Nonequilibrium thermoelectric transport through vibrating molecular quantum dots

We employ the functional renormalization group to study the effects of phonon-assisted tunneling on the nonequilibrium steady-state transport through a single level molecular quantum dot coupled to electronic leads. Within the framework of the spinless Anderson-Holstein model we focus on small to intermediate electron-phonon couplings and we explore the evolution from the adiabatic to the antiadiabatic limit and also from the low-temperature nonperturbative regime to the high-temperature perturbative one. We identify the phononic signatures in the bias-voltage dependence of the electrical current and the differential conductance. Considering a temperature gradient between the electronic leads we further investigate the interplay between the transport of charge and heat. Within the linear response regime we compare the temperature dependence of various thermoelectric coefficients to our earlier results obtained within the numerical renormalization group [Phys. Rev. B 96, 195156 (2017)]. Beyond the linear response regime in the context of thermoelectric generators we discuss the influence of molecular vibrations on the output power and the efficiency. We find that in the antiadiabatic limit the thermoelectric efficiency can be significantly enhanced.

Figure 1

Charge current $J^{\mathrm{c}}$ vs bias-voltage $V$ at particle-hole symmetry ${\tilde{\varepsilon }}_0=0$, for the listed electron-phonon couplings $\lambda /{\omega }_0$ (a) in the crossover regime of phonon frequency ${\omega }_0/\mathrm{\Gamma }=1$ and (b) in the antiadiabatic limit ${\omega }_0/\mathrm{\Gamma }=10$. Inset to (a): $J^{\mathrm{c}}$ vs $V$ in the adiabatic limit ${\omega }_0/\mathrm{\Gamma }=0.1$. Inset to (b): $J^{\mathrm{c}}$ vs $V$ at finite gate voltage ${\tilde{\varepsilon }}_0=-E_{\mathrm{p}}$ in the antiadiabatic limit (${\omega }_0/\mathrm{\Gamma }=10$). For $\lambda /{\omega }_0=0,0.5,1,\sqrt{2}$, we find ${\mathrm{\Gamma }}_{\mathrm{eff}}/\mathrm{\Gamma }=1,0.92,0.70,0.46$ for the crossover regime in (a), while for the antiadiabatic regime in (b), we find ${\mathrm{\Gamma }}_{\mathrm{eff}}/\mathrm{\Gamma }=1,0.83,0.45,0.19$. The temperature is $k_{\mathrm{B}}T/\mathrm{\Gamma }=0.1$ in all plots.

Figure 2

(a), (b) Differential conductance $G/G_0$ as a function of bias voltage and gate voltage (stability diagram) in the antiadiabatic limit ${\omega }_0/\mathrm{\Gamma }=5$, at $k_{\mathrm{B}}T/\mathrm{\Gamma }=0.1$, for (a) $\lambda /{\omega }_0=0$ and (b) $\lambda /{\omega }_0=0.5$. (c), (d) The bias-voltage dependence of $G$ for various gate voltages for $\lambda /{\omega }_0=0,0.5$, respectively.

Figure 3

(a) Electrical current and (b) molecular dissipation rate as a function of temperature for varying bias voltage and coupling $\lambda /{\omega }_0=0.5$, in the antiadiabatic limit ${\omega }_0/\mathrm{\Gamma }=10$, and at particle-hole symmetry ${\tilde{\varepsilon }}_0=0$.

Figure 4

(a) Normalized electrical conductance $G/G_0$, (b) the Seebeck coefficient $S$ (in units of $k_{\mathrm{B}}/e$), and (c) the normalized electronic contribution to the thermal conductance ${\kappa }_e/\mathrm{\Gamma }$ (in units of $k_{\mathrm{B}}^2/e^2$) vs reduced temperature for various electron-phonon coupling strengths in the antiadiabatic limit (${\omega }_0/\mathrm{\Gamma }=5.0$) and for a fixed gate voltage ${\tilde{\varepsilon }}_0/\mathrm{\Gamma }=-1.0$. The solid lines represent the NRG data and the circles are obtained within FRG by calculating the charge and heat current for an infinitesimal temperature gradient and bias voltage (calculating the transport coefficients by taking the derivatives numerically).

Figure 5

Sketch of a molecular quantum dot with vibrational degrees of freedom trapped between two electronic leads held at different temperatures and chemical potentials. The dashed lines indicate the transport of charge against the bias voltage by means of the temperature gradient, i.e., charging of a battery.

Figure 6

(a) Output power, (b) rescaled efficiency, and (c) molecular energy dissipation rate as a function bias voltage for various couplings and phonon frequencies, at temperatures $k_{\mathrm{B}}{T}_{\mathrm{L}}/\mathrm{\Gamma }=0.1$ and $k_{\mathrm{B}}{T}_{\mathrm{R}}/\mathrm{\Gamma }=2.1$, and for gate voltage $\widetilde{{\varepsilon }_0}/\mathrm{\Gamma }=2$. The inset shows the efficiency as a function of output power for $\lambda =0,k_{\mathrm{B}}{T}_{\mathrm{L}}/\mathrm{\Gamma }=1,k_{\mathrm{B}}{T}_{\mathrm{R}}/\mathrm{\Gamma }=90$, and ${\tilde{\varepsilon }}_0/\mathrm{\Gamma }=40$.

Figure 7

(a) Comparison of FRG (dashed lines) and NRG (solid lines) for the frequency dependence of the equilibrium molecular spectral function for two different coupling strengths in the antiadiabatic limit ${\omega }_0/\mathrm{\Gamma }=10$ at $k_{\mathrm{B}}T/\mathrm{\Gamma }=0.1$. The inset illustrates $\pi \mathrm{\Gamma }A\left(\omega \right)$ as a function of $\omega /\mathrm{\Gamma }$ within NRG using two different broadening parameters, for $\lambda /{\omega }_0=1$. (b), (c) The evolution of the frequency dependence of the molecular spectral function and the nonequilibrium distribution function Eq. (8) as increasing the bias voltage for ${\omega }_0/\mathrm{\Gamma }=10$ and $k_{\mathrm{B}}T/\mathrm{\Gamma }=0.1$. Note the logarithmic $x$ axis in (a) and (b). In (b) the curve corresponding to $eV/{\omega }_0=0.5$ is almost indistinguishable from the equilibrium case $eV=0$, and also $eV=4.0$ from $eV/{\omega }_0=5.0$.