Size effects in superconducting thin films coupled to a substrate

Aurelio Romero-Bermúdez and Antonio M. García-García

Phys. Rev. B 89, 064508 – Published 18 February 2014

Abstract

Recent experimental advances in surface science have made it possible to track the evolution of superconductivity in films as the thickness enters the nanoscale region where it is expected that the substrate plays an important role. Here, we put forward a mean-field, analytically tractable, model that describes size effects in ultrathin films coupled to the substrate. We restrict our study to one-band, crystalline, weakly coupled superconductors with no impurities. The thin-film substrate/vacuum interfaces are described by a simple asymmetric potential well and a finite quasiparticle lifetime. Boundary conditions are chosen to comply with the charge neutrality condition. This model provides a fair description of experimental results in ultrathin lead films: on average, the superconducting gap decreases with thickness and it is always below the bulk value. Clear oscillations, remnants of the shape resonances, are still observed for intermediate thicknesses. For materials with a weaker electron-phonon coupling and negligible disorder, a modest enhancement of superconductivity seems to be feasible. The relaxation of the charge neutrality condition, which is in principle justified in complex oxide heterostructures and other materials, would lead to a much stronger enhancement of superconductivity by size effects.

Received 17 January 2014
Revised 5 February 2014

DOI:https://doi.org/10.1103/PhysRevB.89.064508

Authors & Affiliations

Aurelio Romero-Bermúdez¹ and Antonio M. García-García^1,2

¹University of Cambridge, Cavendish Laboratory, JJ Thomson Avenue, Cambridge, CB3 0HE, United Knigdom
²CFIF, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal

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Vol. 89, Iss. 6 — 1 February 2014

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Images

Figure 1
Asymmetric finite well. The values of $V_{s}$ and $V_{0}$ are discussed in Sec. 3a.
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Figure 2
Pb band diagram in the $[111]$ direction [23].
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Figure 3
Superconducting order parameter $Δ$ for a Pb thin film coupled to a Si substrate. $Δ_{0}$ is the bulk Pb gap. The film/interface coupling is modeled by an asymmetric well potential with $V_{s} = E_{F} + 1.70$ eV, $V_{0} = E_{F} + 4.25$ eV. Level broadening is assumed to be negligible. The dimensionless coupling constant is $ρ = 0.385$ and the Debye energy $ℏ ω_{D} = 9.048$ meV. The band-structure parameters are given in Eq. (23). The upper plot does not satisfy the charge neutrality condition, while the lower plot does it with $b$ obtained from Eq. (10). Lines show the evolution of the order parameter as a function of the thickness. Dots correspond to the estimate positions of Pb monolayers. The continuous and dashed lines show the difference between the $k$ -dependent gap with $Δ = {min}_{n} Δ (k_{n})$ from Eq. (3) and the $k$ -independent $Δ$ from Eq. (4), respectively. We note that the pattern of shape resonances is more intricate than in Ref. [1] as a consequence of the interplay between the finite number of states in the asymmetric well potential and the more realistic dispersion relation.
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Figure 4
Superconducting order parameter $Δ$ for a Pb film coupled to a Si substrate with no level broadening. $V_{s} = E_{F} + 0.9$ eV, $V_{0} = E_{F} + 4.25$ eV. $Δ_{0}$ is the bulk Pb gap. The parameters correspond to those of Fig. 3 except for the smaller step $V_{s}$ and $b$ obtained from Eq. (10). The continuous line corresponds to the $k$ -dependent gap with $Δ = {min}_{n} Δ (k_{n})$ from Eq. (3) and the dashed line corresponds to the $k$ -independent $Δ$ from Eq. (4). The $k$ -independent approximation becomes less accurate as the step height decreases. As was expected, reducing the height of the film/substrate interface increases the leaking of probability out of the film which, reduces both the superconducting gap $Δ$ and the effect of the charge neutrality condition measured by $b$ . Moreover, the pattern of shape resonances is richer as the number of bound states is smaller in this case.
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Figure 5
Superconducting order parameter $Δ$ at $T = 0$ for Pb films on a Si substrate for different quasiparticle lifetimes. $Δ_{0}$ is the bulk Pb gap, $ρ = 0.385$ , and $ℏ ω_{D} = 9.048$ meV. Results include the charge neutrality condition. The gray line corresponds to the limit of no level broadening ( $τ \to \infty$ ) [Eq. (4)] (green) $τ (fs) = τ_{0} + 3 N$ , (red) $τ (fs) = τ_{0} + 1.5 N$ , and (blue) $τ (fs) = τ_{0} + 0.7 N$ where $N$ is the number of ML. These values are chosen in order to estimate the range in which finite- $τ$ corrections are relevant. For less than 10 ML this occurs for $τ \leq 10 τ_{0}$ , with $τ_{0}$ given by Eq. (27). Dots correspond to the exact position of the Pb monolayers.
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Figure 6
Superconducting gap $Δ$ in units of the bulk gap $Δ_{0}$ for $τ \to \infty$ . All parameters are equal to those of Fig. 3 except the dimensionless coupling constant $ρ = 0.180$ and the Debye energy $ℏ ω = 33.882$ meV. Results in the upper plot do not satisfy the charge neutrality condition, while the lower plot does include it with $b$ obtained from Eq. (10). The gap $Δ = {min}_{n} Δ (k_{n})$ (continuous line) shows a moderate enhancement of superconductivity, even when charge neutrality is imposed. By contrast, no enhancement is observed (dashed line) in the approximate solution (4).
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