Evidence for current suppression in superconductor–superconductor bilayers

One possibility to achieve surface magnetic fields beyond $B_{\mathrm{sh}}$ of Nb is to use a different superconducting material with a greater $B_\mathrm{sh}$ (e.g., $\mathrm{Nb}_\mathrm{3}$Sn or $\mathrm{Nb_{1-x}Ti_{x}N}$) [9]; however, there is no viable replacement with a $B_{\mathrm{c1}}$ exceeding that of $\mathrm{Nb}$. This is problematic, as all real SRF cavities possess both surface defects and topographic imperfections, facilitating vortex penetration below $B_\mathrm{sh}$. Vortices that penetrate at these "weak points" often evolve into a thermomagnetic flux avalanche, quenching superconductivity at SRF cavity operating temperatures ($T \lesssim 4 \; \mathrm{K}$) [10–12].

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 $B_\mathrm{sh}$ (see e.g., [10, 12–14]). 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 $\mathrm{Nb}$ 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 ($\lambda_{\mathrm{L}}$) of their material, nucleation of parallel vortices will only become energetically favorable at larger fields than $B_\mathrm{c1}$ 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 $\mathrm{Nb}$ cavity can also increase the field of first vortex penetration ($B_{\mathrm{vp}}$) 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 [16].

In summary, the maximum field in superconducting heterostructures that can be sustained while remaining in the Meissner state $(B_\mathrm{max})$ 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].

Recall that, for a bulk superconductor in the "local" London limit (see e.g., [17]) with an ideal flat surface, the Meissner screening profile, $B(z)$, is given by [18]:

$$\begin{align} B(z) = B_{0} \times \begin{cases} 1, & z < 0 , \\[8pt] \exp \left( - \dfrac{z}{\lambda_{\mathrm{L}}} \right), & z \unicode{x2A7E} 0 , \end{cases} \end{align} \tag{ 1 }$$

where $B_{0}$ is the (effective) applied magnetic field, $z$ is the depth below the superconductor's surface, and $\lambda_{\mathrm{L}}$ 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., $\lambda_{\mathrm{L}}$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:

$$\begin{align} B(z) = B_{0} \times \begin{cases} 1 , & z \lt 0 , \\[8pt] \exp \left( - \dfrac{z}{\lambda_{\mathrm{s}}} \right) ,& 0 \unicode{x2A7D} z \lt d_{\mathrm{s}} , \nonumber\\[8pt] \exp \left( - \dfrac{d_{\mathrm{\mathrm{s}}}}{\lambda_{\mathrm{s}}} \right) \exp \left( -\dfrac{z - d_{\mathrm{s}}}{\lambda_{\mathrm{sub}}} \right ) , & z \unicode{x2A7E} d_{\mathrm{s}} , \nonumber\\ \end{cases} \end{align} \tag{ 2 }$$

where $d_{\mathrm{s}}$ is the thickness of the top superconducting layer, and the $\lambda_{i}$ denote the penetration depth in the surface ($i = \mathrm{s}$) and substrate ($i = \mathrm{sub}$) 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. Notably, the substrate layer significantly influences the screening properties of the surface layer superconductor when their penetration depths differ. This occurs because an SS bilayer's electromagnetic (EM) response depends on satisfying the boundary and continuity criteria for both 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 $d_{\mathrm{s}} \sim \lambda_{\mathrm{s}}$ [14]. For example, when the surface layer penetration depth is larger than the substrate's (i.e., $\lambda_{\mathrm{s}} \gt \lambda_{\mathrm{sub}}$), the Meissner current in the surface layer is suppressed by the substrate layer's counter-current (i.e., a counterflow current generated by the substrate in a multilayer superconductor [10–12, 19]) to satisfy the boundary and continuity condition at the interface. This results in a higher $B$-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, $B_{0}$ and current density, $J$ (or equivalently vector potential, $A$ ). For a SS structure this yields [10, 12, 14, 19]:

$$\begin{align} B(z) = B_{0} \times \begin{cases} 1,& z \unicode{x2A7D} 0 , \\[8pt] \displaystyle D_{\mathrm{S-S}}^{-1} \left[ \cosh \left( \frac{d_{\mathrm{s}} - z}{\lambda_{\mathrm{s}}} \right) + \left( \frac{\lambda_{\mathrm{sub}} }{\lambda_{\mathrm{s}} } \right) \sinh \left( \frac{d_{\mathrm{s}} - z}{\lambda_{\mathrm{s}}} \right ) \right] , & 0 \lt z \unicode{x2A7D} d_{\mathrm{s}} , \\[8pt] \displaystyle D_{\mathrm{S-S}}^{-1} \left[ \exp \left( - \frac{z - d_{\mathrm{s}} }{\lambda_{\mathrm{sub}} } \right) \right ], & z > d_{\mathrm{s}} , \end{cases} \end{align} \tag{ 3 }$$

where the symbols have the same meaning as in equation (2), and, the common factor $D_{\mathrm{S-S}}$ is given by:

$$\begin{align*} D_{\mathrm{S-S}} = \cosh \left( \frac{d_{\mathrm{s}}}{\lambda_{\mathrm{s}}} \right ) + \left( \frac{\lambda_{\mathrm{sub}} }{\lambda_{\mathrm{s}} } \right ) \sinh \left( \frac{d_{\mathrm{s}}}{\lambda_{\mathrm{s}}} \right ) .\end{align*}$$

The current density distribution, $J(z)$ can be obtained from the field screening profiles using the expression $J(z) = -\frac{1}{\mu_{0}}\frac{\mathrm{d}B(z)}{\mathrm{d}z}$. Both equations (2) and (3) are essentially forms of exponential decay; however, the screening behavior is significantly 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 $\mathrm{Nb_{1-x}Ti_{x}N}~({50}\; \mathrm{nm})/\mathrm{Nb}$ with assumed penetration depths of $\lambda_{\mathrm{Nb_{1-x}Ti_{x}N}} = 200 \; \mathrm{nm}$ and $\lambda_{\mathrm{Nb}} = 50 \; \mathrm{nm}$ for $\mathrm{Nb_{1-x}Ti_{x}N}$ and $\mathrm{Nb}$, respectively. As alluded to above, the two models have qualitatively similar behavior; $B(z)$'s decay rate in the $\mathrm{Nb}$ 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 $J(z)$ in the $\mathrm{Nb}$ substrate. Conversely, a notable difference is apparent in the top $\mathrm{Nb_{1-x}Ti_{x}N}$ 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.

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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. Perhaps the best way of achieving this through low-energy muon spin rotation (LE-µSR) [20–24], which uses implanted positive muons to probe the sub-surface field distributions in the material under investigation. The key feature of this approach is the ability to control the energy at which the muons are implanted, conferring depth-resolution to the measurements. This is in contrast to other surface-sensitive techniques (e.g., magnetic force microscopy [25–29], scanning tunnelling microscopy [30, 31], etc), which provide lateral resolution across the surface, but lack the depth sensitivity to directly view the Meissner screening profile across a buried interface.

To this end, we measured the Meissner screening profile and observed suppression of screening current in the surface layer in $\mathrm{Nb_{1-x}Ti_{x}N}/\mathrm{Nb}$ samples with different alloy thicknesses using the LE-µSR technique. These measurements were conducted under applied fields $(15 \lesssim B_{0} \lesssim 25)$ mT. By globally fitting the field profiles of all the samples, we quantitatively determined the common magnetic penetration depths of $\mathrm{Nb_{1-x}Ti_{x}N}$, $\lambda_{\mathrm{Nb_{1-x}Ti_{x}N}}$, and $\mathrm{Nb}$, $\lambda_{\mathrm{Nb}}$. This quantitative assessment involves comparing Kubo's counter-current-flow model (i.e., London theory with appropriate boundary and continuity conditions) with a simple London model without appropriate boundary conditions. The resultant comparison highlights the significant suppression of the Meissner current in the surface $\mathrm{Nb_{1-x}Ti_{x}N}$ layer in $\mathrm{Nb_{1-x}Ti_{x}N}/\mathrm{Nb}$ samples with film thicknesses shorter but close to the London penetration depth of $\mathrm{Nb_{1-x}Ti_{x}N}$.

The LE-µSR experiments were performed at the Paul Scherrer Institute's (PSI) Swiss Muon Source located in Villingen, Switzerland, using the µE4 beamline [32]. In the beamline a thin film of condensed noble gas [33] is used to reduce the energy of a "surface" muon beam of ${\sim}4 \; \mathrm{MeV}$ down to around $15 \; \mathrm{eV}$. Following that, the muons are accelerated to create a beam with an adjustable energy $E \unicode{x2A7D} 30 \; \mathrm{keV}$ which corresponds to an implantation depth of ${\lesssim}150 \; \mathrm{nm}$ in $\mathrm{Nb}$ and $\mathrm{Nb}$-based alloys. These low energy positive muons ($\mu^{+}$) are ${\sim}100 \; \mathrm{\%}$ spin-polarized. The $\mu^+$ are implanted into a sample one at a time using a (quasi-)continuous beam [34], wherein they quickly thermalize in the target and their spins precess around the local magnetic field at the Larmor frequency, $\omega_{\mu}$ permitting depth-resolved measurements of the field screening profile in surface-parallel applied fields up to ${\sim}30 \; \mathrm{mT}$ [35].

When a $\mu^+$ 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 $\mu^{+}$ decay is determined in a transverse field arrangement wherein a magnetic field is applied perpendicular to the initial direction of muon spin-polarization and parallel to the sample surface. The positron event rate in one (or more) "counters" $i$, is given by:

$$\begin{align} N_{i}(t) = N_{0,i} \exp \left( - \frac{t}{\tau_{\mu}} \right ) [ 1 + A_{i}(t) ] + b_{i} , \end{align} \tag{ 4 }$$

where $\tau_\mu = {2.2}\;{\unicode{x03BC}\mathrm{s}}$ is the muon lifetime, $N_{0,i}$ represents the total number of "good" decay events (i.e., decays from muons stopped in the sample), $b_{i}$ is the time-independent rate from uncorrelated "background" events, and $A_{i}(t)$ represents the time-evolution of the muon ensemble asymmetry:

$$\begin{align} A_{i}(t) = A_{0,i} P(t), \end{align} \tag{ 5 }$$

where $A_{0,i}$ is the experimental decay asymmetry and $P(t)$ is the polarization of the muon ensemble.

In a transverse-field experiment, the time-evolution of $P(t)$ is given by:

$$\begin{align} P(t) = \int_{0}^{\infty} p(B) \cos ( \gamma_{\mu} B t + \phi ) \, \mathrm{d}{\text{B}}, \end{align} \tag{ 6 }$$

where $p(B)$ is the internal magnetic field distribution sensed by the muons, $\gamma_\mu = 2 \pi \times {135.54}\;\mathrm{MHzT}^{-1}$ is the gyromagnetic ratio of the muon, $B$ is the magnitude of the local magnetic field at the muon site, $t$ is the time after implantation, and $\phi$ is the phase factor (i.e., angle between the initial muon spin-polarization and the effective symmetry axis of a positron detector).

As mentioned in section 2.1, LE-µSR 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 (${{\sim}2}\;\mathrm{keV}\ \text{to}\ {{\sim}30}\;\mathrm{keV}$) were used (see figure 2), providing depth-resolution on the nm scale (i.e., ${{\sim}10}\;\mathrm{nm}\ \text{to}\ {{\sim}150}\;\mathrm{nm}$).

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The stopping profile of muons can be accurately simulated [20, 37, 38] using the TRIM.SP code (a Monte Carlo code) [36], which treats all collisions within the target using the binary collision approximation. Simulation results for $\mu^{+}$ implanted in a $\mathrm{Nb_{1-x}Ti_{x}N}~(50 \; \mathrm{nm})/\mathrm{Nb}$, and a $\mathrm{Nb_{1-x}Ti_{x}N}~(160 \; \mathrm{nm})/\mathrm{Nb}$ SS bilayer are shown in figure 2, illustrating LE-µSR's typical range of spatial sensitivity. For our analysis (see section 3), it was convenient to have the ability to describe these profiles at arbitrary $E$, which can be accomplished by fitting the simulated profiles and interpolating their "shape" parameters [20]. Empirically, we found the $\mu^{+}$ stopping probability, $\rho(z)$, at a given $E$ can be described by:

$$\begin{align} \rho (z) = \sum_{i}^{m} f_{i} p_{i} (z) , \end{align} \tag{ 7 }$$

where $p_{i}(z)$ is a probability density function, $f_{i} \in [0, 1]$ is the $i^{\mathrm{th}}$ stopping fraction, constrained such that

$$\begin{align*} \sum_{i}^{m} f_{i} \equiv 1 , \end{align*}$$

and $z$ is the depth below the surface. For our SS bilayers, the stopping data are well-described using $m=2$ and a $p(z)$ is given by a modified beta distribution. Explicitly,

$$\begin{align} p (z) = \begin{cases} 0, & \text{for } z < 0, \nonumber\\ \dfrac{ ( z / z_{0} )^{\alpha -1} (1 - z / z_{0} )^{\beta - 1} }{z_{0} \, B ( \alpha, \beta ) } , & \text{for } 0 \unicode{x2A7D} z \unicode{x2A7D} z_{0}, \nonumber\\ 0, & \text{for } z > z_{0}, \end{cases} \end{align} \tag{ 8 }$$

where $z \in [0, z_{0}]$ is the depth below the surface and $B ( \alpha, \beta )$ is the beta function:

$$\begin{align*} B ( \alpha, \beta ) \equiv \frac{\Gamma (\alpha ) \Gamma (\beta) }{\Gamma ( \alpha + \beta ) } , \end{align*}$$

with $\Gamma (s)$ denoting the gamma function. Further details of the stopping profile simulation can be found elsewhere [20, 37].

In this study, $\mathrm{Nb_{1-x}Ti_{x}N}/\mathrm{Nb}$ SS bilayer samples were prepared by growing thin films of $\mathrm{Nb_{1-x}Ti_{x}N}$ on "bulk" $\mathrm{Nb}$ substrates using direct current (DC) magnetron reactive sputtering (R-DCMS) in a vacuum chamber with a base pressure of low $1 \times 10^{-10}\;\mathrm{mbar}$. The sputtering target consisted of 80/20 (wt %) Nb/Ti alloy, was used within an Ar and $\mathrm{N}_2~(\mathrm{P_{\mathrm{N}_2}}/\mathrm{P_{\mathrm{Ar}}})$ gas mixture at a pressure of $2 \times 10^{-3}\;\mathrm{mbar}$. Films with nominal thicknesses of $50 \; \mathrm{nm}$, $80 \; \mathrm{nm}$, and $160 \; \mathrm{nm}$ were deposited at $450 \;{}^\circ\mathrm{C}$ on $3 \;\mathrm{mm}$ thick bulk $\mathrm{Nb}$ substrates, with respective $T_{\mathrm{c}}$ values of $15.8 \; \mathrm{K}$, $16.3 \; \mathrm{K}$ and $16.3 \; \mathrm{K}$[46]. Following standard practice for preparing the surface of a $\mathrm{Nb}$ SRF cavity, the substrates were prepared by mechanical polishing (MP) followed by $5 \;\unicode{x03BC}\mathrm{m}$ cold electropolishing (EP) or by $50 \;\unicode{x03BC}\mathrm{m}$ buffered chemical polishing (BCP) (see e.g., [47]). Specifically, the $50 \; \mathrm{nm}$ sample was prepared using EP and the others using BCP. Prior to film growth, the substrates were baked at $600 \;{}^\circ\mathrm{C}$ for 24 h under vacuum and the $\mathrm{Nb_{1-x}Ti_{x}N}$ films were annealed at $450 \;{}^\circ\mathrm{C}$ after deposition. The typical surface roughness of the $\mathrm{Nb_{1-x}Ti_{x}N}$ layer is similar to the original substrate roughness ($1 \; \mathrm{nm}$ for MP+ EP substrates and microns for BCP substrates [48]). All film depositions were performed at Thomas Jefferson National Accelerator Facility (JLab) and further details on deposition technique can be found in [39]. Note that, unlike the elemental superconductors, the magnitude of superconducting properties (such as the penetration depth and the coherence length) of $\mathrm{Nb_{1-x}Ti_{x}N}$ are not robust. This is due to the fact that $\mathrm{Nb_{1-x}Ti_{x}N}$ is not a "natural" compound [39]. Therefore, the superconducting properties of some $\mathrm{Nb_{1-x}Ti_{x}N}$ films prepared using different target stoichiometries, deposition techniques, and preparation methods have been reviewed from the literature and are listed in table 1 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 3 as well as for the prediction of critical fields in section 4.1.

Table 1. Superconducting properties of $\mathrm{Nb_{1-x}Ti_{x}N}$ films from several literatures [9, 39–44]. Here, $T_\mathrm{c}$ is the critical temperature, $B_\mathrm{c}$ is the thermodynamic critical field, $B_\mathrm{c1}$ is the lower critical field, $B_\mathrm{sh}$ is the superheating field, $B_\mathrm{c2}$ is the upper critical field, $\lambda$ is the penetration depth, and $\xi$ is the BCS [45] coherence length.

Sample	$T_\mathrm{c}~(\text{K})$	$B_\mathrm{c}~(\text{mT})$	$B_\mathrm{c1}~(\text{mT})$	$B_\mathrm{sh}~(\text{mT})$	$B_\mathrm{c2}~(\text{mT})$	$\lambda~({\text{nm}})$	$\xi~({\text{nm}})$	Reference
$\mathrm{Nb_{1-x}Ti_{x}N}/\mathrm{Nb}$	15.97		35					[39]
$\mathrm{Nb_{1-x}Ti_{x}N}$/$\mathrm{Al_2O_3}$	17.3		30		15 000	150–200		[9]
$\mathrm{Nb_{1-x}Ti_{x}N}$/$\mathrm{Al_2O_3}$	15.8		25	186		208		[40]
$\mathrm{Nb}_{0.62}\mathrm{Ti}_{0.38}\mathrm{N}/\mathrm{Si}$	${\sim}15.0$						2.4(3)	[41]
$\mathrm{Nb_{1-x}Ti_{x}N}/\mathrm{MgO}$	${\sim}15.0$							[42]
$\mathrm{Nb_{1-x}Ti_{x}N}$/$\mathrm{Al_2O_3}$	${\sim}13.1$							[43]
$\mathrm{Nb}_{0.62}\mathrm{Ti_{0.38}N}/\mathrm{Si}$	${\sim}16.0$					200(20)		[44]

As discussed in sections 1 and 3, 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, $B_{\mathrm{sh}}$ and Ginzburg–Landau (GL) parameter, $\kappa \equiv \lambda / \xi_{\mathrm{GL}}$ (i.e., the ratio between the magnetic penetration depth $\lambda$ and the GL coherence length $\xi_{\mathrm{GL}}$), of each layer are required, which we consider below.

Through linear stability analysis using GL theory (strictly valid at $T \simeq T_{\mathrm{c}}$) $B_{\mathrm{sh}}$ for  $\kappa > 1.1495$ was derived to be [8]:

$$\begin{align} B_{\mathrm{sh}} \approx B_{\mathrm{c}} \left( \frac{\sqrt{20}}{6} - \frac{0.55}{\sqrt{\kappa}} \right) , \end{align} \tag{ 18 }$$

where $B_{\mathrm{c}}$ is the thermodynamic critical field.

Following that, $\kappa$ is calculated for each material from experimentally measured penetration depth with the literature value of the London penetration depth $\lambda_{\mathrm{L}}$ and Bardeen–Cooper– Schrieffer (BCS) [45] coherence length $\xi_{0}$:

$$\begin{align} \kappa = \frac{\lambda}{\xi_{\mathrm{GL}}} = \frac{2 \sqrt{3}}{\pi}\frac{\lambda^2}{\xi_0 \lambda_\mathrm{L}}, \end{align} \tag{ 19 }$$

using the fact that $\xi_{0}$ and $\xi_{\mathrm{GL}}$ both are correlated to the magnetic flux quantum $\Phi_{0}$ [65]. Here, $\lambda_{\mathrm{L}}$ and $\xi_{0}$ are the fundamental properties of the metal defined by the clean stoichiometric material.

The next quantity is the lower critical field $B_{\mathrm{c1}}$, which is derived for both materials from [66]:

$$\begin{align} B_{\mathrm{c1}} = \frac{{\Phi_0}}{4 \pi \lambda^2} \ln(\kappa+0.497). \end{align} \tag{ 20 }$$

Now, $B_{\mathrm{c}}$ of equation (18) needs to be determined, which is well-defined for $\mathrm{Nb}$ [7]. However, since $\mathrm{Nb_{1-x}Ti_{x}N}$ is not a natural compound $B_{\mathrm{c}}$ is not readily available from the literature, which we can be self-consistently evaluated when $B_{\mathrm{c1}}$ is known [65]:

$$\begin{align} B_{\mathrm{c}} = \frac{\sqrt{2}\kappa B_{\mathrm{c1}}}{\ln \kappa}, \end{align} \tag{ 21 }$$

To summarize the results of these calculations, the values of $\kappa, B_{\mathrm{c1}}, B_{\mathrm{c}}$, and $B_{\mathrm{sh}}$ for $\mathrm{Nb_{1-x}Ti_{x}N}$ and $\mathrm{Nb}$ are presented in table 4.

Table 4. Superconducting parameters GL parameter $\kappa$, thermodynamic critical field $B_{\mathrm{c}}$, lower critical field $B_{\mathrm{c1}}$, and superheating field $B_{\mathrm{sh}}$ were calculated from the measured penetration depths of $\lambda_{\mathrm{Nb_{1-x}Ti_{x}N}} = 182.5(31)\;\mathrm{nm}$ and $\lambda_{\mathrm{Nb}} = 43.3(19)\; \mathrm{nm}$. $B_\mathrm{c}$ for Nb and BCS coherence length $\xi_{0}$ for both materials are taken from literature.

Material	$\lambda_{\mathrm{L}}~(\text{nm})$	$\xi_{0}~(\text{nm})$	$\kappa$	$B_\mathrm{c1}~(\text{mT})$	$B_\mathrm{c}~(\text{mT})$	$B_\mathrm{sh}~(\text{mT})$
$\mathrm{Nb_{1-x}Ti_{x}N}$	150 [9]	2.4(3) [41]	102(17)	22.9(11)	710(40)	570(40)
$\mathrm{Nb}$	28.0(15) [20, 24]	40.3(35) [20]	1.83(25)	74(11)	199(1) [7]	229(6)

Finally, the maximum field for which the SS bilayer can remain in the Meissner state $B_{\mathrm{max}}$ 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]:

$$\begin{align} B_{\mathrm{max}} = \mathrm{min} \left\{\gamma_1^{-1} B_{\mathrm{sh}}^{(\mathrm{s})}, \gamma_2^{-1} B_{\mathrm{sh}}^{(\mathrm{sub})} \right\}, \end{align} \tag{ 22 }$$

where $B_{\mathrm{sh}}^{(\mathrm{s})}$ and $B_{\mathrm{sh}}^{(\mathrm{sub})}$ are the superheating fields of the surface and substrate layers, respectively, and the terms $\gamma_{1}$ and $\gamma_{2}$ arise as coefficients while solving the relation for $B_{\mathrm{max}}$ (see [10] for details). Explicitly, the $\gamma_{i}$ s are:

$$\begin{align} \gamma_1 = \dfrac{\sinh\dfrac{d_{\mathrm{s}}}{\lambda_{\mathrm{s}}}+\dfrac{\lambda_{\mathrm{sub}}}{\lambda_{\mathrm{s}}}\cosh\dfrac{d_{\mathrm{s}}}{\lambda_{\mathrm{s}}}}{\cosh\dfrac{d_{\mathrm{s}}}{\lambda_{\mathrm{s}}}+\dfrac{\lambda_{\mathrm{sub}}}{\lambda_{\mathrm{s}}}\sinh\dfrac{d_{\mathrm{s}}}{\lambda_{\mathrm{s}}}} , \end{align} \tag{ 23 }$$

and

$$\begin{align} \gamma_2 = \dfrac{1}{\cosh\dfrac{d_{\mathrm{s}}}{\lambda_{\mathrm{s}}}+\dfrac{\lambda_{\mathrm{sub}}}{\lambda_{\mathrm{s}}}\sinh\dfrac{d_{\mathrm{s}}}{\lambda_{\mathrm{s}}}}. \end{align} \tag{ 24 }$$

In equation (22), the term $\gamma^{-1}_{1} B_{\mathrm{sh}}^{(\mathrm{s})}$ is related to the maximum applied field for the surface layer, whereas the term $\gamma_2^{-1} B_{\mathrm{sh}}^{(\mathrm{sub})}$ corresponds to the substrate. As $B_\mathrm{max}$ is a function of the surface layer thickness, $d_{\mathrm{s}}$, there exists an optimum where its value is maximized [10, 12, 19]

$$\begin{align} B_{\mathrm{max}}^{\mathrm{opt}} = \sqrt{\left(B_{\mathrm{sh}}^{(\mathrm{s})} \right)^2 + \left[ 1 - \dfrac{\lambda_{\mathrm{sub}}^2}{\lambda_{\mathrm{s}}^2} \right]\left(B_{\mathrm{sh}}^{(\mathrm{sub})} \right)^2 }. \end{align} \tag{ 25 }$$

$B_\mathrm{max}$ is plotted in figure 7 as a function of $d_{\mathrm{s}}$, wherein the entire $\mathrm{Nb_{1-x}Ti_{x}N}/\mathrm{Nb}$ 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 $253(5) \; \mathrm{mT}$, $276(5) \; \mathrm{mT}$ and, $377(5)\; \mathrm{mT}$, indicated in figure 7 by "plus" (, , and ) symbols, respectively. Clearly, these values exceed the intrinsic field limit of the $\mathrm{Nb}$ 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 $B_{\mathrm{sh}}$ ($229(6) \; \mathrm{mT}$ for our $\mathrm{Nb}$ substrates). Upon increasing the $d_{\mathrm{s}}$, $B_\mathrm{max}$ is initially increased, as the applied field is shielded by the surface superconductor before it reaches the SS interface. $B_\mathrm{max}$ reaches its optimum (i.e., $B_{\mathrm{max}}^{\mathrm{opt}} = 610(40) \; \mathrm{mT}$) for a surface layer thickness, $d_{\mathrm{m}} \sim 1.4 \lambda_{\mathrm{s}} = 261(14)\;\mathrm{nm}$ according to equation (25).

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Note that a surface layer thicker than $\lambda_\mathrm{s}$ can only remain in the Meissner state above $B_\mathrm{c1}$ 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 $B_{\mathrm{max}}^{\mathrm{opt}}$.