Is There a Lower Size Limit for Superconductivity? - ACS Publications

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Is there a lower size limit for superconductivity? Subhrangsu Sarkar, Nilesh Kulkarni, Ruta Kulkarni, Krishnamohan Thekkepat, Umesh Vasudeo Waghmare, and Pushan Ayyub Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b03659 • Publication Date (Web): 05 Oct 2017 Downloaded from http://pubs.acs.org on October 10, 2017

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Is there a lower size limit for superconductivity? Subhrangsu Sarkar,1 Nilesh Kulkarni,1 Ruta Kulkarni,1 Krishnamohan Thekkepat,2 Umesh Waghmare2 and Pushan Ayyub1* 1

Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental Research, Mumbai 400005, India 2

Theoretical Science Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore-560 064, India

KEYWORDS: superconductivity, superconducting materials, Tantalum, density functional theory, ab initio calculations, quantum size effects, nanoparticles, nanostructured materials.

TABLE OF CONTENTS GRAPHICS

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ABSTRACT

The ultimate lower size limit for superconducting order to exist is set by the ‘Anderson criterion’ – arising from quantum confinement – that appears to be remarkably accurate and universal. We show that carefully grown, phase-pure, nanocrystalline bcc-Ta remains superconducting (with ordering temperature, TC ≈0.9K) down to sizes 40% below the conventional estimate of the Anderson limit of 4.0nm. Further, both the TC and the critical magnetic field exhibit an unusual, non-monotonic size dependence, which we explain in terms of a complex interplay of quantum size effects, surface phonon softening and lattice expansion. A quantitative estimation of TC within first-principles density functional theory shows that even a moderate lattice expansion allows superconductivity in Ta to persist down to sizes much lower than the conventional Anderson limit, which can be traced to anomalous softening of a phonon due to its coupling with electrons. This appears to indicate the possibility of bypassing the Anderson criterion by suitable crystal engineering and obtaining superconductivity at arbitrarily small sizes, an obviously exciting prospect for futuristic quantum technologies. We take a critical look at how the lattice expansion modifies the Anderson limit, an issue of fundamental interest to the study of nanoscale superconductivity.

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Superconductors are one of the earliest group of materials in which size effects were studied systematically (see Ref. 1 for a recent review). Recent developments in nano-electronics and quantum computation based on Josephson junction arrays comprising of superconducting quantum dots are expected to provide a huge impetus for a deeper understanding of the size dependence of the superconducting transition temperature (TC) and the upper critical magnetic field (HC2). Interestingly, the length scales most relevant to superconductivity, namely, the coherence length (ξ0) and London penetration depth do not necessarily govern the size dependence of superconducting properties. Destabilization of the superconducting state at low-dimensions is ultimately controlled by the quantum confinement scale defined by the size at which the Kubo gap (mean electronic energy level spacing near EF, the Fermi energy) exceeds the bulk, zerotemperature superconducting energy gap (∆(0)). This remarkably accurate thumb rule, first suggested by Anderson, has been experimentally validated in virtually all elemental superconductors investigated. Here, we examine the universality of the criterion and show that an isotropic lattice expansion caused by size reduction in nanocrystalline Ta allows it to superconduct down to sizes much lower than predicted by the usual expression for the Anderson limit. Previous results on the general nature of the size dependence of TC and/or HC2 in elemental superconductors such as Nb 2, Pb 3, Al 4,5,6, Sn 7,8, In 9 and V 10 can be summarized as follows. In weakly coupled superconductors 11 (𝜅𝜅 = 2Δ(0)⁄𝑘𝑘𝐵𝐵 𝑇𝑇𝐶𝐶 ≈ 3.5), such as Al, Sn, and In, TC initially increases with decreasing size but finally decreases as d→dC, the Anderson limit. The initial enhancement is usually attributed to surface phonon effects. 12 and – to a lesser extent – Shell Effects, arising from size-dependent oscillations in the energy gap 13. Size-dependent changes in the TC may also occur due to Parity Effect.5 In contrast, TC falls monotonically with size reduction over a comparatively wide range down to dC in intermediate-coupling superconductors such as Nb 3 ACS Paragon Plus Environment

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(𝜅𝜅 = 3.8). In strongly coupled superconductors such as Pb (𝜅𝜅 = 4.5), TC remains almost constant

for sizes down ≈dC, and then falls rapidly to zero. In moderate to strongly coupled systems, the size dependence of TC and HC is determined by a competition between quantum size effect (QSE) and phonon softening. In contrast, granular ultrathin films with poor inter-grain coupling (island structure) may lose superconducting correlations due to large fluctuations in the phase of the superconducting order parameter between neighbouring grains 14. We report certain unprecedented features in the superconducting behavior of nanocrystalline Ta. Not only does the TC show an unusually complicated size dependence, it remains non-zero (≈0.9K) down to our lowest particle size of ≈2.4nm, 40% lower than the dC predicted for Ta. Bulk Ta is a weak-coupling (𝜅𝜅 = 3.6) BCS, Type-I superconductor 15 with TC= 4.48K and HC(0) = 0.083T,

though even small concentrations of defects and dopants drive it into the Type-II state 16. We show that the apparent violation of Anderson’s prediction is connected to another singular feature of nanocrystalline Ta: an expansion of its crystallographic unit cell (by ≈4%) with size reduction. A similar lattice expansion is also observed in nanocrystalline Nb 17. We determined – within the first-principles density functional theory (DFT) – the effect of the size-induced lattice strain on the electronic density of states (DOS) around EF, the phonons, and the TC (from McMillan’s equation). Our analysis provides a semi-quantitative understanding of the unusual size dependences of TC and HC, and indicates the physical basis for the modification of the Anderson limit. Nanocrystalline thin films of phase-pure α-Ta (bcc) were sputter-deposited on Si substrates (see Supporting Information for details of synthesis, characterization and measurements). Since earlier studies of superconductivity in size-confined Ta were compromised by the samples being either mixed-phase or structurally disordered 18,19,20, we made a systematic study to identify the narrow 4 ACS Paragon Plus Environment

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parameter window that yields single phase, crystalline Ta. Twelve samples with average crystallite size (dXRD) in the range 2.4-59nm range were selected after analysis. The x-ray diffraction (XRD) data (Fig. 1(a) shows five representative samples) confirmed that all the as-prepared samples were crystalline (mainly -oriented), monophasic α-Ta. We define dXRD as the instrumentcorrected, coherently diffracting domain size obtained from line profile analysis of the powder XRD scans. A cross-sectional scanning transmission electron microscopy (STEM) image of the sample with the smallest particle size (dXRD = 2.4nm) confirms that even this sample is almost completely (nano-)crystalline (Fig. 1(b)) with close to symmetrical particles. The size distribution obtained for a statistically significant number of particles (Fig. 1(c)) is less than ±15%, as typically observed in high-pressure sputtered samples. Supporting Information also includes an electron diffraction pattern and high resolution STEM image of a typical sample (dXRD = 7nm) that supports their crystalline nature, and scanning electron micrographs showing the surface microstructure. A loosely-connected nanoparticle ensemble has a well-defined, macroscopic 𝑇𝑇𝐶𝐶 (close to that of

the individual grains) provided the nanoparticles form a weakly coupled Josephson junction array,

one of whose features is a hysteresis in the I-V characteristics. 21, 22 Previous studies confirm that the sputter-deposited, nanocrystalline thin films of the type studied here satisfy this condition.2,3 We used electrical transport measurements down to 1.8 K (or 50 mK, for samples with TC < 2.2 K) to determine the TC, which we define as the highest temperature at which the resistivity is zero. HC was determined from magneto-resistance measurements and defined as the magnetic field at which the resistance =0.9RN, where RN is the normal state resistance at that temperature. We also measured the residual resistivity ratio, 𝑅𝑅𝑅𝑅 = �𝑅𝑅300𝐾𝐾 − 𝑅𝑅𝑇𝑇𝐶𝐶 (onset)+𝛿𝛿 �⁄𝑅𝑅300𝐾𝐾 , where δ ≈ step size

for temperature measurement. We observed TC = 4.17K for ‘bulk’ Ta (dXRD = 59nm), in reasonable

agreement with the reported value of 4.48K. With a reduction in the crystallite size (Fig. 2(a)), TC 5 ACS Paragon Plus Environment

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gradually decreases to 3.84K at 31nm, thereafter rapidly falling to a minimum value of 0.85K at 8.4nm (Region-I). Further reduction in dXRD causes an interim increase in the TC up to 1.41K at 7nm (Region-II) followed by a decrease and an apparent saturation (Region-III). Remarkably, the TC remains almost constant (=0.94K) down to our lowest crystallite size (2.4 nm). The residual resistivity ratio, RR, decreases monotonically with reducing size (Fig. 2(c)), which can be ascribed to larger scattering of charge carriers from the intergranular regions. For dXRD ≤ 5nm, RR becomes negative, suggesting a weak metal-insulator transition driven by QSE 23. Note that even these samples continue to superconduct. HC too follows a non-monotonic size dependence (Fig. 2(b)), reaching a maximum of 0.96T at ≈10nm. Then it drops rapidly to 0.37T at 8.4nm, before rising again and saturating near 0.75 T with decreasing size. The Anderson Limit 24, dC, for Ta was estimated from the equation:5 ∆(0) = 32𝜋𝜋ℎ2 ⁄2𝑚𝑚𝑘𝑘𝐹𝐹 𝑑𝑑𝐶𝐶3 ,

where 𝑚𝑚 = electron mass and 𝑘𝑘𝐹𝐹 = Fermi wave vector. Using values appropriate for Ta, we obtain: dC ≈ 4nm. Thus, unlike most previous observations on weak as well as strong coupling BCS

systems, superconductivity in nanocrystalline Ta apparently does not get destabilized at the Anderson Limit and persists till at least 0.6dC. While discussing the size dependence of the TC, we can rule out Parity Effect5 and Shell Effect,13 because the small but non-negligible size distribution (≈±15%) in our samples would smear out the size-dependent fluctuations in TC. In fact, a recent STM study 25 of single Pb nanoparticles reports a suppression of Cooper pairing when the mean electronic level spacing exceeds the superconducting gap, reconfirming the validity of the Anderson criterion. The clear correlation between the width and position (2θ) of the line in the XRD spectra (Fig. 1(a)) implies a size-induced lattice expansion. Fig. 3(a) shows that the lattice constant (a) of

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α-Ta undergoes a monotonic, non-linear expansion with decreasing dXRD. We now attempt to investigate the effect of lattice expansion on the electronic properties relevant to superconductivity. Since DFT calculations on structures containing more than a few hundred atoms are highly resource-intensive, we performed the calculations on bulk Ta by freezing the lattice constant, a, at the experimentally observed (expanded) values for nano-Ta. We first obtained the electronic DOS and integrated it in the range 𝐸𝐸𝐹𝐹 ± ∆(0)⁄2, with ∆(0) =1.4 meV for bulk Ta15. This was multiplied

by the Avogadro Number to get an approximate population of states per mole (N) in the energy window 𝐸𝐸𝐹𝐹 ± ∆(0)⁄2. A comparison of the calculated DOS for bulk (dXRD =59nm, a =3.31Å) and

nanocrystalline Ta (dXRD =2.4nm, a =3.45Å) shows a clear enhancement in the DOS near EF with

decreasing size (Fig. 3(b)). Fig. 3(c) gives the variation in N with lattice constant (which is correlated with dXRD) and indicates about 20% increase in N between the largest and the smallest particles. (Details in Supporting Information, Sec. 4.) To analyze the effect of lattice expansion on TC, we determined the phonon spectra as a function of lattice constant using first-principles density functional perturbation theory (DFPT), and calculated the TC from the modified McMillan formula 26,27:

𝑇𝑇𝐶𝐶 =

𝜔𝜔𝑙𝑙𝑙𝑙𝑙𝑙 1.2

exp �

−1.04(1+𝜆𝜆)

2

�, where 𝜔𝜔𝑙𝑙𝑙𝑙𝑙𝑙 = exp � ∫ 𝜆𝜆(1−0.62𝜇𝜇 ∗ )−𝜇𝜇∗ 𝜆𝜆

𝛼𝛼 2 𝐹𝐹(𝜔𝜔) log(𝜔𝜔) 𝜔𝜔

𝑑𝑑𝑑𝑑�

(1)

Here the Coulomb pseudopotential26,28,29:

𝜇𝜇∗ =

0.26𝑁𝑁(𝐸𝐸𝐹𝐹 ) 1+𝑁𝑁(𝐸𝐸𝐹𝐹 )

(2).

The renormalized electron-phonon mass enhancement parameter: 𝜆𝜆 = 2 ∫[𝛼𝛼 2 𝐹𝐹(𝜔𝜔)⁄𝜔𝜔] 𝑑𝑑𝑑𝑑 is expressed as an integral over phonon frequency, 𝜔𝜔, and the Eliashberg function, 𝛼𝛼 2 𝐹𝐹(𝜔𝜔). For a

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fully relaxed Ta lattice at equilibrium with a = 3.303 Ǻ (bulk value) we obtained TC ≈4.5K, in excellent agreement with the value reported for bulk Ta. We then performed simulations with expansive (tensile) strain up to 4%, and found that the strain had significant effect on the electronic DOS and band structure (Fig. 4 (a,b)), as well as the phonon DOS and spectrum (Fig. 4 (c,d)). We now consider the effect of lattice expansion on the different terms that control TC in McMillan’s equation. (i) The term 𝜔𝜔𝑙𝑙𝑙𝑙𝑙𝑙 is analogous to the Debye frequency, Θ𝐷𝐷 , which decreases

with increasing expansion, as indicated by the phonon dispersion (Fig. 4(d)). (ii) Since 𝑁𝑁(𝐸𝐸𝐹𝐹 )

increases significantly with lattice expansion, 𝜇𝜇 ∗ too should increase, albeit more slowly (see eqn. (2)). As a result of lattice expansion: 𝜔𝜔𝑙𝑙𝑙𝑙𝑙𝑙 would tend to reduce the TC, while 𝜇𝜇 ∗ would tend to

increase it, though their influence is rather limited27. (iii) The phonon dispersion curves (Fig. 4(d))

show that the tensile strain is accompanied by the emergence of a soft phonon mode at N-point. Clearly, the N-point phonon softening for strain ≥ 3% is correlated with the low-frequency shoulder in the phonon-DOS (Fig. 4(c)) and the low-frequency peak in the Eliashberg function (circled in Fig. 4(e)). (iv) The tensile strain causes the Eliashberg function to increase in magnitude and most of its peaks shift to lower frequencies. The red-shift (shown by a blue dashed line) is particularly marked for the first low frequency feature at 55 cm-1. What is the origin of the feature in the Eliashberg function, as well as the N-point soft phonon, both of which move sharply from 60 cm-1 (3% strain) to 25 cm-1 (4% strain)? Such a dramatic softening of only the N-point phonon arises from the nesting effect: electronic bands at about −1 eV near the N-point are essentially flat (along P-N-H lines in Fig. 4(b)) giving a peak in the density of states (Fig. 4(a)). The N-point phonon couples electrons in these occupied states to the unoccupied states at ~ +0.8 eV near the Γ point (H-Γ-N). Due to the large deformation potential 8 ACS Paragon Plus Environment

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(sensitivity to strain) of the flat bands (Fig. 4(b)), the N-phonon frequency changes sharply with strain. Fig. 4(f) summarizes the relative changes in TC, 𝜆𝜆 and 𝜔𝜔𝑙𝑙𝑙𝑙𝑙𝑙 due to lattice expansion. Although 𝜔𝜔𝑙𝑙𝑙𝑙𝑙𝑙 decreases with strain, TC is more strongly influenced by 𝜆𝜆, which occurs in the

exponent in eqn. (1). Thus, lattice expansion causes a pronounced softening of the bulk phonon modes: a reduction in 〈𝜔𝜔〉 enhances 𝜆𝜆, thereby increasing TC, as observed by us in Regions II-III.

We now provide a comprehensive explanation of the experimentally observed size dependence of TC and HC in nano-Ta. The initial monotonic decrease in TC in Region I (60nm—9nm) – also seen in nanocrystalline Nb2 – is due to QSE, possibly aided by the effect of disorder at the inter-granular regions. The critical field in the dirty limit (in which the electronic mean free path, 𝑙𝑙 ≪ 𝜉𝜉0 , the coherence length of the

pure material) is described by the WHH theory 30 as: 𝐻𝐻C (0) ~(2.76𝑒𝑒𝑒𝑒𝑘𝑘𝐵𝐵 ⁄π)𝑇𝑇C 𝑁𝑁(0)ρN , where 𝜌𝜌𝑁𝑁

denotes the normal state resistivity. In this size range, 𝑇𝑇𝐶𝐶 𝑁𝑁(0) decreases by a factor of ≈2.2, which

is much smaller than the 20-fold increase in 𝜌𝜌𝑁𝑁 (Fig.2(c)). Thus, 𝐻𝐻𝐶𝐶 (0) increases substantially

with decreasing size mainly due to disorder. However, TC drops sharply below 10nm due to QSE, and so does HC – because disorder effects saturate out below 8nm, as shown by the gradual

flattening out of RR. The effect of lattice expansion is most dominant in Region II (8nm—6nm), leading to a slight enhancement in TC by ≈1K due to bulk phonon softening. In this size range, the contribution of 𝜌𝜌𝑁𝑁 to HC is increasingly compensated by that of N(0). Down to 8nm, the behavior

of HC is similar to that in Nb 31 and Pb3. Below 8nm, TC and HC show almost identical size

dependence (Fig. 2, inset), since both are expected to be controlled by a similar interplay of QSE, phonon softening and lattice expansion. Below 6nm (Region III), the lattice expansion vs. dXRD curve flattens out slightly (Fig. 3(a)) and quantum confinement effects prevail. Nevertheless, why does superconductivity persist even at the lowest size investigated? The explanation is depicted 9 ACS Paragon Plus Environment

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schematically in Fig. 5. We usually assume the superconducting energy gap, ∆(0), to be sizeindependent (solid red line) and estimate the Anderson limit as the particle size (𝑑𝑑𝐶𝐶0 ) at which the

Kubo gap (∼d−3) intersects ∆(0). In the case of Ta: 𝑑𝑑𝐶𝐶0 = 4nm. However, we must also take account

of the size-dependent lattice expansion in Ta, which leads to an enhancement in λ, and hence ∆(0), with decreasing size. The modified ∆(0) is represented by the dashed red curves. On the other hand, the small, size-dependent increase in the DOS at EF implies that the Kubo gap would rise less slowly than ∼d−3 (dashed blue lines). Consequently, the Kubo gap may either intersect ∆(0) at some lower size, 𝑑𝑑𝐶𝐶1 , or may not even intersect it at all. This is also very significant as it may

explain earlier observations of superconductivity in amorphous Ta.18,19,20 Amorphous Ta samples

have usually been prepared by reducing the film thickness to below 5 nm (see, e.g., Ref. 20), and show a progressive broadening of the XRD line with reducing thickness, indicating a decrease in the crystallite size. Such “amorphous” samples may show superconductivity because they are actually nanocrystalline with extremely small crystallite size. In conclusion, the unusual, non-monotonic size dependence of TC and HC in nanocrystalline Ta, as well as the persistence of superconductivity till very small sizes (much lower than dC) can be explained in terms of a complicated interplay between quantum size effects, surface phonon softening and a size dependent lattice expansion. In addition to enhancing the DOS near EF, the latter tends to sustain superconductivity in nanoparticles by causing anomalous softening of a bulk zone-boundary phonon which nests and couples electronic states in flat energy bands just above and below EF. Finally, we have no reason to suspect the validity of the Anderson criterion, per se, i.e., destabilization of superconductivity would occur at that size for which the Kubo gap exceeds superconducting energy gap. However, in the presence of a size-dependent lattice expansion, the intersection point may get shifted arbitrarily downward, even below absolute zero. In such cases, 10 ACS Paragon Plus Environment

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the answer to the question posed in the title would be “No”. Also, it would not be possible to obtain the intersection point (Anderson limit) from a simple analytical equation. This work opens up the interesting possibility of utilizing nanostructured superconductors (say, in quantum circuits) even when their characteristic dimensions fall below the nominal Anderson Limit. This can be achieved by creating appropriate expansive stresses in the crystal structure via epitaxial growth.

SUPPORTING INFORMATION 1. Synthesis of nanocrystalline α-Ta, 2. Measurement techniques, 3. Microstructure of representative nanocrystalline Ta samples. 4. Computational Methods: First-principles Calculations of Electronic DOS and Superconducting Parameters.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] ORCID: Pushan Ayyub 0000-0002-4423-9131

Acknowledgements: We thank R. Bapat, S. C. Purandare, B. A. Chalke and J. Parmar for their contributions to the microscopic analysis of Ta films, and A. Thamizhavel for his cooperation and advice with regard to low temperature measurements. We also thank A. P. Shah for his help in growing PECVD coated SiO2 layers on Si wafers.

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24. Anderson, P. W. J. Phys. Chem. Solids 1959, 11, 26‒30. 25. Vlaic, S.; Pons, S.; Zhang, T.; Assouline, A.; Zimmers, A.; David, C.; Rodary, G.; Girard, J.-C.; Roditchev, D.; Aubin, H. Nat. Commun. 2017, 8, 14549. 26. McMillan, W. L. Phys. Rev. 1968, 167, 331‒344. 27. Allen, P.B. Phys. Rev. B, 1972, 6, 2577‒2579. 28. Asvini meenaatci, A.T.; Rajeswarapalanichamy, R.; Iyakutti, K. Thin Solid Films 2012, 525, 200‒ 207. 29. Bennemann, K. H.; Garland, J. W. AIP Conf. Proc. 1972, 4, 103‒137. 30. Werthamer, N. R.; Helfand, E.; Hohenberg P. C. Phys. Rev. 1966, 147, 295‒302. 31. Bose, S.; Raychaudhuri, P.; Banerjee, R.; Ayyub P. Phys. Rev. B 2006, 74, 224502.

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FIGURE CAPTIONS Figure 1: (a) Powder XRD data for nanocrystalline α-Ta with crystallite size: dXRD = 2.4nm, 6.8nm, 10nm, 18nm and 59nm. Note the consistent, size-dependent shift of the line towards lower 2θ. (b) High resolution STEM image of the cross section of nanocrystalline Ta sample with 𝑑𝑑𝑋𝑋𝑋𝑋𝑋𝑋 = 2.4nm. (c) Particle size distribution obtained from several STEM images of the sample

with 𝑑𝑑𝑋𝑋𝑋𝑋𝑋𝑋 = 2.4nm. The mean size obtained from an analysis of 100 nanoparticles is 2.14±0.28nm.

Figure 2: Crystallite size dependence of (a) superconducting transition temperature, TC, (b) critical magnetic field, HC (at T = 0.7×TC), and (c) residual resistivity ratio, RR. Inset shows the variation of TC and HC(0) (calculated by extrapolation to 0K) for dXRD < 10nm. Note that there are actually two data points at 𝑑𝑑𝑋𝑋𝑋𝑋𝑋𝑋 = 6.8 and 6.9nm with identical 𝑇𝑇𝐶𝐶 (1.4K), which are not distinguishable at this scale. In all plots, the measurement errors are no larger than the size of the data points.

Figure 3: (a) Variation of the lattice constant, a, with particle size (dXRD). (b) Electronic DOS for bulk Ta with lattice constants corresponding to samples with dXRD =59nm (solid line) and 2.4nm (dotted line); inset shows the data magnified around Fermi energy, EF. (c) Variation of N with lattice constant (N = number of states around EF in an energy window ≈ superconducting gap). Corresponding particle sizes are indicated in some cases. Figure 4: (a) Electronic DOS for bulk Ta with relaxed and expanded (1-4%) lattice; insets show the data magnified around EF and the variation of 𝜇𝜇 ∗ with lattice expansion. (b) Electronic band

structure of Ta with relaxed and expanded lattice. (c) Phonon DOS for relaxed and expanded Ta lattice. (d) Phonon band structure with relaxed and expanded Ta lattice. (e) Variation of Eliashberg function with lattice expansion. (f) Relative changes in 𝑇𝑇𝐶𝐶 , λ, and 𝜔𝜔𝑙𝑙𝑙𝑙𝑙𝑙 with lattice constant.

Figure 5: Schematic representation of the mechanism by which the Anderson Limit may be modified in case there is a size-dependent lattice expansion. The Anderson Limit is normally estimated as 𝑑𝑑𝐶𝐶0 (= 4nm for Ta), the size at which the Kubo gap (∼d−3, shown by the solid blue curve) overtakes the size-independent superconducting energy gap (solid red line). In case there is

a size-dependent lattice expansion (a ∼ f(d)), the Anderson Limit (defined by the intersection point) may be shifted to lower values (𝑑𝑑𝐶𝐶1 ). In extreme cases, the intersection may not occur at all. 14 ACS Paragon Plus Environment

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Figure 1: (a) Powder XRD data for nanocrystalline α-Ta with crystallite size: dXRD = 2.4nm, 6.8nm, 10nm, 18nm and 59nm. Note the consistent, size-dependent shift of the line towards lower 2θ. (b) High resolution STEM image of the cross section of nanocrystalline Ta sample with 𝑑𝑑𝑋𝑋𝑋𝑋𝑋𝑋 = 2.4nm. (c) Particle size distribution obtained from several STEM images of the sample with 𝑑𝑑𝑋𝑋𝑋𝑋𝑋𝑋 = 2.4nm. The mean size obtained from an analysis of 100 nanoparticles is 2.14±0.28nm.

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Figure 2: Crystallite size dependence of (a) superconducting transition temperature, TC, (b) critical magnetic field, HC (at T = 0.7×TC), and (c) residual resistivity ratio, RR. Inset shows the variation of TC and HC(0) (calculated by extrapolation to 0K) for dXRD < 10nm. Note that there are actually two data points at 𝑑𝑑𝑋𝑋𝑋𝑋𝑋𝑋 = 6.8 and 6.9nm with identical 𝑇𝑇𝐶𝐶 (1.4K), which are not distinguishable at this scale. In all plots, the measurement errors are no larger than the size of the data points.

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Figure 3: (a) Variation of the lattice constant, a, with particle size (dXRD). (b) Electronic DOS for bulk Ta with lattice constants corresponding to samples with dXRD =59nm (solid line) and 2.4nm (dotted line); inset shows the data magnified around Fermi energy, EF. (c) Variation of N with lattice constant (N = number of states around EF in an energy window ≈ superconducting gap). Corresponding particle sizes are indicated in some cases.

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Figure 4: (a) Electronic DOS for bulk Ta with relaxed and expanded (1-4%) lattice; insets show the data magnified around EF and the variation of 𝜇𝜇 ∗ with lattice expansion. (b) Electronic band

structure of Ta with relaxed and expanded lattice. (c) Phonon DOS for relaxed and expanded Ta lattice. (d) Phonon band structure with relaxed and expanded Ta lattice. (e) Variation of Eliashberg function with lattice expansion. (f) Relative changes in 𝑇𝑇𝐶𝐶 , λ, and 𝜔𝜔𝑙𝑙𝑙𝑙𝑙𝑙 with lattice constant. 18 ACS Paragon Plus Environment

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Figure 5: Schematic representation of the mechanism by which the Anderson Limit may be modified in case there is a size-dependent lattice expansion. The Anderson Limit is normally estimated as 𝑑𝑑𝐶𝐶0 (= 4nm for Ta), the size at which the Kubo gap (∼d−3, shown by the solid blue curve) overtakes the size-independent superconducting energy gap (solid red line). In case there is

a size-dependent lattice expansion (a ∼ f(d)), the Anderson Limit (defined by the intersection point) may be shifted to lower values (𝑑𝑑𝐶𝐶1 ). In extreme cases, the intersection may not occur at all.

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