Cooling mechanisms in molecular conduction junctions

While heating of a current carrying Ohmic conductors is an obvious consequence of the diffusive nature of the conduction in such systems, current-induced cooling has been recently reported in some molecular conduction junctions. In this paper, we demonstrate by simple models the possibility of cooling molecular junctions under applied bias, and discuss several mechanisms for such an effect. Our model is characterized by single electron tunneling between electrodes represented by free electron reservoirs through a system characterized by its electron levels, nuclear vibrations and their structures. We consider cooling mechanisms resulting from (a) cooling of one electrode surface by tunneling-induced depletion of high-energy electrons; (b) cooling by coherent sub resonance electronic transport analogous to atomic laser-induced cooling and (c) the incoherent analog of process (b)—cooling by driven activated transport. The non-equilibrium Green function formulation of junction transport is used in the first two cases, while a master equation approach is applied in the analysis of the third.

Figure 1

(Color online) The effective nonequilibrium temperature $T_p$ vs applied bias. Parameters of the calculation are $T=300\text{K}$, $d=5\text{A}$, $U=0.2\text{eV}$, and ${\omega }_p=0.1\text{eV}$. Shown are (a) Temperatures of the nonequilibrium parts of the electrodes in the M-I-M junction for the cases $T_{LL}=T_{RR}=T_{LR}^0$ (left electrode—solid line, blue; right electrode—dashed line, red) and $T_{LL}=T_{RR}=10\cdot T_{LR}^0$ (left electrode—dotted line, blue; right electrode–dash-dotted line, red) and (b) Temperature of the molecule asymmetrically coupled to the electrodes in a M-molecule-M junction. The molecular parameters of this calculation are ${\epsilon }_0=0.3\text{eV}$, ${\Gamma }_{ML}=0.009\text{eV}$, ${\Gamma }_{MR}=0.001\text{eV}$, ${\omega }_0=0.1\text{eV}$ [the same value of ${\omega }_0$ is used to normalize the voltage in Fig. 1a], $M=0.2\text{eV}$, and $\Omega =0$. The metallic rate constants $T_{LL}=T_{RR}=T_{LR}^0$ (full line; blue) and $T_{LL}=T_{RR}=10\cdot T_{LR}^0$ (dotted line; blue).

Figure 2

(Color online) Effective molecular temperature, Eq. (14), vs applied bias for metal-molecule-metal junction. The junction parameters were taken as in Fig. 1: ${\epsilon }_0=0.3\text{eV}$, ${\omega }_0=0.1\text{eV}$, $M=0.2\text{eV}$, $d=5\text{A}$, $U=0.2\text{eV}$, ${\omega }_p=0.1\text{eV}$, and $\Omega =0$. The metal-molecule coupling parameters were chosen ${\Gamma }_{ML}^0=0.009\text{eV}$ and ${\Gamma }_{MR}^0=0.001\text{eV}$ (such asymmetric coupling is typical to STM setups), and the metallic rate constants were taken $T_{LL}=T_{RR}=4\cdot {10}^{-4}$ (full line; blue) and $T_{LL}=T_{RR}=4\cdot {10}^{-3}$ (dotted line; blue).

Figure 3

(Color online) Effective temperature [determined from Eq. (23)] of a molecular vibration, displayed vs applied bias of a molecular vibration in a silicon-molecule-silicon junction.

Figure 4

(Color online) Effective molecular temperature vs applied bias for three-site bridge model. See text for parameters.

Figure 5

(Color online) Time evolution of local temperatures $T_1$ and $T_2$ for $\left({\mu }_L,{\mu }_R\right)=\left(1.0,0\right)$. The inset shows the time evolution of the heat currents ${\dot{Q}}_j$ that vanish in the long time limit. This automatically determines the local temperature $T_j$ at the steady state. The temperature on both leads are $k_{B}T/\omega =1$.

Figure 6

(Color online) Local temperatures, $T_j-T$; $j=1,2$ displayed against the bias voltage ${\mu }_L-{\mu }_R$ $\left({\mu }_R=0\right)$. Also shown is electronic current $J_c$. Cooling is indicated by negative values of $T_j-T$. Dashed line (purple): $T_1-T$. Dashed-dotted line (blue): $T_2-T$. Solid line (red)–$J_c$ [$\times 100$ in units of $e^{2}V$ $\left(\hslash =1\right)$]