Nonrenewal statistics in quantum transport from the perspective of first-passage and waiting time distributions

The waiting time distribution has, in recent years, proven to be a useful statistical tool for characterizing transport in nanoscale quantum transport. In particular, as opposed to moments of the distribution of transferred charge, which have historically been calculated in the long-time limit, waiting times are able to detect nonrenewal behavior in mesoscopic systems. They have failed, however, to correctly incorporate backtunneling events. Recently, a method has been developed that can describe unidirectional and bidirectional transport on an equal footing: the distribution of first-passage times. Rather than the time between successive electron tunnelings, the first passage refers to the first time the number of extra electrons in the drain reaches $+1$. Here, we demonstrate the differences between first-passage time statistics and waiting time statistics in transport scenarios where the waiting time either cannot correctly reproduce the higher-order current cumulants or cannot be calculated at all. To this end, we examine electron transport through a molecule coupled to two macroscopic metal electrodes. We model the molecule with strong electron-electron and electron-phonon interactions in three regimes: (i) sequential tunneling and cotunneling for a finite bias voltage through the Anderson model, (ii) sequential tunneling with no temperature gradient and a bias voltage through the Holstein model, and (iii) sequential tunneling at zero bias voltage and a temperature gradient through the Holstein model. We show that for each transport scenario, backtunneling events play a significant role; consequently, the waiting time statistics do not correctly predict the renewal and nonrenewal behavior, whereas the first-passage time distribution does.

Figure 1

(a) The exact sequential and cotunneling current $\langle I\rangle =(\mathbf{I},({\mathbf{J}}_F-{\mathbf{J}}_B)\overline{\mathbf{P}})$ (black) compared to the respective predictions from the FPTD $\frac{1}{{\langle \tau \rangle }^{*}}$ (red) and WTD $\frac{1}{{\langle \tau \rangle }_F}-\frac{1}{{\langle \tau \rangle }_B}$ (blue). Current is measured in units of $\frac{\gamma }{2}$ and voltage is measured in meV. In (b) the randomness parameter $R^{*}$ calculated from Eq. (45) is compared to the Fano factor $F$ defined in Eq. (15). The energies of the spin-split electronic levels are ${\varepsilon }_{\uparrow }=0.5$ meV and ${\varepsilon }_{\downarrow }=-1.5$ meV; the Coulomb repulsion is $U=4$ meV, $T=75\mu \text{eV}$, and $\gamma =0.5$ T. The source and drain Fermi energies are symmetric about zero: ${\mu }_S=-{\mu }_D=V_{SD}/2$.

Figure 2

The Pearson correlation coefficient for sequential tunneling and cotunneling, defined in Eq. (46), calculated from the FPTD only. The energies of the spin-split electronic levels are ${\varepsilon }_{\uparrow }=0.5$ meV and ${\varepsilon }_{\downarrow }=-1.5$ meV; the Coulomb repulsion is $U=4$ meV, $T=75\mu \text{eV}$, and $\gamma =0.5$ T.

Figure 3

(a) The exact current for equilibrium and nonequilibrium phonons $\langle I\rangle =(\mathbf{I},({\mathbf{J}}_F-{\mathbf{J}}_B)\overline{\mathbf{P}})$ (black) compared to the respective predictions from the FPTD $\frac{1}{{\langle \tau \rangle }^{*}}$ (red) and WTD $\frac{1}{{\langle \tau \rangle }_F}-\frac{1}{{\langle \tau \rangle }_B}$ (blue). In (b) the randomness parameters $R^{*}$ calculated from Eq. (45) and $R$ calculated from Eq. (28) are compared to the Fano factor $F$ defined in Eq. (15). Note that here the WTD current prediction is $\frac{1}{{\langle \tau \rangle }_F}-\frac{1}{{\langle \tau \rangle }_B}$, but $R$ is calculated only from the forward WTD. The vibrationally adjusted energy level is $\varepsilon =0$, the vibrational frequency is $\omega =1$, $T_S=T_D=0.05$, the electron-phonon coupling strength is $\lambda =4$, ${\gamma }_{\alpha }=\frac{\gamma }{2}=0.01$, and the equilibrium phonons are kept at a vibrational temperature of $T_V=0.05$. The source and drain chemical potentials are shifted symmetrically about zero: ${\mu }_S=-{\mu }_D=V_{SD}/2$. We use units of $\omega$ for $\langle I\rangle$ ($e\omega$ if we reintroduce $\hslash \text{and}e$) and units of $\omega$ for all energy parameters as well (or $\hslash \omega /e$).

Figure 4

The Pearson correlation coefficient for equilibrium and nonequilibrium vibrations, defined in Eq. (29) and Eq. (46), calculated from the WTD and the FPTD, respectively. The vibrationally adjusted energy level is $\varepsilon =0$, the vibrational frequency is $\omega =1$, $T_S=T_D=0.5$, the electron-phonon coupling strength is $\lambda =4$, ${\gamma }^{\alpha }=\frac{\gamma }{2}=0.01$, and the equilibrium phonons are kept at a vibrational temperature of $T_V=0.05$. Again, the electrode chemical potentials are shifted symmetrically about zero ${\mu }_S=-{\mu }_D=V_{SD}/2$, and the energy units are in $\omega$.

Figure 5

Current (a) and Fano factor (b) predicted from the WTD in (blue) and FPTD (red) and compared to the exact results (black) for equilibrium and nonequilibrium vibrations, as a function of the temperature gradient $\mathrm{\Delta }T$. Again, the WTD prediction is $\frac{1}{{\langle \tau \rangle }_F}-\frac{1}{{\langle \tau \rangle }_B}$, but $R$ is calculated only from the forward WTD. The vibrationally adjusted energy level is $\varepsilon =1$, the vibrational frequency is $\omega =1$, the electron-phonon coupling strength is $\lambda =1$, and the voltage is $V_{SD}=0$, ${\gamma }^{\alpha }=\frac{\gamma }{2}=0.01$. The average temperature across the molecule is $\overline{T}=\frac{T_S+T_D}{2}=0.75$ and the electrode temperatures are symmetric around $\overline{T}$: $T_S=\overline{T}+\mathrm{\Delta }T/2$ and $T_D=\overline{T}-\mathrm{\Delta }T/2$. The equilibrium phonons are kept at a vibrational temperature of $T_V=\overline{T}$. Again, energy and current units are in $\omega$.