Efficiency at Maximum Power in a Parallel Connected Two Quantum Dots Heat Engine

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Introduction
Te concept of thermodynamics has been developed from the analysis of heat engines' performance. Carnot invented an idealized mathematical model of heat engines called the Carnot cycle and proved that there exists a maximum effciency of all heat engines, which is given by Carnot efciency. Tis efciency is a central cornerstone of thermodynamics. It states that a reversible Carnot engine's efciency attains the maximum possible work for a given temperature of the hot (T h ) and cold (T c ) reservoirs but generates zero power because it is an infnitely slow operation. Te efciency (η c � 1 − T c /T h ) of the Carnot cycle is the upper bound on the efciency at which real heat engines are unrealistically high. Te practical implications are more limited since the upper limit η c is only reached for reversible engine. One of the important questions is what will be the efciency at maximum power of a system that operates in fnite time. In a groundbreaking work, Curzon and Ahlborn [1] obtained this efciency for the Carnot engine by optimizing the Carnot cycle with respect to power rather than efciency, which is given by Curzon-Ahlborn efciency, η CA Tis efciency is used to seek a more realistic upper bound on the efciency of a heat engine in the endoreversible approximation [1,2] (taking into account the dissipation only in the heat transfer process). Currently, it has been shown that the Curzon-Ahlborn efciency is an exact consequence of linear irreversible thermodynamics when operating under conditions of strong coupling between the heat fux and the work [3][4][5]. Te value of 1/2 for the linear coefcient in equation (1) is therefore universal for such systems. Furthermore, the diverse system in nature has been found investigating efciency at maximum power such as Brownian particle undergoing a Carnot cycle through the modulation of a harmonic potential [6], Feynman Ratchet and Powl model [7], and quantum dots connected in two leads with diferent temperatures and chemical potential [8]. Te result of efciency at maximum power for all models mentioned previously agreed up to quadratic order in η c .
In thermoelectricity devices, the phenomena of the Seebeck efect, Peltier efect, and Tomson efect have been described by the temperature or potential diference. Snyder and Toberer [9] have discovered the thermoelectric material with signifcantly higher thermodynamic yields in the early 1990s. Te development in the feld of nanostructured materials is particularly intriguing [10]. Since then, thermoelectric experiments on silicon nanowires [11], individual carbon nanotubes [12], and molecular junctions [13] have been reported. Such thermoelectric devices can be used as an energy converter, i.e., heat to work. In particular, Humphrey et al. [14,15] reported that Carnot efciency could be reached for electron transport between two leads at diferent temperatures and chemical potentials, by connecting them through a channel sharply tuned at the energy for which the electron density is the same in both leads.
Recently, a tiny heat engine (with a single level quantum dots) in contact with hot and cold heat reservoirs with diferent chemical potentials has been proposed by Esposito et al. [8]. Tey studied how the device operates and determined the efciency at maximum power and compared their value with that of the Curzon-Ahlborn efciency. Besides, the thermoelectric properties of two quantum dots connected in parallel have been studied in Reference [16].
In this paper, we introduce a detailed thermodynamic analysis of electron transport through parallelly connected two identical quantum dots connecting two leads at diferent temperatures and chemical potentials. Due to the temperature gradient, the electrons transport from hot lead to cold lead through a dot; in contrast, electrons transport from cold lead to hot lead through a dot due to chemical potential. Te temperature and chemical potential gradient cause the electron transport through quantum dots to act as a thermoelectric engine.
Te rest of this paper is organized as follows: in Section 2, the model is introduced and thermodynamic quantities are determined. Section 3 evaluates and explores the behaviors of heat fux and power as a function of energy parameter, a scaled ∆/ϵ characterizing the energy diference between the two levels. In Section 4, we evaluate the efciency at maximum power by using a perturbation solution and numerical solution, and the result that we get for the efciency at maximum power lies between the Carnot efciency and Curzon-Ahlborn efciency. In Section 5, we summarize and conclude.

The Model and Derivation of the Thermodynamic Quantities
We consider a model that is two quantum dots parallelly connected into hot and cold reservoirs of temperatures T r and T l and chemical potentials μ r and μ l , respectively, as shown in Figure 1. Due to their size variation, the quantum dots have diferent single energy levels associated with each of them. Accordingly, we consider the single energy level of the frst quantum dot ϵ 1 to be ϵ + ∆, while that of the second quantum dot ϵ 2 has to be ϵ − ∆.
We assume that the electrons thermalize instantaneously to the temperature of the leads upon tunnelling to the reservoirs and the electron transports through quantum dots with a sharply defned energy. If the level remains occupied by an electron while it is lowered (raised), power is extracted from (injected into) the system, W < 0(W > 0), respectively. If the level remains empty while energy changes, neither power nor heat fux is produced. When the empty (flled) level at energy ϵ + ∆ and ϵ − ∆ is flled (emptied) by an electron, an amount of heat fux Q r (Q l ) enters the system, respectively.
Te dots with energy levels ϵ + ∆ and ϵ − ∆ exchange electrons with leads as shown in Figure 1. Te quantum dot is either empty (state 1) or flled (state 2) for the frst quantum dot and the second quantum dot either empty (state 3) or flled (state 4). Te crucial variables of the problem are the scaled energy barriers with (k B � 1) of the frst and second quantum dots, respectively, which are given by where v � l, r Te master equation [17][18][19] describes the probabilities of the dots being in a particular state change in time as where P 1 , P 2 , P 3 , and P 4 are the probability of the quantum dots in state 1, state 2, state 3, and state 4, respectively.
left lead right lead Figure 1: Sketch of nanothermoelectric engine consisting of two quantum dots embedded between two leads at diferent temperatures and chemical potentials. We choose by convention T l < T r .
Here, the transitional rate in the frst quantum dot is given by and the transitional rates in the second quantum dot are given by Here, a v and b v are the Einstein coefcients which are independent of the dots' energy and f v and g v are the Fermi distributions given by In this paper, we focus on the steady state properties of the device. Te steady state distributions for both dots' occupation follow from W 21(43) P ss 1(3) � W 12(34) P ss 2(4) with P ss 1(3) + P ss 2(4) � 1. Te resulting probability current from the lead v to the frst and second dot is respectively. Using I r � − I l , J r � − J l , W 12 + W 21 � a r + a l , and W 34 + W 43 � b r + b l , we can rewrite the result for the fux from the right lead as where α � (a r a l )/(a r + a l ) and c � (b r b l )/(b r + b l ). Equation (8) is the Landauer formula for a single infnitely sharp resonance (i.e., without broadening). Te steady state heat per unit time for the frst quantum dot _ Q r ′ and the second quantum dot _ Q r ″ extracted from the lead r is, respectively, given by Te total heat fux enters into the quantum dots from lead r.
Te net power output by both quantum dots is the sum of the total heat fux getting into it and dissipates cold into cold reservoir which is given by _ W � αT r x r f r − f l + y r g r − g l +αT l x l f r − f l + y l g r − g l .
Te corresponding thermodynamic efciency reads Te entropy production associated with the master equation (3) is given as follows [20][21][22][23] for the frst quantum dot: where i, j � 1, 2. Noting that InW 12 /W 21 � x v , one fnds, in agreement with standard irreversible thermodynamics [19], the following expression for the entropy production: Termodynamics forces for matter and energy fow, F m and F e , are given by We stress that the corresponding matter and heat fow are given by In the same spirit, the entropy production, thermodynamics forces for matter and energy fow, and their corresponding matter and fow of the second quantum dot can expressed as respectively. Te matter and heat fow are perfectly coupled and the condition for attaining both Carnot and Curzon-Ahlborn efciency, namely, that the determinant of the corresponding Onsager matrix be zero, is fulflled [1,24].
Likewise [8], we frst discuss the case of equilibrium. It can control thermodynamic quantities by controlling the current because of the perfect coupling. Under this condition, detailed balance is valid, I v � 0 and Jv � 0, valid if and only if f l � f r , g l � g r or equivalently, x l � x r , y l � y r . Te efciency and the entropy production then become equal to the Carnot efciency, cf.equation (12), and vanishes (cf. equation (13) and cf. equation (17)), respectively. We note that x l � x r , y l � y r do not require that the thermodynamic forces F m and F e vanish separately, i.e., this singular balancing point equilibrium does not require temperature and Journal of Engineering chemical potential to be identical in both reservoirs [3-5, 14, 15, 24].

Thermodynamic Properties of the Model
In this section, we evaluate and explore the behaviours of thermodynamic quantities such as heat fux and power as a function of a scaled energy parameter ∆/ϵ, which characterizes the energy diference between the two levels of the model.

Heat Flux.
Te rate of heat energy transferred through a given surface of a system can be described by the thermodynamic quantity, which is called heat fux. Substituting equations (2) and (6) where x � 1 − μ r /ϵ, y � 1 − μ l /ϵ. Te steady state heat per unit time with ∆ � 0 becomes Te scaled heat per unit time of the thermodynamic quantity (( _ Q r (∆/ϵ))/( _ Q r (∆ � 0))) can be exploited in Figure 2 as a function of ∆/ϵ. Figure 2 depicts the ratio of total heat fux as a function of ∆/ϵ. Te heat fux getting into the quantum dots increases when y increases. Tis means that when y increases, the chemical potential μ l decreases. If x and y are comparable, this means that the chemical potentials μ l and μ r are equal and the power output will be zero. We note that μ l is diferent from μ r since the thermoelectric engine operates under an irreversible process.
In Figure 3, the scaled net power that delivers from the quantum dots increases when y increases. In general, the scaled input heat fux increases if the scaled net power output also increases. In Section 4, we study the efciency at maximum power of the thermoelectric heat engine.

Thermoelectric Efficiency at Maximum Power
In this section, we determine the efciency at maximum power of parallelly connected quantum dots (i.e., having the same dot energy) thermoelectric heat engine. To obtain the maximum power condition for a given temperature T l and T r , we search for the values of the scaled electron energy, i.e., the frst derivative of power barriers x l , x r , y l , and y r that maximize _ W becomes zero. We fnd the following four equations for both quantum dots. Te frst two equations corresponding to the frst quantum dot are given by and the second two equations corresponding to the second quantum dot are given by g l − g r + y r − 1 − η c y l g 2 r exp y r � 0, Equations (24) and (25) depend on the ratio of the two temperatures. From the two simultaneous equations (equations (24) and (25)), we fnd a transcendental equation which is expressed as and respectively. Since an analytic solution of this equation is not possible, we frst turn to perturbative solutions for η c close to the limiting values 0 (reservoirs of equal temperatures) and 1 (cold reservoir at zero temperature) and also fnd the numerical values of x mp l and x mp r for any values of η c . For the case η c ≈ 0, we substitute x r � a 0 + a 1 η c + a 2 η 2 c + O(η 3 c ) and (27) and (28), respectively, and expand the resulting equation in η c . Te coefcients, a 0 , a 1 , a 2 , b 0 , b 1 , b 2 , and others, are found recursively by solving order by order in η c . At order zero, we fnd an identity. At the frst order, we fnd the transcendental equations a 0 � 2 coth(a 0 /2) and b 0 � 2 coth(b 0 /2). Te numerical solution is a 0 � b 0 � 2.39936. At second order and third order in η 0 , we fnd that a 1 � b 1 � − a 0 /4 and a 2 � b 2 � sinh(a 0 )/(b(1 − cosh(a 0 ))). As we obtain from the perturbative solution, the values of a 0 � b 0 , a 1 � b 1 , and a 2 � b 2 , then x r � y r and the Fermi distribution becomes f r � g r and f l � g l . Ten, substituting all terms in equation (12) gets reduced into Te efciency at maximum power in the regime of small η c : Before we evaluate, numerically, let us defne a dimensionless quantity τ which is the diference between the hot and cold reservoir temperature with respect to the cold reservoir, i.e., From equations (27) and (28) in equation (12) to fnd the efciency at maximum power.
From Figure 4 which is the numerical solution for the efciency at maximum power versus τ, we can observe that as τ goes to zero, the efciency at maximum power becomes zero. Tis is because when τ becomes very small, the heat fux that is getting into the quantum dot becomes very small; hence, the efciency is zero and as τ becomes large, the efciency at maximum power approaches to one because as τ increases, the input power that the quantum dot receives increases which leads to an increase in efciency.
We are solving the efciency at maximum power as η c runs from 0 to 1 numerically. First, we are solving the roots of equations (27) and (28) by varying η c from 0 to 1 and substituting the values into equation (26) to fnd x l and y l . Ten, we substitute all values into equation (12), and fnally, we get Figure 5. Figure 5 shows that the efciency at maximum power increases monotonically when we drive out of equilibrium. It is bounded from above by Carnot efciency η C , while the Curzon-Ahlborn efciency η CA provides a rather tight lower bound. Te deviation between efciency at maximum power and the Curzon-Ahlborn efciency observes the values of η c which goes to the maximum value.

Summary and Conclusion
In this work, we have taken a simple model of single-level two quantum dots connected between two heat reservoirs working as a heat engine. Te model's simplicity enabled us to get analytic solutions for important quantities, such as efciency at maximum power. To analyse the way energy is utilized by the engine, we started from the master equation and derived the efciency, η, for the heat engine by frst evaluating the heat fux extracted from the hot reservoir, _ Q r , heat fux dissipated into the cold reservoir, _ Q l , and delivered by the quantum dot. Maximizing the efciency with respect to our power, _ W, free parameters are the scaled electron energy barriers, x r , x l , y r , and y l . Te efciency at maximum power is evaluated by two approaches such as analytical solution and numerical solution. When the temperature of thermal reservoirs gets closer to each other, the coefcient of the linear therm for the efciency at maximum power is 1/2.
In conclusion, since the quantum dots are connected in parallel, the resulting efciency at maximum power is the same as that obtained from one quantum dot. Tis model introduces another transport mechanism that prevents the energy conversion of thermoelectric devices.

Data Availability
No data were used to support the fndings of this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.  Journal of Engineering