Charge-State Dependent Vibrational Relaxation in a Single-Molecule Junction

Xinya Bian, Zhixin Chen, Jakub K. Sowa, Charalambos Evangeli, Bart Limburg, Jacob L. Swett, Jonathan Baugh, G. Andrew D. Briggs, Harry L. Anderson, Jan A. Mol, and James O. Thomas

Phys. Rev. Lett. 129, 207702 – Published 9 November 2022

Abstract

The outcome of an electron-transfer process is determined by the quantum-mechanical interplay between electronic and vibrational degrees of freedom. Nonequilibrium vibrational dynamics are known to direct electron-transfer mechanisms in molecular systems; however, the structural features of a molecule that lead to certain modes being pushed out of equilibrium are not well understood. Herein, we report on electron transport through a porphyrin dimer molecule, weakly coupled to graphene electrodes, that displays sequential tunneling within the Coulomb-blockade regime. The sequential transport is initiated by current-induced phonon absorption and proceeds by rapid sequential transport via a nonequilibrium vibrational distribution of low-energy modes, likely related to torsional molecular motions. We demonstrate that this is an experimental signature of slow vibrational dissipation, and obtain a lower bound for the vibrational relaxation time of 8 ns, a value dependent on the molecular charge state.

Received 15 March 2022
Accepted 20 October 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.207702

Physics Subject Headings (PhySH)

Carrier dynamics Coulomb blockade Electron-phonon coupling Quantum transport

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Xinya Bian¹, Zhixin Chen¹, Jakub K. Sowa², Charalambos Evangeli ¹, Bart Limburg³, Jacob L. Swett¹, Jonathan Baugh ⁴, G. Andrew D. Briggs ¹, Harry L. Anderson ³, Jan A. Mol ⁵, and James O. Thomas ^1,*

¹Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom
²Department of Chemistry, Rice University, Houston, Texas 77005, USA
³Department of Chemistry, University of Oxford, Oxford OX1 3TA, United Kingdom
⁴Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
⁵School of Physical and Chemical Sciences, Queen Mary University, London E1 4NS, United Kingdom

^*james.thomas@materials.ox.ac.uk

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Vol. 129, Iss. 20 — 11 November 2022

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Images

Figure 1
(a) Device schematic. (b) False-color SEM image of graphene (Gr) transferred onto source ( $S$ ) and drain ( $D$ ), and local-gate electrode (LG). Gr is patterned into a bowtie shape to produce nanoelectrodes via electroburning. (c) Molecular structure of FP2, Ar: 3,5-bis(trihexylsilyl)phenyl. (d) Conductance map ( $d I_{sd} / d V_{sd}$ ) of the FP2 device; transitions studied are outlined in green and blue.
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Figure 2
(a) Temperature-dependent conductance maps of the $N - 4 / N - 3$ transition, showing quasiperiodic resonant transport features. The onset of the second sideband is indicated (orange arrow). (b) Conductance measured along the first [between gray markers, (a)] and second (pink markers) sidebands shows suppression below $e V_{sd} < n ℏ ω_{q}$ . (c) Negative differential resistance (gray band) between the first two sidebands. (d) Temperature dependence of the Coulomb peak (CP) and first sideband (SB), [orange and blue circles, (a)]. The ground state shows a simple $1 / T$ relationship (black line). The first sideband is suppressed, increasing due to thermal population of $q = 1$ , fitted by Eq. (4) (black line). (e) Temperature-dependent conductance through the first and second sideband [square markers on (a)] at $e V_{sd} = ℏ ω_{q}$ ; the conductance of the first sideband is high and decays as $1 / T$ (black line) whilst the second grows with temperature. Error bars are from a fit to the conductance trace around the SB or CP, and gray areas are confidence bands on the fits.
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Figure 3
(a) Onset of first sideband (red dot, panel (d)). Blue $N - 4$ parabolas correspond to an electron at the Fermi level of left $| μ_{L}, N - 4, 0 ⟩$ or right $| 0, N - 4, μ_{R} ⟩$ electrode. The green parabola corresponds to the electron being on the molecule: $| 0, N - 3, 0 ⟩$ . $N - 4$ parabola have the same $x$ -position, but are offset from $N - 3$ by $\pm l_{o s c} \sqrt{2} λ_{q}$ to visualise electron flow from right to left lead. Inelastic cotunneling promotes the molecule to $q = 1$ . If relaxation ( $1 / τ_{N - 4}$ ) is smaller than $W_{1, 0}^{N - 4, N - 3}$ , sequential tunneling proceeds via excited states of FP2, leading to sidebands. (b) Parabolas at the onset of the first $N - 3$ sideband (grey dot) are not observed as $1 / τ_{N - 3}$ out-competes sequential tunneling. (c) Fits to zero-bias gate traces yield $λ_{q}$ and $\bar{Γ}$ values, here $λ_{q} = 2.7$ , $\bar{Γ} = 40 μ eV$ . Traces are offset for clarity. (d) Modelled conductance map at 5 K, using $ℏ ω_{q} = 9 meV$ , $λ_{q} = 2.7$ , $\bar{Γ} = 40 μ eV$ , $1 / τ_{N - 4} = 1 / 100 \times W_{1, 0}^{N - 4, N - 3}$ , $1 / τ_{N - 3} = 100 \times W_{1, 0}^{N - 3, N - 4}$ ; sequential tunneling rates are calculated at the red and grey points respectively.
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Figure 4
(a) Example vibrational mode around 9 meV. (b) DFT-calculated dihedral angle $θ$ [shown in (a)] in each charge state. For $N - 2$ , singlet and triplet (the ground state) are labeled by $(s)$ and $(t)$ .
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