Josephson current through a molecular transistor in a dissipative environment

Tomáš Novotný, Alessandra Rossini, and Karsten Flensberg

Phys. Rev. B 72, 224502 – Published 6 December 2005

Abstract

We study the Josephson coupling between two superconductors through a single correlated molecular level, including Coulomb interaction on the level and coupling to a bosonic environment. All calculations are done to the lowest, i.e., the fourth, order in the tunneling coupling and we find a suppression of the supercurrent due to the combined effect of the Coulomb interaction and the coupling to environmental degrees of freedom. Both analytic and numerical results are presented.

Received 2 August 2005

DOI:https://doi.org/10.1103/PhysRevB.72.224502

Authors & Affiliations

Tomáš Novotný^1,2,*, Alessandra Rossini^1,3, and Karsten Flensberg¹

¹Nano-Science Center, Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
²Department of Electronic Structures, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 121 16 Prague, Czech Republic
³Dipartimento di Fisica, Università di Milano, via Celoria 16, 20133 Milano, Italy

^*Electronic address: novotny@fys.ku.dk

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Vol. 72, Iss. 22 — 1 December 2005

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Images

Figure 1
(Color online) Schematic picture of the physical setup described by the model. The central region (molecule/quantum dot) is coupled to two superconducting leads and can be independently gated by a gate. Electrons entering the central region experience mutual Coulomb interaction and interact with oscillatory modes—either one distinguished mode such as center-of-mass vibration of the molecule as a whole or many modes due to intramolecular vibrations or substrate phonons acting as a heat bath.Reuse & Permissions
Figure 2
(Color online) Josephson current dependence on the Coulomb interaction strength at zero temperature. Shown are the critical Josephson current through an Anderson level with no Coulomb interaction $U = 0$ (full line), with moderate interaction $U = Δ$ (dash-dotted line), and with infinite repulsion $U \to \infty$ (dashes). The interacting cases exhibit the phenomenon of the $π$ junction for $- U < ξ < 0$ , but the overall magnitude of the current decreases largely with the increasing interaction strength.Reuse & Permissions
Figure 3
(Color online) Josephson current dependence on the temperature for $U \to \infty$ . We show the temperature dependence of the critical Josephson current for $∣ ξ ∣ < Δ$ for three different temperatures $T = 0 (β \to \infty)$ (full line), $β Δ = 40$ (dash-dotted line), and $β Δ = 10$ (dashed line with asterisks). In the last case which illustrates roughly the highest temperature achievable within the approximations used the analytic and numerical results are compared.Reuse & Permissions
Figure 4
(Color online) Josephson current for low-frequency phonons at $T = 0$ , $U \to \infty$ . As long as the spectrum of the phonon mode(s) is well below the superconducting gap, i.e. $ω_{0, c} ⪡ Δ$ , the critical Josephson current only depends on the integral $ε_{cl} = \int_{0}^{\infty} d ω J (ω) ∕ ω = g ω_{0, c}$ via the shifted dissipationless function $N (ξ + ε_{cl} sgn ξ)$ (dash-dotted line). Numerical results are shown for a single phonon mode with $ω_{0} = 0.5 Δ$ , $g = 2$ (dots), $ω_{0} = 0.1 Δ$ , $g = 10$ (asterisks), $ω_{0} = 0.05 Δ$ , $g = 20$ (pluses), and for the Ohmic bath with $ω_{c} = Δ$ , $g = 1$ (crosses). All of these cases yield basically the same result very well captured by the analytic expression.Reuse & Permissions
Figure 5
(Color online) Josephson current for high-frequency single oscillator at $T = 0$ , $U \to \infty$ , $ω_{0} = 20 Δ$ . For large enough $ω_{0} ⪢ Δ, ξ$ the Josephson current is expected to be suppressed just by the factor $\exp (- 2 g)$ compared to the case without the phonon. This is shown in the figure where the numerical results for several values of the coupling constant $g = 0.1$ (asterisks), $g = 0.5$ (pluses), $g = 1$ (crosses), and $g = 2$ (dots) are compared with the analytic expressions given by the corresponding lines.Reuse & Permissions
Figure 6
(Color online) Josephson current for high-frequency-cutoff Ohmic bath at $T = 0$ , $U \to \infty$ and for several values of the coupling constant $g$ and cutoff frequency $ω_{c}$ . We see a significant suppression of the critical current already for intermediate coupling $g = 1$ .Reuse & Permissions

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