Thermoelectric energy conversion in molecular junctions out of equilibrium

TL;DR

The paper develops an iterated generalized Kadanoff-Baym ansatz ($i$GKBA) within non-equilibrium Green's functions to model time-resolved thermoelectric transport in nano- and molecular junctions beyond the wide-band limit. By employing Lorentzian lead couplings and a Meir-Wingreen formalism for currents, it demonstrates that finite-bandwidth effects are essential to avoid unphysical divergences inherent to WBLA and achieves accurate benchmarking against full Kadanoff-Baym equations. The authors apply the method to a two-terminal quantum dot and a cyclobutadiene molecular junction, showing that $i$GKBA consistently improves upon GKBA and reproduces full KBE results, including transient buildup of thermoelectric currents and energy conversion metrics. They extract time-resolved Seebeck voltages and compute the instantaneous thermoelectric figure of merit $ZT(t)$ and efficiency, revealing transient windows of enhanced performance that surpass stationary limits and highlighting the potential for ultrafast, nanoscale energy harvesting technologies.

Abstract

Understanding time-resolved quantum transport is crucial for developing next-generation quantum technologies, particularly in nano- and molecular junctions subjected to time-dependent perturbations. Traditional steady-state approaches to quantum transport are not designed to capture the transient dynamics necessary for controlling electronic behavior at ultrafast time scales. In this work, we present a non-equilibrium Green's function formalism, within the recently-developed iterated generalized Kadanoff-Baym ansatz ($i$GKBA), to study thermoelectric quantum transport beyond the wide-band limit approximation (WBLA). We employ the Meir-Wingreen formula for both charge and energy currents and analyze the transition from Lorentzian line-width functions to the WBLA, identifying unphysical divergences in the latter. Our results highlight the importance of finite-bandwidth effects and demonstrate the efficiency of the $i$GKBA approach in modeling time-resolved thermoelectric transport, also providing benchmark comparisons against the full Kadanoff-Baym theory. We exemplify the developed theory in the calculation of time-resolved thermopower and thermoelectric energy conversion efficiency in a cyclobutadiene molecular junction.

Figures (11)

• Figure 1: Molecular junction model considered in this work. Central system (a cyclobutadiene molecule) is coupled to two leads ($\alpha=L,R$) via the frequency-dependent tunneling rates $\Gamma_{\alpha}$, Eq. \ref{['eq:lorentzian']}, characterized by the energy centroids $\epsilon_\alpha$ and bandwidths $\Omega_\alpha$. The leads are held at different temperatures $\beta_\alpha$, and external time-dependent voltages $V_\alpha(t)$ are applied.
• Figure 2: Derivation of the EOMs for the $i$GKBA correlators. Correlators that originate from the self-energy terms in Eqs. \ref{['eq:eom:GRA']} are framed in red. Correlators that appear already at the GKBA level are shaded.
• Figure 3: Time-dependent charge density $\rho$ (a), charge current $J_L$ (b), and energy current $J_L^E$ (c) at the left-lead interface with the quantum dot being weakly coupled to the leads, $\gamma=\varepsilon/10$, and driven by a sudden gate voltage $u(t)=(5\varepsilon/2)\theta(t)$. Inset in panel (c) displays the equilibrium energy current, before the sudden gate voltage is switched on, in terms of the number of poles. Other parameters are $\mu_L=\mu_R=0$, $\beta_L=\beta_R=10/\varepsilon$.
• Figure 4: Same as Fig. \ref{['fig:weak']} but the quantum dot being strongly coupled to the leads, $\gamma=\varepsilon$.
• Figure 5: Time-dependent energy current $J_L^E$ at the left-lead interface with the quantum dot being coupled to the leads with $\gamma=\varepsilon/2$, and driven by a sudden gate voltage $u(t)=(5\varepsilon/2)\theta(t)$. (a) High temperature $\mu_L=\mu_R=0$, $\beta_L=\beta_R=1/\varepsilon$; (b) chemical potential drop $\mu_L=-\mu_R=\varepsilon/2, \beta_L=\beta_R=10/\varepsilon$; and (c) temperature gradient $\mu_L=-\mu_R=\varepsilon/2, \beta_L=1/\varepsilon, \beta_R=10/\varepsilon$. Insets display the equilibrium energy current, before the sudden gate voltage is switched on, in terms of the Lorentzian bandwidth $\Omega$.