Nonlinear effects for three-terminal heat engine and refrigerator

The three-terminal heat device that consists of an electronic cavity and couples to a heat bath is studied both as a heat engine and as a refrigerator. We investigate the characteristic performance in the linear and nonlinear regime for both setups. It is our focus here to analyze how the efficiency of the heat engine and coefficient of performance of the refrigerator are affected by the nonlinear transport. With such considerations, the maximum efficiency and power are then optimized for various energy levels, temperatures and other parameters.

electronical load by a tunneling junction. However, the cavity in ref. 17 is to be cooled, whereas it performs as an auxiliary component in our set-up.
In the following, we calculate the energy-conversion efficiency, electric power, electric current and the transport heat in both linear and nonlinear regime. For the heat engine, the nonlinear efficiency and output power are enhanced compared to the linear transport. While for the refrigerator, the nonlinear efficiency and cooling power are reduced to nearly half of the linear ones. We optimize the maximum efficiency and power by tuning the energy levels, temperatures, and other parameters. Our results show that nonlinear effects can improve the maximum efficiency of the heat engine to 25% of the Carnot efficiency (with parasitic heat leakage included) and the maximum power to more than an order of the linear counterpart.
Model and Formalism. The three-terminal thermoelectric device we consider is illustrated in Fig. 1. The left (right) quantum dot is directly in contact with the left (right) electronic reservoir. It describes an electron leaves the source into the QD1, and hops to the QD2 through the cavity, which is thermalized by the phonon bath. Then, it finally tunnels into the drain. The electronic reservoirs, i = S, D, are characterized by the Fermi-Dirac function ( ) . We assume that strong electron-electron and electron-phonon interactions relax the electron energies as they enter and leave the electronic cavity. Hence, the occupation function of the cavity can also be described by the Fermi-Dirac function, , completely characterized by a chemical potential μ cav and temperature T cav . 27 . To reach steady state, the cavity must exchange energy with the phonon bath (denoted by a brown arrow in Fig. 1). We assume that the thermal conduction between the phonon bath and the cavity is efficient so that the temperature gradient is considerably small. In this way, one can approximately treat that T cav = T ph .
The Hamiltonian of the electronic system is described as , with m * the effective mass and k the wave vector of the charge carrier. The Hamiltonian of quantum dots is shown as where d d i i † denotes particle number operators for the dots, respectively, with i = L,R representing left and right quantum dots. The interaction Hamiltonian which describes the hybridization of the QD states with the states in the source, drain and cavity is given by Figure 1. Schematic view of a three-terminal thermoelectric system. The three-terminal device is composed by two electronic reservoirs (characterized by their temperatures T ele and chemical potential S D ( ) μ ) and a phonon bath, which is held at temperature T Ph . The central cavity, which is thermalized by the phonon bath, is connected to two electrodes via two quantum dots at energy E l r ( ) . with α → V i k , , the interaction strength between the ith dot and αth bath. To capture the transport effects, we apply a bias V = μ/e to this system. The chemical potential of the source and the drain are set anti-symmetrically, i.e., /2 Generally, the electron current through the left (right) electrode into cavity can be evaluated by Landauer-Buttiker formalism 44 denotes the particle occupation of the left and right electrodes. To conserve the electron current, the chemical potential of the cavity can be determined as e l e r , , While for the heat current flowing from the source (drain) to the cavity, I Q,j has two contributions, i.e., is the ideal transmission function for phonons. To consider the low-frequency phnonons, which dominate the steady state behavior, it can be ideally expressed as − with α a dimensionless constant and E cut the cut-off energy of the phonons. Phonons with energy lower than E cut can spring out of the bath and interact with electrons, while the higher energy phonons are bounded in the bath. Moreover, the conservation of energy results in stands for the total energy in the source (drain), and   = E Q P P is the energy flow in the phonon bath.  = − N I e / S D e l r ( ) , ( ) (e < 0) represents the particle current flowing into the source (drain). Combining Eqs (4), (5) and (7), we obtain the heat injected into the system from the phonon bath as Then, the total entropy production rate 46 of the system is contributed from three currents and corresponding thermodynamic forces Specifically, the first term is the electron current driven by chemical potential bias between electrodes, shown as The second term is the energy current of electrons under the temperature bias of two electrodes While the third term originates from the heat flow of phonons from the thermal bath In our work, we set T S = T D = T ele , T P = T Ph . Thus, we can simplify the forces as In the linear regime ( , the thermodynamic fluxes and forces follow the Onsager relations 47 where only the lowest order of thermodynamics biases need to be considered. Specifically,  Generally, the linear transport coefficients can be obtained by calculating the ratios between currents and affinities in the linear response regime with very small voltage bias and temperature difference 43 .

Nonlinear transport effects on a three-terminal heat engine
The three-terminal device can be tuned into a heat engine by setting T ele = T c , T ph = T h . The heat engine has the ability to convert the absorbed heat into the electric power, which is expressed as P out = −I e V. Here I e is the net electrical current through the system as the charge conservation implies I I I e el er , , = = − . The energy-conversion efficiency is then defined by the ratio of the injected heat and the output power where Q in = I Q,P is the heat current flowing from the phonon bath due to the temperature difference between the electrode and the heat bath. Considering the physical significance, the efficiency is well-defined only in the regime with P > 0 and Q in > 0. Consequently, the Carnot efficiency for heat engine is defined by the temperature of the electrode and the phonon bath We firstly analyze the efficiency and output power for a three-terminal heat engine in both linear and nonlinear regimes. At a fixed temperature T h = 1.5T c , the nonlinear transport yields significant improvement of the maximum efficiency and power, as shown in Fig. 2(a) and (b). The maximum efficiency under small voltage bias is 12.8% of the Carnot efficiency, while the full calculation (including the nonlinear transport effect) reaches 14.7% of the Carnot efficiency, which is 1.2 times of the linear counterpart. Moreover, the maximum power in linear regime is 1.6 nW, whereas it increases to 2.6 nW in the nonlinear regime. Hence, we conclude that the nonlinear effect significantly enhances the thermoelectric performance.
To better understand the enhancing mechanism of the maximum efficiency and power, we then investigate how the electrical and heat currents are affected by the nonlinear transport. Figure 2(c) shows that the electrical current is considerably enhanced due to the nonlinear effect. The short-circuit current I SC (the electrical current at zero voltage V = 0) is increased by 1.3 times. We can interpret this from current formulas Eqs (16)  . The nonlinear V oc raises to 1.3 times of that in the linear response regime. The product of the short-circuit current and open-circuit bias gives a nonlinear output power P more than 1.6 times as large as the one in the linear regime, which agrees well with the improvement of the maximum power. Besides, we also examine how the input heat Q in is affected by the nonlinear transport effect. Figure 2(d) reveals that the maximum heat input at V = 0 increases to about 1.5 times as large as that obtained in the linear limit. Hence, the increase of the output power exceeds that of the input heat, which clearly unravels the improvement of the maximum efficiency.
Then, we turn to explore the nonlinear effects in the thermoelectric transport. We plot the short-circuit electrical, heat currents and the open-circuit voltage as functions of Th/Tc, by fixing Tc. As presented in Fig. 3 h c , the "nonlinear" currents and voltage are more than 10% larger than the linear ones. Such enhancement is mainly due to the multichannel induced electron transport.
Furthermore, we study the effect of nonlinear transport on thermoelectric energy by modulating temperatures and QD energies. The ratio of the maximum efficiency over the Carnot efficiency η η / nl max C in Fig. 4(a) and the maximum power in Fig. 4 While the ratio η η / C nl max can directly reflect enhancement of the nonlinear transport effect. Fig. 4(a) shows that the η η / C nl max can increase from 5% to 25%, which is more than twice of the linear counterpart. Fig. 4(d) demonstrates that the enhancement factor P P / nl max li max can be as large as 14, which shows acute dependence on the temperature   Fig. 4(f), we can find that the nominal power P nl nom strongly depends on the temperature ratio and the QD energy. The maximum value appears at T h = 2T c and ≅ − . nom is a constant of 1/4 over all temperature range and different energy levels. Therefore, the nonlinear effect enhances the useful power by more than one order-of-magnitude compared to the linear limit.

Nonlinear transport effects on a three-terminal refrigerator
The three-terminal device can be tuned to be a refrigerator, by exchanging temperatures of the electrode and the phonon bath, i.e., T T S D h ( ) = , T ph = T c , with T h > T c . Then, the phonon bath can be cooled, and heat Q out is transferred to the cavity. Here, we use the invested work as the chief power supplier, P in = IV. The cooling efficiency is defined by the ratio of the cooling heat Q out and the input power P in .
out in and the Carnot efficiency for which the process is reversible is given by We first study the COP and input power for a three-terminal refrigerator by the same method as done at Eqs (11), (16), (18) and (20). The temperatures are selected as T h = 405 K and T c = 347K. Figure 5(a) indicates that the nonlinear transport effect reduces the coefficient of performance, with maximum COP η η / nl C three-fifths of the linear one, which is contrary to the Fig. 2(a) (the heat engine case). The electric power injected into the system is shown in Fig. 5(b). P max nl is 41.7% lower than P max li , which indicates that under the same voltage bias, the nonlinear transport effect consumes much less electric power.
To find out why the cooling efficiency is reduced in the nonlinear regime, we study how the electrical and heat currents are affected by the nonlinear transport, as presented in Fig. 5(c) and (d). As the Eq. 18(a) indicates, the negative current increases with the voltage bias in the linear regime. While the nonlinear one, which contains the contribution of the multichannel transport, does not increase as fast as the linear one. Specifically, the negative maximum electric current via linear-approximation calculation is 1.71 times as large as the full calculation one, which is in accordance with the input power. Figure 5(d) shows that the maximum cooling heat with nonlinear effect firstly increases and then decreases with the voltage bias. While the linear one grows continually with the bias. The trend of the nonlinear cooling heat is determined by the Fermi-Dirac distribution factor , which saturates at large voltage bias. Moreover, lower production lagging behind the consumption leads to the deterioration of the cooling efficiency. The value of the bias at which the cooling heat current starts to flow is called the threshold V C . Figure 5(d) indicates that the "working regime" (the voltage range over which cooling is possible) of the linear effect is slightly extended than the nonlinear effect. This is in consistent with Fig. 5(a) that the cooling efficiencies appear only when the bias exceed a certain value. Figure 5(e) makes it clear that how the threshold bias is determined by the temperature ratio T h /T c . The V C via linear approximation increases with the temperature ratio, while the nonlinear one end abruptly when T h reaches  ≈ . , which indicates an energy balance between source and drain. We can conclude that the cooling power for nonlinear transport effect is limited to low voltage bias.
We then turn to analyze the comprehensive effect of the dot energy and the temperature, shown in Fig.6. We set the QD energy difference − = E E r l 5k T B c and vary the temperature from T c to 2T c . It is found that the optimi- l r B c , which testifies that the "particle-hole symmetric" configuration is also best for a cooling machine. Figure 6(a) also presents that the cooling efficiency diminishes gradually with the increasing temperature ratio T h /T c , where Fig. 5(f) may account for this diverting phenomenon. The cooling heat decreases with increasing T h at fixed energy, and reaches zero at T T 1 5 h c ≈ . , which leads to the zero cooling efficiency. Figure 6(b) gives the cooling efficiency enhancement factor η η / nl max li max under the same parameters as in Fig. 6(a). It is exhibited that the maximum nonlinear efficiency nl max η can reach 90% of the linear one when the temperature bias is very small.

Conclusions
In summary, we have investigated the influence of nonlinear response of three-terminal setup on the thermoelectric performance, including efficiency, power, electric and heat currents. We find that the nonlinear effect can significantly improve the performance of the three-terminal heat engine. When the temperatures of the electrodes and phonon bath are interchanged, the device turns to be a refrigerator. Unlike the heat engine, the nonlinear transport effect considerably reduces the efficiency and cooling power of the three-terminal refrigerator. We also optimize the efficiency and power at different parameters, in which the optimal values can be reached as the device becomes "particle-hole symmetric", with the dot energy E 1 = − E r . From the practical view, three-terminal thermoelectric devices have already been fabricated in experiments where the electron cavity is made of GaAs/