Optimal energy quanta to current conversion

We present a microscopic discussion of a nano-sized structure which uses the quantization of energy levels and the physics of single charge Coulomb interaction to achieve an optimal conversion of heat flow to directed current. In our structure the quantization of energy levels and the Coulomb blockade lead to the transfer of quantized packets of energy from a hot source into an electric conductor to which it is capacitively coupled. The fluctuation-generated transfer of a single energy quantum translates into the directed motion of a single electron. Thus in our structure the ratio of the charge current to the heat current is determined by the ratio of the charge quantum to the energy quantum. An important novel aspect of our approach is that the direction of energy flow and the direction of electron motion are decoupled.

Figure 1

Energy to current converter. The conductor, a quantum dot open to transport between two fermionic reservoirs at voltages $V_1$ and $V_2$ and temperatures $T_1$ and $T_2$, is coupled capacitively to a second dot which acts as a fluctuating gate coupled to a reservoir at voltage $V_g$ and temperature $T_g$. Here we discuss the case $T_1=T_2=T_s$.

Figure 2

Schematic representation of the four occupation states relevant for our system. The electron tunneling processes introducing or extracting an electron in the quantum dot system are described by the rates ${\Gamma }_{\ln }^{\pm }$ through each barrier $l$, which depend on the charge occupation of the other quantum dot, $n=\{0,1\}$. Every time a clockwise (anticlockwise) cycle ${\mathcal{C}}_{+}$ (${\mathcal{C}}_{-}$) is completed, a quantum of energy $E_C$ is transferred from the gate into the conductor (and viceversa).

Figure 3

(a) Energy diagram of the energy converter (when $V_1=V_2$ and $T_g>T_s$) showing the tunneling sequences that contribute to charge transport in the unbiased conductor. (b) If the temperature gradient is reversed ($T_g<T_s$), the fluctuation-generated current will flow in the opposite direction. The colored areas represent the Fermi-Dirac distribution function of each reservoir. To each process that is represented exists a process in the opposite direction that is, however, exponentially suppressed since it decreases the entropy. Tunneling events in each system differ by a charging energy $E_C$ when the other quantum dot either is empty or occupied. Thus, sequences involving the empty state and the simultaneous occupation of the two dots lead to the transfer of the energy $E_C$ and to directed electron motion in the conductor.

Figure 4

Charge and heat currents as functions of temperature and voltage differences. (a) The direction of the currents can be tuned by changing the sign of the temperature gradient (left panel) or the applied bias voltage (right panel). At the points marked with $*$ and $**$, the conditions $E_{s0}={\mathit{qV}}_2$ and $E_{s0}={\mathit{qV}}_1$ are satisfied. Between them, heat flows in opposite directions in the two terminals of the conductor, so one of its reservoirs is cooled down. Beyond them, Joule heat becomes dominant in the conductor. (b) Driving the conductor far enough from equilibrium, heat will flow from the coldest to the hottest system. In the delimited regions, heat flows into the gate ($J_g>0$) being $T_g>T_s$, and viceversa. Parameters: ${\Gamma }_{\ln }=\Gamma$, except ${\Gamma }_{11}=0.1\Gamma$, ${\mathit{kT}}_1={\mathit{kT}}_2=5\hslash \Gamma$, $q^2/C_i=20\hslash \Gamma$, $q^2/C=50\hslash \Gamma$, ${\varepsilon }_{\mathrm{u}}={\varepsilon }_{\mathrm{d}}=0$, $V_g=V_1$.

Figure 5

Efficiency. (a) Power (solid lines) of the heat generated current and efficiency (dashed lines) of the heat to charge current conversion as a function of voltage for ${\mathit{kT}}_s=5\hslash \Gamma$ for different gate temperatures. The efficiency grows linearly up to the stopping potential $V_0$, where Carnot efficiency is achieved. In the right panel, the efficiency at maximum power is plotted and compared with the Carnot and the Curzon-Ahlborn efficiencies. Same parameters as in Fig. 4, except ${\Gamma }_{11}={\Gamma }_{20}=0$. (b) Triple quantum dot system proposed to work as an optimal heat to charge current converter. The side quantum dots act as energy filters.

Figure 6

(a) Charge current when applying finite voltages to the conductor and to the gate. Dashed lines are plotted denoting the coincidence of the energies $E_{\alpha n}$ with the Fermi energies of the leads. The areas enclosed by them have a well defined charge occupation. (b) Differential conductance, $G=\partial I/\partial (V_1-V_2)$. In the region $V_1>V_2$ and $E_{g0}<{\mathit{qV}}_g<E_{g1}$, the occupation of the gate quantum dot induces negative differential conductance. The width of such a plateau is determined by the quantum of transferred energy, $E_C$. Here, $C_s=C_1+C_2$. Parameters: ${\Gamma }_{\ln }=\Gamma$, except ${\Gamma }_{11}=0.1\Gamma$, ${\mathit{kT}}_s={\mathit{kT}}_g=5\hslash \Gamma$, $q^2/C_i=20\hslash \Gamma$, $q^2/C=50\hslash \Gamma$, ${\varepsilon }_s={\varepsilon }_g=0$, $V_g=V_1$.