Universality and quantization of the power-to-heat ratio in nanogranular systems

We study heating and dissipation effects in granular nanosystems in the regime of weak coupling between the grains. We focus on the cotunneling regime and solve the heat-dissipation problem in an array of grains exactly. We show that the power to heat ratio has a universal quantized value, which is geometrically protected: It depends only on the number of grains.

Figure 1

Sketch of the system under consideration in this work: a chain of weakly coupled nanograins. The red curved arrows indicate an inelastic cotunneling process leaving behind an electron-hole (e-h) pair in a grain. These processes are responsible for the electron transport in the system ($I$). The energy of this e-h pair is then dissipated into the bath by $\dot{Q}$. The form factor $n_{\Omega >0}^{\left(\mathrm{LR}\right)}$ in Eq. (3) can be interpreted as the distribution function of electron-hole pairs left behind by a tunneling event through the system, where the electron sits on the left and the hole on the right lead, respectively. The form factor $n_{\Omega >0}^{\left(\mathrm{RL}\right)}$ corresponds to the opposite situation.

Figure 2

The difference of the heat rates dissipated in the first and second grains of a two-grain junction shown in the inset with the different grain temperatures, $T_{g1}\ne T_{g2}$. The curves from top to bottom correspond to temperatures $T=T_{g1}=T_{\mathrm{lead}}$ and $T_{g2}/T=1.2,1.1,1.05,0.95,0.09,0.8$.

Figure 3

The ratio of the heat $\dot{Q}$ dissipated in the grain to the total power $IV$, $2\dot{Q}/IV$ for a single grain junction with the grain temperature $T_g$ and the lead temperature $T_{\mathrm{lead}}$. The curves from top to bottom correspond to $T_g/T_{\mathrm{lead}}=0.6,0.8,0.9,0.99,1.01,1.1,1.2,1.4$. The inset shows a density plot of $2\dot{Q}/IV$ as the function of $T_g/T_{\mathrm{lead}}$ and $V/T_{\mathrm{lead}}$ with $V$ being the bias voltage.

Figure 4

The ratio $IV/\dot{Q}$ for single mode environment. This environment produces large dissipation when $V\sim {\omega }_0$. As follows, there is no universal quantization of the power to heat ration in general.

Figure 5

(a) The diagrams for $P\left(\Omega \right)$ in the first, second, and partially in the third order over the electron-environment interaction parameter $\rho$ in the ultrasmall tunnel junction. (b) All the diagrams for $P\left(\Omega \right)$ for the two-granular junction. The difference between the two cases is apparent from the symmetry structure of the diagrams.