⇦Chinese Physics Letters, 2021, Vol. 38, No. 8, Article code 088801 Coulomb Thermoelectric Drag in Four-Terminal Mesoscopic Quantum Transport Mengmeng Xi (席萌萌)1, Rongqian Wang (王荣倩)2, Jincheng Lu (陆金成)2,3*, and Jian-Hua Jiang (蒋建华)2* Affiliations 1Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China 2School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China 3Center for Phononics and Thermal Energy Science, China-EU Joint Lab on Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China Received 10 April 2021; accepted 7 June 2021; published online 2 August 2021 Supported by the Science and Technological Fund of Anhui Province for Outstanding Youth (Grant No. 1508085J02), the National Natural Science Foundation of China (Grant Nos. 61475004, 11675116, and 12074281), the Chinese Academy of Sciences (Grant No. XDA04030213), the Jiangsu Distinguished Professor Funding and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), and the China Postdoctoral Science Foundation (Grant No. 2020M681376).
*Corresponding authors. Email: jincheng.lu1993@gmail.com; jianhuajiang@suda.edu.cn Citation Text: Xi M M, Wang R Q, Lu J C, and Jiang J H 2021 Chin. Phys. Lett. 38 088801    Abstract We show that the Coulomb interaction between two circuits separated by an insulating layer leads to unconventional thermoelectric effects, such as the cooling by thermal current effect, the transverse thermoelectric effect and Maxwell's demon effect. The first refers to cooling in one circuit induced by the thermal current in the other circuit. The middle represents electric power generation in one circuit by the temperature gradient in the other circuit. The physical picture of Coulomb drag between the two circuits is first demonstrated for the case with one quantum dot in each circuit and it is then elaborated for the case with two quantum dots in each circuit. In the latter case, the heat exchange between the two circuits can vanish. Finally, we also show that the Maxwell's demon effect can be realized in the four-terminal quantum dot thermoelectric system, in which the quantum system absorbs the heat from the high-temperature heat bath and releases the same heat to the low-temperature heat bath without any energy exchange with the two heat baths. Our study reveals the role of Coulomb interaction in non-local four-terminal thermoelectric transport. DOI:10.1088/0256-307X/38/8/088801 © 2021 Chinese Physics Society Article Text Thermoelectric transport is a useful tool to study the fundamental properties of quasiparticles in mesoscopic systems.[1] Abundant information of quasiparticle dynamics and fluctuations, phase coherence and statistics can be revealed in thermoelectric transport.[2–4] Most existing research has focused on elastic thermoelectric transport, such as ballistic transport and resonant tunneling in mesoscopic systems.[5–13] There has only recently been a surge of interest in studying thermoelectric transport involving inelastic processes where electron energy is not conserved during the transport. It was found that inelastic thermoelectric devices can provide higher efficiency and larger output power than conventional elastic thermoelectric devices made of the same material.[14–31] Furthermore, several unconventional effects, such as the cooling by heating effect[32–34] and the linear-response thermal transistor effect,[35–39] have been discovered in inelastic thermoelectric devices. It was also proven that these effects cannot take place in an elastic thermoelectric device. Several types of inelastic thermoelectric devices have been studied, including electron-phonon[40–43] or electron-electron interaction.[41,44–52] In general, electron-phonon interaction is more important at high temperatures, whereas electron-electron interactions are important at very low temperatures. In this work, we will show the inelastic thermoelectric transport at the low-temperature regime and discuss the key role of the electron-electron interaction in it. It has been shown that in a three-terminal device with one quantum dot (QD), electronic noises (essentially heat energy) from the third terminal can induce electrical power generation for the source and drain terminals.[14,53] The underlying mechanism is the inelastic Coulomb scattering, which induces a directional transport when the left-right inversion symmetry is broken as in other quantum ratchets. In these studies, heat in the third electronic terminal is harvested to produce electrical energy across the source and the drain. Recently, Whitney et al. found that in a four-terminal QD thermoelectric device, electrical energy across the source and the drain can be produced by harvesting heat from two additional capacitively coupled terminals.[54] Remarkably, the produced electrical energy is finite when the total heat injected into the QD system vanishes. In this work, we show that cooling of the source in a four-terminal mesoscopic system can be achieved by passing a perpendicular heat current between the two additional terminals, even if these two terminals do not exchange charge with the source or the drain. The cooling by heat current effect is driven by the Coulomb drag between electrons in the two circuits even though they are separated by a charge insulation layer. This effect may happen in a number of four-terminal mesoscopic thermoelectric systems when the Coulomb interaction between two circuits is strong. Here, we focus on illustration of the effect via correlated transport through Coulomb coupled QDs with particular spatial symmetries. These symmetries are realized by engineering the energy dependent coupling between the QDs and the four terminals. With a simple two-QD setup, the cooling by heat current effect can be realized. In a setup with two pairs of double QDs, the cooling by heat current effect can be realized with the total heat exchanged between the two circuits vanishes. This limit is consistent with the thermodynamic analysis that the cooling of the source is driven by the heat flow in the other circuit from the hot terminal to the cold terminal, instead of the total heat flux exchanged between the two circuits. Four-Terminal Quantum-Dots Thermoelectric Systems. We first consider a four-terminal double QDs thermoelectric system, which is illustrated in Fig. 1. The system can be regarded as two independent circuits without Coulomb interaction. The source, the drain and the upper quantum dot form a circuit, while the other two electrodes and the lower quantum dot form another circuit. The two circuits are separated by an insulating layer. Thermoelectrics in the two circuits are correlated once the interaction between electrons in the two QDs are considered. In general, Coulomb interaction between the two QDs can cause drag effect and inelastic thermoelectric transport in the two circuits, which leads to unconventional thermoelectric phenomena. To demonstrate the underlying physics, we will consider the Coulomb interaction of the following form,[14] $$ H_{\rm Coulomb} = E_{\rm C} n_1 n_2 ,~~ \tag {1} $$ where $n_1$ and $n_2$ are the numbers of electrons in the two QDs, respectively. $E_{\rm C}$ is the strength of Coulomb interaction. We will show in this work that such a simple interaction leads to novel thermoelectric phenomena in four-terminal systems. Beside the source and the drain, there are other two electrodes (electronic reservoirs). In our setup, their electrochemical potentials are set to the equilibrium value so that they serve as “heat baths”; i.e., charge motions between them do not produce or consume any electrical energy. We thus consider only the heat current flowing out of the $H$ reservoir, $J_{\rm H}$, and the heat current flowing into the $C$ reservoir, $J_{\rm C}$ (we suppose $T_{\rm H}>T_{\rm C}$, where $T_{\rm H}$ and $T_{\rm C}$ are the temperature of the heat bath H and C, respectively.). In contrast, the source and the drain can have potentials differing from the equilibrium value, leading to charge motion under (self-consistent) electrical fields. We hence consider both the charge current between the source and the drain, $J_{\rm e}$, and the heat current flowing out of (into) the source (drain) terminal, $J_{\rm S}$ ($J_{\rm D}$).
cpl-38-8-088801-fig1.png
Fig. 1. (a) Schematic of the four-terminal QDs thermoelectric system. Beside the source and the drain, there are two other electrodes, H and C. Only the heat currents flowing out/into these electrodes are considered because their electrochemical potentials are set to the equilibrium value. In experiments, these two electrodes (termed as “heat baths”) and the QD $d1$ are fabricated in a lower layer. In the upper layer, there are the source, the drain, and the QD $u1$. The lower and upper layers are separated by an insulating layer (depicted as the gray shadow). Electrons in the two QDs interact with each other via Coulomb (repulsive) interaction. (b) Illustration of correlated electron transport through the QDs $u1$ and $d1$. An electron tunnels between the source and the QD $u1$ only at energy $E_{u1}$ (i.e., when there is no electron in the QD $d1$). The electron tunnels between the drain and the QD $u1$ can only take place at energy $E_{u1}+E_{\rm C}$ (i.e., with an electron in the QD $d1$). For the QD $d1$, electron tunnels between the heat bath $H$ and the QD only at energy $E_{d1}+E_{\rm C}$ (i.e., when there is an electron in the QD $u1$). The electron tunnels between the QD $d1$ and the heat bath $C$ only at energy $E_{d1}$. The interaction between electron, depicted as the wavy lines, is the crucial element for the correlated inelastic transport through the QDs $u1$ and $d1$. For the electron motion indicated by the arrows, an energy $E_{\rm C}$ is pumped into the source and drain subsystem from the H and C subsystem.
Although there are four heat currents, energy conservation imposes the following restriction,[55] $$ J_{\rm S} + \frac{\mu_{\scriptscriptstyle {\rm S}}}{e} J_{\rm e} + J_{\rm H} = J_{\rm C} + J_{\rm D} + \frac{\mu_{\scriptscriptstyle {\rm D}}}{e}J_{\rm e} ,~~ \tag {2} $$ where $\mu_{\scriptscriptstyle {\rm S}}$ and $\mu_{\scriptscriptstyle {\rm D}}$ are the electrochemical potentials of the source and the drain, respectively. It is useful to consider the following combinations $$ J_{\rm in} = J_{\rm H} - J_{\rm C},~~~J_q = \frac{1}{2}(J_{\rm H}+J_{\rm C}) .~~ \tag {3} $$ Here $J_{\rm in}$ can be understood as the total heat injected into the QDs system from the two heat baths H and C. $J_q$ is regarded as the heat flow (exchange) between the reservoirs H and C. We will choose the independent heat currents as three of the four heat currents; e.g., $J_{\rm S}$, $J_{\rm in}$ and $J_q$. The affinities corresponding to the three heat currents and the electrical current can be found by analyzing the total entropy production,[55–57] $$\begin{alignat}{1} &\partial_t S_{\rm tot} = \sum_i J_i A_i,~~ \tag {4a}\\ &A_{\rm e} = \frac{\mu_{\scriptscriptstyle {\rm S}}-\mu_{\scriptscriptstyle {\rm D}}}{eT_{\rm D}},~~ A_{\rm S} \equiv \frac{1}{T_{\rm D}} -\frac{1}{T_{\rm S}},~~ \tag {4b}\\ &A_{\rm in} \equiv \frac{1}{T_{\rm D}} - \frac{1}{2T_{\rm H}} - \frac{1}{2T_{\rm C}}, ~~ A_q \equiv \frac{1}{T_{\rm C}}-\frac{1}{T_{\rm H}} .~~ \tag {4c} \end{alignat} $$ To demonstrate the underlying physics, in the following we will consider the simple situation where there is only a single energy level that allows tunneling from each electrode to the connected QD. We now present the microscopic theory for thermoelectric transport in the QDs system. Explicitly, we shall construct the expressions for the electrical and heat currents as functions of the electrochemical potentials, $\mu_{\scriptscriptstyle {\rm S}}$ and $\mu_{\scriptscriptstyle {\rm D}}$, as well as the temperatures, $T_{\rm S}$, $T_{\rm D}$, $T_{\rm H}$ and $T_{\rm C}$. As shown in Fig. 1(b), we consider particular configurations where QDs $u1$ and $d1$ are capacitively coupled, whereas there is no other coupling between QD $u1$ and other QDs. We also assume that the intradot Coulomb repulsion is so strong that double occupancy is forbidden. In these configurations, the energies for electrons in the QDs are given by $$\begin{alignat}{1} &E_{u1,n_{d1}} = E_{u1} + n_{d1} E_{\rm C},~~ \tag {5a}\\ &E_{d1,n_{u1}} = E_{d1} + n_{u1} E_{\rm C},~~ \tag {5b}\\ &n_{u1}, n_{d1}=0,1 .~~ \tag {5c} \end{alignat} $$ Here, $E_{u1}$ and $E_{d1}$ are the electron energies for QDs $u1$ and $d1$ if there is no interaction between QDs; $n_{u1}$ and $n_{d1}$ denote the numbers of electrons in the $u1$ and $d1$ QDs, respectively. To realize the cooling by heat current effect through the Coulomb interaction, we consider the four-terminal four-QD thermoelectric system; as shown in Fig. 2(a). In this particular configuration, QDs $u2$ and $d2$ are capacitively coupled, whereas there is no other coupling between QD $u2$ and other QDs [see Fig. 2(b)]. Similarly, the expressions hold for QDs $u2$ and $d2$ are $$\begin{align} &E_{u2,n_{d2}} = E_{u2} + n_{d2} E_{\rm C},~~ \tag {6} \end{align} $$ $$\begin{align} &E_{d2,n_{u2}} = E_{d2} + n_{u2} E_{\rm C},~~ \tag {7} \end{align} $$ $$\begin{align} &n_{u2}, n_{d2}=0, 1 .~~ \tag {8} \end{align} $$ For simplicity, we have assumed that the charge interaction strength between the QDs $u2$ and $d2$ are the same as that for the QDs $u1$ and $d1$. Transport through QDs system can actually be divided into two parts: the four-terminal transport through the pair of QDs $u1$ and $d1$ and that through the pair of QDs $u2$ and $d2$. These two channels are independent of each other.
cpl-38-8-088801-fig2.png
Fig. 2. (a) Schematic of the four-terminal QDs thermoelectric system. Beside the source and the drain, there are two other electrodes. Only the heat currents flowing out/in these electrodes are considered because their electrochemical potentials are set to the equilibrium value. In experiments, these two electrodes (termed as “heat baths”) and the two QDs $d1$ and $d2$ can be fabricated in a lower layer. The source and drain electrodes together with the QDs $u1$ and $u2$ can be fabricated in the upper layer. The lower layer and the upper layer are separated by an insulating layer (depicted as the gray shadow). However, electrons in the QDs can interact with each other via the long range Coulomb interaction. (b) Correlated electron tunneling through the QDs $u2$ and $d2$. Similar to electron tunneling through the $u1$ and $d1$ QDs, with different energy configurations for electron tunneling. For electron motion indicated by the arrows in the figure, an energy $E_{\rm C}$ is provided by the source and drain subsystem to the H and C subsystems. The arrows indicate one possible direction of electron motion, which can be reversed, depending on the voltage and temperature biases. The numbers in the brackets indicate the sequences of the correlated tunneling processes.
The electrical and heat currents can be expressed as (see the Supplemental Material) $$\begin{alignat}{1} &J_{\rm e} = e \sum I_i,~~ \tag {9a} \end{alignat} $$ $$\begin{alignat}{1} &J_{\rm S} = E_{u1} I_1 + (E_{u2}+E_{\rm C}) I_2,~~ \tag {9b}\\ &J_{\rm H} = (E_{d1}+E_{\rm C}) I_1 + E_{d2} I_2 ,~~ \tag {9c}\\ &J_{\rm C} = E_{d1} I_1 + (E_{d2}+E_{\rm C}) I_2 ,~~ \tag {9d}\\ &J_{\rm in} = E_{\rm C} (I_1 - I_2),~~ \tag {9e}\\ &J_q=\Big(E_{d1} + \frac{1}{2}E_{\rm C}\Big) I_1 + \Big(E_{d2}+\frac{1}{2} E_{\rm C}\Big) I_2 ,~~ \tag {9f} \end{alignat} $$ where $$\begin{aligned} &I_1 = \gamma_1^{-1} (k^+_{u1,0} k^+_{d1,1} k^-_{u1,1} k^-_{d1,0} -k^+_{d1,0} k^+_{u1,1} k^-_{d1,1} k^-_{u1,0}) , \\ &I_2 = \gamma_2^{-1} (k^+_{d2,0}k^+_{u2,1}k^-_{d2,1}k^-_{u2,0} -k^+_{u2,0}k^+_{d2,1}k^-_{u2,1}k^-_{d2,0}) , \end{aligned}~~ \tag {10} $$ and $$\begin{align} \gamma_1={}&\sum_{\alpha=u1,d1}\sum_{i=\pm} \sum_{n=0,1} k^{-i}_{\alpha, n} (k^i_{\alpha,1-n}k^i_{-\alpha,n} \\ &+ k^{-i}_{-\alpha,n}\sum_j k^j_{-\alpha,n}\delta_{|1+i|,2n}) , \\ \gamma_1={}&\sum_{\alpha=u2,d2}\sum_{i=\pm} \sum_{n=0,1} k^{-i}_{\alpha, n} (k^i_{\alpha,1-n}k^i_{-\alpha,n} \\ &+ k^{-i}_{-\alpha,n}\sum_j k^j_{-\alpha,n}\delta_{|1+i|,2n}) ,~~ \tag {11} \end{align} $$ where $I_{1(2)}$ are the electron number currents through the QD $u1(2)$ [or $d1(2)$], $-u1=d1$ and $-u2=d2$, and $k^{i/j}_{\alpha,n}$ are the transition rates (see the Supplemental Material). Cooling by Thermal Current Effect. In Ref. [58] we have studied the “cooling by transverse heat current” effect, which is a cooling mode in mesoscopic four-terminal thermoelectric devices with two electrodes (the source and the drain) and two heat phononic baths, and the source can be cooled by driving a heat current between the phononic heat baths. This is different from the previous “cooling by heating” effect where cooling is driven by a photonic heat flow injected into the double-QD quantum system.[33] In the “cooling by thermal current” effect, there is no need to inject the energy into the quantum system because the driving force of refrigeration is the energy exchange between the two heat baths H and C. We now study the cooling by thermal current effect in the Coulomb coupled QD system, i.e., cooling the source (i.e., $J_{\rm S}>0$ even though $T_{\rm S} < T_{\rm D}$ and $A_{\rm S} < 0$) as driven by the thermal current $J_q$ when $$\begin{alignat}{1} &\mu_{\scriptscriptstyle {\rm S}}=\mu_{\scriptscriptstyle {\rm D}}=0, ~~ {\rm i.e.},~ A_{\rm e}=0 ,~~ \tag {12a}\\ &T_{\rm C} = \Big[\frac{2}{T_{\rm D}} - \frac{1}{T_{\rm H}} \Big]^{-1}, ~~ {\rm i.e.},~ A_{\rm in}=0 .~~ \tag {12b} \end{alignat} $$ We consider the situations with $T_{\rm H}>T_{\rm D}>T_{\rm C}$. $T_{\rm D}$ is the reference temperature that may be set by the substrate. This effect can take place only when $J_qA_q>0$, i.e., the negative entropy production (i.e., positive free energy production) during the cooling of the source is compensated by the positive entropy production during the thermal conduction $J_q$ (i.e., free energy consumption, $J_qA_q>0$), so that the total entropy production is non-negative,[55,59] $$ \partial_t S_{\rm tot} = J_q A_q + J_{\rm S} A_{\rm S} \ge 0 .~~ \tag {13} $$ The coefficient of performance (COP) for the cooling by thermal current effect is given by the ratio of the two heat currents $$\begin{alignat}{1} \eta_{\scriptscriptstyle {\rm COP}} = \frac{J_{\rm S}}{J_q} = \frac{ E_{u1} I_1+(E_{u2}+E_{\rm C}) I_2 }{\Big(E_{d1}+\frac{1}{2}E_{\rm C}\Big)I_1+\Big(E_{d2}+\frac{1}{2} E_{\rm C}\Big) I_2}.~~~~~~~ \tag {14} \end{alignat} $$ From Eq. (13) and the definition of the COP we find that the reversible COP is $$ \eta^{\rm rev}_{\scriptscriptstyle {\rm COP}}=\Big(\frac{T_{\rm S}}{T_{\rm C}}-\frac{T_{\rm S}}{T_{\rm H}}\Big)\frac{T_{\rm D}}{T_{\rm D}-T_{\rm S}} = -\frac{A_q}{A_{\rm S}} .~~ \tag {15} $$ We regard $T_{\rm D}$ as the reference temperature fixed by the substrate and $T_{\rm C}$ as determined by Eq. (12b), so that only the temperatures $T_{\rm S}$ and $T_{\rm H}$ are independent variables. The working condition for the cooling driven by the thermal current $J_q$ is restricted by $$ 0\le \eta_{\scriptscriptstyle {\rm COP}} \le -\frac{A_q}{A_{\rm S}} .~~ \tag {16} $$ At reversible COP $\eta^{\rm rev}_{\scriptscriptstyle {\rm COP}} = -\frac{A_q}{A_{\rm S}}$ the cooling power vanishes, since the entropy production and the currents vanish.[55,59,60]
cpl-38-8-088801-fig3.png
Fig. 3. (a) Cooling power $J_{\rm S}$ as functions of temperature ratios $T_{\rm S}/T_{\rm D}$ and $T_{\rm H}/T_{\rm D}$ for $E_{u1}=E_{u2}=E_{d1}=E_{d2}=1$ meV. (b) Cooling power $J_{\rm S}$ as functions of QDs energies $E_{u1}=E_{u2}=\varepsilon_u$ and $E_{d1}=E_{d2}=\varepsilon_d$ for $k_{\scriptscriptstyle {\rm B}}T_{\rm H}=6$ meV and $k_{\scriptscriptstyle {\rm B}}T_{\rm S}=0.6$ meV. (c) The COP ratio $\eta_{\scriptscriptstyle {\rm COP}}/\eta_{\scriptscriptstyle {\rm COP}}^{\rm rev}$ for the cooling by thermal current effect as functions of the temperature ratios $T_{\rm S}/T_{\rm D}$ and $T_{\rm H}/T_{\rm D}$ when QDs energy is the same as in (a). (d) $\eta_{\scriptscriptstyle {\rm COP}}/\eta_{\scriptscriptstyle {\rm COP}}^{\rm rev}$ as functions of QDs energies $\varepsilon_u$ and $\varepsilon_d$ for the same parameters as in (b). Common parameters: $\mu_{\scriptscriptstyle {\rm S}}=\mu_{\scriptscriptstyle {\rm D}}=0$, $k_{\scriptscriptstyle {\rm B}}T_{\rm D}=1$ meV, $T_{\rm C}=1/(2/T_{\rm D}-1/T_{\rm H})$, $\varGamma_0=0.1$ meV, and $E_{\rm C}=2$ meV. The white areas represent the parameter regions where the cooling by thermal current cannot be achieved, i.e., $J_{\rm S} < 0$.
In Fig. 3(a) the cooling power of our nonelectric refrigerator is large when the temperature of the source $T_{\rm S}$ is close to the temperature of drain $T_{\rm D}$ and when the temperature of the heat bath $T_{\rm H}$ is much higher than the temperature of the drain, $T_{\rm H}\gg T_{\rm D}$. From Fig. 3(a) the lowest temperature of the source $T_{\rm S}$ that can be cooled down is about $0.25T_{\rm D}$. In the white region the device cannot function as a refrigerator. As shown in Fig. 3(b), for a given temperature of $T_{\rm H}=6T_{\rm D}$ and $T_{\rm S}=0.6T_{\rm D}$, the cooling power is large for $E_{u1}=E_{u2}=\varepsilon_u$ around $k_{\scriptscriptstyle {\rm B}}T_{\rm D}$ ($k_{\scriptscriptstyle {\rm B}}T_{\rm D}=1$ meV throughout this paper), particularly when $\varepsilon_d\gtrsim \varepsilon_u$. In Fig. 3(c) the ratio of the COP over the reversible COP $\eta_{\scriptscriptstyle {\rm COP}}/\eta_{_{\scriptstyle \rm COP}}^{\rm rev}$ is high when the temperature of the source $T_{\rm S}$ is low. This is the regime where the cooling power is rather low. Similarly, as shown in Fig. 3(d), the COP ratio $\eta_{\scriptscriptstyle {\rm COP}}/\eta_{\scriptscriptstyle {\rm COP}}^{\rm rev}$ is high when $\varepsilon_d$ is small, where the cooling power is very low. The COP can be quite a considerable fraction of the reversible COP, indicating that the coupling is strong and the dissipation is low for such optimized efficiency conditions.[60] This phenomenon that the efficiency is optimized when the output power is low (for small dissipations), known as the efficiency-power trade-off, is consistent with the literature available.[55,60,61]
cpl-38-8-088801-fig4.png
Fig. 4. (a) Cooling power $J_{\rm S}$ and (b) COP ratio $\eta_{\scriptscriptstyle {\rm COP}}/\eta_{\scriptscriptstyle {\rm COP}}^{\rm rev}$ as functions of the QDs energies $E_{u1}=E_{u2}=E_{d1}=E_{d2}=\varepsilon$ and $E_{\rm C}$. Other parameters: $\mu_{\scriptscriptstyle {\rm S}}=\mu_{\scriptscriptstyle {\rm D}}=0$, $k_{\scriptscriptstyle {\rm B}}T_{\rm D}=1$ meV, $k_{\scriptscriptstyle {\rm B}}T_{\rm H}=6$ meV, $k_{\scriptscriptstyle {\rm B}}T_{\rm S}=0.6$ meV, $T_{\rm C}=1/(2/T_{\rm D}-1/T_{\rm H})$, and $\varGamma_0=0.1$ meV.
We now study more on the QDs energy dependence of the cooling power $J_{\rm S}$ and the COP ratio $\eta_{\scriptscriptstyle {\rm COP}}/\eta_{\scriptscriptstyle {\rm COP}}^{\rm rev}$ for several different configurations. We first consider the case with $E_{u1}=E_{u2}=E_{d1}=E_{d2}=\varepsilon$ and study how the energy $\varepsilon$ and the charge interaction energy $E_{\rm C}$ affect the power and the COP. First, the cooling by thermal current effect holds for all $E_{\rm C}>0$ when $\varepsilon>0$. However, as shown in Fig. 4(a), for $\varepsilon < 0$, the cooling by thermal current effect holds only when $E_{\rm C} < -\varepsilon$. This is due to $M_{{\rm S},q} = e^{-2}TG[p_1 \varepsilon (\varepsilon + \frac{1}{2}E_{\rm C}) + p_2 (\varepsilon + E_{\rm C})(\varepsilon + \frac{1}{2}E_{\rm C}) ]$ in this regime. The condition for positive $M_{{\rm S},q}$ holds when $E_{\rm C} < -\varepsilon$. From Figs. 4(a) and 4(b), we note that the optimal parameters for both COP and cooling power can be $\varepsilon\sim k_{\scriptscriptstyle {\rm B}}T_{\rm D}$ and $E_{\rm C}\sim k_{\scriptscriptstyle {\rm B}}T_{\rm D}$, or $\varepsilon\sim -2 k_{\scriptscriptstyle {\rm B}}T_{\rm D}$ and $E_{\rm C}\sim k_{\scriptscriptstyle {\rm B}}T_{\rm D}$ ($k_{\scriptscriptstyle {\rm B}}T_{\rm D}=1$ meV). Although the COP ratio increases with decreasing $E_{\rm C}$ for any given $\varepsilon$, the limit with $E_{\rm C}\to 0$ does not validate our theory. Four-Terminal Thermoelectric System as a Maxwell's Demon. Maxwell's demon is a hypothetical demon, which can detect and control the motion of a single molecule. It was conceived by James Clerk Maxwell in 1871 to explain the possibility of violating the second law of thermodynamics. The demon is assumed to control a door, which separates two boxes to realize the non-equilibrium gas distribution.[62] This operation that appeared to be in violation of the second law of thermodynamics, decreasing the entropy of the gas without any work input. Despite the complexity, such a Maxwell's demon has been realized in various systems in different forms.[63–66] For example, the QD multi-terminal systems can be possibly realized as a Maxwell's demon using the inelastic transport without any measurement of individual particles or any feedback involved.[50,58,67,68] As shown in Fig. 5, here we introduce a Maxwell's demon based on two electronic thermal baths (the hot bath $H$ and cold bath $C$) which can reduce the entropy of the working substance (the source and the drain), without changing the energies and particles of the working substance. The nonequilibrium Maxwell's demon is used to reverse the natural direction of the source and drain where the drain has a higher temperature. As we introduced in the previous section, when we realize the cooling by thermal current effect, the entropy of the source and drain is reduced, while the entropy of the whole thermoelectric system is increased. This operation seemingly break the second law of thermodynamics, the entropy production of the two heat baths H and C compensates for that of the source and drain.
cpl-38-8-088801-fig5.png
Fig. 5. Illustration of a four-terminal mesoscopic thermoelectric device as a Maxwell demon. Maxwell's demon does not supply any energy to the working substance, i.e., the total heat current injected into the central quantum system from the two thermal baths is zero, $J_{\rm in}=J_{\rm H}-J_{\rm C}=0$.
The condition at which Maxwell's demon neither injects nor extracts heat or energy into the working substance is[58] $$ J_{\rm in} = 0.~~ \tag {17} $$ The power of the Maxwell demon vanishes when the temperature gradient between the two heat baths H and C is vanishing. In Fig. 6, we show that in our system it is indeed able to realize the cooling by thermal current with vanishing total injected heat, $J_{\rm in}=0$. We find that the cooling by thermal current effect emerges in the whole region of $0\le \varepsilon_2\le 5$ meV, regardless of the sign change and the vanishing of the total heat injected into the QDs $u1$ and $d1$ from the two heat baths H and C. From the illustration in Fig. 6(b), we can find that $J_{\rm S}A_{\rm S}$ is negative and $J_qA_q$ is positive when $\varepsilon_2\approx 2.1\,$ meV, i.e., $J_{\rm in}=0$. Therefore, the entropy of the whole system is still increasing, and the entropy decrease of the working substance is compensated by the entropy increase of the two heat baths. This results are consistent with Ref. [58].
cpl-38-8-088801-fig6.png
Fig. 6. (a) Cooling power $J_{\rm S}$, the total injected heat $J_{\rm in}$, and the thermal current $J_q$ as a function of the QD energy $E_{u2}=E_{d2}=\varepsilon_2$. (b) COP ratio $\eta_{\scriptscriptstyle {\rm COP}}/\eta_{\scriptscriptstyle {\rm COP}}^{\rm rev}$, entropy production rate $\partial _tS_{\rm tot}$ as a function of the QDs energy $E_{u2}=E_{d2}=\varepsilon_2$. Inset: $-J_{\rm S}A_{\rm S}$, $J_qA_q$ as a function of the QDs energy $\varepsilon_2$. Other parameters: $\mu_{\scriptscriptstyle {\rm S}}=\mu_{\scriptscriptstyle {\rm D}}=0$, $k_{\scriptscriptstyle {\rm B}}T_{\rm D}=1$ meV, $k_{\scriptscriptstyle {\rm B}}T_{\rm H}=6$ meV, $k_{\scriptscriptstyle {\rm B}}T_{\rm S}=0.6$ meV, $T_{\rm C}=1/(2/T_{\rm D}-1/T_{\rm H})$, $E_{u1}=E_{d1}=\varepsilon_1=4$ meV, $E_{\rm C}=2$ meV, and $\varGamma_0=0.1$ meV.
Conclusion and Discussions. In this work, we have demonstrated a mode of the cooling by thermal current effect, the transverse thermoelectric effect and Maxwell's demon effect, through a four-terminal four QDs thermoelectric device (the four terminals are the source, the drain and two electronic thermal baths, respectively). We have demonstrated that the Coulomb interaction between the two circuits separated by an insulating layer is the key factor and the inelastic thermoelectric transport in the two circuits leads to these three unconventional thermoelectric effects. The source can be cooled by passing a thermal current between the two electronic heat baths. We have also shown that the two thermal baths can be exploited for power generation and cooling the low-temperature heat bath (or heating the high-temperature heat bath) in a demon-like way, where the working substance does not exchange any energy or particle with the two thermal baths. Our study may be beneficial for autonomous and energy-efficient electric applications on the nanoscale devices. The authors thank Professor Chen Wang for helpful discussion. References Colloquium : Heat flow and thermoelectricity in atomic and molecular junctionsLinear and nonlinear mesoscopic thermoelectric transport with coupling with heat bathsFundamental aspects of steady-state conversion of heat to work at the nanoscaleMultichannel Landauer formula for thermoelectric transport with application to thermopower near the mobility edgeThe best thermoelectric.Thin-film thermoelectric devices with high room-temperature figures of meritIntense terahertz laser fields on a quantum dot with Rashba spin-orbit couplingReversible Thermoelectric NanomaterialsOptimal Bandwidth for High Efficiency ThermoelectricsThermoelectric cooperative effect in three-terminal elastic transport through a quantum dotThree-terminal refrigerator based on resonant-tunneling quantum wellsOptimal energy quanta to current conversionScattering Theory of Nonlinear Thermoelectric TransportVibrational cooling, heating, and instability in molecular conducting junctions: full counting statistics analysisThermoelectric three-terminal hopping transport through one-dimensional nanosystemsPowerful and efficient energy harvester with resonant-tunneling quantum dotsThermoelectric energy harvesting with quantum dotsStaircase Quantum Dots Configuration in Nanowires for Optimized Thermoelectric PowerQuantum efficiency bound for continuous heat engines coupled to noncanonical reservoirsNonlinear effects for three-terminal heat engine and refrigeratorEnhancing Thermoelectric Performance Using Nonlinear Transport EffectsNear-field three-terminal thermoelectric heat engineAbsorption refrigerators based on Coulomb-coupled single-electron systemsThermal drag in electronic conductorsQuantum-dot circuit-QED thermoelectric diodes and transistorsThermoelectric Conversion at 30 K in InAs/InP Nanowire Quantum DotsExperimental Realization of a Quantum Dot Energy HarvesterRectification of thermal fluctuations in a chaotic cavity heat engineEnhancing efficiency and power of quantum-dots resonant tunneling thermoelectrics in three-terminal geometry by cooperative effectsCooling by Heating: Very Hot Thermal Light Can Significantly Cool Quantum SystemsCooling by Heating: Refrigeration Powered by PhotonsBrownian thermal transistors and refrigerators in mesoscopic systemsThermal Diode: Rectification of Heat FluxNegative differential thermal resistance and thermal transistorPhonon thermoelectric transistors and rectifiersQuantum Thermal TransistorAll-thermal transistor based on stochastic switchingThree-terminal thermoelectric transport through a molecular junctionThermoelectric transport with electron-phonon coupling and electron-electron interaction in molecular junctionsHopping thermoelectric transport in finite systems: Boundary effectsFull counting statistics of vibrationally assisted electronic conduction: Transport and fluctuations of thermoelectric efficiencyCoulomb dragMesoscopic Coulomb Drag, Broken Detailed Balance, and Fluctuation RelationsVoltage Fluctuation to Current Converter with Coulomb-Coupled Quantum DotsThree-terminal quantum-dot refrigeratorsThermoelectrics with Coulomb-coupled quantum dotsThermal transistor and thermometer based on Coulomb-coupled conductorsAutonomous conversion of information to work in quantum dotsPerformance of the T -matrix based master equation for Coulomb drag in double quantum dotsAndreev-Coulomb Drag in Coupled Quantum DotsCorrelations of heat and charge currents in quantum-dot thermoelectric enginesThermoelectricity without absorbing energy from the heat sourcesThermodynamic bounds and general properties of optimal efficiency and power in linear responsesEfficiency Statistics and Bounds for Systems with Broken Time-Reversal SymmetryPower-Efficiency-Dissipation Relations in Linear ThermodynamicsUnconventional four-terminal thermoelectric transport due to inelastic transport: Cooling by transverse heat current, transverse thermoelectric effect, and Maxwell demonDegree of coupling and its relation to efficiency of energy conversionEfficiency and dissipation in a two-terminal thermoelectric junction, emphasizing small dissipationOptimal efficiency and power, and their trade-off in three-terminal quantum thermoelectric engines with two output electric currentsExperimental Observation of the Role of Mutual Information in the Nonequilibrium Dynamics of a Maxwell DemonOn-Chip Maxwell’s Demon as an Information-Powered RefrigeratorExperimental realization of a Szilard engine with a single electronPower generator driven by Maxwell’s demonNonequilibrium System as a DemonMaxwell's demon in a double quantum dot with continuous charge detection
[1]Chen G 2005 Nanoscale Energy Transport and Conversion (Oxford: Oxford University Press)
[2] Dubi Y and Di Ventra M 2011 Rev. Mod. Phys. 83 131
[3] Jiang J H and Imry Y 2016 C. R. Phys. 17 1047
[4] Benenti G, Casati G, Saito K, and Whitney R S 2017 Phys. Rep. 694 1
[5] Sivan U and Imry Y 1986 Phys. Rev. B 33 551
[6] Mahan G D and Sofo J O 1996 Proc. Natl. Acad. Sci. USA 93 7436
[7]Goldsmid H J 2010 Introduction to Thermoelectricity (Berlin: Springer)
[8] Venkatasubramanian R, Silvola E, Colpitts T, and O'Quinn B 2001 Nature 413 597
[9] Jiang J H, Weng M Q, and Wu M W 2006 J. Appl. Phys. 100 063709
[10] Humphrey T E and Linke H 2005 Phys. Rev. Lett. 94 096601
[11] Zhou J, Yang R, Chen G, and Dresselhaus M S 2011 Phys. Rev. Lett. 107 226601
[12] Lu J, Wang R, Liu Y, and Jiang J H 2017 J. Appl. Phys. 122 044301
[13] Lin Z, Yang Y Y, Li W, Wang J, and He J 2020 Phys. Rev. E 101 022117
[14] Sánchez R and Büttiker M 2011 Phys. Rev. B 83 085428
[15] Sánchez D and López R 2013 Phys. Rev. Lett. 110 026804
[16] Simine L and Segal D 2012 Phys. Chem. Chem. Phys. 14 13820
[17] Jiang J H, Entin-Wohlman O, and Imry Y 2012 Phys. Rev. B 85 075412
[18] Jordan A N, Sothmann B, Sánchez R, and Büttiker M 2013 Phys. Rev. B 87 075312
[19] Sothmann B, Sánchez R, and Jordan A N 2015 Nanotechnology 26 032001
[20] Li L and Jiang J H 2016 Sci. Rep. 6 31974
[21] Agarwalla B K, Jiang J H, and Segal D 2017 Phys. Rev. B 96 104304
[22] Wang R, Lu J, Wang C, and Jiang J H 2018 Sci. Rep. 8 2607
[23] Jiang J H and Imry Y 2017 Phys. Rev. Appl. 7 064001
[24] Jiang J H and Imry Y 2018 Phys. Rev. B 97 125422
[25] Erdman P A, Bhandari B, Fazio R, Pekola J P, and Taddei F 2018 Phys. Rev. B 98 045433
[26] Bhandari B, Chiriacò G, Erdman P A, Fazio R, and Taddei F 2018 Phys. Rev. B 98 035415
[27] Lu J, Wang R, Ren J, Kulkarni M, and Jiang J H 2019 Phys. Rev. B 99 035129
[28] Prete D, Erdman P A, Demontis V, Zannier V, Ercolani D, Sorba L, Beltram F, Rossella F, Taddei F, and Roddaro S 2019 Nano Lett. 19 3033
[29] Jaliel G, Puddy R K, Sánchez R, Jordan A N, Sothmann B, Farrer I, Griffiths J P, Ritchie D A, and Smith C G 2019 Phys. Rev. Lett. 123 117701
[30] Sothmann B, Sánchez R, Jordan A N, and Büttiker M 2012 Phys. Rev. B 85 205301
[31] Jiang J H 2014 J. Appl. Phys. 116 194303
[32] Mari A and Eisert J 2012 Phys. Rev. Lett. 108 120602
[33] Cleuren B, Rutten B, and van den Broeck C 2012 Phys. Rev. Lett. 108 120603
[34] Lu J, Wang R, Wang C, and Jiang J H 2020 Phys. Rev. B 102 125405
[35] Li B, Wang L, and Casati G 2004 Phys. Rev. Lett. 93 184301
[36] Li B, Wang L, and Casati G 2006 Appl. Phys. Lett. 88 143501
[37] Jiang J H, Kulkarni M, Segal D, and Imry Y 2015 Phys. Rev. B 92 045309
[38] Joulain K, Drevillon J, Ezzahri Y, and Ordonez-Miranda J 2016 Phys. Rev. Lett. 116 200601
[39] Sánchez R, Thierschmann H, and Molenkamp L W 2017 Phys. Rev. B 95 241401
[40] Entin-Wohlman O, Imry Y, and Aharony A 2010 Phys. Rev. B 82 115314
[41] Ren J, Zhu J X, Gubernatis J E, Wang C, and Li B 2012 Phys. Rev. B 85 155443
[42] Jiang J H, Entin-Wohlman O, and Imry Y 2013 Phys. Rev. B 87 205420
[43] Agarwalla B K, Jiang J H, and Segal D 2015 Phys. Rev. B 92 245418
[44] Narozhny B N and Levchenko A 2016 Rev. Mod. Phys. 88 025003
[45] Sánchez R, López R, Sánchez D, and Büttiker M 2010 Phys. Rev. Lett. 104 076801
[46] Hartmann F, Pfeffer P, Höfling S, Kamp M, and Worschech L 2015 Phys. Rev. Lett. 114 146805
[47] Zhang Y, Lin G, and Chen J 2015 Phys. Rev. E 91 052118
[48] Thierschmann H, Sánchez R, Sothmann B, Buhmann H, and Molenkamp L W 2016 C. R. Phys. 17 1109
[49] Yang J, Elouard C, Splettstoesser J, Sothmann B, Sánchez R, and Jordan A N 2019 Phys. Rev. B 100 045418
[50] Sánchez R, Samuelsson P, and Potts P P 2019 Phys. Rev. Res. 1 033066
[51] He W X, Cao Z, Li G Y, Li L, Lü H F, Li Z, and Luo H G 2020 Phys. Rev. B 101 035417
[52] Tabatabaei S M, Sanchez D, Yeyati A L, and Sanchez R 2020 Phys. Rev. Lett. 125 247701
[53] Sánchez R, Sothmann B, Jordan A N, and Bttiker M 2013 New J. Phys. 15 125001
[54] Whitney R S, Sánchez R, Haupt F, and Splettstoesser J 2016 Physica E 75 257
[55] Jiang J H 2014 Phys. Rev. E 90 042126
[56] Jiang J H, Agarwalla B K, and Segal D 2015 Phys. Rev. Lett. 115 040601
[57] Proesmans K, Cleuren B, and van den Broeck C 2016 Phys. Rev. Lett. 116 220601
[58] Lu J, Jiang J H, and Imry Y 2021 Phys. Rev. B 103 085429
[59] Kedem O and Caplan S R 1965 Trans. Faraday Soc. 61 1897
[60] Entin-Wohlman O, Jiang J H, and Imry Y 2014 Phys. Rev. E 89 012123
[61] Lu J, Liu Y, Wang R, Wang C, and Jiang J H 2019 Phys. Rev. B 100 115438
[62]Maxwell J C 1871 Theory of Heat (London: Longman)
[63] Koski J V, Maisi V F, Sagawa T, and Pekola J P 2014 Phys. Rev. Lett. 113 030601
[64] Koski J V, Kutvonen A, Khaymovich I M, Ala-Nissila T, and Pekola J P 2015 Phys. Rev. Lett. 115 260602
[65] Koski J V, Maisi V F, Pekola J P, and Averin D V 2014 Proc. Natl. Acad. Sci. USA 111 13786
[66] Chida K, Desai S, Nishiguchi K, and Fujiwara A 2017 Nat. Commun. 8 15310
[67] Sánchez R, Splettstoesser J, and Whitney R S 2019 Phys. Rev. Lett. 123 216801
[68] Annby-Andersson B, Samuelsson P, Maisi V F, and Potts P P 2020 Phys. Rev. B 101 165404