Evidence for current suppression in superconductor-superconductor bilayers

Superconducting radio frequency (SRF) cavities, which are critical components in many particle accelerators, need to be operated in the Meissner state to avoid strong dissipation from magnetic vortices. For a defect-free superconductor, the maximum attainable magnetic field for operation is set by the superheating field, $B_{\mathrm{sh}}$, which directly depends on the surface current. In heterostructures composed of different superconductors, the current in each layer depends not only on the properties of the individual material, but also on the electromagnetic response of the adjacent layers through boundary conditions at the interfaces. Three prototypical bilayers [$\mathrm{Nb_{1-x}Ti_xN}$(50 nm)/Nb, $\mathrm{Nb_{1-x}Ti_xN}$(80 nm)/Nb, and $\mathrm{Nb_{1-x}Ti_xN}$(160 nm)/Nb] are investigated here by depth-resolved measurements of their Meissner screening profiles using low-energy muon spin rotation (LE-$\mu$SR). From fits to a model based on London theory (with appropriate boundary and continuity conditions), a magnetic penetration depth for the thin $\mathrm{Nb_{1-x}Ti_xN}$ layers of $\lambda_\mathrm{Nb_{1-x}Ti_xN} =$ 182.5(31) nm is found, in good agreement with literature values for the bulk alloy. Using the measured $\lambda_\mathrm{Nb_{1-x}Ti_xN}$, the maximum vortex-free field, $B_{\mathrm{max}}$, of the superconductor-superconductor (SS) bilayer structure was estimated to be 610(40) mT. The strong suppression of the surface current in the $\mathrm{Nb_{1-x}Ti_xN}$ layer suggests an optimal thickness of $\sim 1.4 \lambda_{\mathrm{Nb_{1-x}Ti_xN}} =$ 261(14) nm.


I. INTRODUCTION
A large accelerating gradient ( acc ) (energy gain per unit length) is required for high energy accelerators to limit their length and therefore their cost [1,2].Currently, the highest  acc values are achieved using normal conducting radio frequency (RF) cavities, some exceeding 100 MV m −1 [3,4].In the case of field-emission-free superconducting radio frequency (SRF) cavities, the maximum  acc is proportional to the highest sustainable vortex-free surface magnetic field, which is presently achieved by cavities made from niobium sheets.Some of these cavities have produced  acc values as high as ≈49 MV m −1 [5], corresponding to surface magnetic fields on the order of ∼210 mT, exceeding "clean" Nb's lower critical field,  c1 ≈ 170 mT at 2 K [6,7].While this achievement is commendable, it remains below the ultimate limit for bulk Nb, which is set by its superheating field,  sh ≈ 240 mT [8].While Nb cavity operating conditions continue to approach this material limit, substantial advances in accelerator technology necessitate finding alternative materials.

A. SRF materials beyond niobium
One possibility to achieve surface magnetic fields beyond  sh of Nb is to use a different superconducting material with a greater  sh (e.g., Nb 3 Sn or Nb 1-x Ti x N) [9]; however, there is no viable replacement with a  c1 exceeding that of Nb.This is problematic, as all real SRF cavities possess both surface defects and topographic imperfections, facilitating vortex penetration below  sh .Vortices that penetrate at these "weak points" often evolve into a thermomagnetic flux avalanche, quenching superconductivity at SRF cavity operating temperatures ( ≲ 4 K) [10][11][12].
To overcome this, a different approach has been proposed, wherein superconducting multilayers are used to push the field of first-flux penetration beyond Nb's intrinsic  sh (see e.g., [10,[12][13][14]).Gurevich [13] was the first to suggest the use of multilayer structures as a means of preventing thermomagnetic avalanches induced by vortex penetration at defects before they become predominant.The approach is to coat a conventional Nb cavity with several thin superconducting and insulating layers, the simplest version of which is one superconducting and one insulating layer on Nb, referred to as a superconductor-insulator-superconductor (SIS) structure.The insulating layer decouples the superconducting layers and if the layers are thinner than the London penetration depth ( L ) of their material, nucleation of parallel vortices will only become energetically favorable at larger fields than  c1 of layer material.Kubo [10] suggested that a simpler structure containing only a single superconducting layer with a larger penetration depth on top of a Nb cavity can also increase the field of first vortex penetration ( vp ) due to the presence of an energy barrier at the superconductor-superconductor (SS) interface analogous to the vacuum-superconductor interface (i.e., the Bean-Livingston (BL) barrier [15]).Experimental evidence for this interface barrier has been reported in Ref. 16.
In summary, the maximum field in superconducting heterostructures that can be sustained while remaining in the Meissner state ( max ) depends on the thickness and superconducting properties of all individual layers in a correlated way.This is a direct consequence of Maxwell's equations with continuity conditions enforced at interface boundaries [10].

B. Magnetic screening and current in superconducting heterostructures
Recall that, for a bulk superconductor in the "local" London limit (see e.g.[17]) with an ideal flat surface, the Meissner screening profile, (), is given by [18]: ,  ≥ 0, (1) where  0 is the (effective) applied magnetic field,  is the depth below the superconductor's surface, and  L is the London penetration depth.Equation ( 1) is well-known for its applicability to semi-infinite superconductors; however we are interested in SS bilayers comprised of dissimilar layers whose materials have different screening properties (i.e.,  L s).Considering a naive exponential London decay in each component of the SS bilayer by treating the screening properties independently, the field screening profile is given by: where  s is the thickness of the top superconducting layer, and the   denote the penetration depth in the surface ( = s) and substrate ( = sub) layers, respectively.While Equation ( 2) is both conceptually simple and qualitatively correct in its form, it does not consider any "coupling" between the adjacent layers.The substrate having a substantial influence on the surface superconductor's screening properties, when the surface layer superconductor penetration depth differs from the substrate.Since an SS bilayer's electromagnetic (EM) response depends on the boundary/continuity requirements for the magnetic field and vector potential.Recently, it has been predicted that this coupling depends also on the surface layer's thickness and is most effective when  s ∼  s [14].For example, when the surface layer penetration depth is larger than the substrate's (i.e.,  s >  sub ), the Meissner current in the surface layer is suppressed by the substrate layer's countercurrent (i.e., a counterflow current generated by the substrate in a multilayer superconductor [10][11][12]19]) to satisfy the boundary and continuity condition at the interface.This results in a higher -field for vortex entry in the outer layer with a correspondingly a reduced shielding of the substrate (higher field at the substrate interface).This effect is expected for all superconducting heterostructures with and without insulting interlayers.Quantitatively, the field screening considering counter-current-flow induced by the substrate is derived by solving the relation between the applied field,  0 and current density,  (or equivalently vector potential, ).For a SS structure this yields [10,12,14,19]: where the symbols have the same meaning as in Equation ( 2), and, the common factor  S−S is given by: The current density distribution,  () can be obtained from the field screening profiles using the expression  () = − 1  0 d() d .Both Equations ( 2) and (3) are essentially forms of exponential decay; however, the screening behavior is sig-nificantly modified in the surface layer.Figure 1 presents a comparison of the magnetic field profiles and current density distributions, with (a) showing normalized field screening behavior and (b) representing normalized current density distributions.In both figures, the solid red curve describes the London screening behavior in the absence of a "coupling" between the superconducting layers [Equation (2)], whereas the blue dashed curve corresponds to screening according to Kubo's counter-current-flow model [Equation (3)].Here, the SS bilayer is Nb 1-x Ti x N (50 nm)/Nb with assumed penetration depths of  Nb 1-x Ti x N = 200 nm and  Nb = 50 nm for Nb 1-x Ti x N and Nb, respectively.As alluded to above, the two models have qualitatively similar behavior; ()'s decay rate in the Nb substrate is identical, with the two curves differing only in their amplitudes at the SS boundary.This similarity is also observed in the decay rate of  () in the Nb substrate.Conversely, a notable difference is apparent in the top Nb 1-x Ti x N layer, where the decay rate is substantially reduced in Kubo's model.This is the effect of the reduced current in the surface layer as seen in Figure 1(b) due to the counter-current induced by the substrate.In order to observe the effect of counter-current in SS bilayers, it is necessary to investigate the field screening profiles experimentally and quantify the penetration depths using an appropriate model.For the first time, we measured the Meissner screening profile and observed suppression of screening current in the surface layer in Nb 1-x Ti x N/Nb samples with different alloy thicknesses using the low-energy muon spin rotation (LE-SR) technique.These measurements were conducted under applied fields (15 ≲  0 ≲ 25) mT.The benefit of LE-SR is that it allows a direct measurement of the magnetic flux profile locally across the sample, providing information about the field screening profile.By fitting the field profile, the magnetic penetration depths of Nb 1-x Ti x N,  Nb 1-x Ti x N and Nb,  Nb are quantified comparing Kubo's counter-current-flow model (i.e., London theory with appropriate boundary and continuity conditions) and a simple London model without appropriate boundary conditions.The resultant comparison highlights the significant suppression of the Meissner current in the surface layer, while also experimentally validating counter-current-flow model.

A. The LE-𝜇SR technique
LE-SR experiments were performed at the Paul Scherrer Institute's (PSI) Swiss Muon Source located in Villingen, Switzerland, using the E4 beamline [20].The muon beamline is used to reduce the energy of a "surface" muon beam of ∼4 MeV down to around 15 eV.Following that, the muons are accelerated to create a beam with an adjustable energy  ≤ 30 keV which corresponds to an implantation depth of ≲ 150 nm in Nb and Nb-based alloys.These low energy positive muons ( + ) are ∼100 % spin-polarized.The  + are implanted into a sample one at a time using a (quasi-)continuous beam [21], wherein they quickly thermalize in the target and their spins precess around the local magnetic field at the Larmor frequency,   permitting depth-resolved measurements of the field screening profile in surface-parallel applied fields up to ∼30 mT [22].
When a  + decays, it emits a positron preferentially along its spin direction at the moment of decay.The emitted positrons are detected as a function of time in a set of positron detectors symmetrically placed surrounding the sample.This allows for the temporal evolution of the muon's spin orientation to be deduced, and consequently, the properties of the magnetic fields it experiences.
In this experiment, the asymmetry of  + decay is determined in a transverse field arrangement wherein a magnetic field is applied perpendicular to the initial direction of muon spinpolarization and parallel to the sample surface.The positron event rate in one (or more) "counters" , is given by: where   = 2.2 µs is the muon lifetime,  0, represents the total number of "good" decay events (i.e., decays from muons stopped in the sample),   is the time-independent rate from uncorrelated "background" events, and   () represents the time-evolution of the muon ensemble asymmetry: where  0, is the experimental decay asymmetry and () is the polarization of the muon ensemble.
In a transverse-field experiment, the time-evolution of () is given by: where () is the internal magnetic field distribution sensed by the muons,   = 2 × 135.54 MHz T −1 is the gyromagnetic ratio of the muon,  is the magnitude of the local magnetic field at the muon site,  is the time after implantation, and  is the phase factor (i.e., angle between the initial muon spin-polarization and the effective symmetry axis of a positron detector).

B. Muon Stopping Profiles
As mentioned in Section II A, LE-SR has the ability to explore the local field in a depth resolved manner.Muons of a particular energy stop over a specific range distribution when implanted into a sample.In this experiment, a range of implantation energies (∼2 keV to ∼30 keV) were used (see Figure 2), providing depth-resolution on the nm scale (i.e., ∼10 nm to ∼150 nm).
The stopping profile of muons can be accurately simulated [24][25][26] using the TRIM.SP code (a Monte Carlo code) [23], which treats all collisions within the target using the binary collision approximation.Simulation results for  + implanted in a Nb 1-x Ti x N(50 nm)/Nb, and a Nb 1-x Ti x N(160 nm)/Nb SS bilayer are shown in Figure 2, illustrating LE-SR's typical range of spatial sensitivity.For our analysis (see Section III), it was convenient to have the ability to describe these profiles at arbitrary , which can be accomplished by fitting the simulated profiles and interpolating their "shape" parameters [26].Empirically, we found the  + stopping probability, (), at a given  can be described by: where   () is a probability density function,   ∈ [0, 1] is the  th stopping fraction, constrained such that and  is the depth below the surface.For our SS bilayers, the stopping data are well-described using  = 2 and a () is given by a modified beta distribution.Explicitly, where  ∈ [0,  0 ] is the depth below the surface and (, ) is the beta function: Typical stopping profiles for  + implanted in (a) Nb 1-x Ti x N(50 nm)/Nb, and (b) Nb 1-x Ti x N(160 nm)/Nb SS bilayer, simulated using the Monte Carlo code TRIM.SP [23].The densities of Nb 1-x Ti x N and Nb are 6.6223 g cm −3 , and 8.57 g cm −3 , respectively.The light gray color in the first 50 nm of figure (a) and 160 nm of figure (b) refers to the Nb 1-x Ti x N film thickness on bulk Nb substrate (i.e., dark gray color).The normalized stopping distribution  of  + is plotted against the depth  below the surface.The black solid curves are fits to the stopping profile (represented as a histogram) using Equations ( 7) and (8).These fits clearly capture all features of the stopping profiles.
with Γ() denoting the gamma function.Further details of the stopping profile simulation can be found elsewhere [24,26].

C. Sample Preparation
In this study, Nb 1-x Ti x N/Nb SS bilayer samples were prepared by growing thin films of Nb 1-x Ti x N on "bulk" Nb substrates using direct current (DC) magnetron reactive sputtering (R-DCMS) in a vacuum chamber with a base pressure of low 1 × 10 −10 mbar.The sputtering target consisted of 80/20 (wt.%) Nb/Ti alloy, was used within an Ar and N 2 (P N 2 /P Ar ) gas mixture at a pressure of 2 × 10 −3 mbar.Films with nominal thicknesses of 50 nm, 80 nm, and 160 nm were deposited at 450 °C on 3 mm thick bulk Nb substrates, with respective  c values of 15.8 K, 16.3 K and 16.3 K [35].The substrates were prepared by mechanical polishing (MP) followed by 5 µm cold electropolishing (EP) or by 50 µm buffer chemical TABLE I. Superconducting properties of Nb 1-x Ti x N films from several literatures [27][28][29][30][31][32][33].Here,  c is the critical temperature,   is the thermodynamic critical field,  c1 is the lower critical field,  sh is the superheating field,  c2 is the upper critical field,  is the penetration depth, and  is the BCS [34]  polishing (BCP) (see e.g., Ref. 36).Specifically 50 nm sample was prepared using MP and others using BCP.Prior to film growth, the substrates were baked at 600 °C for 24 h under vacuum and the Nb 1-x Ti x N films were annealed at 450 °C after deposition.The typical surface roughness of the Nb 1-x Ti x N layer is similar to the original substrate roughness (1 nm for MP+EP substrates and microns for BCP substrates).All film depositions were performed at Thomas Jefferson National Accelerator Facility (JLab) and further details on deposition technique can be found in Ref. 27.Note that, unlike the elemental superconductors, the magnitude of superconducting properties (such as the penetration depth and the coherence length) of Nb 1-x Ti x N are not robust.This is due to the fact that Nb 1-x Ti x N is not a "natural" compound [27].Therefore, the superconducting properties of some Nb 1-x Ti x N films prepared using different target stoichiometries, deposition techniques, and preparation methods have been reviewed from the literature and are listed in Table I for reference.Although the tabulated values for various samples show considerable variation, the attributes derived from all reviewed research are in fair agreement with one another.These will be used to compare our measured penetration depths in Section III as well as for the prediction of critical fields in Section IV A.

III. RESULTS
Typical muon spin-precession signals are shown in Figure 3(a) for the normal conducting state (20 K) and in Figure 3(bf) for the Meissner state (2.7 K) in Nb 1-x Ti x N (50 nm)/Nb.In the normal state, there is no substantial energy dependence to the time evolution of the muon ensemble polarization.This means muons implanted at different depths experience almost the same local field.By contrast, in the Meissner state the temporal evolution of () varies as the implantation energy increases, wherein the  + spin-precession rate is greatly reduced, and the signal is more strongly damped at high implantation energies.
Figure 4 shows the Fourier amplitude (i.e., √︁ (Fourier power) [37]) of the LE-SR time spectra depicted in Figure 3  7 K) at an applied magnetic field of ∼25 mT parallel to the sample surface.In the normal state (gray shaded background panel), there is no substantial energy dependence to the time evolution of the muon ensemble polarization, meaning the implanted muons experience the same local field.By contrast, it is evident that the temporal evolution of () varies in the Meissner state (plain white background panels).As the implantation energy increases, the  + spin-precession frequency is reduced, and the signal is more strongly damped.The solid red lines denote fits to all of the data (i.e., a global fit) using Equations ( 11) to (13).Clearly, the model captures all the data's main features.
fields) [see Figure 4(b-f)].For energies above ∼14.5 keV, the Fourier spectra show two distinct peaks, implying at least two unique field regions are sensed, consistent with the different materials in the SS bilayer.
The measured internal field distribution, (), in the Meissner state depends on energy via the muon implantation depth profile and the magnetic screening due to the Meissner current.We will now consider how to approximate () in Equation ( 6) for our analysis.In the Meissner state, the applied field decays to zero monotonically below the sample surface and the field screening is expected to be intrinsically asymmetric.For the Nb 1-x Ti x N/Nb samples, it is found that a sum of two skewed Gaussian (SKG) components (i.e., one for each material) gives a good fit describing the data in all measurement conditions.Because each layer in the SS has a different screening properties, the SKG distribution function is defined as [38]: where  p is the "peak" field (i.e., not the mean) and  ± denotes the distribution's "width" on either side of  p .
By substituting Equation ( 9) into Equation ( 6) for (), the polarization formula can be written as: where where Erfi() is the imaginary error function.Therefore, the total asymmetry signal () yields: where  reflects the fraction of muons stopping in each component of the SS bilayer, constrained such that To fit the data, the program musrfit was used [39].The red lines in Figure 3 are fits to all the data (i.e., a global fit) of the 50 nm sample using Equations ( 9) to (13), where the phase,  is shared as a common parameter.The imposition of this restriction is necessary because in situations where () is significantly damped at high implantation energies in the Meissner state, the phase becomes poorly defined, and only a few complete precession periods can be resolved.The fit was constrained such that for  ≤ 14.5 keV (i.e., mean stopping depths ≤ 50 nm) we assumed  = 1 in Equation ( 12) and used  = 2 at higher implantation energies.This choice gave the best fit to the data at all measurement conditions, as evidenced by the goodness of fit criterion (i.e., reduced- 2 = 1.06).
In order to construct the Meissner screening profile in the Nb 1-x Ti x N (50 nm)/Nb sample, the mean field, ⟨⟩, needs to be derived from () for each implantation energy .The ⟨⟩ is a convenient means of encapsulating the ()'s shift to lower fields as the  increases.The ⟨⟩ is derived using the fit parameters  p, ,  +, , and  −, (see Appendix A) of Equation ( 12): The field screening profile of Nb 1-x Ti x N (50 nm)/Nb at an applied field of  0 ∼ 25 mT as a function of energy  (bottom scale) and corresponding mean implantation depth ⟨⟩ (top scale) in the Meissner ( = 2.7 K) and normal state ( = 20 K) is shown in Figure 5.The closed circles and open squares in Figure 5(a) and (b) represent the mean field ⟨⟩ of the same data.In the normal state the ⟨⟩ is not screened and in the Meissner state ⟨⟩ decays with increasing  as expected.
In order to fit ⟨⟩, we shall consider a model that describes all essential features of the data.In Equations ( 6) and ( 9) to ( 14), ⟨⟩ is derived by fitting a field distribution () at a given energy .At specific , muons sample over a range of depths (i.e., distribution) which is simulated and quantified by () as discussed in Section II B. The quantities () and () are both energy dependent.Hence, ⟨⟩ depends on the Meissner screening and the  + implantation distribution ().The mean field ⟨⟩ as a function of  is therefore: where the dependence on  is accounted for implicitly by (, ) which is predetermined from fits to simulated implantation profiles (see Figure 2).The screening profile () is derived from Equations ( 2) and (3).Note that the applied magnetic field,  applied in both Equations ( 2) and ( 3 3) in Nb 1-x Ti x N (50 nm)/Nb as a function of field (note   =   ), in the normal (20 K) and Meissner (2.7 K) state with an applied magnetic field of ∼25 mT.The red lines are skewed Gaussian fits corresponding to the field distribution described by Equations ( 11) to (13).Above ∼14.5 keV two distinct peaks are observed indicating that muons of a single implantation energy sense the field in both layers of the SS bilayer.
the Meissner state due to the sample geometry, which needs to be accounted.This is done by using: where the demagnetization factor  depends on the geometry of the sample [40][41][42].To compare our measured penetration depths with literature values (see Table I), the  dependence of  was assumed to follow the phenomenological power law [43]: where (0) is the magnetic penetration depth at 0 K.The fits to the normal state data in Figure 5 are represented by dashed red curves.The solid red curves denote fits to the Meissner state data using Equation (15) and one of the screening models introduced in Section I B (i.e., counter-current-flow or simple London model).Figure 5(a) is fitted with a simple London model (Equation ( 2)), and the counter-current-flow model field distribution (Equation ( 3)) is used to fit the data in Figure 5(b).It can be seen that both models capture all physically meaningful details of the data and give excellent fits.The fit parameters for both models are tabulated in Table II.The values of the extracted parameters ,  Nb , and  Nb 1-x Ti x N are almost identical in the two models.However, a large discrepancy exists between the determined values of  Nb 1-x Ti x N .The simple London model gives  Nb 1-x Ti x N = 498(34) nm, while Kubo's counter-current-flow model gives  Nb 1-x Ti x N = 185(7) nm.Interestingly, the value determined using Kubo's counter-currentflow model [Equation (3)] is in good agreement with literature estimates (see Table I), whereas the expression in Equation ( 2) overestimates  Nb 1-x Ti x N by a factor of ∼2.5.This observation strongly supports the predictions of the counter-current-flow theory [10] and suggests that Equation ( 2) is not appropriate for quantifying () in superconducting heterostructures.
To be more conclusive about this observation, we measured the field screening profile in three samples with different Nb 1-x Ti x N thicknesses (50 nm, 80 nm, and 160 nm) deposited on Nb substrates, see Figure 6.Using the counter-current-flow model, the field screening profiles were fitted simultaneously (i.e., global fit) with the penetration depth values at 0 K of TABLE II.Fit results of the Nb 1-x Ti x N (50 nm)/Nb bilayer with a counter-current-flow (i.e., Equation ( 3)) and a naive bi-exponential model (i.e., Equation ( 2)).Here,  applied is the applied magnetic field,  is the demagnetization factor,  Nb 1-x Ti x N is the thickness of Nb 1-x Ti x N layer, and  Nb 1-x Ti x N ,  Nb are the penetration depths of Nb 1-x Ti x N and Nb at 0 K. where () is the field screening formula, i.e., Equation (3).In the normal state, there is no energy or depth dependence to ⟨⟩, which represents the strength of the applied magnetic field.However, in the Meissner state, ⟨⟩ decays with increasing .The apparent difference in ⟨⟩ at  ∼ 0 keV between the Meissner and normal state is due to the field "enhancement" in the Meissner state.The fit parameters are shown in the Table III.

Model
Nb 1-x Ti x N and Nb as shared fit parameters, this is justified by the fact that when the profiles for each sample were fit separately, identical  values were obtained.Other fit parameters were the thickness of each film and individual demagnetization factors.The thickness of the Nb 1-x Ti x N (160 nm)/Nb sample cannot be determined from the fit as all muons are stopped in the Nb 1-x Ti x N layer, see Figure 2(b).This parameter was therefore directly measured using transmission electron microscopy (TEM) and found to be 168 nm [44].
The best fit parameters were determined to be:  Nb 1-x Ti x N (0 K) = 182.5 (31) nm (using  c of Nb 1-x Ti x N mentioned in Section II C) and  Nb (0 K) = 43.3(19) nm (using  c = 9.25 K for Nb [7]).All fit parameters can be found in Table III.Although the magnetic screening is very different for each sample, the fact that the global fit gives excellent agreement with the entire data, with the penetration depths of the layer and the substrate as common fit parameters, further confirms the applicability of the counter-current-flow model to the data.

IV. DISCUSSION
From Figure 6, it is obvious that in both the 50 nm and 80 nm samples the decay of () is weaker in the Nb 1-x Ti x N layers, whereafter it is attenuated strongly in the Nb substrate.This bipartite screening represents the presence of two distinct penetration depths (i.e.,  Nb 1-x Ti x N and  Nb ), each associated with a distinct region in the SS bilayer.This spatially segregated response is directly resolved by the raw LE-SR data, as evidenced by the Fourier spectra in Figure 4.Note that a low-temperature baked [45] Nb was considered an "effective" bilayer due to the anomalous Meissner screening [46] near the surface, however, more recent analysis evidenced that this bipartite screening profiles is absent (i.e., there is no evidence for an effective SS bilayer) [26].The analysis shown in Figure 6, gives  values that are independent of the particular sample used and measurement conditions, implying that the measured quantities are intrinsic to the individual materials (originating from this batch of "stocks" and the coating procedure).The experimentally obtained  Nb 1-x Ti x N in Table III agree with the literature values shown in Table I, highlighting the suppression of the Meissner current in the surface layer.The obtained  Nb = 43.3(19) nm exceeds the average literature estimate of Nb's penetration depth in the "clean" limit  = 28.0(15)nm [26,47].We propose that this increased  Nb is due to the suppression of the electron mean free path, ℓ.There might be some impurity added to the Nb substrate due to the material "doping" while exposing its surface to the Ar/N 2 mixture during sputtering at 450 °C.Commonly, low temperature baking of Nb in N 2 reduce ℓ [48][49][50] and consequently increase  based on the Pippard's approximation [51].TABLE III.Individual parameters derived from a global fit to the counter-current-flow model of three Nb 1-x Ti x N/Nb samples.The magnetic penetration depths at 0 K of the Nb 1-x Ti x N layer and the Nb substrate were derived as global fit parameters, using the analysis approach described in Section III.Here,  applied is the strength of the magnetic field applied parallel to the sample surface,  is the demagnetization factor, and  Nb 1-x Ti x N is the thickness of the Nb Note that, Pippard's nonlocal electrodynamics [51] were not considered in describing the superconducting properties of Nb.Our previous LE-SR investigation on "bare" and "N 2 doped" Nb samples [26] have shown that the London model sufficiently describes these properties, suggesting that the effects of nonlocal electrodynamics could be even more prominent in higher purity samples [52].
Regarding the field screening in the Nb 1-x Ti x N (160 nm)/Nb sample the field decays far more rapidly in the first few nanometers than in the other samples but the whole data can be fitted with a single  Nb 1-x Ti x N value.Agreement of the film thicknesses extracted as fit parameters with the nominal thicknesses of the films for different measurement conditions further confirm that the counter-current-flow model can very well describe the material properties.Also, the different magnitude of  applied for the 50 nm sample does not have any effect on the other fit parameters.
An (apparent) difference in applied fields,  applied for measurements in the normal and Meissner state is observed in Figure 6. applied is used as a shared fit parameter between Meissner and normal state data for all the samples, while for the Meissner state the magnetic field enhancement is accounted for by the demagnetization factor,  as an individual fit parameter (see Equation ( 16)).From the fit, the extracted value of  applied agrees with the nominal applied fields of the samples.
We also find that a non-superconducting layer (i.e., "dead layer") at the surface is absent in our model.While such feature is often found in "real" superconductors, its absence here is not unexpected, given the surface roughness of our samples and the chemical stability of Nb 1-x Ti x N. The 50 nm sample is mirrored surface finished and others are prepared by BCP however, we did not observe any effect of surface roughness in the field screening profile.The surface of Nb 1-x Ti x N oxidizes on exposure to the ambient atmosphere (forming NbO x and TiO x ), with the thickness of the oxide layer saturating quickly to ∼1.3 nm [53].This layer is too thin for observation by LE-SR at the implantation energies used here.Thus, while we can not completely rule out the existence of a thin ∼1 nm non-superconducting region at the surface of our samples, we assert that such a feature is too small to meaningfully impact the material quantities reported here.

A. Predictions of critical fields
As discussed in Sections I and III, the counter-current-flow model predicts that multilayer superconductors can maximize the field of first-flux entry beyond the individual superheating field of its layers and substrate.For a crude estimate of this quantity, the superheating field,  sh and Ginzburg-Landau (GL) parameter,  ≡ / GL (i.e., the ratio between the magnetic penetration depth  and the GL coherence length  GL ), of each layer are required, which we consider below.
Through linear stability analysis using GL theory (strictly valid at  ≃  c )  sh for  > 1.1495 was derived to be [8]: where  c is the thermodynamic critical field.Following that,  is calculated for each material from experimentally measured penetration depth with the literature value of the London penetration depth  L and Bardeen-Cooper-Schrieffer (BCS) [34] coherence length  0 : using the fact that  0 and  GL both are correlated to the magnetic flux quantum Φ 0 [54].Here,  L and  0 are the fundamental properties of the metal defined by the clean stoichiometric material.
The next quantity is the lower critical field  c1 , which is derived for both materials from [54]: Now,  c of Equation ( 18) needs to be determined, which is well-defined for Nb [7].However, since Nb 1-x Ti x N is not a natural compound  c is not readily available from the literature, which we can be self-consistently evaluated when  c1 is known [54]: To summarize the results of these calculations, the values of ,  c1 ,  c , and  sh for Nb 1-x Ti x N and Nb are presented in Table IV.Prediction of the maximum applied field  max where the Meissner state can be sustained for an SS bilayer as a function of thickness of the top Nb 1-x Ti x N superconducting layer,  s (i.e.,  Nb 1-x Ti x N ) in Nb 1-x Ti x N/Nb.The orange curve starting from the left represents  max of the substrate Nb layer and the curve starting at right corresponds to the surface Nb 1-x Ti x N layer.Here the measured penetration depths of  s =  Nb 1-x Ti x N = 182.5 (31) nm and  sub =  Nb = 43.3(19) nm were used to find the magnitude of  1 and  2 using Equations ( 23) and (24).The predicted values of the superheating field of Nb 1-x Ti x N and Nb are  Finally, the maximum field for which the SS bilayer can remain in the Meissner state  max is derived by solving the relation between applied field and screening current density in the London model, with appropriate boundary and continuity conditions [10,12,19]: where  (s) sh and  (sub) sh are the superheating fields of the surface and substrate layers, respectively, and the terms  1 and  2 arise as coefficients while solving the relation for  max (see Ref. 10 for details).Explicitly, the   s are: and In Equation ( 22), the term  −1 1  (s) sh is related to the maximum applied field for the surface layer, whereas the term  −1 2  (sub) sh corresponds to the substrate.As  max is a function of the surface layer thickness,  s , there exists an optimum where its value is maximized [10,12,19] max is plotted in Figure 7 as a function of  s , wherein the entire Nb 1-x Ti x N/Nb SS structure remains in the Meissner state.The predicted maximum applied fields for our different film thicknesses (50 nm, 80 nm, and 160 nm) were found to be 253(5) mT, 276(5) mT and, 377(5) mT, indicated in Figure 7 by "plus" (+, +, and +) symbols, respectively.Clearly, these values exceed the intrinsic field limit of the Nb substrate.The orange curve in Figure 7 represents the criteria for the surface and substrate layer to remain in the Meissner state.For zero film thickness, the substrate can sustain its Meissner state up to the superheating field of the substrate  sh (229(6) mT for our Nb substrates).Upon increasing the  s ,  max is initially increased, as the applied field is shielded by the surface superconductor before it reaches the SS interface. max reaches its optimum (i.e.,  opt max = 610(40) mT) for a surface layer thickness,  m ∼ 1.4 s = 261(14) nm according to Equation (25).
Note that a surface layer thicker than  s can only remain in the Meissner state above  c1 in the presence of a BL barrier [15] just like a bulk superconductor of same material.The strong suppression of the screening current by the counter-current-flow between substrate and surface layers therefore suggests that multilayer structures with several interlayers to stop vortices are necessary in order to achieve largest  opt max .

V. SUMMARY
In conclusion, the depth-dependent field screening profile in SS bilayers composed of Nb 1-x Ti x N films (50 nm, 80 nm, and 160 nm) deposited on Nb substrates were measured using LE-SR.A fit of the magnetic screening profile to a countercurrent-flow model yielded a penetration depth for Nb 1-x Ti x N of 182.5 (31) nm in agreement with literature values.This is contrasted by fits to a naive biexponential model, which was found to overestimate  by a factor of ∼2.5.For the Nb substrates, a common  of 43.3(19) nm was found.This comparison highlights the pronounced suppression of the Meissner current within the surface layer and serves as an experimental validation of the counter-current-flow model.Using these quantities, the optimum maximum field that can be sustained before first-flux entry by a Nb 1-x Ti x N/Nb heterostructure with these material properties was predicted to be 610 (40) mT.This study emphasizes the necessity of considering counter-current-flow model in SS and SIS structures to accurately predict the optimal layer thicknesses.

FIG. 1 .
FIG.1.Magnetic field profiles given by Equations (2) and (3) in (a) and the current density distributions of those equations normalized to the current density at interface in (b).The used magnetic penetration depths are  Nb 1-x Ti x N = 200 nm and  Nb = 50 nm for the Nb 1-x Ti x N and Nb layers, respectively.The thickness of the Nb 1-x Ti x N layer is 50 nm.Comparing the two field profiles in (a), the strongest effect on field screening is observed in the Nb 1-x Ti x N layer due to the suppressed Meissner current in the Nb 1-x Ti x N layer as seen in (b).
FIG. 2.Typical stopping profiles for  + implanted in (a) Nb 1-x Ti x N(50 nm)/Nb, and (b) Nb 1-x Ti x N(160 nm)/Nb SS bilayer, simulated using the Monte Carlo code TRIM.SP[23].The densities of Nb 1-x Ti x N and Nb are 6.6223 g cm −3 , and 8.57 g cm −3 , respectively.The light gray color in the first 50 nm of figure (a) and 160 nm of figure (b) refers to the Nb 1-x Ti x N film thickness on bulk Nb substrate (i.e., dark gray color).The normalized stopping distribution  of  + is plotted against the depth  below the surface.The black solid curves are fits to the stopping profile (represented as a histogram) using Equations (7) and(8).These fits clearly capture all features of the stopping profiles.

FIG. 3 .
FIG.3.Asymmetry as a function of time for different implantation energies (given in the panel's inset) in Nb 1-x Ti x N (50 nm)/Nb in both the normal (20 K) and Meissner state (2.7 K) at an applied magnetic field of ∼25 mT parallel to the sample surface.In the normal state (gray shaded background panel), there is no substantial energy dependence to the time evolution of the muon ensemble polarization, meaning the implanted muons experience the same local field.By contrast, it is evident that the temporal evolution of () varies in the Meissner state (plain white background panels).As the implantation energy increases, the  + spin-precession frequency is reduced, and the signal is more strongly damped.The solid red lines denote fits to all of the data (i.e., a global fit) using Equations (11) to(13).Clearly, the model captures all the data's main features.

FIG. 5 .
FIG. 5.Nb 1-x Ti x N (50 nm)/Nb field profile: Plot of the mean magnetic field, ⟨⟩, sensed by  + at different implantation energies, , in a Nb 1-x Ti x N (50 nm)/Nb sample at an applied field ( 0 ∼ 25 mT) parallel to the sample surface in the Meissner ( = 2.7 K) and normal state ( = 20 K).The closed circles and open squares are the data points in the Meissner state and normal state, respectively.The implantation energy  is related to the mean implantation depth ⟨⟩ as shown in the top -axis.The solid red lines are fits to the data in the Meissner state and the dashed red lines are fits to the normal state data.Both figures represent the same data points fitted to different models.In the Meissner state ⟨⟩ decays with increasing  as expected.The fit to Figure5(a) represents the field screening using Equation (2) i.e., a simple London model with fit parameters  Nb 1-x Ti x N = 498(34) nm and  Nb = 42.9(30)nm. Figure 5(b) is fitted with the Equation (3) which considers counter-current-flow induced by the substrate layer and the extracted fit parameters are  Nb 1-x Ti x N = 185(7) nm and  Nb = 43.6(29)nm.
FIG. 7.Prediction of the maximum applied field  max where the Meissner state can be sustained for an SS bilayer as a function of thickness of the top Nb 1-x Ti x N superconducting layer,  s (i.e.,  Nb 1-x Ti x N ) in Nb 1-x Ti x N/Nb.The orange curve starting from the left represents  max of the substrate Nb layer and the curve starting at right corresponds to the surface Nb 1-x Ti x N layer.Here the measured penetration depths of  s =  Nb 1-x Ti x N = 182.5(31)nm and  sub =  Nb = 43.3(19)nm were used to find the magnitude of  1 and  2 using Equations (23) and(24).The predicted values of the superheating field of Nb 1-x Ti x N and Nb are sh = 229(6) mT, respectively.The +, + and + are the position of maximum fields for each of the 50 nm, 80 nm, and 160 nm samples.

FIG. 8 .
FIG.8.Plot of the fit parameters  0 ,  p ,  ± , and  of Equations (9) to(13) as a function of  in Nb 1-x Ti x N (50 nm)/Nb in the Meissner state (2.7 K) at an applied magnetic field of ∼25 mT.For  ≤ 14.5 keV the fit is constrained such that  = 1 in Equation (12) indicating the  + sample is only implanted in the Nb 1-x Ti x N layer.(a) The blue and orange closed circles are the asymmetry,  0 data points corresponding to the Nb 1-x Ti x N and Nb layer, respectively.(b) the peak field,  p of Nb 1-x Ti x N and Nb layer are denoted by the blue and orange closed circles, (c) the distribution's "width" on either side of  p ,  ± is plotted for both Nb 1-x Ti x N and Nb layers indicated by colored closed circles shown in the figure inset, and (d) represents the shared parameter, phase .
(15)6.Plot of the mean magnetic field, ⟨⟩, sensed by  + at different implantation energies, , in Nb 1-x Ti x N/Nb samples with different Nb 1-x Ti x N thicknesses (i.e., 50 nm, 80 nm, and 160 nm) at applied fields of 15.0 ≲  0 ≲ 25.0 mT, parallel to the sample surface in the Meissner state ( ≤ 2.8 K) and normal state ( ≥ 20 K).The mean implantation depth ⟨⟩ corresponding to  of each sample is shown in the top -axis on each panel.The colored closed circles and open squares are the data derived from the LE-SR measurements.The solid and dashed lines represent a (global) fit to the data using Equation(15) 1-x Ti x N layer.

TABLE IV .
(19)rconducting parameters GL parameter , thermodynamic critical field  c , lower critical field  c1 , and superheating field  sh calculated from the measured penetration depths of  Nb 1-x Ti x N = 182.5(31)nmandNb= 43.3(19)nm. c for Nb and  0 for both materials are taken from literature.