Spin-vibron coupling effects in single-molecule magnets grafted to a nanoelectromechanical system

V. Moldoveanu and R. Dragomir

Phys. Rev. B 104, 075441 – Published 23 August 2021

Abstract

We present a theoretical analysis of the interplay between the spin-vibron and electron-vibron interactions in a hybrid system made of a single-molecule magnet and a suspended conductor. The latter is coupled to particle reservoirs and supports quantized vibrational modes which, once activated, interact with the localized magnetic moment $S$ of the nanomagnet. The dynamics of the molecular spin, the average vibron number, and the transient currents are calculated from the reduced density operator of the hybrid system. We focus on the effect of the vibron-assisted transitions from the lowest energy spin doublet $S_{z} = \pm S$ to higher energy excited states. The numerical simulations performed for the simplest case $S = 2$ prove that the vibron-assisted spin transitions and dynamics can be described in terms of a three-level $Λ$ model borrowed from quantum optics. In particular we predict the existence of Rabi oscillations of the transient currents as fingerprints of the spin-vibron coupling. The role of symmetric or asymmetric bias configurations in setting different mixtures of molecular spin states in the steady-state regime is also emphasized.

Received 7 June 2021
Revised 26 July 2021
Accepted 12 August 2021

DOI:https://doi.org/10.1103/PhysRevB.104.075441

Physics Subject Headings (PhySH)

Molecular magnetism Quantum transport Transport phenomena

Molecular magnets

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

V. Moldoveanu and R. Dragomir

National Institute of Materials Physics, Atomistilor 405A, Magurele 077125, Romania

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Issue

Vol. 104, Iss. 7 — 15 August 2021

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Images

Figure 1
Spin-split $Λ$ -type configuration of the single-particle states of the $S = 2$ hybrid system. The horizontal solid lines represent energy levels of the hybrid system alone (i.e., without vibrons) and the arrows indicate the vibron-assisted transitions between states with dominant spin components ${\bar{S}}_{z} = 0$ and ${\bar{S}}_{z} = \pm 2$ . The resonant frequencies are $ω_{\pm}$ . If $ω_{0} = ω_{-}$ the blue/red arrows mark the resonant/off-resonant transitions.
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Figure 2
Dynamics of the average molecular spin for both resonant frequencies $ω_{\pm}$ in the symmetric and asymmetric bias configurations. (a) $ω_{0} = ω_{-}$ ; (b) $ω_{0} = ω_{+}$ . The chemical potentials are as follows: $μ_{L} = 0.485 meV, μ_{R} = - 0.6 meV$ for the configuration $B_{out}, μ_{L} = 1.3 meV, μ_{R} = 0.2 meV$ for $B_{in}$ , and $μ_{L} = 1.3 meV, μ_{R} = - 0.6 meV$ for the symmetric bias $B_{sym}$ . Other parameters: $α = 0.15, D = 0.056 meV, E / D = 1 / 5$ , and $V_{L} = V_{R} = 15 μ eV$ . The leads are not spin-polarized.
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Figure 3
(a) Short time dynamics of the molecular spin for asymmetric bias configurations $B_{in}$ at both resonant frequencies $ω_{0} = ω_{\pm}$ . The period of the oscillations depends on $α$ . (b) Rabi-like oscillations of the transient current $J_{R}$ in the drain lead for two values of the spin-vibron coupling in the $B_{in}$ configuration. (c) The average vibron number. The initial state is $| ϕ_{0, - 2}; 0 〉$ . Other parameters: $μ_{L} = 1.3 meV, μ_{R} = 0.2 meV, D = 0.056 meV, E / D = 1 / 5$ , and $V_{L} = V_{R} = 15 μ eV$ .
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Figure 4
Short time occupation $P_{Q = 1, S_{z}, N}$ of the lowest energy $N$ -vibron states $| ϕ_{Q = 1, S_{z}}; N 〉$ for asymmetric bias configuration $B_{in}$ at resonant frequency $ω_{0} = ω_{-}$ . (a) $P_{Q = 1, - 2, 1}$ and $P_{Q = 1, 0, 0}$ ; (b) $P_{Q = 1, - 2, 2}$ and $P_{Q = 1, 0, 1}$ . Other parameters: $α = 0.15, μ_{L, in} = 1.3 meV, μ_{R, in} = 0.2 meV, D = 0.056 meV, E / D = 1 / 5$ , and $V_{L} = V_{R} = 15 μ eV$ .
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Figure 5
Time-dependent populations $P_{Q, S_{z}}$ of empty ( $Q = 0$ , dashed lines) and single-electron states ( $Q = 1$ , solid lines) associated to a dominant component $S_{z}$ of the molecular spin. (a) The asymmetric bias configuration $B_{in}, ω_{0} = ω_{-}$ . (b) The symmetric configuration $B_{sym}, ω_{0} = ω_{-}$ . (c) The asymmetric bias configuration $B_{in}, ω_{0} = ω_{+}$ . In all figures the initial state is $| ϕ_{0, - 2}; 0 〉$ . Other parameters are the same as in Fig. 2.
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Figure 6
(a) Average molecular spin for different values of the ratio $E / D$ . (b) The time-dependent current $J_{R}$ . The solid lines correspond to the spin-vibron coupling $α = 0.15$ and the dashed lines to the noninteracting case $α = 0$ . The system is in the asymmetric bias configuration $B_{in}$ at frequency $ω_{0} = ω_{-}$ .
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