Dynamical symmetry breaking in vibration-assisted transport through nanostructures

A theoretical model of a single molecule coupled to many vibronic modes is presented. At low energies, transport is dominated by electron-vibron processes where transfer of an electron through the dot is accompanied by the excitation or emission of quanta (vibrons). Because the frequency of the $n$th mode is taken as an $n$th multiple of the frequency of the fundamental mode, several energetically degenerate or quasidegenerate vibronic configurations can contribute to transport. We investigate the consequences of strong electron-vibron coupling in a fully symmetric setup. Several striking features are predicted. In particular, a gate asymmetry and pronounced negative differential conductance features are observed. We attribute these features to the presence of slow channels originating from the interplay of Franck-Condon suppression of transport channels with spin and/or orbital degeneracies.

Figure 1

(a)–(c) Plots of the numerical differential conductance $dI/dV$(arbitrary units) of the system for coupling constants $\lambda =0.68$, 0.83, and 1.18, respectively. The charge excitation energy is ${\varepsilon }_0=1.4\mathrm{meV}$ and the energy of the lowest vibron mode is ${\varepsilon }_1=0.04\mathrm{meV}$. Additional parameters are a thermal energy of $k_{B}T=0.8\mu \mathrm{eV}$, orbital mismatch ${\varepsilon }_{\Delta }=0$, and ${\gamma }_{sl}={\gamma }_{dl}=0.02\mu \mathrm{eV}$ for $l=1,2$. The black lines running parallel to the Coulomb diamond edges correspond to negative differential conductance (NDC). Notice that here and in the following figures $dI/dV$(arbitrary units) is normalized to the maximum of $dI/dV$(arbitrary units) in the considered parameter range. The gate voltage is set to zero by convention at the degeneracy point.

Figure 2

(a) The low-bias transition regions of the stability diagram are labeled as A, B, C, D. (b) Energy-level scheme for the relevant transitions in the stability diagram involving regions A–D. Above the dashed line region D is activated with two vibron modes being in the transport window. The energy of the lowest vibron mode is $\hslash \omega =0.04\mathrm{meV}$. The number of degenerate states is indicated in brackets.

Figure 3

In the table for each of the different regions A, B, C in the stability diagram, relevant transitions, population of states, and current are given. $P_0^g$,$P_0^e$ represent the population of the 0-particle ground and first excited states, respectively, and $P_1^g$,$P_1^e$ the population of the 1-particle ground and first excited states. $I$ is the corresponding current in each region. In the transition scheme, the black arrows represent the drain and the dashed red arrows the source transitions. ${\Gamma }_{00}$ denotes the transition rate from 0-particle ground state to the 1-particle ground state, while ${\Gamma }_{01}$ denotes the transition rate from the 0-particle ground state to a 1-particle first excited state.

Figure 4

(a) Energy-level scheme for transitions in region B and (b) in region C. Importantly, because the bias voltage is too low, the transition $|1_k,0\rangle \to |0,1\rangle$ in region B and the transition $|0,0\rangle \to |1_k,1\rangle$ in region C is not allowed.

Figure 5

Populations of the low-energy states corresponding to the stationary density matrix calculated for different electron vibron coupling $\lambda$ and different gate-bias ranges. The first column corresponds to the case $\lambda =1$, thus ${\Gamma }_{11}=0$, while in the second column $\lambda =0.83$. The letters A, B, C labeling the rows refer to the stability diagram regions defined in Fig. 2. The states are ordered in energy. The rest of the parameters are the same as used for Fig. 1.

Figure 6

Plots of the differential conductance for a coupling constant $\lambda =0.68$ with two different approximations: (a) neglecting the higher harmonics of the system vibrations, (b) keeping coherences between the degenerate states with different vibronic configurations. The rest of the parameters are the same as used for Fig. 1a.

Figure 7

(a)–(c) Stability diagrams for coupling constants $\lambda =0.68$, 0.83, and 1.18, respectively. Additional parameters are a thermal energy of $k_{\mathrm{B}}T=0.8\mu \mathrm{eV}$, orbital mismatch ${\varepsilon }_{\Delta }=0.016$, ${\varepsilon }_0=0.56{\varepsilon }_1$, and ${\gamma }_s={\gamma }_d=0.02\mu \mathrm{eV}$ while for (d)–(f) ${\varepsilon }_{\Delta }=0.006{\varepsilon }_0$. The rest of the parameters are the same as used for Fig. 1.

Figure 8

(a) The low-bias transition regions are labeled as A, B, C, D, E, F, G. (b) Energy-level scheme for the transitions relevant in the low-bias regions of the stability diagram. As in Fig. 2, $\hslash \omega =0.04\mathrm{meV}$. The degeneracy of each state has been shown in brackets.

Figure 9

(a) Transition scheme for region D at ${\varepsilon }_{\Delta }=0.016{\varepsilon }_0$. (b) Transition scheme for region F at ${\varepsilon }_{\Delta }=0.016{\varepsilon }_0$.

Figure 10

(a)–(c) Stability diagrams for a molecule for the case of a coupling to the leads which depends on the orbital degree of freedom. All the parameters are the same as used in Figs. 1 and 7. The asymmetry is $a=1/45$ with orbital mismatch ${\varepsilon }_{\Delta }=0.016{\varepsilon }_0$.

Figure 11

(a), (b) Stability diagrams for both an orbital and left and right asymmetry. All the parameters are the same as used in Fig. 10c. The asymmetry with respect to the left and right lead is $b=45$ in (a) while it is $b=1/45$ in (b).