Confinement effects on electron and phonon degrees of freedom in nanofilm superconductors: A Green function approach

R. Saniz, B. Partoens, and F. M. Peeters

Phys. Rev. B 87, 064510 – Published 25 February 2013

Abstract

The Green function approach to the Bardeen-Cooper-Schrieffer theory of superconductivity is used to study nanofilms. We go beyond previous models and include effects of confinement on the strength of the electron-phonon coupling as well as on the electronic spectrum and on the phonon modes. Within our approach, we find that in ultrathin films, confinement effects on the electronic screening become very important. Indeed, contrary to what has been advanced in recent years, the sudden increases of the density of states when new bands start to be occupied as the film thickness increases, tend to suppress the critical temperature rather than to enhance it. On the other hand, the increase of the number of phonon modes with increasing number of monolayers in the film leads to an increase in the critical temperature. As a consequence, the superconducting critical parameters in such nanofilms are determined by these two competing effects. Furthermore, in sufficiently thin films, the condensate consists of well-defined subcondensates associated with the occupied bands, each with a distinct coherence length. The subcondensates can interfere constructively or destructively giving rise to an interference pattern in the Cooper pair probability density.

Received 1 June 2012

DOI:https://doi.org/10.1103/PhysRevB.87.064510

Authors & Affiliations

R. Saniz, B. Partoens, and F. M. Peeters

Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium

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Vol. 87, Iss. 6 — 1 February 2013

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Figure 1
(a) Temperature dependence of the four gap values of a 13.3 $a_{0}$ Al nanofilm [three (111) monolayers thick]. The critical temperature is $T_{c} = 2.8 T_{c}^{b}$ . (b) Dependence of $T_{c}$ on film thickness. Confinement effects on the coupled electron and phonon systems have a strong impact (solid line); as $d$ increases, $T_{c}$ drops when a new electron band starts to be occupied but rises when the number of phonon modes (given by the number of monolayers) contributing to the pairing increases. The filled circles indicate the predicted $T_{c}$ as a function of the number of monolayers in the film (cf. upper scale). In the contact-potential case (dotted line), $T_{c}$ depends on $d$ only through the total electron density of states per spin at the Fermi level. The diamonds indicate the $T_{c}$ as a function on number of monolayers in the contact-potential case.Reuse & Permissions
Figure 2
(a) Diagonal elements of the long-wavelength limit of the static screened Coulomb interaction, $v^{scr} = ɛ^{- 1} v$ , as a function of film thickness. Sharp drops occur whenever the number of occupied bands increases. The drops are due to the resulting increase of screening in the nanofilm. (b) Fermi level $μ$ (solid line) and total electron density of states per spin at the Fermi level, $N (0) = \sum_{n} N_{n} (0),$ as a function of $d$ (dashed line). The kinks in $μ$ and sharp peaks in $N (0)$ indicate when a new band starts to be occupied as $d$ increases.Reuse & Permissions
Figure 3
(a) Dependence of $T_{c}$ and of $N (0)$ on the number of (111) monolayers. Only for films thicker than a critical number of monolayers do $T_{c}$ and the total density of states correlate. In the present example, this occurs above $\sim$ 20 monolayers. (b) The dependence of the energy gap values on $d$ . As $d$ increases, a new gap value appears as a new band is occupied, i.e., a new subcondensate is formed (cf. Sec. 5). The trends in the dependence on $d$ are similar to those of $T_{c}$ [compare with Fig. 1b]. The spread in gap values rapidly decreases with film thickness.Reuse & Permissions
Figure 4
Typical dependence of the Cooper pair wave-function probability density on the relative coordinate $u$ . The plot is for $Z = d / 2$ and $ζ = d / 2$ , with $d = 13.3$ $a_{0}$ . The amplitude of the oscillations as a function of interparticle distance $u$ is clearly modulated. The condensate tends to be depleted of Cooper pairs for interparticle distances close to those indicated by the vertical arrows. The condensate being composed by distinct subcondensates, the regions of low amplitude are due to interference between the wave functions associated with each of the subcondensates.Reuse & Permissions
Figure 5
(a) Coherence lengths as a function of center of mass coordinate for nanofilms for thicknesses corresponding to four, five, and six (111) monolayers. The coherence lengths are shorter than in bulk Al ( $\sim$ $30236 a_{0}$ ).[28] (b) The mean-square radius of a subcondensate is determined mainly by how fast the functions $I_{n}^{2} (u)$ [cf. Eq. (43)] decay with $u$ . The decay rates for the four subcondensates in a 3 (111) monolayer thick nanofilm are clearly different. The plot shows that the different functions $I_{n} (u)$ oscillate with different wavelengths.Reuse & Permissions

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