# Structural and Superconducting Proximity Effect of SnPb Bimetallic Nanoalloys

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## Abstract

**:**

_{r}Pb

_{1-r}nanoalloys with various compositions r were synthesized, and their superconducting properties were measured using superconducting quantum interference devices (SQUIDs) magnetometer. Our results reveal a superconducting proximity effect (SPE) between immiscible Sn and Pb granules in the range of r = 0.2~0.9, as a weak superconducting coupling can be established with the coexistence of phonon hardening and increased Ginzburg–Landau coherence length. Furthermore, our results provide new insights into improving the study of the superconducting proximity effect introduced by Sn doping.

## 1. Introduction

_{2}Se

_{3}interface junctions down to 250 mK and in different magnetic fields [3]. As a result, a proximity-effect-induced chiral superconducting phase is found and formed at the interface between the superconducting Sn and the strong spin-orbit coupling material Bi

_{2}Se

_{3}.

_{2}Te

_{3}surface, demonstrating that superconducting quantum interference devices can be constructed based on proximity-effect-induced superconductivity [4,5]. The interplay of BCS superconductivity and nontrivial band topology is expected to give rise to the search for Majorana fermion quasiparticles in condensed matter systems. Therefore, knowledge of the precise electron–phonon and superconducting coupling strength of Pb/Sn is essential in explaining the proximity effect.

_{r}Pb

_{1-r}(0.01 ≤ r ≤ 0.99) bimetallic nanoalloys. Granular Sn can be randomly distributed within the Pb matrix during the alloying process. In this scenario, if the grain size is smaller than the superconducting characteristic length scales, then studying SPE can give more insight into the superconducting properties of immiscible bimetallic nanoalloys.

## 2. Methods

_{r}Pb

_{1-r}(0.01 ≤ r ≤ 0.99) superconductors were prepared using a physical solid-state reaction. This Sn/Pb concentration tuning alters the crystal structure, thereby changing the superconducting properties. Surface morphological analysis and atomic percentage calculation of all the samples were performed by field-emission scanning electron microscopy (FE-SEM) using a JEOL JSM-6500F microscope (JEOL Ltd., Tokyo, Japan). Energy dispersive spectroscopy (EDS; Inca x-sight model 7557, Oxford Instruments, Abingdon, Oxfordshire, UK) was utilized to estimate the atomic percentages of the constituent elements. Energy-dispersive spectroscopy (EDS) is a valuable technique for estimating the samples’ atomic rates of constituent elements. To study the alloying effect of Sn and Pb on structural properties, synchrotron radiation X-ray diffraction (SR-XRD) of the nanoalloys was carried out using synchrotron radiation beamline BL-01C2 at the National Synchrotron Radiation Research Center (NSRRC) in Taiwan with an incident wavelength λ = 0.7749 Å.

## 3. Results

#### 3.1. Elemental Analysis and EDS Mapping of SnPb

_{r}Pb

_{1-r}(0.01 ≤ r ≤ 0.99) bimetallic nanoalloys, which are shifted vertically for clarity. The series of Sn and Pb constituent elements observed in EDS spectra are assigned to Pb-Mα

_{1}, Sn-Lα, Sn-Lβ

_{1}, and Sn-Lβ

_{2}. The weak peak of C-Kα1 and O-Kα1 in the low energy regions originated from the carbon tape used for mounting the sample and surface oxygen, respectively. Figure S1a–k (Supporting Information) presents the EDS mapping with SEM images of Sn

_{r}Pb

_{1-r}bimetallic nanoalloys, where red and green represent the atomic percent of Pb and Sn, respectively. Figure 1b shows the plot of atomic % of Sn concerning composition r, which increases linearly with a slope of 89 (10) atomic percentage with Sn composition (at. %/r). The observed slight discrepancy of at. % with initial composition r could be due to the inhomogeneous distribution of constituent elements. The phase diagram is very sensitive to preparation conditions, so slight changes in the temperature, pressure, or initial composition can change the weight percent of the constituent phases for each nanoalloy. The EDS mapping images of Sn

_{r}Pb

_{1-r}nanoalloys show the distribution of segregated Sn (bright green color) and Pb (red color) elements with grain size varying from <d> = 121 (10)~46 (3) nm. The observed discrepancy of atomic % with initial composition is due to three-dimensional inhomogeneous distributions of segregated grains of Sn and Pb. In such a scenario magnetization, M(T) can be fitted with the London equation of granular sized <d>, which will be discussed in the magnetization section.

#### 3.2. Crystal Structural and X-ray Refinement Analysis of SnPb

_{r}Pb

_{1-r}(0.01 ≤ r ≤ 0.99) nanoalloys over a narrow scattering range of 2θ is shown in Figure 2a, where the vertical axis represents the composition r. The “contour color fill” function provided in the Origin software was utilized to draw a 2D plot in which different colors were used to differentiate the peak intensities of the diffraction pattern. From the above 2D plot, for r = 0.01~0.1, nuclear peaks (111) and (200) indexed based on Fm-3m (No. 225) become visible, indicating the formation of a Pb-Sn solid solution having the same structure as that of virgin Pb, as shown in Figure 2b. Since the atomic radii of Sn (140 pm) are smaller than Pb (175 pm), Sn can occupy interstitial sites in the Pb lattices resulting in the formation of the interstitial solid solution, as shown in Figure 2c [9,10,11,12,13,14]. Sn

_{r}Pb

_{1-r}nanoalloys that contain 0~2 at. % Sn behave like the copper-nickel alloys; a single solid solution α-Pb phase forms during solidification [12,13]. These nanoalloys are strengthened by solid-solution strengthening, strain hardening, and controlling the solidification process to refine the grain structure. A previous comprehensive study

^{11}reported that the growth mechanism of SnPb solid solution is due to the grain boundary migration and sliding occurring systematically, giving rise to a series of migration markings on the surface of deformed specimens. However, for r ≥ 0.2, the Pb phase becomes unstable and shows additional diffraction picks (101) and (200) indexed based on I4

_{1}/amd (No. 141) of the Sn phase, as shown in Figure 2d. For further detailed structural analysis, Rietveld refinement of the XRD pattern of Sn

_{r}Pb

_{1-r}bimetallic nanoalloys was carried out using the GSAS software package [15,16]. All the fitting parameters, including the lattice constant and the weight percent of constitute phases, are tabulated in Table 1. The refined patterns of the nanoalloys are shown in Figure 3a (spectra are shifted vertically for clarity). From refined XRD spectra of Sn

_{r}Pb

_{1-r}nanoalloys, immiscible phases of Sn and Pb were observed for all compositions except the region of r = 0.01~0.1, which is in good agreement with the reported phase diagram of the Sn-Pb system [17]. From the fitted values of the lattice constant of the Pb-phase and the Sn Phase vs. r value shown in Figure 3b,c respectively, it can be observed that the Sn doped Pb (r = 0.01) results in a lattice expansion of 0.19% (Figure 3b). For the highest Sn concentration (r = 0.99), both Sn (0.19% along basal plane and c-axis) and Pb (0.06%) show lattice expansion (Figure 3c). The observed expansions from both Sn and Pb phases in nanoalloys could be because of the strain effect, as the thermal expansion coefficient for Sn (22 × 10

^{−6}m/mK) is lower than Pb (28.9 × 10

^{−6}m/mK). The crystallite size of both Sn and Pb phases were calculated using the Scherrer method: $D=\frac{\mathrm{k}\mathsf{\lambda}}{\mathsf{\beta}\mathrm{cos}\mathsf{\theta}}$ nm, where k is constant, θ is the angle of diffraction, λ is the incident wavelength (λ = 0.7749 Å), and β is the full width at half maximum. The obtained crystallite sizes are in nanometers as shown in Table 1, and it can be observed that the Sn-doped Pb (r = 0.01 to 0.99) results in a change in crystallite size (Figure 3b).

#### 3.3. Temperature Dependence of Magnetization

_{a}= 100 Oe in the ZFC and FC modes for r = 0.01, 0.7, 0.8, 0.9, and 0.99 SnPb nanoalloys are shown in Figure 4a–e, where r = 0.1~0.6 are shown in Figure S2 (Supporting Information). The observed step-like behavior of M(T) curves for r = 0.01 indicates that magnetic flux cannot penetrate the materials in a small external field of 100 Oe. However, with an increased Sn concentration, the broadening of M(T) was observed, which could be because of the inhomogeneous distribution of Pb in the Sn matrix. Interestingly, a one-step-like transition was observed from nanoalloys with initial composition r = 0.01–0.7. However, nanoalloys with r ≥ 0.8 show two-step-like transitions. The M(T) curves can be described effectively using a modified London equation with free fitting parameter p, assuming that alloying of Sn and Pb resulted in the formation of SC grains of size <d> obtained from the EDS mapped image of Sn

_{r}Pb

_{1-r}nanoalloys [18,19]. The analysis of mean granular size <d> of segregated Sn (bright green color) and Pb (red color) elements obtained from the EDS mapping results are tabulated in Table 2. From the modified London equation, the DC magnetization can be written as:

^{3}) and Sn (7.3 g/cm

^{3}) concerning initial composition, and λ

_{L}is the London penetration depth defined as

_{C1}(100 Oe) = 6.8 (2) K (maximum, r = 0.7). At higher Sn concentration (r ≥ 0.8), a slight kink occurs in the low-temperature region at T

_{C2}, signaling the onset of a small superconductivity gap contributed from the Sn phase. The observed two-step behavior in r ≥ 0.8 nanoalloys can be described using a superposition of the London equation for Pb(M1) and Sn(M2), showing two transitions, T

_{C1}and T

_{C2}. The fitted value of T

_{C1}(0 Oe) (for further details see Section 3.5) and T

_{C2}(10 Oe) concerning initial composition r is plotted in Figure 4f. Detailed fitting parameters to M(T) curves are summarized in Table 2. The T

_{C1}and T

_{C2}values at 0 and 10 Oe magnetic fields are slightly higher than that of the of reported values from bulk Pb and Sn (7.2 K and 3.7 K, respectively), possibly because of the strain effect observed from the X-ray diffraction refinement [20,21].

#### 3.4. Critical Fields and Theoretical Analysis

_{r}Pb

_{1-r}bimetallic nanoalloys, the field dependence of magnetization M(H

_{a}) below T

_{C}over ± 1200 Oe field was carried out. Figure 5a–d shows the four selected M(H

_{a}) loops measured at 2 K for r = 0.01, 0.7, 0.9, and 0.99 nanoalloys, respectively. Details of the M(H

_{a}) loops measured at 2 K for r = 0.1 to 0.6 nanoalloys are shown in Figure S3 (Supporting Information). Type-I-like M(H

_{a}) loops were observed for r = 0.01 and 0.99, whereas M(H

_{a}) loops of all remaining nanoalloys show type-II-like behavior reflecting the character of the magnetic flux penetration into the superconducting nanoalloys. The Meissner effect was observed in the low field region from M(H

_{a}). It deviated from linearity after reaching the lower critical field H

_{C1}, as shown in Figure 5a–d, where solid lines represent the linear fit. Above the upper critical field H

_{C2}(pointed by the arrow in Figure 5a–d), magnetization M(H

_{a}) eventually turns to M = 0 states. To estimate the temperature dependency of H

_{C1}and H

_{C2}, M(H

_{a}) loops were measured using a protocol of H

_{a}= 0 Oe → +900 Oe → 0 Oe at various temperatures, as shown in Figure 5e–h for r = 0.01, 0.7, 0.9, and 0.99 nanoalloys, respectively. Details of the M(H

_{a}) loops measured at various temperatures for r = 0.1 to 0.6 nanoalloys are shown in Figure S4 (Supporting Information). Figure 5i,j shows the T/T

_{C1}dependency of H

_{C1}and H

_{C2}, which can be described using

_{C1}(0) and H

_{C2}(0) are the lower and upper critical fields at zero temperature, respectively. The fit by using the above former expression yields a maximum field of H

_{C1}(0) = 596(7) Oe from Sn

_{0.2}Pb

_{0.8}nanoalloy and the minimum of H

_{C1}(0)~240(5) Oe from Sn

_{0.99}Pb

_{0.01}nanoalloy. The H

_{C1}(0) values for other nanoalloys lie between these two critical fields and are tabulated in the supporting information of Table 2. We also note that for r = 0.8, 0.9, and 0.99 nanoalloys, due to the appearance of the second superconducting gap of Sn, it is difficult to estimate H

_{C1}(0) values. Contrary to H

_{C1}, H

_{C2}for Sn

_{r}Pb

_{1-r}nanoalloys shows typical type-II superconductor temperature dependency with linear variation near T

_{C}(0) and tends to saturate at low temperatures. The fit by using the above later expression yields a maximum field of H

_{C2}(0) = 886(16) Oe from Sn

_{0.2}Pb

_{0.08}nanoalloy and a minimum value of H

_{C2}(0) = 824(8) Oe from Sn

_{0.01}Pb

_{0.99}nanoalloy.

_{C1}(0) and H

_{C2}(0), in the extreme type-II limit, within the framework of Ginzburg–Landau (GL) theory [23], the GL superconducting coherence ξ

_{GL}length and magnetic penetration depth λ(0) can be estimated by using ${\mathsf{\xi}}_{\mathrm{GL}}\left(0\right)=\sqrt{{\mathsf{\Phi}}_{0}/2{\mathsf{\pi}\mathrm{H}}_{\mathrm{C}2}\left(0\right)}$ and $\mathsf{\lambda}\left(0\right)=\mathsf{\kappa}\left(0\right){\mathsf{\xi}}_{\mathrm{GL}}\left(0\right)$, where ${\mathsf{\Phi}}_{\mathrm{o}}=\frac{\mathrm{h}}{2\mathrm{e}}=2.0678\times {10}^{9}\mathrm{Oe}$Å

^{2}is the quantum flux and $\mathsf{\kappa}\left(0\right)=\sqrt{\raisebox{1ex}{${\mathrm{H}}_{\mathrm{C}2}\left(0\right)$}\!\left/ \!\raisebox{-1ex}{$2{\mathrm{H}}_{\mathrm{C}1}\left(0\right)$}\right.}$. The small value of H

_{C2}(0) implies a long superconducting GL coherence length ξ

_{GL}(0) = 632 Å, λ(0) = 541 Å (r = 0.01), and the similarly large value of H

_{C2}implies a short ξ

_{GL}(0) = 609 Å, λ(0) = 525 Å (r = 0.2). The estimated GL parameter $\mathsf{\kappa}\left(0\right)$ for corresponding r = 0.01 and 0.2 nanoalloys are 0.856 and 0.862, indicating type-II superconductivity. In GL theory, a superconductor is called a type-I superconductor if $\mathsf{\kappa}\left(0\right)<\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.$ and type-II superconductor if $\mathsf{\kappa}\left(0\right)>\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.$. When the order parameter throughout the sample is constant, the GL model reduces to the London model. Furthermore, the value of λ(0) shows increasing behavior with the increase of r, possibly due to an increase in disorder. The observed enhanced value of λ(0) = 828 Å for r = 0.99 indicates the highest degree of disorders in this series of nanoalloys. Magnetic measurement is performed with an applied magnetic field H

_{a}lower than the value of H

_{C1}(0), shielding current flows around the sample surface to exclude H

_{a}. In this case, penetration depth λ is obtained as the penetrating length of H

_{a}at the sample edge. However, magnetic measurements were performed with H

_{a}larger than H

_{C1}(0) (mixed state), and the field penetrated the sample as vortices. In this case, λ is given as a decay length of H

_{a}from the center of the vortex. In conjunction with increasing λ(0) carrier mean free path, $\ell $ at low-temperature T = 2 K can be estimated from GL relation

_{o}is the BCS coherence length [20,24]. The lowest value of $\ell =$ 1260 Å was observed for r = 0.2 nanoalloy. In addition, the calculated values of the thermodynamic critical field, ${\mathrm{H}}_{\mathrm{TC}}\left(0\right)\approx \sqrt{{\mathrm{H}}_{\mathrm{C}1}\left(0\right){\mathrm{H}}_{\mathrm{C}2}\left(0\right)},$ shows a maximum value of ~727 Oe for r = 0.2 nanoalloy. The estimated superconducting parameters for all the SnPb nanoalloys are summarized in Table 2. The observed value of $\ell $ is higher than that of the atomic spacing of Pb and Sn, indicating that the observed slight enhancements in the values of T

_{C}could be due to the strain effect [21,22,25,26,27,28]. Furthermore, note that estimated values of ξ

_{GL}(0) and $\ell $ for both high or lower (i.e., Sn concentration) are close to the superconductor parameter of pure Pb, which could either be due to segregating of a fraction of Pb within Sn matrix or superconducting the proximity effect. Therefore, with the increase of Sn concentration, coupling strength could show decreasing behavior [29].

#### 3.5. Superconducting Coupling Strength

_{a}varying from 10 to 700 Oe and fitted with the modified London equation as shown in Figure 6a–e. More details of the M(T) measured at various applied magnetic fields for r = 0.1 to 0.6 nanoalloys are shown in Figure S5 (Supporting Information). The H

_{a}dependency of T

_{C1}shown in Figure 6f can be fitted using ${\mathrm{T}}_{\mathrm{C}1}\left({\mathrm{H}}_{a}\right)={\mathrm{T}}_{\mathrm{C}1}\left(0\right){\left[1-{\mathrm{H}}_{a}/{\mathrm{H}}_{\mathrm{C}}\left(0\right)\right]}^{\gamma}$, where $\gamma $ is a fitting parameter. The obtained highest critical field H

_{C}(0) at T = 0 K, zero-field critical temperature T

_{C1}(0), and the fitting parameter γ are tabulated in Table 3. The fitted value of H

_{C}(0) represents the upper critical field, and its value matches very well with the fitted value of H

_{C2}(0) (Equation (4); Table 2). The relative coupling strength of a superconducting system can be revealed in the deviation of H

_{C}(T)/H

_{C}(0) from the parabolic dependence of $1-{\left(\mathrm{T}/{\mathrm{T}}_{\mathrm{C}1}\right)}^{2}$ [30]. It is known that weakly coupled systems yield negative deviations, while strongly coupled systems yield positive deviations (marled as red color) [31]. A negative deviation (marked as blue color) was observed for all Sn

_{r}Pb

_{1-r}nanoalloys, as shown in Figure 7, which can be described using the α-model defined as $\frac{{\mathrm{H}}_{\mathrm{C}}\left(\mathrm{T}\right)}{{\mathrm{H}}_{\mathrm{C}}\left(0\right)}=\left[1-{\left(\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{C}1}}\right)}^{\mathsf{\alpha}}\right]$, where the fitting parameter $\mathsf{\alpha}={\mathsf{\Delta}}_{o}/{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{C}1}$ represents the coupling strength (summarized in Table 3). From Figure 7, we note that coupling strength α subverts with an increase of Sn concentration and shows a minimum of α = 1.821(3), T

_{C1}(0) = 7.272(3) K for r = 0.9 nanoalloy, which is more vital than BCS coupling strength of 1.747 [32,33]. The observed subverted coupling strength as compared to strong-coupled Pb is possibly due to the propagation of SPE through the weak-coupled Sn matrix. The observed results suggest that the coupling strength of strong-coupled Pb can be tuned through SPE in Sn

_{r}Pb

_{1-r}bimetallic nanoalloys. A slight enhancement of superconducting transition temperature (T

_{C}) was observed from the magnetization, revealing a main diamagnetic Meissner state below T

_{C}= 7.338(2) K and a critical field of H

_{C}= 852(5) Oe from Sn

_{0.7}Pb

_{0.3}mixed-phase nanoalloy associated with strain effect.

_{ln}and the electron–phonon coupling constant λ

_{ep}, we utilized the Eliashberg theory-based McMillian formulation [34]. The corrections of the BCS values by strong electron–phonon interactions have been deduced in the following approximate analytic formulas that link ω

_{ln}/T

_{C1}(0) to experimental thermodynamic quantities:

_{C}versus estimated ω

_{ln}from Equation (6), where the green colors represent weak-coupled strength α. The value of ω

_{ln}is helpful for the description of the superconducting properties of conventional superconductors. The most weak-coupled Sn

_{0.9}Pb

_{0.1}nanoalloy (sample #10) has α = 1.821(3) and T

_{C1}(0) = 7.272(3) K and shows the maximum phonon energy of ω

_{ln}= 241 K (phonon hardening). An electron–phonon coupling constant λ

_{ep}can be estimated from the McMillan equation,

_{ep}; we set μ* = 0.11 [34]. The obtained values of λ

_{ep}are tabulated in Table 3. From calculated values of ${\mathsf{\omega}}_{\mathrm{ln}}$ and λ

_{ep}

_{,}it becomes clear that T

_{C1}is a combined effect of both electron–phonon coupling strength and phonon energy. The minimum λ

_{ep}= 0.685 and maximum ω

_{ln}= 241 K were observed for Sn

_{0.9}Pb

_{0.1}bimetallic nanoalloy. The above value of λ

_{ep}is smaller than that of pure Pb and Sn, λ

_{ep}= 1.580 and 0.761, respectively. From the above finding, it is clear that the propagation of superconductivity through SPE weakened the electron–phonon coupling strength and hardened the phonon of the Pb phase.

## 4. Discussions

_{GL}(0) and the average logarithmic phonon energy ω

_{ln}are linked with the coupling strength of the superconductor. Figure 9 depicts the initial composition r dependence of calculated values of ξ

_{GL}(0) and ω

_{ln}from Sn

_{r}Pb

_{1-r}(0.01 ≤ r ≤ 0.99) bimetallic nanoalloys. For an initial Sn concentration from r = 0.01 to 0.2, a steep drop behavior of ξ

_{GL}(0) was observed (the dashed red line is guided for the eyes), while the two elements of Pb and Sn form a solid solution of the α-Pb phase. The resultant phase diagram is thus completely different from those of other composition regions r = 0.2~0.99, typical immiscible bimetallic superconductors of Sn and Pb compounds, in which ξ

_{GL}(0) and ω

_{ln}increase when Sn is substituted for Pb in the strong-coupled superconducting “parent” compounds. Notably, the solid solutions in the superconducting range exhibit ξ

_{GL}(0) anomalies at r = 0.01–0.1. The behavior is reminiscent of the disorder, and substitution in weak coupling superconductivity could enhance the electron–phonon coupling constant. A simplified physical model of electron–phonon coupling has been developed by Gao et al. to allow heat transfer from phonons to electrons and applied to study defects or disorders as a function of the strength of electron–phonon coupling [37]. The number of point defects produced in the primary damage state increases with the strength of electron–phonon coupling, signaling that the defect or disorder/substitution plays a role in the increasing strength of the electron–phonon coupling constant. A maximum of λ

_{ep}= 1.046 from r = 0.01 and λ

_{ep}= 0.97 from r = 0.1 (as shown in Table 3) indicates that the similarity suggests the enhancement of the electron–phonon coupling in the present case also has a substitutional origin. On the other hand, ω

_{ln}shows a linear increasing behavior (the solid blue line is guided for eyes) with the decrease of coupling strength from r = 0.01 to 0.9, above which fluctuations were observed. The observed results agree with theoretical findings except for the high Pb concentration (r = 0.01 and 0.1) region.

## 5. Conclusions

_{r}Pb

_{1-r}(0.01 ≤ r ≤ 0.99) bimetallic nanoalloys were prepared using the simple physical annealing system. The observed discrepancy between the estimated atomic % using EDS and initial composition could be because of segregated grains of Sn and Pb. EDS-mapped images show the inhomogeneous distribution of nano to micrometer size Sn and Pb segregated grains. SR-XRD refinement carried out using Rietveld analysis reveals the formation of the pure Pb-solid solution and immiscible mixed-phase nanoalloys. The effect of strain was observed on the lattice constants of Sn and Pb phases in all immiscible nanoalloys. Magnetization measurements revealed slight enhancement in the superconducting transition T

_{C}for strained nanoalloy when fitted to modify the London equation. The coupling strength α, electron–phonon coupling constant λ

_{ep}, and average logarithmic phonon energy ω

_{ln}were estimated for all nanoalloys using the α-model and McMillian formulation. The estimate α, λ

_{ep}showed subverted behavior and phonon hardening due to the propagation of SPE through the Sn matrix. The estimated sizes of the Sn phase increased from 50~60 nm as the composition r increased from 0.2~0.9, reflecting the excess surface contributions of the Sn matrix giving rise to enhanced propagation of SPE. In the immiscible regime, the behavior of ξ

_{GL}(0) is the same as that of the phonon energy, ω

_{ln,}i.e., both physical parameters increase with the decrease of coupling strength α, revealing a superconducting proximity effect between immiscible Sn and Pb granules in the range of r = 0.2~0.9.

## Supplementary Materials

_{a}) loops; Figure S4: The M(H

_{a}) loops measured using protocol 0 Oe → +900 Oe → 0 Oe field; Figure S5: Temperature dependence of ZFC magnetization of Sn

_{r}Pb

_{1-r}bimetallic nanoalloys.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Typical EDS elemental spectra reveal a series of peaks associated with Sn and Pb elements, verifying that the Sn

_{r}Pb

_{1-r}bimetallic nanoalloys contain only Sn and Pb elements; (

**b**) atomic % of Sn with respect to the initial composition of Sn

_{r}Pb

_{1-r}bimetallic nanoalloys.

**Figure 2.**(

**a**) a 2D plot of XRD patterns of Sn

_{r}Pb

_{1-r}bimetallic nanoalloys over a narrow range of scattering angle 2θ; the crystalline structure of (

**b**) α-Pb; (

**c**) α-Pb/Sn solid solution; and (

**d**) β-Sn, respectively.

**Figure 3.**(

**a**) observed (colored crosses) and Rietveld refined (solid black lines) X-ray diffraction patterns of Sn

_{r}Pb

_{1-r}nanoalloys; (

**b**,

**c**) the effect of strain on the lattice constant of Pb and Sn is plotted with respect to initial composition r. Lattice expansion was observed for both Pb and Sn phases; (

**d**) the variation of crystallite size with composition r for Pb and Sn phases.

**Figure 4.**(

**a**–

**e**) temperature dependence of magnetization measured between 2 to 8 K in an external magnetic field of H

_{a}= 100 Oe in the ZFC and FC modes for r = 0.01, 0.7, 0.8, 0.9, and 0.99 nanoalloys. The red and blue solid curve presents the fitted curve of the modified London expression to the data; (

**f**) fitted value of transition temperature T

_{C1}(black sphere, see Section 3.5) and T

_{C2}(blue circle, measured at 10 Oe) as a function of the initial composition. The horizontal dashed lines represent the reported values of T

_{C}from bulk Pb and Sn.

**Figure 5.**(

**a**–

**d**) isothermal magnetization M(H

_{a}) loops measured at 2 K for r = 0.01, 0.7, 0.9 and 0.99 nanoalloys. The solid red line is the linear fit to the first magnetization curve. Arrows mark the observed irreversible field; (

**e**–

**h**) the M(H

_{a}) loops measured using a protocol of H

_{a}= 0 Oe → +900 Oe → 0 Oe at higher temperature up to 7 K for r = 0.01, 0.7, 0.9, and 0.99 nanoalloys; (

**i**,

**j**) lower and upper critical field as a function of temperature. Solid lines fit the power law mentioned in the text.

**Figure 6.**(

**a**–

**e**) the effect of applied field H

_{a}on the transition temperature T

_{C1}of Sn

_{r}Pb

_{1-r}nanoalloys; (

**f**) the H

_{a}dependency of T

_{C1}. The solid line is fitted to the expression given in the text.

**Figure 7.**Deviation of H

_{C}(T)/H

_{C}(0) from the parabolic dependence of 1 − (T/T

_{C1})

^{2}fitted to the expression H

_{C}(T)/H

_{C}(0) = [1 − (T/T

_{C1})

^{α}], where α represents the coupling strength of the superconductor.

**Figure 8.**A 2D plot of observed T

_{C}versus estimated ω

_{ln}from the equation explained in the text where the various green colors represent weak-coupled strength α, respectively.

**Figure 9.**The plot of composition r depends on the GL coherence length and the phonon energy. The red dashed line and solid blue lines are guides for the eyes.

**Table 1.**Summary of best fitted Rietveld refined fitting parameters of the Sn and Pb phases from the Sn

_{r}Pb

_{1-r}nanoalloys using the GSAS software package: lattice constant (a, b, c), weighted residue (Rwp), residual of least square refinement (Rp), a squared ratio between Rwp and Rexp (χ

^{2}), crystalline size, and weight fractions (Wt. fraction).

Sn_{r}Pb_{1-r} | Lattice Constants (Å) | Rwp | Rp | χ^{2} | Crystalline Size (nm) | Wt. Fraction (%) | ||||
---|---|---|---|---|---|---|---|---|---|---|

Pb Phase | β-Sn Phase | Pb Phase | Sn Phase | |||||||

a = b = c | a = b | c | Pb | β-Sn | ||||||

0.01 | 4.9600 (1) | 0.3554 | 0.1881 | 21.45 | 55 | 100 | 0 | |||

0.1 | 4.9567 (1) | 5.8443 (4) | 3.1884 (3) | 0.0265 | 0.0111 | 0.1184 | 53 | 55 | 75.123 | 24.877 |

0.2 | 4.9548 (1) | 5.8422 (3) | 3.1859 (2) | 0.0424 | 0.0215 | 0.2699 | 50 | 54 | 69.128 | 30.872 |

0.3 | 4.9552(1) | 5.8427 (1) | 3.1875 (1) | 0.031 | 0.017 | 0.1587 | 57 | 60 | 68.782 | 31.218 |

0.4 | 4.9526 (1) | 5.8394 (1) | 3.1855 (1) | 0.0344 | 0.018 | 0.1888 | 62 | 66 | 65.555 | 34.445 |

0.5 | 4.9559 (1) | 5.8431 (2) | 3.1877 (1) | 0.0442 | 0.0229 | 0.2843 | 47 | 61 | 50.246 | 49.754 |

0.6 | 4.9549 (1) | 5.8423 (1) | 3.1872 (1) | 0.0555 | 0.0306 | 0.4864 | 57 | 60 | 36.476 | 63.524 |

0.7 | 4.9568 (1) | 5.8433 (1) | 3.1894 (1) | 0.0459 | 0.0246 | 0.3812 | 58 | 61 | 54.848 | 45.152 |

0.8 | 4.9593 (1) | 5.8474 (1) | 3.1894 (1) | 0.0716 | 0.04 | 1.175 | 58 | 65 | 41.589 | 58.411 |

0.9 | 4.9566 (2) | 5.8441 (1) | 3.1877 (1) | 0.0479 | 0.0236 | 0.3648 | 59 | 60 | 34.294 | 65.706 |

0.99 | 4.9540 (9) | 5.8428 (2) | 3.1878 (1) | 0.1424 | 0.0903 | 5.461 | 57 | 53 | 0.775 | 99.225 |

**Table 2.**Summary of obtained physical parameters: mean granular size <d>, critical field H

_{C1}(0), upper critical field H

_{C2}(0), and thermodynamical critical field H

_{C}(0), Ginzburg–Landau coherence length ξ

_{GL}(0), Ginzburg–Landau parameter k, penetration depth λ(0), and mean free path $\ell $ of Sn

_{r}Pb

_{1-r}nanoalloys estimated within the superconducting state.

Sn_{r}Pb_{1-r} | <d> nm | H_{C1}(0) Oe | H_{C2}(0) (Oe) | H_{TC}(0) (Oe) | ξ(0) (Å) | λ (Å) | ξ_{GL}(0) (Å) | $\mathsf{\kappa}\left(0\right)$ | $\mathsf{\ell}\left(\mathsf{\u212b}\right)$ |
---|---|---|---|---|---|---|---|---|---|

0.01 | 121 (10) | 562 (7) | 824 (8) | 681 | 896 | 541 | 632 | 0.856 | 1536 |

0.1 | 105 (9) | 563 (8) | 863 (13) | 697 | 897 | 540 | 618 | 0.875 | 1340 |

0.2 | 80 (5) | 596 (7) | 886 (16) | 727 | 896 | 525 | 609 | 0.862 | 1260 |

0.3 | 79 (4) | 444 (10) | 880 (12) | 625 | 894 | 609 | 612 | 0.995 | 1298 |

0.4 | 77 (5) | 340 (4) | 839 (10) | 534 | 897 | 696 | 626 | 1.111 | 1452 |

0.5 | 70 (3) | 281 (3) | 867 (17) | 494 | 893 | 765 | 616 | 1.242 | 1358 |

0.6 | 81 (4) | 229 (7) | 877 (13) | 448 | 894 | 848 | 613 | 1.384 | 1309 |

0.7 | 75 (5) | 269 (5) | 852 (13) | 479 | 888 | 782 | 622 | 1.258 | 1472 |

0.8 | 75 (4) | 260 | 854 (10) | 471 | 895 | 795 | 621 | 1.281 | 1397 |

0.9 | 69 (3) | 250 | 852 (11) | 462 | 896 | 811 | 622 | 1.305 | 1398 |

0.99 | 46 (3) | 240 | 841 (13) | 449 | 897 | 828 | 626 | 1.323 | 1444 |

**Table 3.**Summary of obtained physical parameters, such as critical temperature T

_{C1}(0), critical field H

_{C}(0), fitting parameters γ, coupling strength α, phonon energy ω

_{ln}, and electron–phonon coupling constant λ

_{ep}of Sn

_{r}Pb

_{1-r}nanoalloys, estimated within the superconducting state.

Sample# | Sn_{r}Pb_{1-r} | T_{C1}(0) (K) | H_{C} (Oe) | γ | α | ω_{ln} (K) | λ_{ep} |
---|---|---|---|---|---|---|---|

1 | 0.01 | 7.271 (9) | 823 (5) | 0.505 (8) | 1.983 (10) | 102 | 1.046 |

2 | 0.1 | 7.263 (16) | 823 (8) | 0.513 (13) | 1.947 (15) | 115 | 0.97 |

3 | 0.2 | 7.273 (16) | 828 (8) | 0.521 (14) | 1.920 (16) | 128 | 0.913 |

4 | 0.3 | 7.290 (25) | 850 (15) | 0.528 (23) | 1.896 (23) | 143 | 0.862 |

5 | 0.4 | 7.266 (13) | 840 (31) | 0.537 (31) | 1.861 (12) | 173 | 0.785 |

6 | 0.5 | 7.297 (22) | 868 (59) | 0.537 (57) | 1.864 (19) | 171 | 0.79 |

7 | 0.6 | 7.289 (22) | 878 (64) | 0.537 (60) | 1.863 (20) | 172 | 0.788 |

8 | 0.7 | 7.338 (2) | 852 (5) | 0.541 (5) | 1.847 (1) | 193 | 0.751 |

9 | 0.8 | 7.280 (10) | 854 (27) | 0.524 (26) | 1.909 (10) | 135 | 0.887 |

10 | 0.9 | 7.272 (3) | 853 (8) | 0.549 (8) | 1.821 (3) | 241 | 0.685 |

11 | 0.99 | 7.267 (14) | 841 (33) | 0.534 (33) | 1.872 (12) | 163 | 0.807 |

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**MDPI and ACS Style**

Gandhi, A.C.; Manjunatha, K.; Chan, T.-S.; Wu, S.Y.
Structural and Superconducting Proximity Effect of SnPb Bimetallic Nanoalloys. *Nanomaterials* **2022**, *12*, 4323.
https://doi.org/10.3390/nano12234323

**AMA Style**

Gandhi AC, Manjunatha K, Chan T-S, Wu SY.
Structural and Superconducting Proximity Effect of SnPb Bimetallic Nanoalloys. *Nanomaterials*. 2022; 12(23):4323.
https://doi.org/10.3390/nano12234323

**Chicago/Turabian Style**

Gandhi, Ashish Chhaganlal, Krishtappa Manjunatha, Ting-Shan Chan, and Sheng Yun Wu.
2022. "Structural and Superconducting Proximity Effect of SnPb Bimetallic Nanoalloys" *Nanomaterials* 12, no. 23: 4323.
https://doi.org/10.3390/nano12234323