Influence of phonon-assisted tunneling on the linear thermoelectric transport through molecular quantum dots

We investigate the effect of vibrational degrees of freedom on the linear thermoelectric transport through a single-level quantum dot described by the spinless Anderson-Holstein impurity model. To study the effects of strong electron-phonon coupling, we use the nonperturbative numerical renormalization group approach. We also compare our results, at weak to intermediate coupling, with those obtained by employing the functional renormalization group method, finding good agreement in this parameter regime. When applying a gate voltage at finite temperatures, the inelastic scattering processes, induced by phonon-assisted tunneling, result in an interesting interplay between electrical and thermal transport. We explore different parameter regimes and identify situations for which the thermoelectric power as well as the dimensionless figure of merit are significantly enhanced via a Mahan-Sofo type of mechanism. We show, in particular, that this occurs at strong electron-phonon coupling and in the antiadiabatic regime.

Figure 1

(Solid lines) NRG results for the temperature dependence of the normalized electric conductance $G/G_0$ ($G_0=e^2/h$), the thermopower $S$ (in units of $k_{\mathrm{B}}/e$), and the normalized electronic contribution to the thermal conductance ${\kappa }_e/\mathrm{\Gamma }$ (in units of $k_{\mathrm{B}}/h$). (a)–(c) Evolution with increasing electron-phonon coupling at a given phonon frequency ${\omega }_0/\mathrm{\Gamma }=5$ and level position ${\tilde{\varepsilon }}_0=-\mathrm{\Gamma }$. Selected FRG results (stars) [using Eq. (8)] at weak to intermediate couplings serve as checks on the NRG results. (d)–(f) Dependence on level position (gate voltage) for a given coupling $\lambda /{\omega }_0=2.0$ and a fixed frequency ${\omega }_0/\mathrm{\Gamma }=5$. (g)–(i) Evolution from the adiabatic to the antiadiabatic limit for $\lambda /{\omega }_0=2.0$ and ${\tilde{\varepsilon }}_0=-\mathrm{\Gamma }$.

Figure 2

(a) The 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 $k_{\mathrm{B}}/h$) vs the dimensionless gate voltage ${\tilde{\varepsilon }}_0/\mathrm{\Gamma }$ for various temperatures in the antiadiabatic limit (${\omega }_0/\mathrm{\Gamma }=20$) for $\lambda /{\omega }_0=0.5$. The solid lines represent the NRG data and the stars are calculated with FRG (using the Sommerfeld expansion).

Figure 3

The power factor $P{F}_0$ (in units of $k_{\mathrm{B}}^2/h$), the Lorenz number $L$ (in units of $k_{\mathrm{B}}^2/e^2$), and the dimensionless figure of merit $Z{T}_0$ vs the reduced temperature $T/{\mathrm{\Gamma }}_{\mathrm{eff}}$: (a)–(c) for different electron-phonon coupling for a given phonon frequency ${\omega }_0/\mathrm{\Gamma }=5$ and gate voltage ${\tilde{\varepsilon }}_0=-\mathrm{\Gamma }$, (d)–(f) at different gate voltages for a given coupling $\lambda /{\omega }_0=2.0$ and a fixed frequency ${\omega }_0/\mathrm{\Gamma }=5$, and (g)–(i) from the adiabatic to the antiadiabatic limit for a $\lambda /{\omega }_0=2.0$ and ${\tilde{\varepsilon }}_0=-\mathrm{\Gamma }$.

Figure 4

(a) $Z{T}_0$ vs the coupling strength and temperature for a fixed gate voltage ${\tilde{\varepsilon }}_0=-\mathrm{\Gamma }$ and phonon frequency ${\omega }_0/\mathrm{\Gamma }=5.0$. (b) $Z{T}_0$ vs the gate voltage and temperature for a fixed coupling strength $\lambda /{\omega }_0=2.0$ and phonon frequency ${\omega }_0/\mathrm{\Gamma }=5.0$. Vertical and horizontal dashed lines in (b) indicate $T={\mathrm{\Gamma }}_{\mathrm{eff}}$ and $|{\tilde{\varepsilon }}_0|={\mathrm{\Gamma }}_{\mathrm{eff}}$, respectively.

Figure 5

The dimensionless figure of merit $Z{T}_0$ vs reduced temperature $T/{\mathrm{\Gamma }}_{\mathrm{eff}}$ for different numbers of bosons and different $\mathrm{\Gamma }/D$ deep in the antiadiabatic regime ${\omega }_0/\mathrm{\Gamma }={10}^3$ for $\lambda /{\omega }_0=2.0$, ${\tilde{\varepsilon }}_0=-\mathrm{\Gamma }$, and illustrating the convergence with respect to the number of bosons kept.

Figure 6

The dimensionless figure of merit $Z{T}_0$ vs the reduced temperature $T/{\mathrm{\Gamma }}_{\mathrm{eff}}$ for different ${\omega }_0/\mathrm{\Gamma }$ at ${\tilde{\varepsilon }}_0=-\mathrm{\Gamma }$ and $\lambda /{\omega }_0=2$. The values of ${\omega }_0/{\mathrm{\Gamma }}_{\mathrm{eff}}$ are also listed. The main panel shows the monotonically increasing behavior of the peak value of $Z{T}_0$ with increasing ${\omega }_0/\mathrm{\Gamma }$ in the extended antiadiabatic regime (${\omega }_0/{\mathrm{\Gamma }}_{\mathrm{eff}}\gtrsim 1$), while the inset shows the opposite behavior for ${\omega }_0/{\mathrm{\Gamma }}_{\mathrm{eff}}\lesssim 1$. The $\lambda =0$ curve in the inset lies on top of the ${\omega }_0/\mathrm{\Gamma }=0.01$ curve, illustrating that the noninteracting and adiabatic limits are almost identical.

Figure 7

The normalized spectral function $\pi \mathrm{\Gamma }A\left(\nu \right)$ vs frequency $\nu$. (a) The coupling strength dependence of the $T=0$ spectral function for ${\tilde{\varepsilon }}_0=-\mathrm{\Gamma }$ and ${\omega }_0=5\mathrm{\Gamma }$. (b) The temperature dependence of the spectral function for ${\tilde{\varepsilon }}_0=-\mathrm{\Gamma }$, ${\omega }_0=5\mathrm{\Gamma }$, and $\lambda /{\omega }_0=2$. (c) The (phonon) frequency dependence of the spectral function for ${\tilde{\varepsilon }}_0=-\mathrm{\Gamma }$, $\lambda /{\omega }_0=2.0$, and $T/\mathrm{\Gamma }=0.4$. (d) The gate voltage dependence of the $T=0$ spectral function for ${\omega }_0=5\mathrm{\Gamma }$ and $\lambda /{\omega }_0=2$. We used NRG parameters $\mathrm{\Lambda }=4$ and $N_z=20$. The spectral sum rule ${\int }_{-\infty }^{+\infty }A\left(\nu \right)d\nu =1$ is satisfied numerically to within a few percent for all parameters and temperatures shown.

Figure 8

Thermopower $S$ vs the normalized gate voltage $-{\tilde{\varepsilon }}_0/{\omega }_0$ for various temperatures, in the extreme antiadiabatic limit ${\omega }_0/\mathrm{\Gamma }=4.0\times {10}^7$ and for a fixed electron-phonon coupling $\lambda /{\omega }_0=2.0$. The inset shows $S$ vs $-{\tilde{\varepsilon }}_0/{\mathrm{\Gamma }}_{\mathrm{eff}}$ for the five lowest temperatures of the main panel. We choose $\mathrm{\Gamma }/D={10}^{-10}$ to resolve high-temperature results more precisely. The renormalized tunneling rate is ${\mathrm{\Gamma }}_{\mathrm{eff}}=1.659\times {10}^{-2}\mathrm{\Gamma }$.