. 3(c)), reflecting the combined effects of substrate and nitrogen doping, which increase disorder and distort the vortex structure,⁴⁴ thereby lowering U_o(B). Here, the obtained activation energies are compatible with values reported for superconducting thin films such as Nb⁴⁴ and NbN.⁴⁵ Furthermore, Fig. 3(d) summarizes the U_o(B) behavior for all films. The dependence follows U_o(B) = B^−a(1 − B/B*)^b,^46–48 with fitted exponents a ∼ 0.5 and b ∼ 1.5, comparable to other superconducting films. The obtained reduction in U_o with decreasing film thickness is attributed to smaller vortex volumes in thinner samples, consistent with FeTeSe films reported elsewhere.⁴⁹

The field-dependent superconducting transitions R_s(B) at fixed temperatures are shown in Fig. 4(a–d) for the WN5, WN10, WN25, and WN40 films. In all cases, the R_s(B) isotherms exhibit superconducting transition towards lower magnetic fields with increasing temperatures. The temperature dependence of the upper critical fields, B_c2(T), extracted using the R_sn/2 criterion, is plotted in the insets of Fig. 4.

Fig. 4 Sheet resistance (R_s) as a function of the magnetic field at fixed temperatures for the WN5 (a), WN10 (b), WN25 (c), and WN40 samples (d). The inset of each panel shows the temperature-dependent upper critical magnetic field, B_c2(T), along with the fitting to the experimental data using WHH eqn (2), as shown by solid red curves.

The zero-temperature upper critical field, B_c2(0), was obtained using the Werthamer–Hohenberg–Helfand (WHH) theory^50,51

where t = T/T_c and are the reduced temperature and magnetic field, respectively. α is the Maki parameter and λ_so denotes the spin–orbit scattering parameter. The best fittings to the experimental data using eqn (2) were obtained for α = 0 and λ_so = 0 for all the samples, in agreement with previous analyses on sputtered nitrogen-doped W thin films.^52,53 The fitted curves are displayed by red color in the insets of Fig. 4(a)–(d) along with the values of B_c2(0) and the zero-temperature coherence lengths ξ(0), as obtained from , with ϕ₀ = h/2e the magnetic flux quantum.

The electron diffusion constant was determined from D = (4k_B/πe)(dB_c2/dT)⁻¹ and the density of states was determined from the Einstein relationship N(0) = 1/(De²R_snd), where e is the electronic charge and d is the film thickness. The extracted D and N(0) are reported in Table 1. Furthermore, the electron–phonon coupling constant λ_ep was estimated from the McMillan formula:⁵⁴

using Θ_D∼400 K for tungsten–rhenium alloys⁵⁵ and μ* = 0.1–0.17.^41,56 For all films, λ_ep ∼ 0.6, similar to tungsten carbide⁵⁶ and consistent with weak-coupling BCS superconductivity.

We also determined the Pippard coherence length from , where Δ = 1.76k_BT_c is the superconducting gap. The extracted ξ_p values (of the order of hundreds of nanometers for all the films) are much larger than ξ(0) and satisfy the relationship

ξ(0) ∼ (ξ_pl_e)^1/2 ≪ ξ_p,

with v_F and l_e taken from Hall measurements. This confirms that WReN films are dirty-limit, type-II superconductors, where disorder strongly contributes to their superconducting behavior. Such systems are well known to display fluctuation-driven phenomena at low temperatures,^{5–8,12–15,43} as discussed in the following sections.

Quantum phase transition (QPT)

Disordered superconductors have long been the subject of intense investigation due to the emergence of QPTs^5–8 governed by QFs. Such transitions can arise from intrinsic disorder, rare-region effects, applied magnetic fields, and temperature-dependent influences on electronic transport. These phenomena are commonly characterized by superconductor–insulator or superconductor–metal transitions (SIT/SMT), exhibiting activated scaling of the resistance with magnetic field and temperature, R(B,T).^5,14 Experimental signatures of these effects have been reported in a variety of superconducting systems, including β-W,⁷ NbN,⁵⁷ Ta₂PdS₅,⁸ MoS₂,⁶ MgTi₂O₄,⁵⁸ and FIB-deposited W.¹⁴

In our magnetoresistance measurements, we observed that the R_s–B(T) isotherms cross at a single point at higher temperatures (5.5 K–6.25 K) near the T_c for the WN40 sample, as shown in Fig. 5(a). Similar crossing behavior can also be observed at lower temperatures (down to the millikelvin range) and over a broad range of temperatures and magnetic fields for all WReN films, which may be indicative of QGS phenomena.^7,8 We denote the crossing point coordinates as R_X = 45.08 Ω and B_X = 6.85 T in Fig. 5(a). For these disordered WReN films, the QPT is predominantly induced by the applied magnetic field and can be analyzed using finite-size scaling (FSS).^7,8,14 According to FSS theory,^7,8,14 the resistance near the transition can be expressed as:

where f(x) is a scaling function with f(0) = 1 and z and v are the dynamical and correlation-length critical exponents, respectively. The correlation length ξ_c diverges near the QPT as ξ_c ∝ δ^−v with δ = |B − B_X|, and the characteristic frequency follows ω ∝ ξ_c^−z under the influence of QFs. These critical exponents are independent of microscopic details and can be evaluated by plotting dR/dB versus T⁻¹ (inset of Fig. 5(a)), where the inverse slope yields the product zv ∼ 0.45 for our WN40 sample. Using eqn (3), we collapse the R(B,T) curves, as shown in Fig. 5(b). The inset clearly demonstrates two distinct behaviors of the scaled resistance above and below unity. The obtained value of zv ∼ 0.45, slightly below 0.66, is consistent with the three-dimensional (3D) nature of the 40 nm WN sample.⁵⁸ We note that our measurements are limited by the maximum achievable magnetic field, which constrains a full exploration of the transition. Extending this study to higher fields and lower temperatures remains a promising avenue for future work.

Fig. 5 (a) Sheet resistance (R_s) as a function of the magnetic field for the WN40 sample (main panel) in the high temperature (4.5–6.25 K) and high magnetic field (6.0–7.0 T) ranges, showing a singular crossing point. The inset shows the corresponding dR/dB versus T⁻¹ behavior. Its linear behavior is shown by the red solid line. (b) FFS scaling of the QPT in the WN40 film obtained using eqn (3) for selected temperatures. Inset: zooming of the data in the low field region shows two different behaviors of the R_s(B,T) curves (see the main text for details).

Magnetoconductivity and relaxation times

It is well established that magnetoconductivity correction (MC) models^4,24–28 play a crucial role in determining τ_s, which are essential parameters for evaluating the suitability of superconductors in SSPDs. According to the MC model, the total change in magnetoconductivity (δG^TOTAL) arises from quantum corrections due to several mechanisms: WL, AL, MT, SFs, and density of states (DOS) fluctuations under the criterion of k_Fl_e > 1. At temperatures T ≫ T_c, the dominant contribution to magnetoconductivity originates from WL, a quantum interference effect that occurs in disordered systems. In the presence of a magnetic field, this interference is modified, particularly by spin–orbit scattering, which influences the phase coherence between different electron trajectories. The WL contribution^9,25 to δG^TOTAL is given bywhere and ψ is the digamma function. The characteristic magnetic fields B_so = ℏ/(4eDτ_so) and B_ϕ = ℏ/(4eDτ_ϕ) correspond to spin–orbit and dephasing scattering, respectively, with τ_so denoting the spin–orbit scattering time. As B increases, the magnitude of δG^WL(B,T) increases, contributing positively to δG^TOTAL.

For temperatures T ∼ T_c, SFs become prominent due to the formation of short-lived Cooper pairs, resulting in the broadening of the superconducting transition. These fluctuations reduce the effective electron density and enhance the MC. The AL correction^1,2 to δG^TOTAL can be expressed as

where B_c = Ck_BT/[πeD ln(T/T_c)] is the characteristic field and C is a numerical factor typically ranging between 2 and 6.

Near T_c, the MT correction becomes significant as SFs and electron pairing effects intensify.^3,39 These fluctuations enhance the mean free path l_e and reduce localization effects, thereby increasing the overall MC. The MT correction,^3,39,59 formulated by Lopes dos Santos and Abrahams (LSA),⁶⁰ is expressed as

where β_LSA(T) = 2πk_BTℏ⁻¹(1/τ_GL − 1/τ_ϕ)⁻¹, and τ_GL = πℏ/[8k_BT ln(T/T_c)], is the Ginzburg–Landau relaxation time. The MT term reflects the interplay between SFs and phase coherence effects.

The final contribution, δG^DOS(B,T), arises from fluctuations in the electronic DOS near T_c.^61,62 These variations stem from the energy dependence of the DOS near the Fermi level, which is modified by SFs. The corresponding correction⁶¹ is given by

where ζ(3) = 1.202. Hence, δG^TOTAL(B,T) can be expressed as the summation of all individual correction terms:

Furthermore, we used the following equation to calculate δG from the experimental data:

where R_S⁻¹(B,T) and R_S⁻¹(0,T) are the inverse resistances at finite and zero fields. Here, we adopted thinner WReN films (WN5, WN10 and WN25) under the criteria of k_Fl_e > 1 for their relaxation time analysis. Note that the values of k_Fl_e were determined from Hall measurements. The extracted experimental data curves obtained using eqn (9) are shown by multiple colors in Fig. 6(a–c), over which well-aligned black curves, derived from the fit of eqn (8), are plotted for all WReN films. From the fits, we extract the τ_ϕ values of 7.75 ps (WN5), 26.25 ps (WN10), and 52.91 ps (WN25), which are shorter than those of WRe films with comparable thicknesses.³⁷ Furthermore, the temperature dependence of τ_ϕ and τ_ϕ⁻¹ is shown in Fig. 6(d–f) for all WReN films.

Fig. 6 Total change in magnetoconductivity (δG) as obtained from eqn (9) at different temperatures for the WN5 (a), WN10 (b) and WN25 (c) films. The black lines are the best fit to the data obtained using eqn (8), which includes corrections in δG(B,T) arising from weak localization (WL), Aslamazov–Larkin (AL) fluctuations, Maki–Thompson (MT) fluctuations, and density of states (DOS) fluctuations. Total phase scattering relaxation rate (τ_ϕ) (insets) and its inverse [τ_ϕ⁻¹(T)] (main panel) variation as a function of the temperature for the WN5 (d), WN10 (e), and WN25 films (f). The black curves are the best fit to the data using eqn (10). The inset of panel (f) shows a comparison of τ_ϕ(T) values for the WN25, WN25/Si,³⁷ and W25/Si films.³⁷

The τ_ϕ can be expressed as

where τ_e–e, τ_e–ph, and τ_e–fl are given by following eqn (11)–(13).

where and α_e–ph and n are fitting parameters. The well-simulated curves over the experimental τ_ϕ(T) data using eqn (10) are shown by black color in Fig. 6(d–f) for all WReN films and the fitting parameters are summarized in Table 2. Furthermore, we also extracted τ_e–ph (T) variation and plotted it in Fig. 7 together with fits (black lines) using eqn (12) for the WN5, WN10 and WN25 films.

Fig. 7 τ_e–ph as a function of the temperature for the WN5, WN10 and WN25 films. The black lines show the best fit to the data using eqn (12).

Table 2 Fitting parameters (α_e–ph and n) and relaxation times [τ_ϕ, (τ_e–ph) and (τ_e–e)] at T ≈ 4–6 K as evaluated from eqn (10)–(13) for the films in the present study, compared with the results from the literature. d is the thickness of the films

Films	d (nm)	αe–ph × 10^−10	n	τϕ (ps)	τe–ph (ps)	τe–e (ps)	τe–ph/τe–e	Ref.
a Present work.
WN5	5	1.06	2.91	7.75	34.84	23.07	1.51	PW^a
WN10	10	0.7	3.25	26.25	49.41	47.06	1.03	PW^a
WN25	25	0.99	2.98	52.91	89.48	97.26	0.92	PW^a
WN25/Si	25	2.38	2.97	45.70	84.01	91.31	0.92	37
W25/Si	25	1.98	2.95	57.80	94.53	116.7	0.81	37
WSi	5	—	—	9	66	47	1.4	24
WSi	5	55	3.00	6.6	92	24.4	3.8	4
NbTiN	6	31	3.5	52	16.9	—	—	25
NbN	5	10.8	3.53	4.51	9.3	10.8	0.86	26
WGe	5	—	—	—	82	32	2.6	28
MoGe	5	—	—	—	32	28	1.2	28
MoSi	5	—	—	6.67	30	30	1	28

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Discussion

We investigated the effect of disorder on the superconducting and normal-state properties of amorphous WReN thin films by combining Hall transport, fluctuation conductivity, magnetoresistance, and quantum scaling analyses. The results establish a consistent picture in which nitrogen incorporation increases disorder, thereby reshaping both the electronic landscape and the superconducting ground state.

Hall measurements demonstrate that nitrogen systematically reduces carrier mobility and enhances scattering, confirming the emergence of a disorder-dominated normal state. This is consistent with the reduced RRR (< 1) and with the non-metallic temperature dependence associated with localization effects. The superconducting transition reflects these modifications: SF analyses (AL and MT terms) show that a higher nitrogen content correlates with enhanced T_c, larger B_c2(0), and correspondingly shorter coherence lengths ξ(0). These trends highlight the role of controlled disorder in strengthening superconducting pairing in amorphous WReN, in contrast to crystalline alloys where disorder is typically detrimental.

Magnetic field-dependent R_s(T) curves reveal thermally activated flux-flow behavior, with U_o(B) that decrease at high magnetic fields. This indicates weakened pinning and enhanced vortex mobility, consistent with increased thermal activation of the vortices in highly disordered films. A slight substrate dependence, larger U_o(B) for films grown on SiO₂/Si substrates, suggests that interface disorder and strain modulate vortex dynamics and can serve as additional tuning parameters for flux motion.

A clear crossing point in the magnetoresistance isotherms indicates a magnetic field-driven QPT chracteristic and may lead to the existence of QGS phenomena. The extracted critical exponent zv ≈ 0.45 lies within the range characteristic of 3D QPTs. Although this value does not conclusively demonstrate a QGS, the trends are compatible with enhanced QFs in the presence of strong disorder, and the full range of zv needs further investigation to clarify. Further measurements in a broad range of temperatures and higher magnetic fields are needed to establish whether WReN hosts a quantum Griffiths-like regime with true SIT/SMT characteristics.

Magnetoconductivity analysis provides additional insight into quasiparticle relaxation relevant for nonequilibrium superconducting dynamics. The disorder enhances electron–electron scattering while modifying electron–phonon coupling, reflected in the shorter τ_ϕ and in the increased ratios τ_e–ph/τ_e–e obtained for the WReN films than pure WRe,³⁷ particularly for WN5 (τ_ϕ − 7.75 ps and τ_e–ph/τ_e–e − 1.51). These values fall within the range reported for well-established SSPD materials, such as WSi, NbN, NbTiN, WGe, and MoSi, as mentioned in Table 2. Although our study focuses on fundamental transport mechanisms rather than device performance, the extracted relaxation parameters indicate that WReN shares key microscopic features with materials employed in SSPDs.

We can now discuss the suitability of WReN thin films for the realization of SSPDs. The first observation is that nitrogen incorporation during deposition leads to disorderness and amorphous characteristics for the WReN films. This represents an advantage for uniformity, reduced constrictions and defects, and higher fabrication tolerance, also in terms of substrate choice. Furthermore, a short order of relaxation times with a larger τ_e–ph/τ_e–e ratio suggests an efficient energy transfer from the photon to the electronic system, with a positive impact on efficiency. This finding is promising also for the realization of micrometric detectors (SMSPDs).⁶³ Interestingly, the τ_e–ph/τ_e–e ratio is compatible with those of other tungsten-based materials, such as WSi and WGe.²⁸ In contrast, the characteristic τ_e–e time scales are shorter than those of amorphous materials, and this can potentially result in devices with improved time performance, in particular in shorter dead times. Regarding the superconducting properties, the values of T_c are very similar to those of amorphous superconductors performing as good SNSPDs (WSi, WGe, and MoSi) and lower than crystalline NbN and NbTiN. In particular, the T_c of WReN is intermediate between Re-based superconductors used in this field, namely NbRe^64,65 and NbReN.⁶⁶ This tunable, relatively low T_c can be suitable for the detection of low-energy infrared photons used in quantum communication, LIDAR, and satellite communications. Finally, the short coherence length also supports ultrathin-film device fabrication while preserving superconducting properties, making it a promising material for efficient SSPDs. The practical implementation of the WReN films in SSPDs in meander form is currently the subject of active research and ongoing investigation.⁶⁷

Overall, our results demonstrate that WReN thin films represent a versatile platform in which disorder can be tuned to manipulate SFs, vortex dynamics, and quantum interference effects. The observation of a field-driven QPT, together with signatures of pronounced QFs, makes WReN a promising system for exploring quantum phase transitions in disordered superconductors. Future work involving broader nitrogen concentrations, lower-temperature scaling, and direct photon-detection experiments will clarify the microscopic mechanisms and further assess the potential of WReN for quantum electronic technologies.

Conclusions

We deposited WReN films of different thicknesses (5–60 nm) on SiO_2/Si substrates by DC magnetron sputtering, which show amorphous and disordered characteristics. The films exhibit superconducting transition temperatures T_c ∼ 4–6 K, upper critical fields B_c2(0) ∼ 8 T, and coherence lengths ξ(0) ∼ 6 nm.

The thermally activated vortex dynamics is observed, with behavior influenced by the film thickness, substrate, and nitrogen content. In the 40 nm WReN film, a disorder- and field-induced QPT characteristic near T_c highlights the role of QFs. Hall measurements (20–300 K) indicate nearly constant normal-state parameters, confirming metallic transport with moderate disorder. Magnetoconductivity analysis of thinner films (5–25 nm) reveals shorter phase relaxation times and enhanced τ_e–ph/τ_e–e ratios than pure WRe, comparable to those of established SSPD materials (WSi, NbN, NbTiN, WGe, and MoSi).

These results establish WReN films as promising candidates for superconducting and quantum devices, particularly single-photon detectors, and provide a foundation for further studies exploring disorder, vortex dynamics, and energy relaxation in these systems.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgements

This research was partially supported by the project IR0000003–IRIS, funded by the NextGeneration EU-funded Italian National Recovery and Resilience Plan with the Decree of the Ministry of University and Research number 124 (21/06/2022) for Mission 4 – Component 2 – Investment 3.1. This research was also partially supported by the project “High-performance Josephson junctions for ferrotransmons – CONJUNCTIONS” in the frame of Partenariato Esteso “NQSTI”, Spoke 5, funded by the Italian National Recovery and Resilience Plan Mission 4, Component 2, Investment 3.1.

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Footnote

† Present address: Technische Universität Braunschweig, 38106 Braunschweig, Germany.

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