With the progress in fabrication and scaling technology, efficient heat harvesting in lower dimensional systems has gained a lot of attention Singha et al. [2015]; Singha [2020a]; Singha and Muralidharan [2017]; Mi et al. [2008]; Li et al. [2010]; Sumithra et al. [2012]; Bahk et al. [2013]; Agarwal and Muralidharan [2014]; Kim and Lundstrom [2012, 2011]; Faleev and Léonard [2008]; Neophytou and Kosina [2013]; Singha [2018]; Singha and Muralidharan [2018]. One of the major issues affecting heat harvesting performance in nano-systems is the drastic lattice heat flux resulting from a small spatial separation between the heat reservoir and the heat sink. The large lattice heat flux severely limits the overall efficiency of the heat engine and poses a major performance issue in cases where supply of heat energy is limited. Tailoring the lattice thermal conductance, in an attempt to gain enhanced harvesting efficiency generally affects the current path, thereby deteriorating the peak harvested power. As such, one of the crucial focus of the modern thermoelectric community is to facilitate an independent optimization of the electron transport path and lattice heat conduction path, by introducing a spatial separation between the current path and the heat source Uchida et al. [2016]; Sinova et al. [2015]; Saitoh et al. [2006]. This phenomenon of harvesting heat from a reservoir, which is spatially separated from the current conduction path, is known as non-local heat harvesting Uchida et al. [2016]; Sinova et al. [2015]; Saitoh et al. [2006]; Walldorf et al. [2017]; Daré [2019]; Zhang and Chen [2019]; Daré and Lombardo [2017]; Zhang et al. [2016]; Sánchez and Büttiker [2011]. Thus, non-local heat engines are three terminal systems where power is delivered between two terminals while extracting heat from the third terminal which is spatially separated from the path of current flow. \colorblack In this case, tailoring the lattice heat transport path, in an attempt to gain enhanced efficiency, can be accomplished without altering the current conduction path. Recently designs and concepts of non-local heat engines and refrigerators using Coulomb coupled quantum dots have been proposed and explored theoretically Walldorf et al. [2017]; Daré [2019]; Zhang and Chen [2019]; Daré and Lombardo [2017]; Zhang et al. [2016]; Sánchez and Büttiker [2011] as well as experimentally Thierschmann et al. [2015] in literature. However, the operation of such non-local heat engines demand a sharp step-like change in the system-to-reservoir coupling around the ground state energy Walldorf et al. [2017]; Daré [2019]; Zhang and Chen [2019]; Daré and Lombardo [2017]; Zhang et al. [2016]; Sánchez and Büttiker [2011], which is impossible to achieve in a practical scenario Thierschmann et al. [2015]. In this paper, I propose a realistic design strategy to accomplish non-local heat harvesting using Coulomb coupled quantum dots. Unlike the optimal non-local heat-engine based on Coulomb coupled systems Walldorf et al. [2017]; Daré [2019]; Zhang and Chen [2019]; Daré and Lombardo [2017]; Zhang et al. [2016]; Sánchez and Büttiker [2011], the proposed design doesn’t demand a change in the system-to-reservoir coupling near the ground state. The performance proposed heat-engine is then evaluated and compared with the optimal set-up. It is demonstrated that the performance of the proposed heat engine hovers around $50\%$ of the optimal set-up. However, the novelty of the proposed set-up is the conjugation of fabrication simplicity along with a reasonable power output. At the end, the processes leading to a performance deterioration of the proposed set-up are discussed and analyzed. This paper is organized as follows. In Sec. II, I illustrate the proposed design strategy and elaborate the transport formulation employed to analyze the thermoelectric performance of the same. Next, Sec III elaborates a detailed analysis on the heat harvesting performance and regime of operation of the proposed heat-engine. A performance comparison between the proposed heat engine and the optimal set-up is also conducted along with a brief discussion on the transport processes leading to a performance deterioration of the proposed heat engine. I conclude this paper briefly in Sec. IV. The Appendix Sec. elaborates some intuitive and conceptual understanding of the proposed heat engine operation, as well as details the derivations of the equations employed to investigate the performance of the heat engine.

The proposed heat engine, schematically demonstrated in Fig. 1, consists of three dots $S_{1},~{}S_{2}$ and $G_{1}$ which are electrically coupled to the reservoirs $L$, $R$ and $G$ respectively. I will now discuss the ground state configuration and other features of the system consisting of the three dots $S_{1},~{}S_{2}$ and $G_{1}$. $S_{1}$ and $S_{2}$ are tunnel coupled to each other, while $G_{1}$ is capacitively coupled to $S_{1}$. The ground states of $S_{1}$ and $S_{2}$ form a stair-case configuration with $\varepsilon_{s}^{2}=\varepsilon_{s}^{1}+\Delta\varepsilon$. Any electronic tunneling between the dots $S_{1}$ and $G_{1}$ is suppressed via suitable fabrication techniques. Energy exchange between the two dots is, however, possible via Coulomb coupling Hübel et al. [2007a]; Chan et al. [2002]; Molenkamp et al. [1995]; Hübel et al. [2007b]; Ruzin et al. [1992]. In the optimal Coulomb-coupled system based heat-engine discussed in literature, an asymmetric system-to-reservoir coupling is required for heat harvesting Sánchez and Büttiker [2011]; Erdman et al. [2018]. In the proposed set-up, instead of the asymmetric system-to-reservoir coupling, the system itself is made asymmetric with respect to the reservoir $L$ and $R$ by choosing an energy difference between the ground states of the adjacent quantum dots $S_{1}$ and $S_{2}$, with $\varepsilon_{s}^{2}=\varepsilon_{s}^{1}+\Delta\varepsilon$. The purpose of such arrangement is to deliver power to an external load connected between $L$ and $R$ while extracting heat from reservoir $G$. Another equivalent realistic set-up, based on Coulomb coupled systems, that can be employed for efficient non-local heat harvesting is demonstrated in Fig. 7 and discussed briefly in Appendix A. \colorblack\colorblack Due to random electron exchange between the dot $G_{1}$ and the reservoir $G$, the total energy of the system fluctuates stochastically with time. For a Coulomb coupled system with constant energy resolved system-to-reservoir coupling and with symmetric ground state configuration with respect to reservoir $L$ and $R$, the stochastic fluctuations result in the same probability of electron transfer from $L$ to $R$ and from $R$ to $L$, making the average current zero, although there is a non-zero component of electronic heat flow between $G$ and $L(R)$ (See Appendix B). However, for a Coulomb coupled system with asymmetric ground state configuration, the same fluctuation can be converted to a directed current flow when $T_{G}\neq T_{L(R)}$ (Appendix C). I will demonstrate via numerical calculations and theoretical arguments (given in Appendix C) that in the system detailed above, thermoelectric power can be delivered between the terminals $L$ and $R$ (with a net electronic flow from $L$ to $R$) by extracting heat energy from the reservoir $G$ when $T_{G}>T_{L}(R)$. Thus, power can be delivered between terminals non-local to the heat source. The excess energy $\Delta\varepsilon=\varepsilon_{s}^{2}-\varepsilon_{s}^{1}$, required for the electrons to tunnel from $S_{1}$ to $S_{2}$ is supplied from the reservoir $G$ via Coulomb coupling. Coming to the realistic fabrication possibility of such a system, due to the recent advancement in solid-state nano-fabrication technology, triple and quad quantum dot systems with and without Coulomb coupling have already been realized experimentally Eng et al. [2015]; Flentje et al. [2017]; Froning et al. [2018]; Noiri et al. [2017]; Hong et al. [2018]; Takakura et al. [2014]. In addition, it has been experimentally demonstrated that quantum dots that are far from each other in space, may be bridged to obtain strong Coulomb coupling, along with excellent thermal isolation between the hot and cold reservoirs Hübel et al. [2007a]; Chan et al. [2002]; Molenkamp et al. [1995]; Hübel et al. [2007b]; Ruzin et al. [1992]. Also, the bridge may be fabricated between two specific quantum dots to drastically enhance their mutual Coulomb coupling, without affecting the electrostatic energy of the other quantum dots Hübel et al. [2007a]; Chan et al. [2002]; Molenkamp et al. [1995]; Hübel et al. [2007b]; Ruzin et al. [1992]. \colorblack Thus, the change in electron number $n_{S_{1}}~{}(n_{G_{1}})$ of the dot $S_{1}$ ($G_{1}$) influences the electrostatic energy of the dot $G_{1}$ ($S_{1}$). In general, the total increase in electrostatic energy $U$ of the configuration demonstrated in Fig. 1 (a), consisting of three dots, due to fluctuation in electron number can be given by (Refer to Appendix D for detailed derivation):

where $n_{x}^{tot}$ is the total electron number, and $U^{self}_{x}=\frac{q^{2}}{C^{self}_{x}}$ is the electrostatic energy due to self-capacitance $C^{self}_{x}$ (with the surrounding leads) of quantum dot ‘$x$’ (Refer to Appendix D for details). $U^{m}_{x_{1},x_{2}}$ is the electrostatic energy arising out of Coulomb coupling between two different quantum dots that are separated in space (Appendix D) and $n_{x}^{eq}$ is the total number of electrons present in dot $x$ in equilibrium at $0K$ and is determined by the minimum possible electrostatic energy of the system. $n_{x}=n_{x}^{tot}-n_{x}^{eq}$ is the total number of excess electrons added in the ground state of the dot $x$ due to stochastic fluctuations from the reservoirs (Refer to Appendix D for details). Here, a minimal physics based model is used to investigate the heat engine performance under the assumption that the change in potential due self-capacitance is much greater than than the average thermal voltage $kT/q$ or the applied bias voltage $V$, that is $U^{self}_{x}=\frac{q^{2}}{C^{self}_{x}}>>(kT,~{}qV)$. Hence, electron occupation probability or transfer rate via the Coulomb blocked energy level, due to self-capacitance, is negligibly small. The analysis of the entire system of dots may hence be approximated by limiting the maximum number of electrons in each dot to one. Thus the analysis of the entire system may be limited to eight multi-electron levels, which I denote by the electron occupation number in the ground state of each quantum dot. Hence, a possible state of interest in the system may be denoted as $\ket{n_{S_{1}},n_{G_{1}},n_{S_{2}}}=\ket{n_{S_{1}}}\mathbin{\mathop{\otimes}\limits}\ket{n_{G_{1}}}\mathbin{\mathop{\otimes}\limits}\ket{n_{S_{2}}}$, where $n_{S_{1}},n_{G_{1}},n_{S_{2}}\in(0,1)$, are used to denote the number of electrons present in the ground-states of $S_{1},~{}G_{1}$ and $S_{2}$ respectively. I also assume that the electrostatic coupling between $S_{1},~{}S_{2}$ and between $S_{2},~{}G_{1}$ are negligible, such that, for all practical purposes under consideration, $U^{m}_{S_{1},S_{2}}\approx 0$ and $U^{m}_{G_{1},S_{2}}\approx 0$. Since, the electronic transport and ground states in $S_{1}$ and $G_{1}$ are mutually coupled, I treat the pair of dots $S_{1}$ and $G_{1}$ as a sub-system ($\varsigma_{1}$), $S_{2}$ being the complementary sub-system ($\varsigma_{2}$) of the entire system consisting of three dots Singha [2020b]. The state probability of $\varsigma_{1}$ is denoted by $P_{i,j}^{\varsigma_{1}}$, $i$ and $j$ being the number of electrons in the ground state of dot $S_{1}$ and $G_{1}$ respectively. $P_{k}^{\varsigma_{2}}$, on the other hand, denotes the probability of occupancy of the dot $S_{2}$ in the sub-system $\varsigma_{2}$. It can be shown that if $\Delta\varepsilon$ is much greater than the ground state broadening due to system-to-reservoir coupling, then the interdot tunneling rate between $S_{1}$ and $S_{2}$ is optimized when $\varepsilon_{s}^{1}+U^{m}_{S_{1},G_{1}}=\varepsilon_{s}^{2}$, that is when $\Delta\varepsilon=U^{m}_{S_{1},G_{1}}$ Singha [2020b] (See Appendix D). To evaluate the optimal performance of the proposed heat-engine, I hence assume $\Delta\varepsilon=U^{m}_{S_{1},G_{1}}$ Singha [2020b]. Henceforth, I would simply represent $U^{m}_{S_{1},G_{1}}$ as $U_{m}$. Under the assumption stated above, the equations governing sub-system state probabilities in steady state can be derived as Singha [2020b] (See Appendix D for detailed derivation):

where $\gamma_{c}=\gamma_{l}(\varepsilon)=\gamma_{r}(\varepsilon)=\gamma_{g}(\varepsilon)$ and $\gamma$ are related to the reservoir-to-system tunnel coupling and the inter-dot tunnel coupling respectively Singha [2020b]; Datta [2005], $\varepsilon$ being the independent energy variable. In the above set of equations, $f_{\lambda}(\varepsilon)$ denotes the probability of occupancy of the reservoir $\lambda$ at energy $\varepsilon$. For the purpose of calculations in this paper, I assume an equilibrium Fermi-Dirac statistics at the reservoirs. Hence, $f_{\lambda}(\varepsilon)$ is given by:

where $\mu_{\lambda}$ and $T_{\lambda}$ respectively denote the quasi-Fermi energy and temperature of the reservoir $\lambda$. From the set of Eqns. (2) and (3), it is clear that an electron in $S_{1}$ can tunnel into $S_{2}$ only when the ground state in the dot $G_{1}$ is occupied with an electron. The set of Eqns. (2) and (3) are coupled to each other and may be solved using any iterative method. Here, I use Newton-Raphson iterative method to solve the steady-state values of sub-system probabilities. On calculation of the sub-system state probabilities $P_{i,j}^{\varsigma_{1}}$ and $P_{k}^{\varsigma_{2}}$, the electron current flow into (out of) the system from the reservoirs $L(R)$ can be given as:

In addition, the electronic component of heat flow from the reservoir $G$ can be given by:

Interestingly, we note that Eqn. (6) is not directly dependent on $\varepsilon_{g}$. This is due to the fact that the net electronic current into or out of the reservoir $G$ is zero (See Appendix D for details). Next, I use the set of Eqns. (2), (3), (5) and (6), to evaluate the thermoelectric generation performance of the the set-up demonstrated in Fig. 1. To analyze the performance of the heat engine, I use a voltage-controlled set-up demonstrated in literature Leijnse et al. [2010]; Sothmann et al. [2014]; Nakpathomkun et al. [2010], where a bias voltage $V$ is applied between $L$ and $R$, with the positive and negative terminals being connected to $L$ and $R$ respectively, to emulate the voltage drop due to thermoelectric current flow across an external load. Thus the bias voltage $V$ mimics the voltage drop between the terminals across which power is to be delivered. Assuming the equilibrium electrochemical potential across the entire set-up is $\mu_{0}$, and a voltage drop $V$ across the external load distributed symmetrically between the left and right system-to-reservoir interfaces, the quasi-Fermi levels at the reservoirs $L$, $R$ and $G$ may be written as $\mu_{L}=\mu_{0}-\frac{V}{2},~{}\mu_{R}=\mu_{0}+\frac{V}{2}$ and $\mu_{G}=\mu_{0}$ respectively. The generated power ($P$) and efficiency ($\eta$) can be defined as:

where $V$ is the applied potential bias in the voltage controlled model and $I_{Q}$ is the sum of electronic and lattice heat flux from the heat source ($G$). In the non-local heat engine the lattice heat flux can be favourably engineered Androulakis et al. [2006]; Hsu et al. [2004]; Pan et al. [2015]; Feser et al. [2012]; Davis and Hussein [2014] without affecting the current conduction path. In addition the lattice heat flux is generally independent of the the electronic heat conductivity and doesn’t change with the system energy configuration Mingo and Broido [2004]; Mingo [2004]; Zhou et al. [2007, 2010]; Boukai et al. [2008]; Hochbaum et al. [2008]; Balandin et al. [1999]; Chen [1998]. Hence, to simplify the calculation, I assume ideal condition by neglecting the lattice heat flux, as done in recent literature Walldorf et al. [2017]; Daré and Lombardo [2017]; Zhang et al. [2016]; Sánchez and Büttiker [2011]; Singha et al. [2015]; Singha and Muralidharan [2017]; Whitney [2014, 2015]. The generation efficiency for our case, can hence be defined as:

where $I_{Qe}$ is the electronic component of the heat current extracted from the heat source $G$ (defined in Eq. 6). It should be noted that the efficiency given by Eq. (8) is the maximum achievable efficiency under ideal conditions and is limited by thermodynamic considerations of heat to work conversion for the given system energy configuration. \colorblack
To understand the operation of the proposed heat engine, let us assume that the system is initially in the vacuum state $\ket{0,0,0}$, where the ground state of all the dots are empty. Now, let us consider a complete cycle $F\Rightarrow\ket{0,0,0}\rightarrow\ket{1,0,0}\rightarrow\ket{1,1,0}\rightarrow\ket{0,1,1}\rightarrow\ket{0,1,0}\rightarrow\ket{0,0,0}$. In this cycle, the system starts from the initial vacuum state $\ket{0,0,0}$. Next an electron enters into $S_{1}$ from $L$, with an energy $\varepsilon_{s}^{1}$, followed by an electron tunneling into $G_{1}$ from $G$ with an energy $\varepsilon_{g}+U_{m}$. Next, the electron in $S_{1}$ tunnels into $S_{2}$ with an energy $\varepsilon_{s}^{2}=\varepsilon_{s}^{1}+U_{m}$ and finally tunnels out of $S_{2}$ into reservoir $R$. The system, at last returns to its initial state when the electron in $G_{1}$, finally tunnels out into $G$ with an energy $\varepsilon_{g}$. In this cycle, an electron is transferred from reservoir $L$ to $R$ while absorbing a heat packet $U_{m}$ from $G$. In the reverse cycle, corresponding to the cycle $F$ described above, that is $R\Rightarrow\ket{0,0,0}\rightarrow\ket{0,1,0}\rightarrow\ket{0,1,1}\rightarrow\ket{1,1,0}\rightarrow\ket{1,0,0}\rightarrow\ket{0,0,0}$, an electron is transferred from reservoir $R$ to $L$, while dumping a heat packet $U_{m}$ into reservoir $G$. It can be shown that when $T_{G}>T_{L(R)}$, under short-circuited condition between $L$ and $R$, the rate of the forward cycle $F$ is greater than the reverse cycle $R$ (See Appendix D), resulting in a net unidirectional flow of electrons from $L$ to $R$ while absorbing heat from reservoir $G$. Also for all other equivalent cycles, resulting in transfer of electrons between $L$ and $R$ under short-circuited condition, it can be shown that a rate of forward sequence that results in electron transfer from $L$ to $R$ while absorbing heat from $G$ is greater than the reverse sequence when $T_{G}>T_{L(R)}$. Thus for $T_{G}>T_{L(R)}$, a net thermoelectric electronic flow occurs from $L$ to $R$ by extracting heat from the spatially separated reservoir $G$ and hence the set-up proposed in this paper constitutes a non-local heat engine. Without loss of generality, I assume that $\gamma_{c}=10^{-5}\frac{q}{h}$ and $\gamma=10^{-4}\frac{q}{h}$ (The effect of the ratio $\gamma/\gamma_{c}$ on the heat harvesting performance of the set-up is discussed in Appendix E). Such values of $\gamma$ and $\gamma_{c}$ limits electronic transport phenomena in the weak coupling regime, where the effects of cotunneling can be neglected and the transport through the entire system can be described in terms of the quantum master equation (QME) approach, given by the set of Eqns. (2)-(3). This simplifies the analysis to a great extent and facilitates an intuitive understanding of the computed results. \colorblack The temperature of the reservoirs $L(R)$ and $G$ are assumed to be $T_{L(R)}=5K$ and $T_{G}=10K$. The average temperature between the hot and the cold reservoirs is, hence, given by $T=\frac{T_{L(R)}+T_{G}}{2}=7.5K$.

To conclude, in this paper I have proposed a realistic design strategy for non-local heat engine based on Coulomb coupled systems. The performance of the proposed design was then theoretically analyzed and compared with the optimal set-up Sánchez and Büttiker [2011] using the QME approach. It was demonstrated that the proposed set-up outputs a maximum power of around $50\%$ of the optimal set-up. However, the crucial advantage of the proposed design strategy is that along with a reasonable output power, it also circumvents the demand for a sharp step-like change in reservoir-to-system coupling, which is required for proper operation of the optimal set-up proposed in literatureSánchez and Büttiker [2011]. In the above discussions, I have limited transport through the quantum dots in the weak coupling regime, so that the effects of co-tunneling can be neglected. The generated power in the proposed system can be increased by a few orders of magnitude by tuning electronic transport in the strong coupling regime, that is by increasing the system-to-reservoir, as well as, the interdot tunnel coupling. It will be interesting to study the effects of cotunneling on heat-harvesting performance of the proposed system as electronic transport is gradually tuned towards the strong coupling regime. \colorblack In addition, I have also assumed ideal conditions by neglecting the lattice thermal conductance. Understanding the impact of lattice heat flux on the generation efficiency and an investigation on the effect electron-phonon interaction on the performance of the proposed design also constitute interesting research directions. Although not shown here, the proposed system can also work as an efficient non-local heat engine when the reservoir $G$ acts as a heat sink (cold) with respect to the reservoirs $L$ and $R$ (hot). In such a case, the direction of thermoelectric current flow is reversed. The different possible design strategies for non-local heat engines and their performance is left for future exploration. Nevertheless, the set-up proposed in this paper can be employed to fabricate high performance non-local heat engines using Coulomb coupled systems.
Acknowledgments: The author would like to thank Sponsored Research and Industrial Consultancy (IIT Kharagpur) for their financial support via grant no. IIT/SRIC/EC/MWT/2019-20/162.
Data Availability: The data that supports the findings of this study are available within the article