Phonon limited superconducting correlations in metallic nanograins

Conventional superconductivity is inevitably suppressed in ultra-small metallic grains for characteristic sizes smaller than the Anderson limit. Experiments have shown that above the Anderson limit the critical temperature may be either enhanced or reduced when decreasing the particle size, depending on the superconducting material. In addition, there is experimental evidence that whether an enhancement or a reduction is found depends on the strength of the electron-phonon interaction in the bulk. We reveal how the strength of the e-ph interaction interplays with the quantum-size effect and theoretically obtain the critical temperature of the superconducting nanograins in excellent agreement with experimental data. We demonstrate that strong e-ph scattering smears the peak structure in the electronic density-of-states of a metallic grain and enhances the electron mass, and thereby limits the highest Tc achievable by quantum confinement.

A recent experiment by Bose et al. 1 revealed material-dependent quantum size effects in superconducting grains which led to reconsider an old but fundamental problem of superconductivity in confined systems. Over the last 50 years the properties of nanoscale superconductors have been investigated extensively, both theoretically and experimentally, due to their general importance for the understanding of the nature of the ground state in confined systems 2 . The question of how the superconducting properties of confined systems are altered with thickness down to the nanometer size range has also significant technological ramifications. Anderson 3 argued that conventional superconductivity should disappear for sample sizes such that the average level spacing δ (inversely proportional to the volume of the grain) is of the order of the BCS gap Δ . Blatt and Thompson in their pioneering work predicted that above the Anderson limit the critical temperature T c exhibits quantum oscillations with remarkable thickness-dependent resonant enhancements 4 . Subsequent experimental studies of superconducting confined systems showed either a decrease or an increase of T c with sample size, depending on the material.
A scanning tunneling microscope (STM) was used to detect the superconducting gap in a single physically isolated ultra-small Pb/Sn grain 1,43,44 . These studies traced the size evolution of superconductivity in isolated nanoparticles that were grown on a substrate. It was demonstrated that, while in both systems (Pb and Sn) superconductivity is ultimately quenched at the Anderson limit, the size dependence of T c before the destruction of superconductivity was different. In a Sn particle oscillations of the superconducting energy gap with particle size were observed and the enhancement of the gap was reported to be as large as 60%. Contrarily, Pb particles exhibited a decrease of the gap with decreasing particle size. The theoretical explanation was based on a model of the density-of-states (DOS), which included a phenomenological broadening parameter due to scattering or recombination processes, escape rates, and instrumental broadening. A recent study was devoted to the role of the substrate in the decay of T c with sample size 45 .
Recently, a series of experimental results obtained with samples made of superconducting Pb islands with thickness between 5 and 60 monolayers and grown on Si(111) substrate were reported 46,47 . These superconducting samples can be considered as quasi-2D superconducting islands with strong effects of level quantization. A reduction of the superconducting gap of ultrathin islands was observed. Layer-dependent ab initio density functional calculations for freestanding Pb films showed that the electron-phonon coupling decreases with decreasing film thickness and hence results in the size-dependent destruction of superconductivity 44 . Interestingly the suppression of superconductivity with size depends to a good approximation only on the volume of the island and is independent of its shape 47 . Therefore, the above overview of the experimental findings over the last 50 years shows that the behavior of T c changes from an increase to a decrease with sample size reduction as the strength of the e-ph interaction increases. In this article we show that the experimental results can be explained by an interplay of quantum confinement for the electronic degrees of freedom and a phonon environment. The key ingredients of the mechanism is the inelastic electron phonon scattering that is sufficiently strong to smear the discrete energy spectrum of the particle into a continuum 48 and the mass renormalization due to the electron-phonon scattering. Thus, in addition to the pair-forming effect due to virtual phonons inherent to the BCS model we take into account also the pair-breaking effect due to thermal phonons inevitably present in any solid-state system. Since the quantum-size effect is inversely proportional to the electron mass it is seen that the effect is due to the e-ph mass-enhancement parameter λ, defined as the frequency derivative of the electron self-energy due to the e-ph interaction.

Results
The theory behind the model. The electronic states in metal structures are subject to different scattering mechanisms. While the influence of imperfections (impurities, surface roughness, layer-width fluctuations) can, at least in principle, be controlled by improved technology, phonon scattering is inherent to the solid state of matter. The general Hamiltonian for a coupled electron-phonon system interacting via a linear interaction is given by , , , Here p is integer numbering the single-particle energy levels ε p , the operator a p,σ ( ) σ , † a p annihilates (creates) an electron in state p with spin σ and the operator b w ( ) † b w annihilates (creates) a phonon in state w. We describe the phonons by a Debye spectrum, i.e., we use ω = c w w , where c denotes the velocity of sound and w ranges from zero to w D , the Debye wave number. The corresponding maximum phonon energy is denoted by ω D . Here coupling V w is related with the e-ph coupling used in superconductivity as Here k F denotes the Fermi wave vector. For our quantitative studies devoted to the explanation of the material and size dependence of the superconducting critical temperature Δ T c we have used bulk phonon parameters, thus neglecting any change of the phonon spectrum due to the presence of the surface. This approach is motivated by the large ratio of electron and phonon quantum-confinement energies which scale as m ion /m e . Nevertheless, the influence of the size-dependent coupling constant on the superconducting characteristics can be essential for ultra-small grains and this issue was studied in ref. 43. The phonon induced contribution to the level broadening can be expressed in terms of the spectral Eliashberg function α 2 F(ω) 49,50 Metal where f(ξ) and n(ħω′ ) are the electron and phonon occupation numbers, respectively. With increasing sample size the energy separation between different single-electron levels decreases and more levels move into the Debye window, thereby opening additional inter-level scattering channels. Therefore, the use of 3D acoustic (LA) e-ph scattering should be adequate for calculations of the acoustic phonon-induced single-electron level broadening when the average energy spacing δ π is sufficiently smaller than the mean LA phonon energies. Here V is the sample volume. Within the 3D Debye model the Eliashberg function is α ω λ ω ω θ ω , which results at T = 0 in the level broadening: At the Fermi level ξ = 0 and finite T we obtain 53 : . This model requires one external parameter, i.e., the mass-enhancement parameter λ.
To describe the superconducting state we consider the reduced Bardeen-Cooper-Schrieffer (BCS) pairing Hamiltonian, where only the time-reversal states are paired where the interaction matrix element g qp is given by with g denoting the coupling constant and ϕ q (r) the single-electron wave function. The first term in Eq. (6) contains the single-electron energies, and the second term is the attractive (when g > 0) pairing interaction due to the exchange of virtual phonons. We have assumed that the electron-electron interaction is unaffected by quantum confinement and it is the same as in the bulk. In the bulk the real inter-electron potential is well approximated by a δ-function pseudopotential. Employing such a simplified interaction requires a regularization, which makes the matrix elements non-zero only between states within the Debye window around the Fermi surface. As shown in the Methods section the pairing gap equation is where E p is the quasi-particle energy given by p p p 2 2 within the pairing interval (Debye window) and is the total width of the q-level, which is approximated by the phonon contribution given in Eq. (3), and the single electron spectrum is described by ξ λ µ ε µ λ ( ) = + ( − )/( + ) 1 . This expression takes into account the modification of the single-electron spectrum due to the electron-phonon interaction, tied to a thin energy shell of approximate width 2ħω D . The function ( ) ξ ξ − , Γ F p p describes the shape of the broadened levels. Since we are dealing with a system at very low temperature the spectrum of a broadened level is asymmetric around ξ p . However, for the purpose of our work this is an unnecessary detail and we assume a symmetric spectral line 54  The solution of Eq. (8) gives From the pairing gap one can find the density gap ∆ p P . This expression is used in our simulations. It accounts for the phonon-induced enhancement of the level density in the quasi-particle spectrum by a factor 1 + λ.
Numerical results. We model grains as nanoparticles with L x = D, L y = D + a and L z = 2 nm (see Fig. 1), where we vary the lateral size D and take a = 0.05 nm (a finite a is added to avoid the strong degeneracy of the single-electron levels typical for highly-symmetric samples). In our numerics we include the level broadening due to the electron-phonon interaction. The calculations were performed with parameters typical for aluminium, tin, niobium, and lead (see Table 2). Figure 2 shows the calculated DOS,     Here T c B is the superconducting onset temperature for a sample with D → ∞. For each sample size the transition temperature, T c , was defined as the temperature which corresponds to steepest descent of the superconducting order parameter as shown in Fig. (4), which displays the values of Δ P , calculated according to Eq.    of the sample is decreased from D ≈ 30 nm to D ≈ 8 nm. A further reduction of the particle size leads to a slight increase of T c before its suppression. Lead nanoparticles exhibit a small decrease of T c with decreasing sample size. The transition temperature decreases by 3-4% when the width of the sample is decreased from D ≈ 30 nm to D ≈ 8 nm. The different behavior of Nb and Pb nanoparticles as compared with Al and Sn is a direct consequence of the fact that quantum confinement is smaller due to (i) the heavier electron mass and (ii) larger broadening of the single electron levels, both as a result of the stronger electron-phonon interaction.

Discussion
We presented a theoretical explanation for the puzzling very different dependence of T c on grain size in different superconductors. To realize this we presented a theory for the superconducting correlations in nanograins coupled to a phonon environment. We showed how environmental entanglement emerges in the ground-state of these systems and why it has a strong influence on the superconducting characteristics. The pair-breaking effect due to thermal phonons was accounted for, which broadens the single electron levels, as well as the virtual phonons that renormalize the band electron mass. Without such level broadening and mass renormalization T c always increases with decreasing sample size above the Anderson limit. In our theory, as corroborated by experimental observations, quantum confinement can either increase or show constant behaviour/slight decrease of the average superconducting critical temperature, depending on the material parameters. Our analysis conclusively shows that a slight decrease of T c for smaller samples is expected for the strongly phonon-coupled nanoparticles of lead while an increase is typical for samples made of weakly phonon coupled superconducting materials, both in accordance with experimental findings.
In the above study, we have not accounted for the influence of disorder 56 . We note that in the presence of disorder the level broadening can be size-dependent, because the effect of disorder typically increases with the reduction of sample size due to e.g. surface-roughness, resulting in a stronger level broadening at small sizes. As shown in ref. 57, disorder also results in a repulsion of the energy levels in small metallic samples making them evenly distributed, which additionally decreases T c when reducing the sample size. As was found by Altshuler and Aronov the Coulomb interaction combined with impurity scattering produces a dip in the density-of-states at the Fermi level, that in turn further decrease the transition temperature with decreasing sample size. To estimate the effect of disorder we use a weak disorder model for homogeneous superconducting samples established by Finkelstein 58 , according to which the suppression of superconductivity is driven by impurities that reinforce Coulomb and spin interactions. According to this model the critical temperature is found from the following expression , γ τ = / ( ) kT 1 ln c0 and the diffusion time we define as τ = D/v F , where v F is the Fermi velocity. Here R s is the sheet resistance, τ 0 is the diffusion time for a sample with large lateral size. Within this model we obtain the results shown as dashed lines in Fig. 3. For the calculations we adopted the following parameters R s (Al) = 14.1Ω , R s (Sn) = 54Ω , R s (Pb) = 110Ω , R s (Nb) = 90Ω .

Methods
In the case of finite superconducting systems the mean field approximation exhibits several drawbacks. The most evident is the presence of sharp phase transitions, which is characteristic of very large systems. This effect is due to statistical fluctuations. The static-path approximation (SPA) provides a microscopic way to include the static fluctuations of the mean field. SPA considers only the static paths in the path integral representation of the partition function. In this paper we have used the reduced SPA formalism within which the statistical sum is written as 59 The minimum of Ω′ (∆) SPA results in Eqs. (8)(9). From the pairing gap one can find the density gap ∆ p P .