Thermoelectric energy harvesting with quantum dots

We consider the setup shown schematically in figure 2. It consists of two quantum dots $\alpha =1,2$ in the Coulomb-blockade regime. Each dot hosts a single level with energy ${{\varepsilon }_{\alpha }}$. Assuming strong onsite Coulomb interaction, the dots can host either 0 or 1 excess electron. In the following, we neglect the electron spin as it will only lead to a renormalization of the tunnel couplings introduced below. The two quantum dots are capacitively coupled to each other such that they can exchange energy but no particles. The strength of the coupling is characterized by the Coulomb energy U that is needed to occupy both quantum dots at the same time and will parametrize the energy exchange between the two dots. The conductor dot, $\alpha =1$, is tunnel coupled to two electronic reservoirs $r={\rm L},{\rm R}$. The reservoirs are in local thermal equilibrium and characterized by chemical potentials ${{\mu }_{\,r}}$ and temperature T_c. The gate dot, $\alpha =2$, is coupled to a single reservoir ${\rm G}$ with chemical potential ${{\mu }_{{\rm G}}}$ and temperature T_h. In order to obtain a finite thermoelectric response through the conductor dot to a temperature difference ${{T}_{h}}-{{T}_{c}}$, we need to have energy-dependent tunnel coupling strengths. Here, we model this energy dependence by choosing the tunnel couplings ${{\Gamma }_{sn}}$ between the dot and its respective reservoir $s={\rm L},{\rm R},{\rm G}$ to depend on the number of electrons, n, on the other quantum dot. There are different ways how to realize this energy dependence in an actual experiment. First of all, the transmission through a tunnel barrier generically depends on energy of the tunneling particle. This effect is most pronounced for energies close to the barrier height. Second, quantum dots with an excited state that couples asymmetrically to the two leads and decays quickly into the ground state can mimic energy-dependent tunneling rates as well. Finally, additional quantum dots can be added between the conductor dot and its two reservoirs. These additional dots serve as energy filters that only transmit electrons at a given energy and thereby allow to achieve an effectively energy-dependent tunnel coupling.

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In order to describe the system, we trace out the noninteracting electronic reservoirs. The remaining strongly interacting quantum dot degrees of freedom are then characterized by a reduced density matrix for the double quantum dot. The time evolution of the system is determined by a master equation ${\bf \dot{P}}={\bf WP}$ for the reduced density matrix elements ${\bf P}=({{P}_{00}},{{P}_{10}},{{P}_{01}},{{P}_{11}})$ describing the probability to find the double dot empty P₀₀, occupied with one electron on the conductor P₁₀ or gate dot P₀₁ and doubly occupied P₁₁. In the sequential tunneling regime ${{\Gamma }_{ri}}\ll {{k}_{{\rm B}}}{{T}_{h,c}}$ the transition rates ${\bf W}$ follow from Fermiʼs golden rule and are given by

$$\begin{eqnarray}\begin{array}{l} {\bf W}=\left( \begin{array}{ccccccccccccccc} -{{\sum }_{r}}W_{r0}^{+}-W_{{\rm G}0}^{+} & {{\sum }_{r}}W_{r0}^{-} & W_{{\rm G}0}^{-} & 0 \\ {{\sum }_{r}}W_{r0}^{+} & -{{\sum }_{r}}W_{r0}^{-}-W_{{\rm G}1}^{+} & 0 & W_{{\rm G}1}^{-} \\ W_{{\rm G}0}^{+} & 0 & -{{\sum }_{r}}W_{r1}^{+}-W_{{\rm G}0}^{-} & {{\sum }_{r}}W_{r1}^{-} \\ 0 & W_{{\rm G}1}^{+} & {{\sum }_{r}}W_{r1}^{+} & -{{\sum }_{r}}W_{r1}^{-}-W_{{\rm G}1}^{-} \\ \end{array} \right). \\ \quad \quad \quad \\ \end{array}\end{eqnarray} \tag{ 1 }$$

The transition rates $W_{sn}^{\pm }$ describe tunneling of electrons onto or off the dot through barrier s when the other dot has n electrons. Specifically, they are written as

$$\begin{eqnarray}&&W_{r0}^{\pm }={{\Gamma }_{r0}}f_{r}^{\pm }({{\varepsilon }_{1}}),\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&W_{{\rm G}0}^{\pm }={{\Gamma }_{{\rm G}0}}f_{g}^{\pm }({{\varepsilon }_{2}}),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&W_{r1}^{\pm }={{\Gamma }_{r1}}f_{r}^{\pm }({{\varepsilon }_{1}}+U),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&W_{{\rm G}1}^{\pm }={{\Gamma }_{{\rm G}1}}f_{g}^{\pm }({{\varepsilon }_{2}}+U),\end{eqnarray} \tag{ 5 }$$

where $f_{r}^{+}(x)=1-f_{r}^{-}(x)$ = ${{\{{\rm exp} [(x-e{{V}_{r}})/({{k}_{{\rm B}}}{{T}_{r}})]+1\}}^{-1}}$ denotes the Fermi function with temperature T_r and bias voltage V_r = μ_r/e.

In order to calculate the charge and heat currents as well as their fluctuations and correlations, we use a full-counting statistics approach. To this end, we multiply the transition rates with a factor ${{{\rm e}}^{{\rm i}{{q}_{s}}{{\chi }_{s}}}}$ where q_s denotes the charge transferred through barrier s in the associated tunneling event. The cumulant generating function $\mathcal{S}({{\chi }_{s}})$ is then given by the eigenvalue of the resulting matrix ${{{\bf W}}_{\chi }}$ that goes to zero as $\chi \to 0$. The charge currents are simply given by the derivative of the cumulant generating function with respect to the counting fields, ${{I}_{s}}={{\left. \partial \mathcal{S}/\partial {{\chi }_{s}} \right|}_{{{\chi }_{s}}=0}}$. Similarly, for the energy currents, we introduce counting factors ${{{\rm e}}^{{\rm i}{{E}_{s}}{{\xi }_{s}}}}$, where E_s denotes the energy transferred through barrier s in the corresponding tunneling event. Higher-order derivatives give the correlations that will be discussed in section 2.1.4. Finally, heat currents follow from charge and energy currents as $J_{s}^{h}=J_{s}^{E}-{{V}_{s}}{{I}_{s}}$. We remark that while charge and energy currents are conserved, ${{\sum }_{s}}{{I}_{s}}={{\sum }_{s}}J_{s}^{E}=0$, heat currents in general are not conserved due to Joule heating.

In the following, we want to drive a charge current through the conductor dot simply by applying a temperature difference between the reservoirs of conductor and gate dot, i.e. without applying a bias voltage across the conductor dot. In order to achieve this goal, two requirements have to be met. First of all, the left–right symmetry of the system has to be broken. This can be achieved by choosing the tunnel couplings of the conductor dot to the left and right reservoirs different. In addition, we also have to break the particle–hole symmetry underlying the system. This is achieved by having energy-dependent tunneling rates for the conductor dot.

In the absence of a bias voltage, the charge current I through the conductor dot can be simply related to the heat current J flowing out of the gate dot [44]

$$\begin{eqnarray}&&I=\frac{e}{U}\frac{{{\Gamma }_{{\rm L}0}}{{\Gamma }_{{\rm R}1}}-{{\Gamma }_{{\rm L}1}}{{\Gamma }_{{\rm R}0}}}{({{\Gamma }_{{\rm L}0}}+{{\Gamma }_{{\rm R}0}})({{\Gamma }_{{\rm L}1}}+{{\Gamma }_{{\rm R}1}})}J.\end{eqnarray} \tag{ 6 }$$

The ratio of heat and charge currents is determined by the ratio of electron charge and charging energy. Furthermore, it depends on the ratio of the different tunnel couplings that enter the problem. The direction of the charge current can be controlled via the asymmetry of tunnel couplings as well as by the direction of the heat current, i.e. by the sign of the temperature bias ${{T}_{h}}-{{T}_{c}}$. The largest current, I = eJ/U, can be achieved in an optimal configuration when the conductor dot couples only to, say, the left lead when the gate dot is empty and only to the right lead, when the gate dot is occupied.

The heat-driven current as a function of the level positions of the two dots is shown in figure 3. As a function of the gate dot level, ${{\varepsilon }_{2}}$, the current exhibits a peak at ${{\varepsilon }_{2}}=-U/2$ with a width given by the temperature of the hot bath, T_h. As a function of the conductor dot level, ${{\varepsilon }_{1}}$, the current has a plateau between ${{\varepsilon }_{1}}=0$ and ${{\varepsilon }_{1}}=-U$. The borders of the plateau are smeared by the temperature of the cold bath, T_h.

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The mechanism giving rise to the heat-driven charge current is the following. Initially, the double dot is completely empty. In a first step, an electron tunnels onto the conductor dot from, say, the left reservoir. Next, an electron tunnels onto the gate dot. This requires the charging energy U which is extracted from the hot reservoir. Afterwards, the electron from the conductor dot tunnels into the right lead, thereby releasing the charging energy into the cold reservoir. In a final step, the gate dot is emptied such that the system returns to its initial state. In one such transport cycle, one quantum of energy, i.e. the charging energy, is transferred from the hot to the cold, giving rise to correlations of the charge and heat currents. We remark that, in the optimal configuration, one electron is transmitted from the left to the right reservoir for every absorbed quantum of heat. This is known as the tight-coupling limit.

So far, we demonstrated that our quantum-dot heat engine can convert a heat current into a directed charge current. In a next step, we want to generate a finite output power by adding a load to the system. Hence, we apply a bias voltage V across the conductor dot against which the heat-driven current can perform work. The output power is then simply given as P = IV. It is shown as a function of the applied bias in figure 4(a). For zero bias, the output power obviously vanishes. Similarly, it vanishes at the so called stopping voltage ${{V}_{{\rm stop}}}$ where the heat- and bias-driven currents compensate each other such that I = 0. In between, the system reaches its maximal output power. For situations close to equilibrium, the maximum power is reached at half the stopping voltage. As the system is taken far away from equilibrium by lowering T_c, the point of maximum power is shifted toward larger bias voltages.

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Another important quantity to characterize the thermoelectric performance of a heat engine is its efficiency of heat to work conversion. It is defined as the ratio between the output power and the input heat which in our case is given by the heat current flowing out of the hot reservoir, i.e., we have $\eta =P/J$. In the tight-coupling limit, the efficiency grows linearly with the applied bias voltage and we simply have $\eta =eV/U$, see figure 4(a). Hence, at the stopping voltage ${{V}_{{\rm stop}}}=U{{\eta }_{{\rm C}}}/e$ the device reaches Carnot efficiency ${{\eta }_{{\rm C}}}=1-{{T}_{c}}/{{T}_{h}}$ indicating that it operates as an optimal heat to current converter. However, at this point, the heat engine operates reversibly and, therefore, does not produce any output power. A more relevant quantity to consider is the efficiency at maximum power. It is shown as a function of the temperature bias in figure 4(b). For a small temperature bias, it increases linearly as ${{\eta }_{{\rm C}}}/2$ in agreement with a general thermodynamic bound for systems with time-reversal symmetry [85]. In the nonlinear regime, it grows faster than ${{\eta }_{{\rm C}}}/2$ and even reaches ${{\eta }_{{\rm C}}}$ for ${{T}_{c}}\to 0$. We remark, however, that at this point our master equation approach is no longer valid since higher order tunneling contributions that give rise to a broadening of the dot level become important.

Up to now, we only discussed the average transport of heat and charge through the quantum-dot heat engine. However, at the nanoscale, fluctuations of average quantities turn out to be important [92]. Indeed, we discussed above how the generation of current depends on the correlation of charge fluctuations in the two dots.

A way to characterize these fluctuations is to look at the full counting statistics $P(N,t)$ which addresses the probability that N electrons have passed through the system in a given time t [65, 93–96]. In a quantum dot system, this quantity can actually be measured by coupling a charge sensor such as a quantum point contact or a single-electron transistor to the quantum dot [97–99]. As the number of electrons on the dot changes, the electrical potential felt by the charge sensor changes, thus leading to a change in the current through it. This allows to measure the charge on the quantum dot in real time.

Interestingly, in the quantum-dot heat engine we can not only access the full counting statistics of charge but also of heat [47]. Measuring the charge of both, the conductor and the gate dot, e.g., via a quantum point contact that couples asymmetrically to the two dots, allows us to reconstruct the counting statistics of heat from the counting statistics of charge due to the intimate relationship between heat and charge transfer in the system. This way, nonequilibrium fluctuation relations can be measured that relate the charge and heat currents [100–102] in terms of electron counting. This is important because in the presence of temperature gradients, fluctuation relations for charge currents [103–105] become configuration-dependent [100, 106] unless one also considers energy currents. Notably, the charge fluctuation theorem becomes universal (only depending on the thermodynamic forces) in the tight-coupling limit [47, 107].

Additional insight of the thermoelectric performance of the quantum dot heat engine can be accessed by investigating the charge and heat current–current correlations [49]. The interest on electronic heat noise has appeared only very recently [108–115]. Already in the linear regime the charge–heat cross-correlations are related to Seebeck-like and Peltier-like coefficients by means of an extended three-terminal fluctuation–dissipation theorem. Far from equilibrium, dimensionless quantities for the auto- and cross-correlations (charge and heat Fano factors, and cross-correlation coefficient) can be defined. Divergences of the charge Fano factor can be used to measure the nonlinear thermovoltage. Most interestingly, the charge–heat cross-correlations are maximal in the tight-coupling limit [116], leading to Carnot-efficient configurations [49].

We consider two open quantum dots i = 1, 2 coupled via a mutual capacitance C. Each cavity is connected to an electronic reservoir $r={\rm L},{\rm R}$ via a quantum point contact, see figure 5. The reservoirs are in local thermal equilibrium and described by a Fermi function ${{f}_{r}}(E)={{\{{\rm exp} [(E-{{\mu }_{r}})/({{k}_{{\rm B}}}{{T}_{r}})]+1\}}^{-1}}$ with chemical potential ${{\mu }_{r}}$ and temperature T_r. Interaction effects are captured by capacitive couplings C_ir between cavity i and reservoir r that screen potential fluctuations.

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We focus on the semiclassical regime where the number of open transport channels N_ir of the quantum point contacts is large. We assume that dephasing destroys phase information but preserves energy. In this situation, the cavities are characterized by distribution functions f_i(E) that depend only on energy. For later convenience, we write these distributions as

$$\begin{eqnarray}&&{{f}_{i}}(E)=\frac{\mathop{\sum }\limits_{r}{{\mathcal{T}}_{ir}}{{f}_{r}}(E)}{\mathop{\sum }\limits_{r}{{\mathcal{T}}_{ir}}}+\delta {{f}_{i}}.\end{eqnarray} \tag{ 7 }$$

The first term corresponds to the average of the reservoir distribution functions weighted with the transmission ${{\tau }_{ir}}$ of the corresponding QPC. The second term describes fluctuations of the distribution function that have to be determined in the following. In addition, the cavities are characterized by their potential U_i and associated fluctuations $\delta {{U}_{i}}$. In order to obtain a finite thermoelectric response, we need to have energy-dependent transmissions ${{\mathcal{T}}_{ir}}$. Here, we model this energy dependence as ${{\mathcal{T}}_{ir}}=\mathcal{T}_{ir}^{0}-e\mathcal{T}{{^{\prime} }_{ir}}\delta {{U}_{i}}$. While the first term, $\mathcal{T}_{ir}^{0}$ describes the energy-independent transmissions that depend linearly on the number N_ir of open transport channels, the second term captures changes in the transmission due to fluctuations of the cavity potential. We remark that the energy-dependent term $\mathcal{T}{{^{\prime} }_{ir}}$ is independent of the number of open transport channels.

The distribution functions of the cavities obey a kinetic equation of the form (see e.g. [117])

$$\begin{eqnarray}&&e{{\nu }_{i{\rm F}}}\frac{{\rm d}{{f}_{i}}}{{\rm d}t}=e{{\nu }_{i{\rm F}}}\frac{\partial {{f}_{i}}}{\partial {{U}_{i}}}{{\dot{U}}_{i}}+\frac{e}{h}\mathop{\sum }\limits_{r}{{\mathcal{T}}_{ir}}({{f}_{ir}}-{{f}_{i}})+\delta {{i}_{\Sigma }},\end{eqnarray} \tag{ 8 }$$

where ${{\nu }_{i{\rm F}}}$ denotes the density of states at the Fermi energy in cavity i. The kinetic equation describes how the charge in a given energy interval changes due to changes of the cavity potential U_i, in- and outgoing electron currents through the quantum point contacts as well as due to fluctuations of these currents $\delta {{i}_{\Sigma }}$. Here, the index Σ indicates that we have to sum over all contacts r of a given cavity i.

In a next step, we obtain a relation between the fluctuations of the cavity distributions $\delta {{f}_{i}}$ and potentials $\delta {{U}_{i}}$ by expressing the charge inside each cavity in terms of an integral over the distribution functions as well as in terms of the various capacitances and potentials. Using this relation between $\delta {{f}_{i}}$ and $\delta {{U}_{i}}$, we can transform the kinetic equation (8) into a Langevin equation for the potential fluctuations $\delta {{U}_{i}}$. Due to the nonlinearity introduced into the problem by the energy-dependent transmission, the Langevin equation has a multiplicative noise term that leads to the Itô–Stratonovich problem in the interpretation of the stochastic integral when converting the Langevin equation into a Fokker–Planck equation [118, 119]. For the problem at hands, it turns out that only the kinetic prescription by Klimontovich [120] provides a meaningful solution that exhibits vanishing heat and charge currents in global equilibrium. From the Fokker–Planck equation, we can obtain the expectation values of the potential fluctuations, $\langle \delta {{U}_{i}}\rangle$ and $\langle \delta {{U}_{i}}\delta {{U}_{j}}\rangle$, which subsequently allow us to evaluate the charge current between cavity 1 and its contact r via the standard scattering matrix expression

$$\begin{eqnarray}&&{{I}_{1r}}=\frac{e}{h}\int {\rm d}E{{\mathcal{T}}_{1r}}({{f}_{1r}}-{{f}_{1}})+\delta {{I}_{r}}.\end{eqnarray} \tag{ 9 }$$

We first focus on the situation where a finite temperature bias ${{T}_{1}}-{{T}_{2}}$ is applied across the system while there is no bias voltage applied across cavity 1, i.e. ${{V}_{1{\rm L}}}={{V}_{1{\rm R}}}$. For the charge current through cavity 1 we find to lowest order in the energy-dependent transmission the compact expression

$$\begin{eqnarray}&&\langle {{I}_{1{\rm L}}}\rangle =\frac{\Lambda }{{{\tau }_{{\rm RC}}}}{{k}_{{\rm B}}}({{T}_{1}}-{{T}_{2}}).\end{eqnarray} \tag{ 10 }$$

The charge current depends on the asymmetry parameter

$$\begin{eqnarray}&&\Lambda =\frac{G{{^{\prime} }_{1{\rm L}}}{{G}_{1{\rm R}}}-G{{^{\prime} }_{1{\rm R}}}{{G}_{1{\rm L}}}}{G_{1\Sigma }^{2}},\end{eqnarray} \tag{ 11 }$$

with ${{G}_{ir}}=({{e}^{2}}/h)\mathcal{T}_{ir}^{0}$, ${{G}_{i\Sigma }}={{G}_{i{\rm L}}}+{{G}_{i{\rm R}}}$ and $G{{^{\prime} }_{ir}}=({{e}^{3}}/h)\mathcal{T}{{^{\prime} }_{ir}}.$ It characterizes both the breaking of left–right symmetry as well as the breaking of particle–hole symmetry due to energy-dependent transmissions. From equation (11), we conclude that in order to have a finite charge current driven by the temperature bias, we need to break both symmetries simultaneously. We remark that a similar antisymmetric combination of energy-dependent transmissions was found in the expression for the charge current through Coulomb-blockaded dots (6), as there we also have to break left–right and particle–hole symmetry at the same time. In addition, the current also depends on the RC time of the cavity, ${{\tau }_{{\rm RC}}}={{C}_{{\rm eff}}}/{{G}_{{\rm eff}}}$. Here, ${{G}_{{\rm eff}}}={{G}_{1\Sigma }}{{G}_{2\Sigma }}/({{G}_{1\Sigma }}+{{G}_{2\Sigma }})$ denotes the effective conductance of the double cavity while ${{C}_{{\rm eff}}}$ is an effective capacitance that characterizes the coupling between the two cavities. For weakly coupled cavities, it grows as ${{C}^{-2}}$.

We now turn to the discussion of how the current depends on the number of open transport channels. As the energy-dependent part of the transmission does not scale with N_ir, we find that the current is independent of the number of open channels. For realistic parameters [121–123] with ${{C}_{{\rm eff}}}=10\;{\rm fF}$, $G^{\prime} =({{e}^{2}}/h)\;{{({\rm mV})}^{-1}}$ and a temperature bias of 1 K, we obtain $I\sim \;0.1\;{\rm nA}$. Hence, the current for the cavity heat engine is about two orders of magnitude larger than for the heat engine operating in the Coulomb-blockade regime. The reason for this enhanced current is simply the difference between transport through a tunnel barrier and a fully open quantum channel.

In order to generate a finite output power, we have to apply a bias voltage ${{V}_{1{\rm L}}}-{{V}_{1{\rm R}}}$ against which the heat-driven charge current through cavity 1 can perform work. The output power is then simply given by $P=\langle {{I}_{1{\rm L}}}\rangle ({{V}_{1{\rm L}}}-{{V}_{1{\rm R}}})$. It vanishes when no bias voltage is applied. Similarly, it also vanishes at the stopping voltage ${{V}_{{\rm stop}}}=\Lambda \;{{k}_{{\rm B}}}({{T}_{1}}-{{T}_{2}})/({{G}_{1}}{{\tau }_{{\rm RC}}})$ (here, ${{G}_{1}}={{G}_{1{\rm L}}}{{G}_{1{\rm R}}}/{{G}_{1\Sigma }}$ denotes the conductance of cavity 1) where heat- and bias-driven charge currents exactly compensate each other such that $\langle {{I}_{1{\rm L}}}\rangle =0$. The maximal power is generated at half the stopping voltage and is given by

$$\begin{eqnarray}&&{{P}_{{\rm max} }}=\frac{{{\Lambda }^{2}}}{4{{G}_{1}}\tau _{{\rm RC}}^{2}}{{[{{k}_{{\rm B}}}({{T}_{1}}-{{T}_{2}})]}^{2}}.\end{eqnarray} \tag{ 12 }$$

Surprisingly, the output power drops inversely as the number of open transport channels. The reason for this lies in the fact that the energy-dependent transmission is independent of the number of open transport channels. While the heat-driven current thus is independent of the number of channels as well, the bias driven current linearly scales with the number of channels. Hence, the more transport channels are open, the smaller the stopping voltage and, hence, the smaller the output power that can be achieved.

We now turn to the discussion of the efficiency η given by the ratio of output power to input heat. The input heat is given by the heat current flowing between the cavities. To leading order in the nonlinearity, it is given by

$$\begin{eqnarray}&&{{J}_{H}}=\frac{1}{{{\tau }_{{\rm RC}}}}{{k}_{{\rm B}}}({{T}_{2}}-{{T}_{1}}),\end{eqnarray} \tag{ 13 }$$

i.e. there is a finite heat current even in the absence of energy-dependent transmissions. As the heat current is independent of the applied bias voltage, we find that the maximal efficiency and the efficiency at maximum power coincide and are given by

$$\begin{eqnarray}&&{{\eta }_{{\rm max} }}=\frac{{{\Lambda }^{2}}}{4{{G}_{1}}{{\tau }_{{\rm RC}}}}{{k}_{{\rm B}}}({{T}_{2}}-{{T}_{1}}).\end{eqnarray} \tag{ 14 }$$

Analyzing the scaling behavior of the efficiency with the number of transport channels, we find that it drops inversely to the square of the number of open channels. This faster decrease as compared to the output power is due to the proportionality of the heat current to the number of transport channels. For realistic parameters, we find that for a few open channels an output power of a few fW can be generated. At the same time, the efficiency reaches at most a few percent of the Carnot efficiency for a device operating a liquid helium temperatures.

We consider a setup consisting of a central cavity connected to two electrodes via quantum dots cf figure 7. The electrodes are assumed to be in local thermal equilibrium characterized by a Fermi distribution with temperature ${{T}_{{\rm R}}}$ and chemical potential ${{\mu }_{{\rm L},{\rm R}}}$. Each quantum dot has a single resonant level relevant for transport. The levels are characterized by their width γ which we assume to be the same for both dots in the following and their level position ${{E}_{{\rm L},{\rm R}}}$. The difference of level positions $\Delta E={{E}_{{\rm R}}}-{{E}_{{\rm L}}}$ characterizes the energy an electron can gain in passing through the cavity. It is different from the level spacing within each dot and can be tuned by applying a gate voltage. The central cavity is in thermal contact with a hot bath. This coupling is treated as a third terminal that injects a heat current J but no charge in the conductor, as in figure 1. For our purposes here, the nature of the hot source needs not to be further specified. In contrast to the previous model, where the cavity was assumed out of equilibrium, here we make the simple assumption that fast relaxation processes via electron-electron and electron–phonon scattering give rise to a Fermi distribution of electrons inside the cavity with temperature ${{T}_{{\rm C}}}$ and chemical potential ${{\mu }_{{\rm C}}}$. The cavity temperature and chemical potential are determined by the conservation of charge and energy flowing into the cavity ${{I}_{{\rm L}}}+{{I}_{{\rm R}}}=0$ and ${{J}_{{\rm L}}}+{{J}_{{\rm R}}}+J=0$, where

$$\begin{eqnarray}&&{{I}_{j}}=\frac{2e}{h}\int {\rm d}E\;{{\mathcal{T}}_{j}}(E)[{{f}_{j}}(E)-{{f}_{C}}(E)]\end{eqnarray} \tag{ 15 }$$

denotes the charge current flowing between the cavity and reservoir $j={\rm L},{\rm R}$ while

$$\begin{eqnarray}&&{{J}_{j}}=\frac{2}{h}\int {\rm d}E\;{{\mathcal{T}}_{j}}(E)E[{{f}_{j}}-{{f}_{C}}]\end{eqnarray} \tag{ 16 }$$

is the energy current flowing between the cavity and reservoir j. In the above expressions for the currents

$$\begin{eqnarray}&&{{\mathcal{T}}_{j}}(E)=\frac{{{\gamma }^{2}}}{{{(E-{{E}_{j}})}^{2}}+{{\gamma }^{2}}},\end{eqnarray} \tag{ 17 }$$

denotes the transmission function of the quantum dot levels [127]. It is a Lorentzian with width γ centered around the level position ${{E}_{{\rm L},{\rm R}}}$. The charge conservation condition can be satisfied by placing the cavity potential symmetrically between the resonant levels of the quantum dots ${{\mu }_{{\rm C}}}=({{E}_{{\rm L}}}+{{E}_{{\rm R}}})/2$ and putting the chemical potentials of the reservoirs symmetrically with respect to ${{\mu }_{{\rm C}}}$, ${{\mu }_{\,r}}={{\mu }_{{\rm C}}}\pm \mu /2$.

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We first discuss the regime $\gamma \ll {{k}_{{\rm B}}}{{T}_{{\rm R}}},{{k}_{{\rm B}}}{{T}_{{\rm C}}}$ in which an analytical solution of the problem can be obtained that provides us with an intuitive understanding of the underlying physics. Afterwards, we will discuss the regime $\gamma \sim {{k}_{{\rm B}}}{{T}_{{\rm R}}},{{k}_{{\rm B}}}{{T}_{{\rm C}}}$ which we numerically find to yield the largest output power.

With narrow resonances, an electron that enters the cavity from the left lead with energy E_L has to gain the precise amount of energy $\Delta E$ in order to be able to leave the cavity into the right lead. In the steady state, rectified charge current $I={{I}_{{\rm L}}}$ and heat current J are therefore proportional to each other with the proportionality constant being given by the ratio between electron charge and difference of level positions,

$$\begin{eqnarray}&&I=\frac{e}{\Delta E}J,\end{eqnarray} \tag{ 18 }$$

again defining a tight-coupling limit. We now assume that no bias voltage is applied between the two electrodes. We inject a heat current J in such a way as to keep the cavity at a given temperature ${{T}_{{\rm C}}}$. As a result, we find that the charge current

$$\begin{eqnarray}&&I=eJ/\Delta E\approx \frac{e\gamma \Delta E}{4\;h}[{{({{k}_{{\rm B}}}{{T}_{{\rm R}}})}^{-1}}-{{({{k}_{{\rm B}}}{{T}_{{\rm C}}})}^{-1}}],\end{eqnarray} \tag{ 19 }$$

flows in response to the temperature difference between the hot cavity and the cold electrodes. The above expression is valid in the limit where ${{k}_{{\rm B}}}{{T}_{{\rm R}}},{{k}_{{\rm B}}}{{T}_{{\rm C}}}\gg \Delta E$. From (19) we conclude that the current and, hence, the output power grow both with the level width as well as with the difference of level positions until these quantities exceed the temperature.

The output power that can be generated against an externally applied bias voltage $V=\mu /e$ is given by

$$\begin{eqnarray}&&P=\frac{\gamma }{4\;h{{k}_{\sf{B}}}{{T}_{{\rm R}}}}\mu ({{\mu }_{{\rm stop}}}-\mu ),\end{eqnarray} \tag{ 20 }$$

for temperatures larger than eV and $\Delta E$. Here, ${{\mu }_{{\rm stop}}}=\Delta E(1-{{T}_{{\rm R}}}/{{T}_{{\rm C}}})$ denotes the stopping voltage at which heat- and bias-driven current compensate each other. At the stopping voltage, the device operates reversibly and reaches Carnot efficiency. At half the stopping voltage, the output power becomes maximal with

$$\begin{eqnarray}&&{{P}_{{\rm max} }}\approx \frac{\gamma \Delta {{E}^{2}}\eta _{{\rm C}}^{2}}{16h{{k}_{{\rm B}}}{{T}_{{\rm R}}}}.\end{eqnarray} \tag{ 21 }$$

At this point, the efficiency of heat to work conversion is given by ${{\eta }_{{\rm C}}}/2$ in agreement with general thermodynamic bounds for time-reversal symmetric systems [85, 86, 128].

We now turn to the situation of arbitrary level width and position. In this case, the integrals in (15) and (16) have to be evaluated numerically. The main results of our analysis are summarized in figure 8 where we defined the average temperature $T=({{T}_{{\rm C}}}+{{T}_{{\rm R}}})/2$ and the temperature bias $\Delta T={{T}_{{\rm C}}}-{{T}_{{\rm R}}}$. Figure 8(a) shows the output power as a function of the energy difference $\Delta E$. For small $\Delta E$ the power grows quadratically with $\Delta E$ in agreement with the analytical result (21). After reaching a maximum at $\Delta E\approx 6{{k}_{\sf{B}}}T$ it decays exponentially for large $\Delta E$ due to the small number of electrons available at very high energies. Figure 8(b) shows the maximal output power we can obtain when optimizing bias voltage, level positions and level width at the same time. The corresponding optimal parameters are depicted in figure 8(d). While the optimal bias voltage grows linearly with the applied temperature bias, the optimal level position and width are nearly independent of it and given by $\Delta E\approx 6{{k}_{\sf{B}}}T$ and $\gamma \approx {{k}_{\sf{B}}}T$. The maximal power is given by ${{P}_{{\rm max} }}\sim 0.4{{({{k}_{{\rm B}}}\Delta T)}^{2}}/h$ which amounts to about 0.1 pW for temperature bias of 1 K. At the same time, the efficiency at maximum power is nearly independent of the temperature bias and given by about $0.2{{\eta }_{{\rm C}}}$. Compared to the Coulomb-blockade regime, we therefore loose a factor of two in efficiency. This is, however, more than compensated by the two orders of magnitude gain in output power. The reason for this dramatic increase in power is the combination of a strong energy dependence of the transmission functions in combination with a large number of electrons that can pass through a fully open quantum channel. We estimate that an area of $1\;{\rm c}{{{\rm m}}^{2}}$ covered with nanoengines that each have an area of $100\;{\rm n}{{{\rm m}}^{2}}$ can produce an output power of 10 W for a temperature bias of 10 K.

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Such a high packing density can be achieved by a strongly parallelized setup shown in figure 9. Here, the electrodes are connected to a large central cavity via layers of self-assembled quantum dots embedded into an insulating matrix. Electrons tunnel through the quantum dots like through holes in a slice of Swiss cheese driven by the thermal bias. Importantly, the positions of the dots in the two layers do not have to match. Apparently, the Swiss-cheese sandwich heat engine outperforms a heat engine based on chaotic cavities discussed above. The reason for this lies in the fact that here we put many optimized channels in parallel while for open quantum dots there are many channels in parallel out of which only a single one is relevant for thermoelectric purposes. We remark that the heat engine based on self-assembled quantum dots offers the additional advantage that the irregular nature of the quantum dots layers can help to increase scattering of phonons at the interface and thereby reduce unwanted leakage heat currents between the hot cavity and the cold reservoirs.

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So far, we considered the ideal situation where all dots have the same, optimal properties. In a real sample, there will be fluctuations of level positions from dot to dot. In order to investigate in how far these imperfections deteriorate the performance of the heat engine, we assume that the level positions are distributed according to a Gaussian with width σ centered around the average level position ${{E}_{{\rm L},{\rm R}}}$. The output power as a function of the distribution width is shown in figure 10. As expected the power drops down as the width of the distribution increases. Importantly, even for a spread of 10% the power decreases only to 90% of its optimal value, i.e. the proposal can tolerate a certain degree of imperfections. Interestingly, for a given degree of fluctuations, choosing nonoptimal values of the level width or bias voltage can even increase the output power.

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We consider a setup similar to the one discussed in section 2.3. It consists of a central cavity in local thermal equilibrium with temperature T_h and chemical potential ${{\mu }_{{\rm C}}}$. It is coupled to cold electronic reservoirs with temperature T_c and chemical potentials ${{\mu }_{r}}$ via quantum wells, see figure 11. We assume that the quantum wells are noninteracting such that charging effects can be neglected. It is an interesting topic of future research to investigate the influence of interactions on the thermoelectric properties of quantum-well heat engines.

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The cavity potential and temperature are determined by the conservation of charge and energy, ${{I}_{{\rm L}}}+{{I}_{{\rm R}}}=0$ and $J_{{\rm L}}^{E}+J_{{\rm R}}^{E}+J=0$, where I_r and $J_{r}^{E}$ denote the charge and energy current flowing between the cavity and reservoir r, respectively. In addition, we have the heat current J that is injected into the cavity from a hot thermal bath and later on serves as a drive for the heat engine.

The charge and energy currents can be derived within a scattering matrix approach as [135]

$$\begin{eqnarray}\begin{array}{rcl} {{I}_{r}} & = & \frac{e{{\nu }_{2}}\mathcal{A}}{2\pi \hbar }\int {\rm d}{{E}_{\bot }}{\rm d}{{E}_{z}}{{\mathcal{T}}_{r}}({{E}_{z}}) \\ {} & {} & \times [{{f}_{r}}({{E}_{z}}+{{E}_{\bot }})-{{f}_{{\rm C}}}({{E}_{z}}+{{E}_{\bot }})], \\ \end{array}\end{eqnarray} \tag{ 22 }$$

and

$$\begin{eqnarray}\begin{array}{rcl} J_{r}^{{\rm E}} & = & \frac{{{\nu }_{2}}\mathcal{A}}{2\pi \hbar }\int {\rm d}{{E}_{\bot }}{\rm d}{{E}_{z}}({{E}_{z}}+{{E}_{\bot }}){{\mathcal{T}}_{r}}({{E}_{z}}) \\ {} & {} & \times [{{f}_{r}}({{E}_{z}}+{{E}_{\bot }})-{{f}_{{\rm C}}}({{E}_{z}}+{{E}_{\bot }})]. \\ \end{array}\end{eqnarray} \tag{ 23 }$$

In these expressions, ${{\nu }_{2}}={{m}_{*}}/(\pi {{\hbar }^{2}})$ denotes the density of states of the two-dimensional electron gas inside the quantum well with an effective electron mass ${{m}_{*}}$. $\mathcal{A}$ is the surface area of the quantum well. The energies associated with the electron motion in the quantum well plane and perpendicular to it are E_z and ${{E}_{\bot }}$, respectively. The transmission of the quantum wells is given by [127]

$$\begin{eqnarray}&&{{\mathcal{T}}_{r}}(E)=\frac{{{\Gamma }_{r1}}(E){{\Gamma }_{r2}}(E)}{{{(E-{{E}_{nr}})}^{2}}+{{[{{\Gamma }_{r1}}(E)+{{\Gamma }_{r2}}(E)]}^{2}}/4},\end{eqnarray} \tag{ 24 }$$

with the (energy-dependent) coupling strengths ${{\Gamma }_{r1}}$ and ${{\Gamma }_{r2}}$ between the quantum well and reservoir r or the cavity, respectively. The energies E_rn denote the subband thresholds at which a new transport channel through the quantum well opens up. For the following discussion, we consider the limit that the quantum wells are only weakly coupled to the reservoirs and the cavity, ${{\Gamma }_{r1}},{{\Gamma }_{r2}}\ll {{k}_{{\rm B}}}{{T}_{c}}{{k}_{{\rm B}}}{{T}_{h}}$. In addition, we restrict ourselves to the case where only the lowest subband is relevant for transport. Due to the large level spacing of narrow quantum wells, this is a reasonable approximation. A discussion of thermoelectric transport through a single quantum well that takes into account higher subbands as well can be found in [136]. Under the above assumptions, the transmission function of the quantum wells simplifies to a delta peak, ${{\mathcal{T}}_{r}}(E)=2\pi {{\Gamma }_{r1}}{{\Gamma }_{r2}}/({{\Gamma }_{r1}}+{{\Gamma }_{r2}})\delta ({{E}_{z}}-{{E}_{1r}})$. In this limit, the integrals in (22) and (23) can be solved analytically and we obtain for the charge and energy currents

$$\begin{eqnarray}\begin{array}{rcl} {{I}_{r}} & = & \frac{e{{\nu }_{2}}\mathcal{A}}{\hbar }\frac{{{\Gamma }_{r1}}{{\Gamma }_{r2}}}{{{\Gamma }_{r1}}+{{\Gamma }_{r2}}} \\ {} & {} & \times \left[ {{k}_{{\rm B}}}{{T}_{c}}{{K}_{1}}\left( \frac{{{\mu }_{r}}-{{E}_{r}}}{{{k}_{{\rm B}}}{{T}_{c}}} \right)-{{k}_{{\rm B}}}{{T}_{h}}{{K}_{1}}\left( \frac{{{\mu }_{{\rm C}}}-{{E}_{r}}}{{{k}_{{\rm B}}}{{T}_{h}}} \right) \right], \\ \end{array}\end{eqnarray} \tag{ 25 }$$

and

$$\begin{eqnarray}\begin{array}{rcl} J_{r}^{{\rm E}} & = & \frac{{{E}_{r}}}{e}{{I}_{r}}+\frac{{{\nu }_{2}}\mathcal{A}}{\hbar }\frac{{{\Gamma }_{r1}}{{\Gamma }_{r2}}}{{{\Gamma }_{r1}}+{{\Gamma }_{r2}}} \\ {} & {} & \times \left[ {{({{k}_{{\rm B}}}{{T}_{c}})}^{2}}{{K}_{2}}\left( \frac{{{\mu }_{r}}-{{E}_{r}}}{{{k}_{{\rm B}}}{{T}_{c}}} \right)-{{({{k}_{{\rm B}}}{{T}_{h}})}^{2}}{{K}_{2}}\left( \frac{{{\mu }_{{\rm C}}}-{{E}_{r}}}{{{k}_{{\rm B}}}{{T}_{h}}} \right) \right], \\ \end{array}\end{eqnarray} \tag{ 26 }$$

respectively. For simplicity, we denoted the energy of the single relevant subband in each quantum well as E_r. Furthermore, we introduced the integrals ${{K}_{1}}(x)=\int _{0}^{\infty }{\rm d}t{{(1+{{e}^{t-x}})}^{-1}}={\rm log} (1+{{e}^{x}})$ and ${{K}_{2}}(x)=\int _{0}^{\infty }{\rm d}tt{{(1+{{e}^{t-x}})}^{-1}}=-{\rm L}{{{\rm i}}_{2}}(-{{e}^{x}})$ with the dilogarithm ${\rm L}{{{\rm i}}_{2}}(z)=\sum _{k=1}^{\infty }\frac{{{z}^{k}}}{{{k}^{2}}}$. From (26) we find that the energy current consists of two different contributions. The first one is proportional to the charge current while the second term breaks this proportionality. It arises due to the transverse degrees of freedom and is absent in the case of quantum dots with sharp levels, see section 2.3.2.

In the following, we analyze the thermoelectric properties of the quantum-well heat engine in the linear response regime. For simplicity, we assume that both quantum wells are intrinsically symmetric, i.e. we have ${{\Gamma }_{r1}}={{\Gamma }_{r2}}$. We parametrize the tunnel couplings as ${{\Gamma }_{{\rm L}1}}={{\Gamma }_{{\rm L}2}}=(1+a)\Gamma$ and ${{\Gamma }_{{\rm R}1}}={{\Gamma }_{{\rm R}2}}=(1-a)\Gamma$, where Γ denotes the total coupling strength while $-1\leqslant a\leqslant 1$ describes the asymmetry between the couplings of the left and right quantum well. We, furthermore, introduce the average temperature $T=({{T}_{h}}+{{T}_{c}})/2$ and the temperature difference $\Delta T={{T}_{h}}-{{T}_{c}}$.

The charge current through the heat engine to linear order in the applied bias voltage $eV={{\mu }_{{\rm R}}}-{{\mu }_{{\rm L}}}$ and in the applied temperature difference $\Delta T$ is given by

$$\begin{eqnarray}\begin{array}{rcl} {{I}_{{\rm L}}} & = & -{{I}_{{\rm R}}}=\frac{e{{\nu }_{2}}\mathcal{A}\Gamma }{2\hbar }{{g}_{1}}\left( \frac{{{E}_{{\rm L}}}}{{{k}_{{\rm B}}}T},\frac{{{E}_{{\rm R}}}}{{{k}_{{\rm B}}}T} \right) \\ {} & {} & \times \left[ -eV-{{k}_{{\rm B}}}\Delta T{{g}_{2}}\left( \frac{{{E}_{{\rm L}}}}{{{k}_{{\rm B}}}T},\frac{{{E}_{{\rm R}}}}{{{k}_{{\rm B}}}T} \right) \right], \\ \end{array}\end{eqnarray} \tag{ 27 }$$

where we introduced the auxiliary functions

$$\begin{eqnarray}&&{{g}_{1}}(x,y)=\frac{1-{{a}^{2}}}{2+(1-a){{e}^{x}}+(1+a){{e}^{y}}}\end{eqnarray} \tag{ 28 }$$

and

$$\begin{eqnarray}\begin{array}{rcl} {{g}_{2}}(x,y) & = & x-y+(1+{{e}^{x}}){\rm log} (1+{{e}^{-x}}) \\ {} & {} & -(1+{{e}^{y}}){\rm log} (1+{{e}^{-y}}). \\ \end{array}\end{eqnarray} \tag{ 29 }$$

From (27) we directly infer that a finite temperature bias $\Delta T$ can drive a charge current in the absence of an applied bias voltage eV. The direction of the charge current can be tuned by adjusting the threshold energies ${{E}_{{\rm L}}}$ and ${{E}_{{\rm R}}}$.

By applying a bias voltage eV against the heat-driven current, the heat engine can perform work. The resulting output power $P={{I}_{{\rm L}}}V$ becomes maximal at half the stopping voltage where it takes the value

$$\begin{eqnarray}&&{{P}_{{\rm max} }}=\frac{{{\nu }_{2}}\mathcal{A}\Gamma }{2\hbar }{{\left( \frac{{{k}_{{\rm B}}}\Delta T}{2} \right)}^{2}}{{g}_{1}}\left( \frac{{{E}_{{\rm L}}}}{{{k}_{{\rm B}}}T},\frac{{{E}_{{\rm R}}}}{{{k}_{{\rm B}}}T} \right)g_{2}^{2}\left( \frac{{{E}_{{\rm L}}}}{{{k}_{{\rm B}}}T},\frac{{{E}_{{\rm R}}}}{{{k}_{{\rm B}}}T} \right).\end{eqnarray} \tag{ 30 }$$

In order to evaluate the efficiency of heat to work conversion, again given by the ratio between output power and input heat, we need to calculate the heat current injected from the hot bath. At half the stopping voltage, it takes the form

$$\begin{eqnarray}&&J=\frac{{{\nu }_{2}}\mathcal{A}\Gamma }{2\hbar }{{({{k}_{{\rm B}}}T)}^{2}}\frac{\Delta T}{T}{{g}_{3}}\left( \frac{{{E}_{{\rm L}}}}{{{k}_{{\rm B}}}T},\frac{{{E}_{{\rm R}}}}{{{k}_{{\rm B}}}T} \right),\end{eqnarray} \tag{ 31 }$$

where the function ${{g}_{3}}(x,y)$ satisfies $0\lt {{g}_{3}}(x,y)\lt 2{{\pi }^{2}}/3$. Its complete analytical expression can be found in [48]. Hence, the efficiency at maximum power is given by

$$\begin{eqnarray}&&{{\eta }_{{\rm maxP}}}=\frac{{{\eta }_{C}}}{4}\frac{{{g}_{1}}\left( \frac{{{E}_{{\rm L}}}}{{{k}_{{\rm B}}}T},\frac{{{E}_{{\rm R}}}}{{{k}_{{\rm B}}}T} \right)g_{2}^{2}\left( \frac{{{E}_{{\rm L}}}}{{{k}_{{\rm B}}}T},\frac{{{E}_{{\rm R}}}}{{{k}_{{\rm B}}}T} \right)}{{{g}_{3}}\left( \frac{{{E}_{{\rm L}}}}{{{k}_{{\rm B}}}T},\frac{{{E}_{{\rm R}}}}{{{k}_{{\rm B}}}T} \right)}.\end{eqnarray} \tag{ 32 }$$

In the following, we discuss the output power and efficiency in more detail, starting with a symmetric setup a = 0. In this case, both power and efficiency are symmetric with respect to an interchange of ${{E}_{{\rm L}}}$ and ${{E}_{{\rm R}}}$, see figure 12. The output power takes its maximal value ${{P}_{{\rm max} }}\approx \frac{{{\nu }_{2}}\mathcal{A}\Gamma }{2\hbar }{{\left( \frac{{{k}_{{\rm B}}}\Delta T}{2} \right)}^{2}}$ when one level is deep below the equilibrium chemical potential while the other is located at approximately $1.5{{k}_{\sf{B}}}T$ above it. An explanation for this behavior will be given below. The efficiency at maximum power takes its maximal value ${{\eta }_{{\rm maxP}}}\approx 0.1{{\eta }_{{\rm C}}}$ when both levels are above the equilibrium chemical potential and satisfy ${{E}_{{\rm L}}}\approx {{E}_{{\rm R}}}\pm 2{{k}_{\sf{B}}}T$. However, for these parameters there is only an exponentially suppressed number of electrons that can contribute to transport such that the output power in this regime becomes vanishingly small. For level positions that optimize the output power, we find an efficiency at maximum power of about ${{\eta }_{{\rm maxP}}}\approx 0.07{{\eta }_{{\rm C}}}$. Hence, the quantum-well heat engine is not as efficient as a quantum-dot based setup in the limit of narrow resonances $\Gamma \ll {{k}_{\sf{B}}}T$. The reason for this lies in the different energy-filtering properties of quantum dots and wells. Quantum dots with narrow resonances transmit energies only at a single energy. Hence, they reach the tight-coupling limit where heat and charge current are proportional to each other. In this situation, the efficiency at maximum power is then given by ${{\eta }_{{\rm maxP}}}={{\eta }_{{\rm C}}}/2$. Quantum wells on the other hand transmit any electron with an energy larger than the threshold voltage as in this case the energy E can be decomposed into a part associated with the motion in the plane of the well and perpendicular to it, $E={{E}_{z}}+{{E}_{\bot }}$. Hence, high-energy electrons can be transmitted through the well if most of their energy is in the perpendicular degrees of freedom such that E_z matches the resonance condition. As a result, quantum wells act as much less efficient energy filters.

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Given the rather weak energy filtering properties of quantum wells, it is quite surprising that the efficiency at maximum power is only a factor of three smaller than for quantum dots with level widths of the order of ${{k}_{\sf{B}}}T$—the configuration that gives the largest output power, as discussed in section 2.3.2. To understand this feature, we consider the situation depicted in figure 11. The right quantum well has a threshold energy slightly above the equilibrium chemical potential. As the number of electrons with energies much larger than ${{E}_{{\rm R}}}$ is exponentially small, it acts as a good energy filter. For the left well, the energy filtering relies on a different mechanism. Electrons with energy E can enter the cavity only if the associated state in the left reservoir is occupied, ${{f}_{{\rm L}}}(E)\gt 0$ and, at the same time, the corresponding state in the cavity is empty, ${{f}_{{\rm C}}}(E)\lt 1$. This defines an energy window of about ${{k}_{\sf{B}}}T$ in which electrons are transmitted through the well. This explains why both quantum-dot and quantum-well based heat engines have similar efficiencies.

We now turn to the asymmetric case $a\ne 0$. In this situation, the output power and efficiency are no longer symmetric with respect to an interchange of ${{E}_{{\rm L}}}$ and ${{E}_{{\rm R}}}$. In fact, as can be seen in figures 12(c) and (d), we find that power and efficiency are strongly suppressed if ${{E}_{{\rm L}}}\lt 0$ and ${{E}_{{\rm R}}}\gt 0$ for an asymmetry $a\gt 0$ (the roles of ${{E}_{{\rm L}}}$ and ${{E}_{{\rm R}}}$ are interchanged for $a\lt 0$). However, for ${{E}_{{\rm L}}}\gt 0$ and ${{E}_{{\rm R}}}\lt 0$ we find that the power can be enhanced by up to 20% for an asymmetry of about $a\approx 0.5$ while the efficiency at maximum power is even nearly doubled compared to the symmetric case.

We finally estimate what output power can be expected for a realistic device. For a GaAs based structure with an effective electron mass ${{m}_{*}}=0.067{{m}_{e}}$, level width of $\Gamma ={{k}_{\sf{B}}}T$ and asymmetry a = 0.5 operating at room-temperature $T=300\;K$, we obtain a maximal output power of ${{P}_{{\rm max} }}=0.18\;{\rm W}\;{\rm c}{{{\rm m}}^{-2}}$ for a temperature bias of $\Delta T=1\;K$. Hence, the quantum-well heat engine is about a factor of two more powerful than the previously discussed quantum-dot heat engine. We remark that materials with higher effective mass can yield even larger output powers. Similarly to the quantum-dot case, we also find that a quantum-well heat engine is robust with respect to fluctuations of the threshold energies. As can be seen in figure 12(c) the output power is hardly affected at all by fluctuations of the right threshold energy as long as $-{{E}_{{\rm R}}}\gg {{k}_{\sf{B}}}T$ is fulfilled. Fluctuations of the left threshold energy are more crucial, but even here we find that variations of as much as ${{k}_{\sf{B}}}T$ reduce the output power by only 20%.

We now turn to a discussion of the thermoelectric performance of our heat engine in the nonlinear regime. Similarly, to the quantum-dot heat engine discussed in section 2.3.2, we optimize the applied bias voltage, the threshold energies of the wells as well as the asymmetry a for a given temperature bias $\Delta T$. The corresponding results are shown in figure 13(b). The optimal bias voltage grows linearly with the temperature bias. The optimal asymmetry $a\approx -0.46$ is independent of $\Delta T$. The optimal threshold energy for the right quantum well decreases only slightly with the temperature bias while the threshold energy of the left well should be chosen as $-{{E}_{{\rm L}}}\gg {{k}_{\sf{B}}}T$ independent of $\Delta T$. The optimized power plotted in figure 13(a) grows quadratically with the applied temperature difference and is approximately given by ${{P}_{{\rm max} }}=0.3\frac{{{\nu }_{2}}\mathcal{A}\Gamma }{2\hbar }{{\left( {{k}_{{\rm B}}}\Delta T \right)}^{2}}$. Due to the quadratic dependence on $\Delta T$ we obtain the same output power for a given value of $\Delta T$ in the linear and nonlinear regime. However, as the efficiency at maximum power grows linearly with the applied temperature bias, the device should be operated as much in the nonlinear regime as possible. In the extreme case of $\Delta T/T=2$, the quantum-well heat engine reaches an efficiency at maximum power of ${{\eta }_{{\rm maxP}}}=0.22{{\eta }_{{\rm C}}}$, i.e. it is about as efficient as the quantum-dot heat engine but delivers twice the power.

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We consider a mesoscopic heat engine as discussed in [56] that consists of one or two quantum dots coupled to electronic reservoirs at temperature T. The electronic reservoirs are connected to each other via an external circuit with impedance $Z(\omega )$ that is kept at temperature ${{T}_{{\rm env}}}$. We assume that the tunnel coupling between the dots and the leads is weak, such that a lowest order description of tunneling is valid. We, furthermore, assume that the relaxation time of the electromagnetic environment is much shorter than the average time between subsequent tunneling events. In this situation, transport through the system can be described by P(E) theory [166]. The rate for an electron transition from system i to j is then given by

$$\begin{eqnarray}\begin{array}{rcl} {{\Gamma }_{i\to j}} & = & 2\pi |t{{|}^{2}}\int {\rm d}{{\varepsilon }_{i}}{\rm d}{{\varepsilon }_{j}}{{\rho }_{i}}({{\varepsilon }_{i}}-{{\mu }_{i}}){{{\bar{\rho }}}_{j}}({{\varepsilon }_{j}}-{{\mu }_{j}})P({{\varepsilon }_{i}}-{{\varepsilon }_{j}}). \\ \end{array}\end{eqnarray} \tag{ 39 }$$

Here, t denotes the tunnel matrix element for a transition from i to j. The density of states for electrons $\rho (\varepsilon )$ is given by $\delta (\varepsilon )$ for a quantum dot with discrete levels and by ${{\nu }_{i}}f(\varepsilon )$ for a metallic island or an electrode, where ${{\nu }_{i}}$ denotes the density of states at the Fermi energy. Similarly, the density of states for holes $\bar{\rho }(\varepsilon )$ is given by $\delta (\varepsilon )$ for quantum dots and ${{\nu }_{i}}[1-f(\varepsilon )]$ for metallic systems. Finally, $P(\varepsilon )$ denotes the probability density that the tunneling electron exchanges the energy with the environment.

For a simple tunnel junction without any embedded quantum dot system, there is no directed charge current in the absence of an applied bias voltage as the rectification of thermal fluctuations requires the presence of a nonlinearity in the heat engine. The simplest way to achieve such a nonlinearity is to add a single quantum dot or metallic island into the junction, see figure 17(b). In the limit of strong Coulomb blockade, the quantum dot or metallic island is either empty or occupied with a single excess electron with energy E. The corresponding occupation probabilities p₀ and p₁ in the stationary state follow from a simple rate equation $0=({{\Gamma }_{{\rm L}-}}+{{\Gamma }_{{\rm R}+}}){{p}_{1}}-({{\Gamma }_{{\rm L}+}}+{{\Gamma }_{{\rm R}-}}){{p}_{0}}$ with ${{p}_{0}}+{{p}_{1}}=1$. Here, ${{\Gamma }_{r\pm }}$ denotes the transition rate for an electron tunneling left (right) through barrier r evaluated according to (39). In the following, we assume that the right tunnel barrier has a much larger transition rate than the left barrier. We, furthermore, assume that its capacitance is larger than that of the left junction such that it effectively decouples from the environment. Under these conditions, we obtain a directed charge current that is simply given by

$$\begin{eqnarray}&&I=f(V-E){{\Gamma }_{{\rm L}+}}-f(E-V){{\Gamma }_{{\rm L}-}},\end{eqnarray} \tag{ 40 }$$

and thus finite even in the absence of an applied bias voltage. We remark that the above expression also describes the current through a junction between two quantum dots with energy levels ${{E}_{{\rm L}}}$ and ${{E}_{{\rm R}}}$ such that $E={{E}_{{\rm L}}}-{{E}_{{\rm R}}}$ if the appropriate definitions for the electron and hole density of states are inserted in (39) and the dot is preferably singly occupied.

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By applying a bias voltage V against the thermally driven current, we can generate a finite output power. As for any Coulomb-blockade heat engine, the output power is limited by the operation temperature. For a device operating at a temperature of about 1 K, one obtains a power in the femtowatt range.

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The efficiency of the heat engine depends strongly on the junction type. For a junction between two quantum dots with discrete energy levels, the tight-coupling limit is reached as every electron that is transferred between the dots has to absorb a photon with energy E. Consequently, the heat engine can reach Carnot efficiency and achieve a large efficiency at maximum power. For junctions between a discrete level and a metal or between two metals, the situation is different. Now photons of different energies can be absorbed, leading to a significantly lower efficiency. Interestingly, the efficiency depends on whether the environment is hotter or colder than the electronic system. For a cold electron system, the sharp Fermi functions help to achieve unidirectional transport against the bias voltage such that decent efficiencies can be achieved. In contrast, for a hot electron system the thermally smeared Fermi function give rise to a current flow with the bias voltage such that only small efficiencies are possible.

As a second example of a photon-driven heat engine we now consider a system where a superconducting microwave cavity connects two mesoscopic conductors cf figure 18. Such hybrid structures that allow one to investigate the interplay of light and matter at the nansocale have recently gained a lot of interest both theoretically [167–177] as well as experimentally [178–186] in the context of circuit quantum electrodynamics. Similar to the previously discussed energy harvester, the hybrid microwave cavity heat engine [57] also allows to separate the hot and the cold part of the engine by a macroscopic distance of the order of a centimeter. This suppresses leakage heat currents far more efficiently than vacuum nanogaps that are only a few nanometer wide [187, 188]. In addition, the heat engine has the advantage that the cavity helps in efficiently transferring photons from the hot to the cold side whereas otherwise photons just get randomly emitted into all directions and can be lost for the energy harvesting purpose.

The heat engine consists of two double quantum dots i = 1, 2, connected to each other via a microwave cavity. Due to strong Coulomb interactions, the double dots can be either empty or occupied with a single electron. In the following, we neglect the electron spin as it only renormalizes the tunnel couplings in a trivial way. The eigenstates of the singly-occupied dot can be expressed as linear combinations of the left $|L{{\rangle }_{i}}$ and right $|R{{\rangle }_{i}}$ double quantum dot states as

$$\begin{eqnarray}&&|+{{\rangle }_{i}}={\rm cos} {{\theta }_{i}}|L{{\rangle }_{i}}-{\rm sin} {{\theta }_{i}}|R{{\rangle }_{i}},\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}&&|-{{\rangle }_{i}}={\rm sin} {{\theta }_{i}}|L{{\rangle }_{i}}+{\rm cos} {{\theta }_{i}}|R{{\rangle }_{i}}.\end{eqnarray} \tag{ 42 }$$

The mixing angle ${{\theta }_{i}}$ that characterizes the hybridization of the dot levels depends on the level positions and the interdot tunnel coupling. It can be controlled by gate voltages applied to the double quantum dot. The energy difference between the eigenstates $|+{{\rangle }_{i}}$ and $|-{{\rangle }_{i}}$ can be tuned independently of ${{\theta }_{i}}$ such that it matches the resonance frequency $\hbar {{\omega }_{0}}$ of the microwave cavity. The coupling between the double dots and the cavity with coupling strength g_e then induces transitions between the ground and excited state of the dot accompanied by the absorption or emission of a microwave photon in the cavity.

Each quantum dot is furthermore tunnel coupled to two electronic reservoirs $\nu ={\rm L},{\rm R}$ at temperature T_i and chemical potential ${{\mu }_{i\nu }}$ with tunnel coupling strength ${{\Gamma }_{i\nu }}$. In the following, we assume for simplicity symmetric tunnel couplings for each double quantum dot, ${{\Gamma }_{i\nu }}={{\Gamma }_{i}}$. We remark that the actual transition rates of the system depend not only on the tunnel coupling ${{\Gamma }_{i}}$ but also on the mixing angle ${{\theta }_{i}}$ that can effectively break the left–right symmetry within a double quantum dot.

We describe transport through the system again within a generalized master equation approach that integrates out the noninteracting electronic reservoirs and characterizes the remaining quantum system consisting of the two double quantum dots and the microwave cavity by its reduced density matrix. In the limit of strong electron–photon coupling, ${{g}_{e}}\gg {{\Gamma }_{i}}$, coherences between different eigenstates of the quantum system can be neglected to lowest order in the tunnel coupling such that a simple rate equation description holds. It is this limit that we will consider first before taking into account the effects of finite electron–photon coupling as well as of relaxation and dephasing within the double quantum dots.

Before discussing the thermoelectric performance of the heat engine, we elucidate the basic operation principle that drives a charge current through the cold dot in the absence of any applied bias voltage. To this end, we choose a situation where the mixing angle of the hot dot is ${{\theta }_{1}}=\pi /4$. In this case, the eigenstates are the bonding and antibonding states. As they do not break left–right symmetry, there is no directed charge current through the double dot. For the cold double dot, we choose the mixing angle ${{\theta }_{2}}\ne \pi /4$ such that the ground state couples more strongly to one electrode while the excited states couples more strongly to the other one. Electrons then tunnel into the ground states of the cold double dot from one side, absorb a photon from the cavity to get into the excited state and then tunnel out to the other side. As there is a net flow of heat (and hence microwave photons) from the hot to the cold double dot, we thus obtain a finite directed charge current through the cold double dot. We remark that the direction of the charge current depends on the mixing angle as it determines the asymmetry of eigenstates. Unlike the energy dependence of tunnel couplings that is required for the heat engines discussed in section 2.1.4, the asymmetry can therefore be changed in a controlled way by manipulating ${{\theta }_{2}}$ via gate voltages.

The thermoelectric performance of the microwave cavity heat engine is summarized in figure 19. In panel (a) we show that charge current I₂ that flows through double quantum dot 2 without an applied bias voltage. It grows linearly with the applied temperature bias and saturates for ${{T}_{2}}\ll {{T}_{1}}$. For ${{\theta }_{2}}\to \pi /2$, the tight-coupling limit is reached where one electron is transferred through the cold double dot for each photon that is transferred through the cavity. As a consequence, we find that the charge current I₂ is proportional to the heat current J₁ flowing out of the hot reservoirs, ${{I}_{2}}/{{J}_{1}}=e/(\hbar {{\omega }_{0}})$, where the ratio of these two currents is determined only by the electron charge and the photon energy. In the regime where the cavity is mostly empty, i.e. ${{\bar{f}}_{i}}={{f}_{{\rm L}i}}(\hbar {{\omega }_{0}}/2)={{f}_{{\rm R}i}}(\hbar \omega /2)\ll 1$ the charge current is approximately given by the analytic expression

$$\begin{eqnarray}&&{{I}_{2}}={\rm cos} (2{{\theta }_{2}})\frac{e{{\Gamma }_{1}}{{\Gamma }_{2}}}{{{\Gamma }_{1}}+{{\Gamma }_{2}}}(\bar{f}_{2}^{2}-\bar{f}_{1}^{2}),\end{eqnarray} \tag{ 43 }$$

that agrees well with the full numerical result, see the dashed lines in figure 19(a). As can be inferred from (43), the current becomes largest when ${{\Gamma }_{1}}={{\Gamma }_{2}}\equiv \Gamma$. We will therefore focus on this limit in the following discussion.

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The output power against an externally applied bias voltage V₂ is depicted in figure 19(b). It shows a typical behavior by growing from P = 0 at ${{V}_{2}}=0$ to a maximal value and dropping to zero at the stopping voltage. For ${{\bar{f}}_{\nu i}}={{f}_{\nu i}}(\hbar {{\omega }_{0}}/2)\ll 1$ it is well approximated by

$$\begin{eqnarray}\begin{array}{rcl} {{P}_{2}} & = & \frac{e\Gamma {{V}_{2}}}{2}\left[ {\rm sin} (2{{\theta }_{2}})\{{{{\bar{f}}}_{{\rm L}2}}-{{{\bar{f}}}_{{\rm R}2}}\}-{\rm cos} (2{{\theta }_{2}})\bar{f}_{1}^{2} \right. \\ {} & {} & \left. +{\rm cos} (2{{\theta }_{2}}){{{\bar{f}}}_{{\rm L}2}}{{{\bar{f}}}_{{\rm R}2}}+\frac{{{{\rm sin} }^{2}}(2{{\theta }_{2}})}{4}\{\bar{f}_{{\rm L}2}^{2}-\bar{f}_{{\rm R}2}^{2}\} \right]. \\ \end{array}\end{eqnarray} \tag{ 44 }$$

The efficiency in general shows behavior qualitatively similar to the output power. Only in the tight-coupling limit it instead grows linearly with the applied bias voltage and reaches Carnot efficiency at the stopping voltage. The results for the maximum power ${{P}_{{\rm max} }}$ and the associated efficiency at maximum power ${{\eta }_{{\rm maxP}}}$ together with the optimizing values of ${{\omega }_{0}}$ and V₂ are shown in figures 19(d)–(f). In the tight-coupling limit, ${{\eta }_{{\rm maxP}}}$ grows as ${{\eta }_{{\rm C}}}/2$ in the linear response regime. In the nonlinear regime, it grows more quickly and reaches ${{\eta }_{{\rm C}}}$ for ${{T}_{1}}\gg {{T}_{2}}$. For ${{\theta }_{2}}\lt \pi /2$, the efficiency at maximum power is slightly reduced but exhibits a qualitatively similar behavior.

In current experiments on circuit quantum electrodynamics relaxation and dephasing within the double quantum dots are the major obstacles in reaching the strong coupling limit. When taking into account a finite value of the electron–photon coupling g_e together with relaxation and dephasing processes with rates ${{\Gamma }_{{\rm R}}}$ and ${{\Gamma }_{{\rm D}}}$, respectively, we find that the charge current through the system is reduced to

$$\begin{eqnarray}&&{{\bar{I}}_{2}}=\frac{4g_{e}^{2}}{(\Gamma +{{\Gamma }_{{\rm R}}})(\Gamma +{{\Gamma }_{{\rm R}}}+4{{\Gamma }_{{\rm D}}})+4g_{e}^{2}}{{I}_{2}}.\end{eqnarray} \tag{ 45 }$$

The corresponding current suppression is shown in figure 20 as a function of ${{\Gamma }_{{\rm R}}}$ and ${{\Gamma }_{{\rm D}}}$. Even when taking these imperfections into account, we estimate that for realistic system parameters one can achieve charge currents of the order of $0.2\;{\rm pA}$ clearly within the reach of current experiments.

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