Reversible Quantum Brownian Heat Engines for Electrons

Brownian heat engines use local temperature gradients in asymmetric potentials to move particles against an external force. The energy efficiency of such machines is generally limited by irreversible heat flow carried by particles that make contact with different heat baths. Here we show that, by using a suitably chosen energy filter, electrons can be transferred reversibly between reservoirs that have different temperatures and electrochemical potentials. We apply this result to propose heat engines based on mesoscopic semiconductor ratchets, which can quasistatically operate arbitrarily close to Carnot efficiency.

Figure 1

Electron transfer between reservoirs: (a) in the presence of a difference in electrochemical potentials, (b) in the presence of a temperature difference, and (c) in the presence of both. (d) The Fermi-Dirac distributions in the energy range around ${\mu }_0=0.5({\mu }_L+{\mu }_R)$ for $T_R=2\mathrm{K}$, $T_L=0.5\mathrm{K}$, and $V=0.1\mathrm{m}\mathrm{V}$.

Figure 2

(a),(b) The energy band structure of this nonlinear rectifier indicates that of an asymmetric quantum dot forming a resonant tunneling structure, with an energy level position that depends on the voltage. Any higher resonant levels are assumed to be out of the reach of thermally excited electrons. $T_R>T_L$ is assumed. (c) A Brownian heat engine consisting of a periodic, static ratchet potential. Particle flow is against a potential gradient of $\Delta E$ per period. With the position of ideal energy filters (indicated as bold horizontal lines) tuned according to Eq. (2), this ratchet works reversibly.

Figure 3

(a) Efficiency (normalized to the Carnot value) of the model device in Figs. 2a, 2b for $T_R=1\mathrm{K}\approx e{V}_0=0.1\mathrm{m}\mathrm{e}\mathrm{V}$, $t_0=1$, and $\delta ={10}^{-4}\mathrm{m}\mathrm{e}\mathrm{V}$, as a function of the position of the resonant level, $\varphi$, and of $T_L$ (see main text for further details). (b) Normalized efficiency for $T_R=1\mathrm{K}$, $T_L=0.5\mathrm{K}$, and $e{V}_0=0.1\mathrm{m}\mathrm{e}\mathrm{V}$. Bold lines (left-hand half) are for a refrigerator, and thinner lines (right-hand half) are efficiencies of a heat engine. The full line is for $\delta \to 0$ [Eq. (5)]. The dashed lines are calculated for, from bottom to top, $\delta ={10}^{-2}$, ${10}^{-3}$, and ${10}^{-4}\mathrm{m}\mathrm{e}\mathrm{V}$. Data are shown only where the efficiencies are defined. (c),(d) The power of the device in Figs. 2a, 2b for the same parameters used in (a) and (b), respectively. To put the numerical values into context, a cooling power of ${10}^7\mathrm{m}\mathrm{e}\mathrm{V}/\mathrm{s}$ corresponds to the heat leaked via phonons to a 2DEG with temperature 0.2 K and area $20\mu {\mathrm{m}}^2$ in a crystal externally cooled to 0.3 K by a ${\mathrm{H}\mathrm{e}}^3$ system (estimate based on Eq. 3.1 in [12]).