Heat-dissipation decomposition and free-energy generation in a non-equilibrium dot with multi-electron states

Abstract

We experimentally demonstrate the decomposition of heat dissipation during free-energy generation in a nanometer-scale dot transitioning to a non-equilibrium steady state via single-electron counting statistics. An alternating-current signal driving a reservoir that injects multiple electrons into the dot makes it non-equilibrium, leading to free-energy generation, heat dissipation, and Shannon-entropy production. By analyzing the time-domain probability distributions of multi-electron states of the dot, we quantitatively decompose the heat dissipation into housekeeping and excess heats, thereby revealing their direct correlation with free-energy generation. This correlation suggests that the ratio of the generated free energy to the work applied to the dot, can potentially reach 0.5 under far-from-equilibrium conditions induced by a large signal, while an efficiency of 0.25 was experimentally achieved. These results establish a quantitative link between decomposed heat dissipation and free-energy generation in a multi-electron stochastic system, providing a thermodynamic framework for non-equilibrium electronic devices.

Figures (6)

• Figure 1: (a) False scanning-electron-microscope image of the device for single-electron counting statistics. The constant ER voltage $V_{\rm ER}$ was 0.5 V. The details of the device structure are explained in Supplemental Material. (b) Energy band diagram between the ER and dot when the excess number of electrons in the dot is $N$. The Fermi energy of the ER is modulated by an AC signal $V(t)$ superimposed on $V_{\rm ER}$. (c) Current flowing through the sense-FET without the AC signal. The sampling rate of the sense-FET was 20 Hz.
• Figure 2: (a) Contour plot of the $\rho_N (t)$ distributions from an equilibrium state to an NESS. The AC signal with amplitude $S_{\rm AC}$ of 100 mV was applied at $t=0$. Experimental data were averaged over 3000 repetitions. (b) $\rho_N (t)$ distributions at $t_1$, $t_2$, and $t_3$ depicted in (a). The solid lines are the Gaussian fits to the experimental results. (c) Transient characteristics of the average $\langle N(t) \rangle$, variance ${\rm Var}(t)$, skewness $\gamma_1(t)$, and kurtosis $\gamma_2(t)$ of $\rho_N (t)$. (d) Transient characteristics of the deviations of the internal energy $\Delta U(t)$, free energy $\Delta F(t)$, and Shannon entropy $k_{\rm B}T\Delta S(t)$ from their initial values at $t=0$. Open marks are the experimental data, and the solid lines are the numerical results of the master equation.
• Figure 3: (a) Schematic of heat dissipation, free-energy generation, and work. The left and right vertical axes represent these rates and the corresponding energy levels (in red), respectively. The horizontal axis indicates “In’’ and “In–Out’’ events, in which $N$ increases by one and in which the same number of electrons enter and leave the dot, respectively. Although these events occur stochastically, they are grouped into two categories. The shaded areas represent the total heat dissipated in each process. (b) $\langle\dot{Q}_{\rm T}\rangle$ and its decomposition into $\langle\dot{Q}_{\rm EX}\rangle$, $\langle\dot{Q}_{\rm HK}\rangle$, $\langle\dot{Q}^{+}\rangle$, and $\langle\dot{Q}^{-}\rangle$. For brevity, the explicit time argument $(t)$ of these quantities is omitted. Scatter points are experimental averages over 3000 repetitions, and solid lines are numerical results from the master equation. The gray region shows $\langle\dot{Q}_{\rm HK}^{\rm SS}\rangle - \langle\dot{Q}_{\rm T}\rangle$. $S_{\rm AC}=100~{\rm mV}$.
• Figure 4: (a) Change in efficiencies $\eta_{\rm EX}$ and $\eta_{\rm W}$. $S_{\rm AC}=100$ mV. Open circles are experimental results averaged over 30000 repetitions. The solid lines are theoretical results obtained from the master equation. (b) $S_{\rm AC}$ dependencies of $\eta_{\rm W}$. Each curve is averaged over 250 repetitions.
• Figure 5: FIG. B1. (a) $\rho_N^F(t)$ at different time in the NESS. Open circles are given by Eq. (\ref{['eq:ME']}) into which Eqs. (\ref{['eq:Gamma_Fourier']}) and (\ref{['eq:Prob_Fourier']}) are substituted. The solid lines are Gaussian curves fitted to the open circles. Change in $\langle N(t) \rangle$ when (b) $\omega_{\rm AC}/2\pi=10^{-6}\varGamma_0$ and (c) $\omega_{\rm AC}/2\pi=10^6 \varGamma_0$. (d) $\omega_{\rm AC}$ dependence of $\Delta \langle N \rangle$, which is a peak-to-peak value of $\langle N(t) \rangle$ over time.