Rectification by charging: Contact-induced current asymmetry in molecular conductors

We outline the qualitatively different physics behind charging-induced current asymmetries in molecular conductors operating in the weakly interacting self-consistent field (SCF) and the strongly interacting Coulomb blockade (CB) regimes. The SCF conductance asymmetry originates in the asymmetric shifts of the closed-shell molecular conducting levels, driven by unequal mean-field potentials for positive and negative biases. A very different current asymmetry arises for CB due to the unequal number of open-shell excitation channels at opposite bias voltages. The CB regime, dominated by single charge effects, typically requires a computationally demanding many-electron or Fock-space description to do justice to its complex excitation spectrum. However, our analysis of molecular CB measurements reveals that many novel signatures can be explained using a simpler orthodox model that involves an incoherent sum of Fock-space excitations and hence treats the molecule as a metallic dot. This also reduces the complexity of the Fock-space description by including charge configurations alone, somewhat underscoring the richness of its electronic structure while retaining the essential single charge nature of the transport process. The inclusion of electronic structure with well-resolved Fock space excitations is, however, crucial in some notable examples.

Figure 1

(Color online) Experiments showing (a) comparable currents reached over unequal voltage widths (Ref. 2) (reprinted with permission) in the SCF limit and (b) unequal currents reached over comparable voltage widths in the CB limit (Ref. 7) (reprinted with permission).

Figure 2

(Color online) Crossover between asymmetries. An SCF current asymmetry arises for ${\gamma }_L\gg {\gamma }_R$ when under positive bias the left contact maintains the HOMO level at neutrality during conduction, while for negative bias, the emptied level is expelled from the bias window by (a) charging and (c) creating unequal plateau widths for opposite bias (stars). In the CB regime, however, charge removal by the negatively biased right contact is rate determining and occurs in two different ways, while for positive bias, (b) charge addition is rate determining and can occur in only one way. (c) This leads to an asymmetry in plateau heights (bold solid line). While the transition between the two limits is hard to model accurately, (c) a phenomenological broadening through an artificial enhanced temperature circles) illustrates how the open-shell CB plateau morphs into a higher effective broadening, restoring the SCF result in the limit of large broadening.

Figure 3

(Color online) Illustration of orthodox model: The orthodox theory parameters define a set of $I\left(V,n_0\right)$ curves, represented by the dashed lines. Starting with $n_0=0$, the current follows the $I\left(V,n_0=0\right)$ curve under bias until $n_0$ changes. If $\left|V_{\text{CB}}\right|<\left|V_{n_0=0}\right|$, as is the case here, the current will rise linearly out of the zero-conductance region; otherwise, there will be a jump onset. For this simulation, $R_1=10\text{M}\Omega$, $R_2=6\text{G}\Omega$, $C_1=8\text{aF}$, $C_2=5.3\text{aF}$, $Q_0=0$, and $T=2\text{K}$.

Figure 4

(Color online) Origin of peak asymmetry and variation with gate voltage. (a) For gate voltages that place the contact Fermi energy ${\mu }_F$ in the $N$ electron blockade region, the levels align such that ${\mu }_F>{ϵ}_{00}^{Nr}$. A state transition diagram shows the addition and removal of up (down) spins resulting in transitions ${ϵ}_{00}^{Nr}$ (bold double arrow) between ground states of neutral and positively charged species (light orange). Also shown in the state transition diagram are transitions (dashed double arrow) between various configurations of neutral excited state (deep orange) and positively charged ground state, labeled ${ϵ}_{10}^{Nr}$. (c) The resulting $I\text{−}V$ shows clear asymmetry in the peak height due to there being more ways to add an electron. (c) Increasing gate voltage increases the number of excitations available, giving a pronounced (d) current height modulation with gate voltage. The inset shows the corresponding $I\text{−}V$ characteristics (Refs. 3, 4, 11).

Figure 5

(Color online) Experiment from Park et al. (Ref. 3) showing gate-rectification properties in the Coulomb blockade regime: (a) experimental traces (reprinted with permission), (b) orthodox fit with parameters $C_1=0.624\text{aF}$, $C_2=0.486\text{aF}$, $C_G=0.0708\text{aF}$, $R_1=1\text{M}\Omega$, $R_2=75\text{M}\Omega$, $Q_0=-0.05e$, and $T=2.2\text{K}$, and (c) fit from Fock-space model.

Figure 6

(Color online) Asymmetric CB results showing (a) experiment (Ref. 5) (reprinted with permission) and (b) orthodox theory with parameters $C_1=3.70\text{aF}$, $C_2=3.24\text{aF}$, $C_G=0.061\text{aF}$, $R_1=2\text{M}\Omega$, $R_2=210\text{M}\Omega$, $Q_0=0.175e$, and $T=4.2\text{K}$.

Figure 7

(Color online) Origin of exchange in asymmetry of conductance peaks: (a) and (b) Schematic of vicinity of charge degeneracy point A in Coulomb diamond. Also shown are the corresponding energy diagrams at threshold. Notice a different set of threshold transport channels between (a) and (b). (c) and (d) State transition diagrams illustrating the different excitation spectra accessed on either side of the charge degeneracy point A. (e) and (f) $G\text{−}V$ plots for scenarios (a) and (b). Notice the clear peak exchange as a result of accessing different excitation spectra in either case.

Figure 8

(Color online) (a) Experimental trace demonstrating peak asymmetry exchange (Ref. 4). Compare with Fock-space model results in Figs. 7e, 7f. (b) Orthodox simulation reproducing peak exchange observed in the experiments. The parameters are $R_1=35\text{M}\Omega$, $R_2=350\text{M}\Omega$, $C_1=0.673\text{aF}$, $C_2=0.612\text{aF}$, $C_G=0.0135\text{aF}$, $Q_0=-0.18e$, and $T=4.2\text{K}$.

Figure 9

(Color online) Limitation of orthodox model: (a) Experimental trace (Ref. 22) (reprinted with permission) showing fine structure. (b) The fine structure in $I\text{−}V$’s, result from keeping track of excitations explicitly within the Fock-space CB model. (c) An orthodox calculation merely maintains the same slope between charge addition jumps, thus may not reproduce any fine structure.

Figure 10

(Color online) Gate-voltage dependence of $I\text{−}V$ curves in the orthodox theory. Blue arrows indicate the direction of movement of the onset voltages with increasing gate voltage. The four plots correspond to the four possible onset combinations: (a) a jump onset at negative bias and a linear onset at positive bias, for which the $I\text{−}V$ conductance gap widens with increasing gate voltage; (b) a linear onset then a jump onset, with the gap narrowing around zero-bias; (c) two linear onsets, which corresponds to a translation of the $I\text{−}V$ curve: and (d) two jump onsets, which correspond to a translation in the opposite direction of (c).