Superconducting properties of thin film Nb_1-x^Ti_x studied via the NMR of implanted 8_Li

Journal of Physics: Condensed Matter PAPER • OPEN ACCESS Superconducting properties of thin film N studied via the NMR of implanted 8Li To cite this article: Md Asaduzzaman et al 2025 J. Phys.: Condens. Matter 37 395701 View the article online for updates and enhancements. You may also like The electronic structure of palladium on magnetoelectric Cr2O3(0001) Takashi Komesu, Shiv Kumar, Prescott E Evans et al. - Investigation of softer lattice dynamics in defect engineered GeTe crystals Saptak Majumder, Pintu Singha, Sharath Kumar C et al. - Fractional magnetization plateaus in the Shastry–Sutherland lattice material Er2Be2GeO7 M Pula, S Sharma, J Gautreau et al. - This content was downloaded from IP address 194.12.132.102 on 09/12/2025 at 16:16 Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 37 (2025) 395701 (16pp) https://doi.org/10.1088/1361-648X/ae0287 Superconducting properties of thin film Nb1−xTixN studied via the NMR of implanted 8Li Md Asaduzzaman1,2,∗, Ryan M L McFadden1,2, Edward Thoeng2,3, Yasmine Kalboussi4, Ivana Curci4, Thomas Proslier4, Sarah R Dunsiger2,5, W Andrew MacFarlane2,6,7, Gerald D Morris2, Ruohong Li2, John O Ticknor6,7, Robert E Laxdal1,2and Tobias Junginger1,2,∗ 1Department of Physics and Astronomy, University of Victoria, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada 2TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3, Canada 3Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia V6T 1Z1, Canada 4Institut des lois fondamentales de l’univers, Commissariat de l’énergie atomique-centre de saclay, Paris-Saclay University, 91191 Gif-sur-Yvette, France 5Department of Physics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada 6Department of Chemistry, University of British Columbia, 2036 Main Mall, Vancouver, BC V6T 1Z1, Canada 7Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada E-mail: [email protected] and [email protected] Received 5 June 2025, revised 14 August 2025 Accepted for publication 2 September 2025 Published 24 September 2025 Abstract We report measurements of the normal-state and superconducting properties of thin-film Nb1−xTixN using 8Li β-detected nuclear magnetic resonance (β-NMR). In these experiments, radioactive 8Li+probes were implanted ∼21 nm below the surface of a Nb0.75Ti0.25N(91 nm) film in Nb0.75Ti0.25N(91 nm)/AlN(4 nm)/Nb and its NMR response recorded (via 8Li’s β-emissions) between 4.6 K and 270 K in a 4.1 T field applied normal to its surface. Resonance measurements reveal wide, symmetric lineshapes at all temperatures, with significant additional broadening below the film’s superconducting transition temperature Tc(0T) = 15.4(7)K due to vortex lattice formation. Fits to a broadening model find a magnetic penetration depth λ(0K) = 180.57(30)nm and upper critical field Bc2(0K) = 18(4)T, consistent with literature estimates. Spin-lattice relaxation (SLR) measurements find a Korringa response at low temperatures, with dynamic (i.e. thermally activated) contributions dominating above ∼100 K. Below Tc, we observe a small Hebel–Slichter coherence peak characterized by a ∗Authors to whom any correspondence should be addressed. Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1 © 2025 The Author(s). Published by IOP Publishing Ltd J. Phys.: Condens. Matter 37 (2025) 395701 M Asaduzzaman et al superconducting energy gap ∆(0K) = 2.60(12)meV and modest Dynes-like broadening. Our measurements suggest a gap ratio 2∆(0K)/kBTc(0T) = 3.92(25), consistent with strong-coupling behavior. Sources for the dynamic high-Trelaxation are suggested. Supplementary material for this article is available online Keywords: NbTiN thin film, superconductor-insulator-superconductor (SIS), superconducting energy gap, penetration depth, coherence peak, critical temperature, upper critical field 1. Introduction Nb1−xTixN is a ternary alloy with a cubic B1 (rocksalt) crystal structure (see figure 1), derived from Group IV and Group V transition metal nitrides [1,2]. It forms a fully miscible, quasibinary solid solution with its end members TiN and NbN over the entire 0 ⩽x⩽1 composition range [3,4], with structural details that closely follow Vegard’s law [5,6]. Similar to its end members, the alloy is a type-II superconductor with a relatively high critical temperature Tc(up to 17 K [7,8]). Thanks to the alloy’s facile synthesis in the form of thin films [1, 9–11], Nb1−xTixN finds use in numerous technical applications that employ superconducting coatings (e.g. tunnel junctions [12,13], THz receivers [14,15] and mixers [16,17], etc). In particular, the alloy has emerged as a promising candidate [18] for coating conventional Nb superconducting radio frequency (SRF) cavities — common components of modern particle accelerators [19] — which we consider in detail below. SRF cavities accelerate charged particle beams via the electric fields created under resonant radio frequency (RF) excitation. Essential for maintaining a high quality factor Q0is low electrical resistivity of the cavity material, making superconductors like Nb ideal [19]. The performance of Nb cavities is ultimately limited by the element’s so-called superheating field Bsh ≈240mT [20,21], beyond which vortex penetration occurs and resistive losses cause Q0to plummet. To address this limitation, coatings with superconductors having higher Tcand Bsh than Nb (e.g. Nb3Sn, Nb1−xTixN) have been proposed [22–24]. Such coatings (with or without an insulating ‘buffer’ layer) are predicted [22–26] to allow such devices to operate in field regimes beyond the capabilities of ‘bare’ Nb cavities, a claim supported by recent experiments [21,27– 31]. Central to modelling the macroscopic behavior of these heterostructures, however, is knowledge of the coating film’s microscopic superconducting properties, which are not wellestablished for Nb1−xTixN. Part of the uncertainty in Nb1−xTixN’s intrinsic properties stems from their compositional ‘tunability’, with a transition temperature Tcthat can be tuned with stoichiometry. For instance, a Tc≈17K has been reported for x =0.34 [32, 33], while a slightly lower value of ∼15 K is observed for x≲0.5 [1,2,34,35]. As x →1, Tcdecreases further to approximately 4 K [34,36]. This trend may be attributed to the properties of the end-member compositions, where NbN exhibits Tc≳16K [37,38], while TiN has a significantly lower Tc≈4K [39,40]. Less well characterized is how stoichiometry affects its other superconducting properties. For example, measurements indicate a magnetic penetration depth λ≳150nm [17,29,34,41–44] for x ⩽0.46, a Ginzburg– Landau (GL) coherence length ξGL ≈4nm [4,11,45,46], a lower critical field Bc1 ≈30mT [35,47,48], and an upper critical field Bc2 ≳15T [1,4,9,36,45,49,50]. Similarly, the alloy’s superconducting energy gap ∆is also known to vary with x, with gap ratios 2∆(0K)/kBTc(0T)ranging from 3.53 to 5 [12–17,42,44,51–53]. This complexity can be further compounded by their variability with other factors such as film thickness, substrate material, and post-deposition annealing [3,10,11,54–57]. As Nb1−xTixN films are most important for technical applications — particularly in form of superconductor-insulator-superconductor (SIS) structures (e.g. Nb1−xTixN/AlN/Nb), which are widely used in tunnel junctions, THz mixers, and SRF cavities — we focus our attention on this material class. One means of elucidating its properties is through the study of the superconductor’s internal magnetic field distribution p(B) in the vortex state [58,59]. Techniques such as nuclear magnetic resonance (NMR) [60,61] and muon spin rotation (µSR) [62,63] are effective approaches for bulk superconductors, but are less suited for thin films or layered heterostructures. On the other hand, closely related techniques based on low-energy implanted spin-probes like low-energy muon spin rotation (LE-µSR) [64,65] and β-detected nuclear magnetic resonance (β-NMR) [66,67] are well-suited to such a situation, enabling spatially resolved measurements at subsurface depths ≲150nm. In this work, we investigate the superconducting and normal-state properties of thin film Nb1−xTixN using 8Li β-NMR spectroscopy [66,67]. Specifically, we report measurements in the normal and vortex states of a Nb0.75Ti0.25N(91 nm)/AlN(4 nm)/Nb heterostructure under a 4.1 T field applied normal to its surface. Resonance measurements find wide 8Li lineshapes that (symmetrically) broaden below the film’s Tcdue to the formation of vortex field lines. Spin-lattice relaxation (SLR) data reveal distinct T-dependent behavior, with a Korringa response [69] below ∼100 K that displays a Hebel–Slichter coherence peak [70,71] below Tc, and relaxation that is dominated by thermally activated fluctuations at higher-T. From an analysis of these features, we 2 J. Phys.: Condens. Matter 37 (2025) 395701 M Asaduzzaman et al Figure 1. Crystal structure of cubic B1 (rocksalt) Nb1−xTixN (space group Fm¯ 3m, number 225). The metal atoms Nb and Ti atoms (blue and red spheres) randomly occupy the Wyckoff 4asite, which forms an face-centred cubic (FCC) sublattice. Similarly, the N atoms (green spheres) occupy the Wyckoff 4bsite, can be viewed as filling the octahedral ‘intersticies’ of the FCC metal sublattice. The gray bonds highlight the octahedral coordination environment of each equilibrium position. Lattice constants a,b,cfor the x =0.25 stoichiometry are indicated in the inset. Created using the VESTA software package [68]. quantify the parameters governing the film’s superconducting properties and compare them against literature values. Our findings provide key insight into the superconducting behavior of Nb1−xTixN thin films in an SIS arrangement. 2. Experiment β-NMR [66,67] experiments were conducted at TRIUMF’s isotope separator and accelerator (ISAC) facility in Vancouver, BC, Canada. The local spin probe 8Li (nuclear spin I=2, radioactive lifetime τ=1.21s (half-life 848 ms), gyromagnetic ratio γ8Li/(2π) = 6.30198(8)MHzT−1, nuclear electric quadrupole moment Q= +32.6mb, and mass m8Li =8.023u) was introduced into the sample by ion-implantation using a beam of 8Li+. The incident ion beam had a typical flux of ∼106ions s−1over a beam spot ∼3 mm in diameter, with a beam implantation energy E=4.85keV, corresponding to a mean stopping depth ⟨z⟩∼21nm, as calculated by the Stopping and Range of Ions in Matter (SRIM) [72] Monte Carlo code (see figure 2). Prior to implantation, the probe was spin-polarized in-flight by collinear optical pumping with circularly polarized laser light [73], achieving a polarization pz≈70%[74]. All β-NMR measurements were performed in an applied field B0=4.1T perpendicular to the sample’s surface. During the measurements, 8Li’s pz, defined by: pz=1 ITr(ρIz).(1) where Izis the operator for the z-component of the nuclear spin and ρis the corresponding density matrix, was monitored Figure 2. Simulated stopping profile for 106 8Li+implanted in Nb0.75Ti0.25N(91 nm)/AlN(4 nm)/Nb at energy E=4.85keV, obtained using the SRIM Monte Carlo code [72]. The target’s layer thicknesses and material densities are indicated in the inset. The stopping profile ρz(E)(i.e. the distribution of implantation depths z) is represented as a histogram, whose mean implantation depth ⟨z⟩ ≈ 21 nm and straggle (i.e. standard deviation) σz≈11 nm. through its β-decay anisotropy. In this process, an electron is preferentially emitted opposite to the direction of the nuclear polarization at the time of decay. This was done using a pair of fast plastic scintillation counters positioned 180◦apart with respect to each other along the beam axis [66,75]. During the data acquisition, the handedness of the laser polarization used during optical pumping was periodically alternated to control the 8Li+beam’s helicity (i.e. positive and negative) [66], with data recorded separately for each polarization sense ±. The four-counter method was used to form the β-decay asymmetry A, which is proportional to pz[76]: A≡r−1 r+1=A0pz,(2) where r≡sN+ F/N+ B N− F/N− B, N± Band N± Fare the beta decay electron events recorded in the forward (F) and backward (B) detectors for the ±polarization senses, and A0≈0.1 is a proportionality factor that depends on the experimental setup (e.g. detection geometry, probe βdecay properties, etc). In this work, two types of experiments were conducted (i) resonance measurements, where the steady-state (i.e. timeintegral) spin-polarization was monitored as a function of the 3 J. Phys.: Condens. Matter 37 (2025) 395701 M Asaduzzaman et al frequency of the small RF magnetic field B1, which is used to map the (static) local field distribution; and (ii) SLR measurements, where the temporal decay of pzcaused by stochastic fluctuations in the probe’s local field is monitored both during and following implantation. In the former, a continuous 8Li+ beam was used with the frequency of the small transverse RF field stepped slowly near 8Li’s Larmor frequency: ω0=γ8LiB0.(3) On resonance, the 8Li spin precesses rapidly due to the RF field, resulting in a loss of the time-averaged asymmetry. Multi-frequency techniques were also used to search for small ‘quadrupolar’ features (see, e.g. [77,78]). For the SLR measurement, a pulsed 8Li+beam was used with a typical duration ∼4s. As the initial state of the probe nuclei are very far from thermal equilibrium, no RF field is required to measure SLR, unlike conventional NMR. During the pulse, the polarization approaches a dynamic equilibrium value, while afterward it relaxes to ∼0. Notably, data acquired in this manner has a characteristic bipartite form with (statistical) error bars that are governed by Poisson statistics. This uncertainty is minimized near the pulse’s trailing edge, but increases exponentially with the 8Li lifetime afterward (see, e.g. [67]). In the present study, the typical duration of either measurement was ∼15 min to ∼30 min. 2.1. Sample preparation First, flat Nb substrates were prepared by cutting fine-grain Nb stock sheets (Wah Chang Corporation) with a residualresistivity ratio (RRR) >150 and machining them into flat plates approximately 12 mm by 8 mm by 0.5 mm. Following machining, the samples underwent buffered chemical polishing (BCP) (see, e.g. [79]) to remove the topmost ∼100 µm of material from the surface. Subsequently, the samples were annealed at 1400 ◦C for 5 h to relieve any remaining mechanical stresses in the metal. After annealing, an additional round of BCP was performed to remove the topmost ∼10 µm of material from the surface, effectively eliminating any contaminants introduced during the annealing process. The Nb1−xTixN/AlN bilayer was deposited on a Nb substrate using thermal atomic layer deposition (ALD) in a custom-built reactor at CEA Saclay, France [80]. The AlN layer was grown using a standard process with AlCl3and NH3 precursors [81], while the Nb1−xTixN film was deposited at 450 ◦C by alternating NbN and TiN cycles. Its composition can be controlled by adjusting the number of TiN and NbN cycles [55–57]. In this work, each Nb1−xTixN supercycle consisted of 4 (TiCl4+NH3) cycles followed by 1 (NbCl5+NH3) cycle, as the subsequent NbCl5pulse etches surface Ti in the form of volatile TiCl4. NH3was pulsed for 0.5 s, TiCl4for 2.5 s, and NbCl5for 1 s, with 10 s purges after each step. This yielded a Ti/Nb ratio of 0.25 (x =0.25), confirmed by xray photoelectron spectroscopy (XPS) [55]. Excess nitrogen incorporated during deposition was effectively removed by high-vacuum annealing at 900 ◦C [55]. Further discussion on the annealing process, along with additional characterization, can be found in section S1 of the supplemental material [82]. The film’s characteristic superconducting transition temperature Tcwas determined to be ∼15 K using a vibrating sample magnetometer (VSM). Point contact tunneling (PCT) [83,84] measurements on a similarly prepared sample show a spatial variation in the density of states (DOS) near the Fermi level, which is encapsulated by an energy gap ∆ = 2.49(29)meV and a Dynes-like [85] broadening parameter ΓD=0.10(6)meV. Full characterization details, along with complementary measurements on similarly prepared samples, are presented in section S2 of the supplemental material [82]. 3. Results and analysis 3.1. Resonance spectra Typical resonance spectra in Nb0.75Ti0.25N are presented in figure 3. At all temperatures, the lineshape consists of a single, broad resonance whose amplitude decreases with decreasing temperature. The simplicity of this spectral shape suggests that all implanted 8Li+stop in high-symmetry lattice positions where the local electric field gradient (EFG) is vanishing [86]. These sites are likely the tetrahedral interstitial positions (e.g. (1/4, 1/4, 1/4) in figure 1), which are vacant in the ideal rocksalt structure. Such a site assignment aligns with the cubic symmetry of Nb1−xTixN (see figure 1), as well as with observations in isostructural compounds like MgO [87]. In the film’s normal state, the resonance’s width is approximately temperature-independent, with a full width at half maximum (FWHM) of 8.126(18) kHz. This large width is likely due to the high natural abundance of spin-active isotopes that make up the film’s elemental composition (see table 1), and is consistent with values predicted from dipolar broadening by host nuclear spins for the likely interstitial or substitutional sites in an ideal, unperturbed host lattice. Below Tc, the resonance broadens substantially, with the linewidth gradually increasing by a factor ∼2. Such a broadening is typical of a superconductor upon formation of the vortex state’s flux-line lattice (FLL) [88]. To quantify these features, the resonances were fitted using a phenomenological pseudo-Voigt (pV) function: pV(ν) = A[αG(ν) + (1−α)L(ν)],(4) which represents a linear combination of Gaussian (G) and Lorentzian (L) components: G(ν)≡exp −(ν−ν0)2 2σ2!, L(ν)≡Γ2 L (ν−ν0)2+ Γ2 L , and it mimics features of the true Voigt function which is a convolution of the two, and provides a good description of the data. Here, Ais the resonance amplitude, α∈[0,1]is a mixing term that defining the lineshape’s Gaussian fraction, ν0is the resonance frequency, σ= Γ/(2√2ln2)is the 4 J. Phys.: Condens. Matter 37 (2025) 395701 M Asaduzzaman et al Figure 3. Typical 8Li NMR lineshapes in Nb0.75Ti0.25N(91 nm) /AlN(4 nm)/Nb, measured in an applied field B0=4.1T perpendicular to the sample’s surface, at select temperatures T (indicated in the figure) above and below the film’s critical temperature Tc≈15K. For T>Tc, the resonance linewidth remains roughly T-independent, but broadens by up to a factor of ∼2 at lower temperatures. The solid black lines represent fits to the data using equation (4). Table 1. Stable spin-active nuclei present in Nb1−xTixN. Here, nAis the natural isotopic abundance, Iis the nuclear spin, γis the gyromagnetic ratio, and Qis the electric quadrupole moment. For comparison, properties of our β-NMR probe 8Li are also listed. Inapplicable properties are marked by an asterisk (∗). Isotope nA(%) Iγ/(2π)(MHz T−1)Q(mb) 14N 99.58 1 3.07 771 +20.44 15N 0.42 1/2 −4.31 727 ∗ 47Ti 7.44 5/2 −2.40 410 +302 49Ti 5.41 7/2 −2.40 475 +247 93Nb 100 9/2 10.43 956 −320 8Li ∗2 6.30 198 +32.6 Gaussian width parameter, ΓL= Γ/2 defines the Lorentzian width, and Γdenotes the line’s (common) FWHM. Consistent with the qualitative description above, the T-dependence of the line’s FWHM remains its most salient feature. Assuming the Lorentzian contribution to the line is systematic (i.e. ‘power broadening’ from the single-tone continuous wave (CW) technique described in section 2) [89], we encapsulate this broadening by σ’s T-dependence, which is shown in figure 4. Although some scatter is evident, σremains approximately constant above Tc, but increases monotonically below the superconducting transition. To understand this broadening, we consider the following model. In the normal conducting (nc) state, the Gaussian width Figure 4. Temperature Tdependence of the Gaussian component σ of the resonance linewidth Nb0.75Ti0.25N(91 nm)/AlN(4 nm)/Nb in an applied field of B0=4.1T perpendicular to its surface. Note that the parameter σis expressed as a magnetic field in units of mT. The solid red line represents a fit to the data for T<35K using the model given by equations (5)–(9), capturing the line broadening below the superconducting transition temperature Tc, with the T-independent (normal state) contribution highlighted as a dotted line. Measured values at temperatures up to 270 K are shown in the inset, where (apart from some small scatter) σ’s T-independence is evident. σcan be approximated by a T-independent constant, whereas in the vortex state the measured value consists of a convolution of the superconducting (sc) and nc contributions: σ2=(σ2 sc +σ2 nc,T<Tc, σ2 nc,T⩾Tc.(5) In an isotropic type-II superconductor with GL parameter κ≫ 1, σsc is related to the material’s effective magnetic penetration depth λeff by the expression [88,90]: σ2 sc (T) = 0.00371 Φ2 0 λ4 eff (T),(6) where Φ0=2.068 ×10−15 Wb is the magnetic flux quantum [91]. In thin-film superconductors subjected to a magnetic field normal to their surface, when the film thickness dis smaller than the material’s (bulk) penetration depth λ,λeff is given by the Pearl length [92]: λeff (T) = λ2(T) d,(7) where we account for λ’s T-dependence using the analytic approximation [93]: λ(T)≈λ(0K) q1−[T/Tc(B0)]2,(8) 5 J. Phys.: Condens. Matter 37 (2025) 395701 M Asaduzzaman et al Table 2. Fit parameters describing the temperature dependence of resonance linewidth’s Gaussian component σ(shown in figure 4) using equations (5)–(9). Here, dNb0.75Ti0.25Nis the thickness of the Nb0.75Ti0.25N thin film, Tc(0T)is the critical temperature at 0 T, λ(0K)is the penetration depth at 0 K, σnc is the Gaussian width in the nc state, B0is the applied magnetic field, and Bc2(0K) represents the upper critical field at 0 K. Both dNb0.75Ti0.25Nand B0 were determined independently and fixed during fitting. Parameter Value Unit Comment dNb0.75Ti0.25N91 nm Fixed (from section 2.1) Tc(0T)15.4(7) K λ(0K)180.57(30) nm σnc 540.1(19) µT B04.1 T Fixed (from section 2) Bc2(0K)18(4) T where λ(0K)is the penetration depth at 0 K and Tc(B0) accounts for the suppression of the transition temperature in an applied field [94]. We treat this suppression empirically by inverting an analytic approximation for the simplest solution to the Werthamer–Helfand–Hohenberg (WHH) [95] expression for Bc2 [96]: h∗(t)≈0.693 B0 Bc2 (0K),(9) where Bc2(0K)is the upper critical field at 0K, h∗(t)≡1−t−0.153(1−t)2−0.152(1−t)4, and t≡Tc(B)/Tc(0T). Further details are given in section S3 of the supplemental material [82]. We note that WHH theory has been used describe the T-dependence of Bc2 in Nb1−xTixN [1,50] and related materials (e.g. NbTi [97] and Nb3Sn [96]), implying the correctness of this choice here, where fitting our data to this expression yields Bc2(0K) = 18(4)T. A fit of equations (5)–(9) to the σ(T)data is shown in figure 4. In the fit, both the film thickness d=91nm and the applied field B0=4.1T were fixed to their known values (see section 2), with all other parameters left free. The fit is in good agreement with the data, with the optimal values for the free parameters summarized in table 2. At this juncture, we note the good agreement of the extracted Tcwith the value identified by magnetometry measurements (see section 2.1), as well as the λ(0K)measured in other films [29]. We shall consider these results further in section 4.1. Besides the changes to the resonance lineshape induced by the superconducting transition, we also quantified 8Li’s NMR shift Kcin the film. Using the resonance position in single crystal MgO with B0∥(100)at 295 K as a reference [87], Kc(in ppm) is obtained using the expression (see, e.g. [98]): Kc=106ν0/ζ0−νMgO/ζMgO νMgO/ζMgO ,(10) where the factor ζi=1+1 3−Niχi(11) corrects for contributions from demagnetization [99], with Nidenoting the demagnetization factor and χidenoting the material’s volume susceptibility. While details of the full calculation can be found in section S4 of the supplemental material [82], we summarize the main results below. We find that Kcvaries between +15 ppm to +35 ppm over the measured Tranges. This magnitude is typical of 8Li in many materials [66, 67], but as is common with small NMR shifts, corrections for demagnetization are a dominant contribution (+25.4 ppm here) [100]. This range of Kcis small compared to other metals (see, e.g. [101]), suggesting weak hybridization with the ternary alloy’s conduction band. We shall consider this quantity further in section 4.1. 3.2. SLR spectra Representative time-differential SLR data at various temperatures are shown in figure 5. At low temperatures the relaxation is very slow, approaching the limit of what is measurable by 8Li due to its radioactive lifetime [66]. As the temperature is raised, so too does the relaxation, though it remains slow compared to other cubic metals (e.g. Ag [102] and Nb [103]), but comparable to other metallic compounds where the probe’s hybridization with the conduction band is weak [104–106]. Interestingly, this increase in relaxation is non-monotonic with temperature, with a local maximum observed near ∼140 K. At all temperatures, the SLR is non-exponential, as might be expected for a host with an abundance of spin-active nuclei (see table 1) and three-dimensional (3D) disorder (see, e.g. [107]). We note that even in the absence of structural disorder from alloying, 93Nb NMR of the end member NbN also displays non-exponential SLR [108], suggesting the behaviour may be intrinsic. To quantify these observations, we adopt a phenomenological approach used to analyze other disordered metal-like compounds [105,106] and fit the SLR data using a stretched exponential model. Explicitly, for an 8Li ion implanted at time t′, the spin polarization at a later time t>t′is given by: R(t,t′) = exp −(t−t′) T1β!,(12) where 1/T1is the SLR rate (i.e. the reciprocal of the signal decays to 1/eof its initial value) and 0 < β ⩽1 is the stretching exponent. This model provides a simple yet effective fit to the data, while minimizing the number of free parameters. To further avoid overparameterization, all SLR data were fit simultaneously using equation (12) convoluted with the 4 s beam pulse and a shared β=0.216(6)[109]. While this approach yields an excellent fit (reduced χ2≈1.02), as pointed out by others [110,111], this choice has the caveat of imparting some temperature dependence to A0. We assert that this choice does 6 J. Phys.: Condens. Matter 37 (2025) 395701 M Asaduzzaman et al Figure 5. 8Li SLR data at various temperatures Tin Nb0.75Ti0.25N(91 nm)/AlN(4 nm)/Nb, measured under a perpendicular applied field of B0=4.1T. The shaded region indicates the duration of the 8Li+beam pulse (4 s). The data exhibit T-dependent relaxation that is non-monotonic with temperature, with a significant non-relaxing or very slow-relaxing component. The solid black lines represent fits to a stretched exponential (equation (12)) convoluted with the 8Li+beam pulse using a common stretching exponent β(described in section 3.2). The displayed data have been binned by a factor of 20 for clarity. not impede the quantitation of 1/T1[112] and that this model provides a simple, accurate fit across all measured conditions. We note that the small βinidicates a very broad distribution of rates, and the relaxation could also be modeled as a sum of exponentials. However, the extra parameters required to define this relaxation lead to overparametrization. The resulting 1/T1 values in both normal and superconducting states are shown in figure 6. Consistent with the above observations, the T-dependence of 1/T1exhibits a rich assortment of behavior. At temperatures below ∼100 K, 1/T1varies linearly with T, typical of metallic systems [69]. Near the film’s Tc, this linear proportionality is modified, revealing a small Hebel–Slichter coherence peak [70,71] that decays exponentially to zero for T≪Tc. Such a feature is expected for an s-wave Bardeen-CooperSchrieffer (BCS) superconductor, but a rare observation by 8Li β-NMR (see, e.g. [113]). Above ∼100 K, additional relaxation contributions appear superimposed on the T-linear contribution. At ∼140 K, 1/T1goes through a local maximum, reminiscent of a Bloembergen-Purcell-Pound (BPP) peak [114]. At higher temperature, the peak vanishes and 1/T1increases exponentially, suggesting another distinct contribution to the SLR. With these qualitative features in mind, we now consider a quantitative model to describe them. We postulate that the distinct T-dependencies of 1/T1correspond to the presence of distinct relaxation mechanisms, with the measured SLR rate Figure 6. Temperature T-dependence of the 8Li SLR rate 1/T1in Nb0.75Ti0.25N(91 nm)/AlN(4 nm)/Nb at B0=4.1T. 1/T1varies nonmonotonically with temperature, with T-linear behavior below ∼100 K that is modified by a Hebel–Slichter coherence peak below the film’s Tc(see inset for a detailed view). Near ∼140 K, a Bloembergen–Purcell–Pound (BPP) peak is observed, while at higher temperatures 1/T1increases exponentially with increasing T. The solid red line shows the fit to equations (9) and (13)–(21) (described in section 3.2), with the individual contributions from the linear slope, BPP peak, and exponential rise shown as dotted lines. The green dash-dotted and blue dashed curves show the expected Hebel–Slichter coherence peak for fixed ΓD=0 and 0.1 meV, respectively, using the same fit parameters. corresponding to their sum. We make the ansatz that the rate contributions can be added linearly: 1 T1 =1 T1e +1 T1BPP +1 T1exp ,(13) where each (1/T1)iterm represents a specific contribution i. Here, we assign i=e to relaxation due to conduction electrons, i=BPP to the BPP peak, and, i=exp to the exponential takeoff at high-T. We now consider the detailed form of each mechanism. The SLR in metallic systems is governed by spin-flip scattering ‘collisions’ between conduction electrons and the probe nuclear spins. The magnitude of this contribution is proportional to the DOS at the Fermi level EF, as well as the probe’s hybridization with the host’s conduction band, yielding a relaxation rate that varies linearly with T[69]. In BCS superconductors below Tc, the condensation of Cooper pairs along with the opening of a gap in the DOS at EFmodifies this linearity, producing a coherence peak [70,71] just below Tcand exponential decay of (1/T1)eto zero as T→0K (i.e. due to the freezing-out of all cross-gap thermal excitations). Quantitatively, this can be described by (see, e.g. [60, 115,116]): 7 J. Phys.: Condens. Matter 37 (2025) 395701 M Asaduzzaman et al 1 T1e =   mT,T>Tc mT ×2 kBTˆ∞ 0 f(E)[1−f(E′)][Ns(E)Ns(E′) + Ms(E)Ms(E′)]dE,T⩽Tc (14) where mis the so-called Korringa slope [69], Eand E′= E+ ¯hω0are the superconducting quasiparticle energies in their initial and final scattering states (the latter defined by the NMR probe’s Larmor frequency), f(E) is Fermi function: f(E) = 1 exp[E/(kBT)] + 1. kBis the Boltzmann constant, Ns(E)denotes the superconducting DOS, and Ms(E)refers to the anomalous quasiparticle density arising from the coherence factor [60,117]. These latter two quantities can be expressed as: Ns(E) = Re(E−iΓD)/h(E−iΓD)2−∆2i1/2,(15) Ms(E) = Re∆/h(E−iΓD)2−∆2i1/2,(16) where ∆is the superconducting energy gap [118]: ∆(T)≈∆(0K)v u u tcos"π 2T Tc2#,(17) ∆(0K)is the gap’s value at 0 K, and ΓDis an empirical broadening parameter (e.g. accounting for finite quasiparticle lifetimes) [85]. Additionally, the dependence of Tc under an applied magnetic field follows the same relationship as described in equation (9), and also quantifies the Bc2 independently. We now consider the SLR contribution from the BPP peak. Quite generally, a local maximum in the SLR rate manifests when the correlation rate τ−1 cof the fluctuating interaction causing the relaxation matches the probe’s Larmor frequency [see equation (3)]. Far away from this ‘resonance,’ the contribution to the relaxation is negligible, leaving a ‘peak’ that is superimposed atop the other SLR contributions. Explicitly, this contribution can be described by [114,119]: 1 T1BPP =c(J1+4J2),(18) where cis a coupling constant proportional to the meansquared transverse fluctuating field, and Jnis the n-quantum NMR spectral density function [119]: Jn=τc 1+ (nω0τc)2.(19) As ω0is fixed in the measurements, the temperature dependence of equation (18) arises from τ−1 c, which we assume follows an Arrhenius form: τ−1 c=τ−1 0exp[−EA/(kBT)],(20) where τ−1 0is the attempt frequency, and EAis the activation energy. Note that these expressions are agnostic the source of the (thermally activated) dynamics causing the relaxation ‘peak.’ Finally, we consider a model for the growth of the SLR observed for T≳200K. Recalling the exponential-like takeoff noted above, we describe this behavior empirically using: 1 T1exp ≈gexp[−Eexp/(kBT)],(21) where gis a prefactor, Eexp is the activation energy. Note that the form of equation (21) is identical to equations (18)–(20) in the ‘slow fluctuating’ limit (i.e. when τ−1 c≪ω0). Combining the above models (equations (9) and (13)–(21)), we fit the 1/T1vs. Tdata shown in figure 6. For consistency with the lineshape analysis in section 3.1, we fixed both Tc(0T)and Bc2(0K)at the values listed in Table 2. Similarly, we fixed ΓDat 0.02 meV to mitigate its strong correlation with ∆(0K)(i.e. the present data lacks to the precision to simultaneously identify both quantities) [120]. We note that this value small compared to what is expected from PCT (see section 2.1); however, it is consistent with the range reported by others [53,121,122]. This (empirical) restriction notwithstanding, our approach enables a robust description of 1/T1’s T-dependence across both Nb1−xTixN’s normal and superconducting states, with the resulting fit in good agreement with the data (see figure 6). Values of the extracted fit parameters are summarized in table 3. 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