Charge correlations in polaron hopping through molecules

In many organic molecules, the strong coupling of excess charges to vibrational modes leads to the formation of polarons, i.e., localized states of charge carriers and molecular deformations. At room temperature, incoherent hopping of polarons along the molecule is the dominant mechanism of charge transport. We study the situation far-from-equilibrium where, due to an applied voltage bias, the induced number of charge carriers on the molecule is high and charge correlations become relevant. We develop a diagrammatic theory that accounts in a finite system for all many-particle correlations and their effect on the incoherent transport. We determine the $I\text{−}V$ characteristics of short sequences of DNA by expanding the diagrammatic theory up to second order in the hopping parameters. Correlations qualitatively modify the results as compared to those obtained in a mean-field approximation.

Figure 1

(Color online) Sketch of charge carriers, hopping rates, and correlations. (a) On a homogeneous chain the rates ${\mathcal{W}}_{nm}$ are all equal. The occupations ${\rho }_l$ are independent of the two-particle correlations ${\rho }_{lm}$ as the corresponding terms in the rate equation cancel out, cf. Eq. (14). (b) On an inhomogeneous chain the rates ${\mathcal{W}}_{nm}\ne {\mathcal{W}}_{mn}$ so the two-particle correlations ${\rho }_{lm}$ affect the occupations ${\rho }_l$. Since the modifications do not remain local, the transport through the entire system is changed.

Figure 2

Schematic drawing of the Keldysh contour and the forward and backward time-evolution operators. The open and crossed circle (the clamp) represent the two operators $a_l$ and $a_l^{†}$, respectively, which are evaluated at time $t$.

Figure 3

Ladder summation of a second-order irreducible block representing two tunneling processes. The irreducible blocks are separated by free sections (indicated by dotted lines). The summation can be recast into a self-consistency equation for the occupation number $\rho$.

Figure 4

(a) Sixth-order diagram in the real-time expansion of the occupation number $⟨U_{{\tilde{H}}_0}^{†}\left(t\right)a_l^{†}{a}_l{U}_{{\tilde{H}}_0}\left(t\right)⟩$ (b) Fourth-order diagram in the real-time expansion of the two-particle correlation function $⟨U_{{\tilde{H}}_0}^{†}\left(t_2\right)a_l^{†}{a}_m{a}_m^{†}{a}_l{U}_{{\tilde{H}}_0}\left(t_2\right)⟩$. Free sections are indicated by the vertical dotted lines.

Figure 5

A second-order hopping diagram. The full lines represent fermion lines. The dashed line represents the sum of all possible vibrational lines arising from the diagrammatic rules, here it has a value $F_l^{+}\left(t_1-t_2\right)F_{m_2}^{+}\left(t_1-t_2\right)+F_l^{+}\left(t_1-t_2\right)+F_{m_2}^{+}\left(t_1-t_2\right)$ [cf. Appendix ].

Figure 6

A second-order hopping diagram. The full lines represent fermions, the dashed line represents the sum of all possible vibrational lines arising from the diagrammatic rules, here it has a value $F_m^{+}\left(t_1-t_2\right)F_{m_3}^{+}\left(t_1-t_2\right)+F_m^{+}\left(t_1-t_2\right)+F_{m_3}^{+}\left(t_1-t_2\right)$.

Figure 7

(Color online) Zero-bias conductance $G_0$ as a function of chemical potential $\mu$ for the DNA molecule with sequence AAAGAAAA with mean-field correlations (black solid line) and exact correlations (red dashed line). The parameters used are ${ϵ}_{\text{A}}=-0.26\text{eV}$, ${ϵ}_{\text{G}}=+0.25\text{eV}$ relative to the zero point of the chemical potential, polaron shifts ${\Delta }_{\text{A}}=0.18\text{eV}$ and ${\Delta }_{\text{G}}=0.47\text{eV}$, symmetric coupling to leads with linewidths ${\Gamma }_{\text{L}}={\Gamma }_{\text{R}}=0.001\text{eV}$, vibrational energies $\hslash {\omega }_{\text{A}}=11\text{meV}$, $\hslash {\omega }_{\text{G}}=16\text{meV}$, and room temperature $k_{\text{B}}T=25\text{meV}$. The inset shows the same plot on logarithmic scale to stress that both models agree for very low and high occupation.

Figure 8

(Color online) Lower graph: difference of the occupation for exact and mean-field correlations for the first (black solid line) fourth (red dashed line) and eighth (blue dashed-dotted line) base of the DNA molecule AAAGAAAA as a function of chemical potential $\mu$. The used transport bias is $V_{\text{b}}=0.01\text{V}$, which is in the linear regime. As reference in the upper graph the plot of Fig. 7 is repeated. Parameters as in Fig. 7.

Figure 9

(Color online) Differential conductance $dI/d{V}_{\text{b}}$ (logarithmic scale) as a function of applied bias $V_{\text{b}}$ for the DNA molecule with sequence AAAGAAAA, for mean-field correlations (black solid line) and exact correlations (red dashed line). Parameters as in Fig. 7 but with the equilibrium chemical potential of the electrodes fixed at $\mu =0\text{eV}$.

Figure 10

(Color online) The zero-bias conductance $G_0$ as a function of chemical potential $\mu$ for the DNA sequence AAAGGAAA (black solid line) with exact correlations. Only a single maximum around the guanine energy is displayed. For the dashed line the hopping rate between the two central guanine bases is enhanced by a factor 100. This shows that the maximum at the adenine energy in the sequence AAAGGAAA is suppressed due to the peculiar hopping parameters relevant for DNA. Other parameters as in Fig. 7.

Figure 11

An example diagram for a second-order hopping process between an electrode and the molecule. It is evaluated in Appendix .