Charging capacitors using diodes at different temperatures. I Theor

Abstract

Nonlinear elements in a rectifying circuit can be used to harvest energy from thermal fluctuations either steadily or transitorily. We study an energy harvesting system comprising a small variable capacitor (e.g., free standing graphene) wired to two diodes and two storage capacitors that may be kept at different temperatures (or at a single one) and use two current loops. The system reaches very rapidly a quasi stationary state with constant overall charge while the difference of the charges at the storage capacitors evolves much more slowly to its stationary value. In this paper, we extract an exponentially small factor out of the solution of the Fokker-Planck equation and use a Chapman-Enskog procedure to describe the long evolution of the marginal probability density for the charge difference, from the quasi stationary state to the final stationary state (thermal equilibrium for equal temperatures). The second paper of this series shows that the results of the perturbation procedure compare well with direct numerical simulations. For a specific form of the diodes' nonlinear mobilities, we can approximate the quasi stationary state by Gaussian functions and further study the evolution of the marginal probability density. The latter adopts the shape of a slowly expanding pulse (comprising left and right moving wave fronts whose fore edges become sharper as time elapses) in the space of charge differences that leaves the final stationary state behind it.

Figures (7)

• Figure 1: Circuit diagram showing the STM tip and sample equivalent to a small capacitor $C_0$, and the opposing diodes D1 and D2, with respective conductances $\mu_1$, $\mu_2\propto R^{-1}$, and storage capacitors $C_1$ and $C_2$. The current-voltage curve of each diode is similar to that of an ideal diode in series with a resistor $R$, and therefore resistances are included in the diodes.
• Figure 2: Function $\hat{E}(\eta,\xi)$ for $\xi=-2,-1,0,1,2$, $\epsilon=0.02$, $T_1= 293$K and $T_2=77$K as (a) numerically evaluated for $w=0.1$, (b) given by Eq. \ref{['eq19a']} in the limit as $w\to 0+$. (c) Comparison between exact and approximate prefactor for $\xi=-2,0,2$ and $w=0.01$. Note that they coincide for $\xi=-2$ but still differ for $\xi=2$.
• Figure 3: Approximate stationary marginal probability density for $\epsilon=0.02$, and different temperature differences showing the departure of a symmetric configuration with increasing $(1-\theta_2)$, which is largest for $T_1= 293$K and $T_2=77$K.
• Figure 4: Rapid slowing down of the advancing front $\Xi(\tau)$ given by Eqs. \ref{['eq23a']} with $w=0.01, 0.1$, by the approximation \ref{['eq23b']} ($w\to 0$) and by Eq. \ref{['eq25']} ($\Xi\gg 1$) for $\tau>20$. The initial condition is $\Xi (0)= 0.1\geq w$. Inset: evolution for $0<\tau<20$.
• Figure 5: (a) Scaled marginal probability density erfc$[(\xi-\Xi(\tau))/\sqrt{2\epsilon\sigma(\tau)}]$ at times 20 and 500 for $\Xi(0)=0.1$. (b) Marginal probability density at times $t=0.1, 1, 20$. The curves at times $t=1,20$ are indistinguishable.