Theory of vibrational absorption sidebands in the Coulomb-blockade regime of single-molecule transistors

Current-driven vibrational nonequilibrium induces vibrational sidebands in single-molecule transistors which arise from tunneling processes accompanied by absorption of vibrational quanta. Unlike conventional sidebands, these absorption sidebands occur in a regime where the current is nominally Coulomb blockaded. Here, we develop a detailed and analytical theory of absorption sidebands, including current-voltage characteristics as well as shot noise. We discuss the relation of our predictions to recent experiments.

Figure 1

(Color online) Schematic vibrational sidebands in $dI∕dV$ vs bias and gate-voltage plane for (a) fast, (b) intermediate, and (c) slow relaxations. The conventional emission sidebands in the sequential tunneling region (colored/shaded) are shown in white. The absorption sidebands discussed in this paper are depicted in red/gray. Horizontal dashed lines appear due to inelastic cotunneling, thick black lines due to sequential tunneling without change of the vibrational state.

Figure 2

Relevant transport processes in the vicinity of the first absorption sideband for $\hslash \omega <eV<2\hslash \omega$. In addition to elastic cotunneling (not shown), the following processes contribute to current flow: (a) inelastic cotunneling with emission of one vibron; (b) population of the molecule by inelastic sequential tunneling with absorption of one vibron; and depopulation by (c) elastic sequential tunneling or (d) inelastic sequential tunneling with emission of at most two vibrons. Molecular states are symbolized by $\mid n,q⟩$, where $n$ refers to the charge state and $q$ to the vibrational state.

Figure 3

The quantity $\tilde{F}=1+I_{\mathrm{seq}}∕I_{\cot }$ as defined in the text vs the relaxation time for (a) weak $(\lambda =0.2)$ and (b) strong $(\lambda =4)$ electron-vibron couplings ($V=1.7\hslash \omega$; ${ϵ}_d=1.6\hslash \omega$).

Figure 4

Relevant states and processes for intermediate relaxation and weak electron-vibron coupling in the vicinity of the $n\text{th}$ absorption sideband. Single solid lines represent cotunneling transitions, double lines sequential tunneling, and dashed lines vibrational relaxation.

Figure 5

Dynamics leading to the first absorption sideband (in the bias range $\hslash \omega <eV<2\hslash \omega$) for intermediate relaxation and strong electron-vibron coupling.

Figure 6

Relevant states and transitions at the second absorption sideband (in the bias range $\hslash \omega <eV<2\hslash \omega$) for slow relaxation and weak electron-vibron coupling.