Electrothermal Transistor Effect and Cyclic Electronic Currents in Multithermal Charge Transfer Networks

A theory is developed to describe the coupled transport of energy and charge in networks of electron donor-acceptor sites which are seated in a thermally heterogeneous environment, where the transfer kinetics are dominated by Marcus-type hopping rates. It is found that the coupling of heat and charge transfer in such systems gives rise to exotic transport phenomena which are absent in thermally homogeneous systems and cannot be described by standard thermoelectric relations. Specifically, the directionality and extent of thermal transistor amplification and cyclical electronic currents in a given network can be controlled by tuning the underlying temperature gradient in the system. The application of these findings toward the optimal control of multithermal currents is illustrated on a paradigmatic nanostructure.

Figure 1

Network graphs for $\mathcal{R}$, $\mathcal{L}$, and $\mathcal{K}$ topologies. Each node represents a donor-acceptor site. As shown explicitly in the $\mathcal{R}$ graph, each site is in contact with an independent thermal bath. The center panel shows a schematic of the energy surfaces $E_a$ (solid curve) and $E_b$ (dashed curve) and ${\mathcal{Q}}_{\mathrm{obt}}$ and ${\mathcal{Q}}_{\mathrm{rel}}$ for the state transition $a\to b$.

Figure 2

Network graph for a three-site ring (${\mathcal{R}}_3$) network.

Figure 3

Cyclic flux ${\mathcal{J}}_c$ (top) and heat current $d{\mathcal{Q}}_1/dt$ (bottom) in the nonadiabatic limit as functions of $T_3$ for a three-site ${\mathcal{R}}_3$ network. Each curve corresponds to the respective value of $E_3^{(0)}$ marked in the legend. In each panel, $E_1^{(0)}=-4$, $E_2^{(0)}=0$, and $T_2=3/2$ with (a),(b) $T_1=3/2$ and (c),(d) $T_1=3/4$. The circular markers denote unithermal points. All reorganization energies are $E_{Rj}=1/2$ for each mode involved in a particular transition and zero otherwise. All quantities are shown in reduced units [53, 60].

Figure 4

(a) Amplification factor $\alpha$ and (b) heat current derivatives $\partial {\dot{\mathcal{Q}}}_s/\partial T_3$ as functions of $T_3$ for a multithermal ${\mathcal{R}}_3$ system with $T_1=3/4$, $T_2=3/2$, and $E_{Rj}=1/2\forall j$ over each state transition. In both panels, $E_1^{(0)}=-4$ and $E_2^{(0)}=E_3^{(0)}=0$.