⇦Chinese Physics Letters, 2020, Vol. 37, No. 12, Article code 120502 Three-Terminal Thermionic Heat Engine Based on Semiconductor Heterostructures Yun-Yun Yang (杨贇贇), Shuai Xu (徐帅), and Ji-Zhou He (何济洲)* Affiliations Department of Physics, Nanchang University, Nanchang 330031, China Received 15 September 2020; accepted 21 October 2020; published online 8 December 2020 Supported by the National Natural Science Foundation of China (Grant No. 11875034).
*Corresponding author. Email: hjzhou@ncu.edu.cn Citation Text: Yang Y Y, Xu S, and He J Z 2020 Chin. Phys. Lett. 37 120502    Abstract We propose a model for three-terminal thermionic heat engines based on semiconductor heterostructures. According to electron transport theory, we drive the formulas for the charge current and energy current flowing from the electron reservoir and we then obtain the power output and efficiency in the linear and nonlinear regimes. Furthermore, we analyze the performance characteristic of the thermionic heat engine and get the maximum power output by optimizing the performance parameters. Finally, we optimize the thermodynamic performance of the thermionic heat engine by maximizing the product of the power output and efficiency. DOI:10.1088/0256-307X/37/12/120502 PACS:05.70.-a, 73.50.Lw, 73.63.Hs, 85.80.Fi © 2020 Chinese Physics Society Article Text Thermoelectric devices can either be used for heat engines, where they convert input thermal energy to electrical power, or for refrigerators, where applied electrical power is used for thermal energy. Recently, interest has grown in investigating thermoelectric devices with high efficiency and high power, including quantum dots,[1–3] nanowires,[4,5] quantum wells[6,7] and superlattices.[8] In 1993, Edwards et al. investigated theoretically the three-terminal quantum-dot refrigerator model based on two coupled discrete quantum dots, and they analyzed the refrigerating performance using a numerical simulation.[9,10] Later, Jordan et al. proposed nanoscale heat engines based on resonant-tunneling quantum dots, and obtained a new high-efficiency energy harvesting device.[11] The thermoelectric performance of the three-terminal heat engine and refrigerator based on resonant-tunneling wells was proposed.[12–14] Similarly, Choi et al. analyzed the nearly rectangular transmission mode of the conduction band in a superlattice and obtained higher power relative to the quantum well.[15] Jiang et al. proposed a quantum-dots resonant tunneling thermoelectric engine in three-terminal geometry, and they enhanced the efficiency and power by cooperative effects.[16,17] The three-terminal thermoelectric devices with energy selective electron or resonant-tunneling quantum dots have been extensively studied in recent years.[18–23] Next, Shakouri et al. and Mahan et al. proposed solid-state thermionic energy conversion.[24–26] As power generators or refrigerators, thermionic devices could be even better than thermoelectric devices.[27–29] Recently, some sold-state thermionic power generators using two-dimensional (2D) heterostructures have been extensively studied.[30–33] The descriptions of thermoelectric devices and thermionic devices are very similar. Both the devices provide a temperature gradient and carry energy by moving electrons. Generally, we distinguish the two devices according to whether the carrier transport mode is ballistic or diffusion. In thermoelectric devices, the motion of electrons is in quasi-equilibrium and is diffusive. In thermionic motion, the device has relatively high efficiency when the electrons ballistically move across the barrier. To meet the ballistic transmission requirements of thermionic devices, we need to control the barrier thickness along the transmission direction to the nanometer level. Based on the past works, we propose a three-terminal thermionic heat engine that is based on two-dimensional semiconductor heterostructures. It is important to note that the two-dimensional energy barrier filters electrons according to their cross-plane momentum rather than the total energy.[34–37] Here, we assume that tunneling is suppressed and that the electrons ballistically cross the barrier. The three-terminal thermionic device that we consider is illustrated in Fig. 1, which consists of a middle cavity connected to the left/right electronic reservoir via 2D semiconductor heterostructures. The electron occupation probability of the electronic reservoirs, $i={\rm L, R}$, is described by the Fermi–Dirac function $f_{i} (E)=\{{\exp[{({E-\mu_{i}})/k_{\rm B} T_{i}}]+1}\}^{-1}$ with temperature $T_{i}$ and chemical potential $\mu_{i}$. We assume that the middle cavity is in thermal equilibrium with the thermal reservoir of temperature $T_{\rm C}$. Considering the strong coupling between electrons, and the strong interaction between electron and phonon, the electron occupation probability of the middle cavity can also be described by the Fermi–Dirac function $f_{\rm C}(E)=\{{\exp[{({E-\mu_{\rm C}})/k_{\rm B} T_{\rm C} }]+1}\}^{-1}$ with temperature $T_{\rm C}$ and chemical potential $\mu_{\rm C}$. The temperature of the left/right reservoir and the middle cavity satisfies ($T_{\rm L} =T_{\rm R} < T_{\rm C}$). A bias voltage $V$ is applied between the left-hand and right-hand reservoirs, and satisfies $eV=\mu_{\rm R} -\mu_{\rm L}$ ($e$ is electronic charge).
cpl-37-12-120502-fig1.png
Fig. 1. Schematic diagram of the three-terminal thermionic heat engine based on semiconductor heterostructures. The black arrows show the path of electron transition, and horizontal bars on the left-hand and right-hand show two level positions $E_{\rm L}$ and $E_{\rm R}$.
Based on the Tsu–Esaki formula,[38] the charge current out of the left/right reservoir can be given by $$ I_{i} =\frac{em^{\ast }A}{2\pi^{2}\hbar ^{3}}\int {[{\zeta_{i}-\zeta_{\rm C}}]}\tau_{i}({E_{x}})dE_{x},~~ \tag {1} $$ where $$ \zeta_{i/{\rm C}} =k_{\rm B}T_{i/{\rm C}} \log\Big[ {1+\exp \Big({\frac{\mu_{i/{\rm C}} -E_{x}}{k_{\rm B}T_{i/{\rm C}}}}\Big)}\Big],~~ \tag {2} $$ and $m^{\ast}$ is the effective mass of electrons, $k_{\rm B}$ is the Boltzmann constant, $A$ is the surface area of 2D semiconductor heterostructures. The transmission function can be expressed as $$ \tau_{i} ({E_{x}})=\alpha_{i} \varTheta ({E_{x}-E_{i}}),~~ \tag {3} $$ where $E_{i} ={({\hbar k_{x}'})} / {2m^{\ast }}$ and $\hbar k_{x}'$ is the minimum cross-plane momentum required to cross the energy barrier, $\alpha_{i}$ is a dimensionless constant. Then, the energy current out of the left/right electronic reservoir is given by $$ J_{i} =\frac{m^{\ast }A}{2\pi^{2}\hbar ^{3}}\int {[ {\varphi_{i} \zeta_{i} -\varphi_{\rm C} \zeta_{\rm C}}]} \tau_{i} ({E_{x} })dE_{x},~~ \tag {4} $$ where $\varphi_{i/{\rm C}} =E_{x} +k_{\rm B} T_{i/{\rm C}}$. The chemical potential $\mu_{\rm C}$ of the middle cavity can be determined by the conservation of charge current, $I_{\rm L} +I_{\rm R} =0$. The heat current $J$ flowing from the heat reservoir into the cavity can be obtained by the energy conservation law $J+J_{\rm L} +J_{\rm R} =0$. When the bias voltage $V$ is applied to the heat-driven charge current, the power output is given by $P=IV$. The efficiency of the thermionic heat engine can be defined as the ratio of the power output to the heat current flowing from the heat reservoir, i.e., $\eta =P/J$. In the following, we analyze the general performance characteristics of the three-terminal thermionic heat engine in the linear and nonlinear regimes, respectively. For simplicity, we introduce the average temperature $T=({T_{i}+T_{\rm C}})/2$ and the temperature difference $\Delta T=T_{\rm C} -T_{i}$. When the bias voltage $eV=\mu_{\rm R} -\mu_{\rm L}$ is applied, we take $\mu_{\rm R} =eV/2$ and $\mu_{\rm L} =-eV/2$. Here, we choose the dimensionless constants $\alpha_{\rm L} =1+a$ and $\alpha_{\rm R} =1-a$, respectively, where $a$ is the asymmetric factor which describes the asymmetry between left and right semiconductor heterostructures and satisfies the bounds $-1 < a < 1$. To get the linear-response system, we need to limit the temperature difference and the chemical potential, i.e., $eV,k_{\rm B} \Delta T\ll k_{\rm B} T$. With the conservation laws, the charge current flowing through the system and the heat current flowing from the thermal reservoir are given by $$ I=I_{\rm L} =-I_{\rm R} =GV+GS\Delta T,~~ \tag {5} $$ $$ J=G\varPi V+({K+GS\varPi})\Delta T,~~ \tag {6} $$ respectively, where $G$ is the electric conductance, $K$ is the thermal conductance, $S$ is the thermopower, $\varPi$ is the Peltier coefficient. $$\begin{align} &G=\frac{-e^{2}m^{\ast }A}{2\pi^{2}\hbar ^{3}}\frac{G_{\rm L1} G_{\rm R1} }{G_{\rm L1} +G_{\rm R1} },~~ \tag {7} \end{align} $$ $$\begin{align} &S=\frac{k_{\rm B} }{e}\Big({\frac{G_{\rm L2} +G_{\rm L3} }{G_{\rm L1} }-\frac{G_{\rm R2} +G_{\rm R3} }{G_{\rm R1}}}\Big),~~ \tag {8} \end{align} $$ $$\begin{align} &\varPi =\frac{k_{\rm B} T}{e}\Big({\frac{G_{\rm R2} }{G_{\rm R1} }-\frac{G_{\rm L2} }{G_{\rm L1} }}\Big),~~ \tag {9} \end{align} $$ $$\begin{align} K={}&\frac{m^{\ast }Ak_{\rm B}^{2} T}{2\pi^{2}\hbar ^{3}}\Big[(G_{\rm L2} +2G_{\rm L3} +G_{\rm L4} +G_{\rm L5} +G_{\rm R2}\\ &+2G_{\rm R3} +G_{\rm R4} +G_{\rm R5}) \\ &-\frac{({G_{\rm L1}+G_{\rm L2}})({G_{\rm L2}+G_{\rm L3}})}{G_{\rm L1}}\\ &-\frac{({G_{\rm R1}+G_{\rm R2}})({G_{\rm R2}+G_{\rm R3}})}{G_{\rm R1} }\Big],~~ \tag {10} \end{align} $$ where we have introduced the auxiliary functions $$\begin{align} &G_{i1} =\int {\frac{1}{1+e^{E_{x} /k_{\rm B} T}}} \tau_{i} (E_{x})dE_{x},\\ &G_{i2} =\int {\frac{E_{x} /k_{\rm B} T}{1+e^{E_{x} /k_{\rm B} T}}} \tau_{i} (E_{x})dE_{x},\\ &G_{i3} =\int {\log[1+e^{-E_{x} /k_{\rm B} T}]} \tau_{i} (E_{x})dE_{x},\\ &G_{i4} =\int {\frac{({E_{x}/k_{\rm B}T})^{2}}{1+e^{E_{x} /k_{\rm B} T}}} \tau_{i} (E_{x})dE_{x},\\ &G_{i5} =\int {\frac{E_{x} }{k_{\rm B} T}\log[1+e^{-E_{x} /k_{\rm B} T}]} \tau_{i} (E_{x})dE_{x}. \end{align} $$ The power output is $$ P=IV=({GV+GS\Delta T})V.~~ \tag {11} $$ The power vanishes either when the bias voltage is not applied or at the stopping voltage $V_{\rm stop} =-S\Delta T$. Furthermore, the bias voltage at maximum power output is at half of the stopping voltage, i.e., $V_{m} =-S\Delta T/2$. Here, the maximum power output is $$ P_{\max } =-G\frac{({S\Delta T})^{2}}{4}.~~ \tag {12} $$ For the bias voltage $V_{m}$ that delivers the maximum power output, the corresponding heat current is given by $$ J_{\rm P}=\Big({\frac{GS\varPi}{2}+K}\Big)\Delta T.~~ \tag {13} $$ The efficiency at maximum power output is simply given by $$\begin{align} \eta_{\rm P} &=\frac{P_{\max } }{J_{\rm P} }=\Big({\frac{-GS^{2}}{2GS\varPi +4\,K}}\Big)\Delta T\\ &\approx \Big({\frac{-GS^{2}T}{2GS\varPi +4\,K}}\Big)\eta_{\rm C},~~ \tag {14} \end{align} $$ where $\eta_{\rm C}$ is the Carnot efficiency. For $\Delta T\ll T$, $\eta_{\rm C} =\Delta T/T_{\rm C} \approx \Delta T/T$.
cpl-37-12-120502-fig2.png
Fig. 2. (a) The maximum power output in units of $\frac{m^{\ast }Ak_{\rm B} T}{2\pi^{2}\hbar ^{3}}\big({\frac{k_{\rm B} \Delta T}{2}}\big)^{2}$ and (b) the efficiency at maximum power output in units of $\eta_{\rm C}$ varying with two level positions for $a=0$. (c) and (d) The same cases as (a) and (b) but for $a=-0.5$.
According to Eqs. (5)-(14) and the numerical simulation, we can plot the three-dimensional graphs of the maximum power output $P_{\max}$ and the efficiency at maximum power output $\eta_{\rm P} /\eta_{\rm C}$ varying with two level positions $E_{\rm L}$ and $E_{\rm R}$, as shown in Figs. 2(a) and 2(b). We will first focus on the symmetric case, $a=0$. In Figs. 2(a) and 2(b), the power output and efficiency are symmetric concerning the exchange of $E_{\rm L}$ and $E_{\rm R}$. As the values of $E_{\rm L}$ and $E_{\rm R}$ decrease continuously, the maximum power output increases. It has a saturation value when one level position is $-E_{\rm L/R} \gg k_{\rm B} T$ and the other level position is $E_{\rm R/L} \approx 0k_{\rm B} T$. In contrast to the maximum power output, the efficiency reduces as the values of $E_{\rm L}$ and $E_{\rm R}$ decrease continuously. It can be seen from Fig. 2 that when the maximum power output reaches the saturation value, the corresponding efficiency is very small. The reason for this is that the 2D semiconductor heterostructures can transmit any electrons whose energy is larger than the level positions. Therefore, the 2D semiconductor heterostructures are low-efficiency energy filters. In an asymmetric case $a=-0.5$, the power output and efficiency are asymmetric concerning the exchange of $E_{\rm L}$ and $E_{\rm R}$. It can be seen from Figs. 2(c) and 2(d) that the maximum power output and the efficiency at maximum power output in the upper left-hand region are obviously very much larger than those in the lower right-hand region. If $a>0$, then the result is the opposite. In particular, the maximum power output and the efficiency at maximum power output in an asymmetric case are larger than those in the symmetric case. Therefore, we can optimize the asymmetric factor $a$ to get the maximum power output. In Fig. 3, we plot three-dimensional graphs of the maximum power output $P_{\max}$ and the efficiency at maximum power output $\eta_{\rm P} /\eta_{\rm C}$ varying with the level position $E_{\rm R}$ and asymmetric factor $a$. It can be seen from Fig. 3 that the maximum power output is $$ P_{\max } \approx 5.7\frac{m^{\ast }Ak_{\rm B} T}{2\pi^{2}\hbar ^{3}}\Big({\frac{k_{\rm B}\Delta T}{2}}\Big)^{2}, $$ where $E_{\rm R} \approx -0.1k_{\rm B} T$ and $a\approx -0.75k_{\rm B} T$ while $-E_{\rm L} \gg k_{\rm B} T$.
cpl-37-12-120502-fig3.png
Fig. 3. (a) The maximum power output in units of $\frac{m^{\ast }Ak_{\rm B} T}{2\pi^{2}\hbar ^{3}}\big({\frac{k_{\rm B} \Delta T}{2}}\big)^{2}$ and (b) the efficiency at maximum power output in units of $\eta_{\rm C}$ varying with the level position $E_{\rm R}$ and asymmetric factor $a$. For both plots, $E_{\rm L} =-40k_{\rm B} T$.
We estimate the maximum power output and the efficiency at maximum power output for the given parameters $m^{\ast }=0.067m_{\rm e}$, $T=300$ K, $\Delta T=1$ K, $a=-0.75k_{\rm B} T$, $E_{\rm L} =-40k_{\rm B} T$, $E_{\rm R} =-0.1k_{\rm B} T$, and obtain that the maximum power output is about $0.3\,{\rm W/cm}^{2}$, the corresponding efficiency is about $0.006\eta_{\rm C}$. In Table 1, we compare the maximum power output and the efficiency at maximum power output for three similar heat engines based on quantum dots,[11] quantum wells[12] and semiconductor heterostructures. We find that the maximum power output of the thermionic heat engine based on semiconductor heterostructure is highest, but the corresponding efficiency is very small.
Table 1. The maximum power output and the efficiency at maximum power output for three similar heat engines based on quantum dots, quantum wells and semiconductor heterostructures in $T=300$ K and $\Delta T=1$ K.
Quantum dots Quantum wells Thermionics
$P_{\max } \,({\rm W/cm^{2}})$ 0.1 0.18 0.3
$\eta_{\rm P}/{\eta_{\rm C}}$ 0.2 0.07 0.006
For the nonlinear regime, we do not need to assume that the values of the temperature difference $\Delta T$ and the bias voltage $V$ are very small. Therefore, the temperature difference $\Delta T$ and the bias voltage $V$ are also the control parameters that we need to optimize. To simplify the power output and the efficiency and clarify the relationship between the various parameters, the power output and the efficiency are rewritten as $$\begin{align} &P=I_{\rm L} V=P\Big({a, \frac{eV}{k_{\rm B} T}, \frac{\Delta T}{T}, \frac{E_{\rm L} }{k_{\rm B} T}, \frac{E_{\rm R} }{k_{\rm B} T}}\Big),~~ \tag {15} \end{align} $$ $$\begin{align} &\eta =\frac{P}{J}=\eta \Big({a, \frac{eV}{k_{\rm B} T}, \frac{\Delta T}{T}, \frac{E_{\rm L} }{k_{\rm B} T}, \frac{E_{\rm R} }{k_{\rm B} T}}\Big).~~ \tag {16} \end{align} $$
cpl-37-12-120502-fig4.png
Fig. 4. (a) The power output in units of $\frac{m^{\ast }A({k_{\rm B} T})^{3}}{2\pi^{2}\hbar ^{3}}$ and (b) the efficiency in units of $\eta_{\rm C}$ varying with two level positions for $a=0$. (c) and (d) The same cases as (a) and (b) but for $a=-0.5$.
According to Eqs. (1)-(4) and (15) and (16), we can plot three-dimensional graphs of the power output $P$ and the efficiency $\eta /\eta_{\rm C}$ varying with two level positions $E_{\rm L}$ and $E_{\rm R}$ at different asymmetric factor $a$ for given $\Delta T/T=0.5$, $eV/k_{\rm B} T=0.5$, as shown in Fig. 4. Since $eV/k_{\rm B} T>0$, the working region concentrates on the upper left one, which is different from that in the linear regime. In Fig. 4, we can get that the power output increases as the level position $E_{\rm L}$ decreases and the power output reaches a maximum value at $-E_{\rm L} \gg k_{\rm B} T$ and $E_{\rm R} \approx 0k_{\rm B} T$. However, the efficiency decreases as the level position $E_{\rm L}$ decreases. Similarly, when $a=-0.5$, the power output and efficiency are larger than the former. Firstly, using Eqs. (15) and (16) and the extremal conditions $$ \frac{\partial P}{\partial E_{\rm L} }=0,~~~\frac{\partial P}{\partial E_{\rm R} }=0,~~ \tag {17} $$ the maximum power output $P_{\max}$ and the efficiency at maximum power output $\eta_{\rm P}$ can be calculated. Thus, we can plot curves of the maximum power output $P_{\max}$ and the efficiency at maximum power output $\eta_{\rm P}$ varying with the bias voltage at different asymmetric factor $a$, as shown in Fig. 5. It is obvious that the maximum power output $P_{\max}$ increases at first and then decreases as the bias voltage increases. When the bias voltage $eV\approx 0.5k_{\rm B} T$, the maximum power output reaches its maximum. In addition, the maximum power output increases as the asymmetric factor decreases. Similarly, the efficiency at maximum power output increases as the asymmetric factor decreases. Thus, to achieve the maximum power output and the efficiency at maximum power output, we should choose the bias voltage $eV\approx 0.5k_{\rm B} T$ and the asymmetric factor $a$ should be as small as possible.
cpl-37-12-120502-fig5.png
Fig. 5. (a) The maximum power output in units of $\frac{m^{\ast }A({k_{\rm B} T})^{3}}{2\pi^{2}\hbar ^{3}}$ and (b) the efficiency at maximum power output in units of $\eta_{\rm C}$ varying with the bias voltage at different asymmetric factor $a$.
Next, we turn to numerically optimize the bias voltage $V$, the asymmetric factor $a$ as well as the level position $E_{\rm L/R}$ simultaneously to maximize the power output. The curves of the optimized parameters that vary with the temperature difference $\Delta T$ are shown in Fig. 6. We can see that the optimal asymmetric factor $a\approx -0.75$ is an invariant value, the optimal bias voltage appears in linear growth as $\Delta T$ increases, and the level position $E_{\rm R}$ increases slightly as $\Delta T$ increases. The level position $E_{\rm L}$ is considered as $-E_{\rm L} \gg k_{\rm B} T$ because the maximum power output reaches its saturation value when $E_{\rm L}$ is a large negative value. The maximum power output increases nonlinearly as the temperature difference increases. The efficiency at maximum power output increases slowly as the temperature difference increases, and its value is very small. By analyzing the data, for a small value of $\Delta T/T$, we find that the maximum power output in the nonlinear regime returns one in the linear regime, and the maximum power output can be approximately given by $$ P_{\max } \approx 1.42\frac{m^{\ast }A({k_{\rm B} T})^{3}}{2\pi^{2}\hbar ^{3}}\Big({\frac{\Delta T}{T}}\Big)^{2}. $$
cpl-37-12-120502-fig6.png
Fig. 6. (a) The maximum power output in units of $\frac{m^{\ast }A({k_{\rm B} T})^{3}}{2\pi^{2}\hbar ^{3}}$ and the efficiency at maximum power output varying with temperature difference $\Delta T$. (b) The level positions, bias voltage and asymmetric factor at maximum power output varying with the temperature difference $\Delta T$.
In addition to the maximum power output as optimization criteria, the efficiency is also a very important performance parameter for the heat engine. However, the maximum efficiency is achieved under conditions that are different from those required for maximum power output, and a larger power output often means smaller efficiency. So, we use the product of power output $P$ and efficiency $\eta$ as optimization criteria, i.e., maximum $\chi =\eta P$.[36,39–42] We turn to numerically optimize the bias $eV$, the asymmetric factor $a$, as well as the level position $E_{\rm L/R}$ simultaneously at maximizing the product of power output and efficiency. The optimization results are shown in Fig. 7. In Fig. 7(b), the optimal asymmetric factor $a\approx -0.93$, the optimal bias voltage appears in linear growth as $\Delta T$ increases, and the level position $E_{\rm R/L}$ decreases slightly as $\Delta T$ increases. In Fig. 7(a), the maximum power output and the efficiency at maximum power output increase nonlinearly as the temperature difference increases. In Table 2, we list the results of the two kinds of optimization criteria. It is found from Table 2 that the power output at maximizing $\chi$ is lower than the maximum power output, while the efficiency at maximizing $\chi$ is greatly higher than the efficiency at the maximum power output. The latter optimization criteria solve the problem of low efficiency of the thermionic heat engine and are more conducive to the practical application.
cpl-37-12-120502-fig7.png
Fig. 7. (a) The power output in units of $\frac{m^{\ast }A({k_{\rm B} T})^{3}}{2\pi^{2}\hbar ^{3}}$ and efficiency at maximizing $\chi$ varying with the temperature difference $\Delta T$. (b) The level positions, bias voltage and asymmetric factor at maximizing $\chi$ varying with the temperature difference $\Delta T$.
Table 2. The comparisons of the power output in units of $\frac{m^{\ast }A({k_{\rm B} T})^{3}}{2\pi^{2}\hbar ^{3}}$ and the efficiency in units of $\eta_{\rm C}$ are listed in two kinds of optimization criteria.
$\Delta T$ Maximize $P$ Maximize $\chi$
$P_{\max}$ $\eta_{\rm P}$ $P_{\chi}$ $\eta_{\chi}$
0 0 0.0057 0 0.1222
0.5 0.3494 0.0071 0.1234 0.1401
1 1.3558 0.0084 0.4892 0.1586
In summary, we have studied the thermodynamic performance of the three-terminal thermionic heat engine based on semiconductor heterostructures. In the linear regime, compared with other thermoelectric three-terminal heat engines based on quantum dot and quantum well, the three-terminal thermionic heat engine has the larger power output but the efficiency at maximum power output is smaller. In the nonlinear regime, we analyze the optimal performance characteristics of the thermionic heat engine at maximizing power output, and maximizing the product of power output and efficiency. The maximum power output grows quadratically with temperature difference while the efficiency at maximum power output increases slowly. Using the product of power output and efficiency as optimization criteria, we can greatly improve the efficiency of the thermionic heat engine at the expense of a small amount of power output. References Quantum-dot Carnot engine at maximum powerStochastically driven single-level quantum dot: A nanoscale finite-time thermodynamic machine and its various operational modesQuantum dot refrigerator driven by photonSilicon nanowires as efficient thermoelectric materialsOptimal performance of three-terminal nanowire heat engine based on one-dimensional ballistic conductorsEffect of quantum-well structures on the thermoelectric figure of meritUse of quantum‐well superlattices to obtain a high figure of merit from nonconventional thermoelectric materialsThin-film thermoelectric devices with high room-temperature figures of meritA quantum‐dot refrigeratorCryogenic cooling using tunneling structures with sharp energy featuresPowerful and efficient energy harvester with resonant-tunneling quantum dotsPowerful energy harvester based on resonant-tunneling quantum wellsA Three-Terminal Quantum Well Heat Engine with Heat LeakageThree-terminal refrigerator based on resonant-tunneling quantum wellsThree-terminal heat engine and refrigerator based on superlatticesThermoelectric three-terminal hopping transport through one-dimensional nanosystemsThree-terminal semiconductor junction thermoelectric devices: improving performanceAn electronic cooling device with multiple energy selective tunnelsPerformance improvement of a four-terminal thermal amplifier with multiple energy selective tunnelsOptimal performance region of energy selective electron cooling devices consisting of three reservoirsOptimal performance regions of an irreversible energy selective electron heat engine with double resonancesThermodynamic Performance of Three-Terminal Hybrid Quantum Dot Thermoelectric Devices *Thermodynamic Performance and Optimal Analysis of A Multi-Level Quantum Dot Thermal AmplifierHeterostructure integrated thermionic coolersMultilayer thermionic refrigerator and generatorMultilayer Thermionic RefrigerationThe B factor in multilayer thermionic refrigerationComparison of solid-state thermionic refrigeration with thermoelectric refrigerationSolid-State Thermionic Power Generators: An Analytical Analysis in the Nonlinear RegimeThermoelectric transport across graphene/hexagonal boron nitride/graphene heterostructuresFirst principles calculations of solid-state thermionic transport in layered van der Waals heterostructuresThermionic Energy Conversion Based on Graphene van der Waals HeterostructuresHigh-Performance Solid-State Thermionic Energy Conversion Based on 2D van der Waals Heterostructures: A First-Principles StudyPower optimization in thermionic devicesElectronic and thermoelectric transport in semiconductor and metallic superlatticesA theoretical study on the performances of thermoelectric heat engine and refrigerator with two-dimensional electron reservoirsThermoelectric efficiency at maximum power in low-dimensional systemsCoefficient of performance under maximum χ criterion in a two-level atomic system as a refrigeratorOptimal low symmetric dissipation Carnot engines and refrigeratorsLow-dissipation heat devices: Unified trade-off optimization and boundsCoefficient of performance at maximum figure of merit and its bounds for low-dissipation Carnot-like refrigerators
[1] Esposito M, Kawai R, Lindenberg K and Van D B C 2010 Phys. Rev. E 81 041106
[2] Esposito M, Kumar N, Lindenberg K and Van D B C 2012 Phys. Rev. E 85 031117
[3] Li W, Fu J, Yang Y Y and He J Z 2019 Acta Phys. Sin. 68 220501 (in Chinese)
[4] Boukai A I, Bunimovich Y, Tahir-Kheli J, Yu J K, Goddard I W A and Heath J R 2008 Nature 451 168
[5] Yang Y Y, Xu S, Li W and He J Z 2020 Phys. Scr. 95 095001
[6] Hicks L D and Dresselhaus M S 1993 Phys. Rev. B 47 12727
[7] Hicks L D, Harman T C and Dresselhaus M S 1993 Appl. Phys. Lett. 63 3230
[8] Venkatasubramanian R, Siivola E, Colpittes T and O'Quinn B 2001 Nature 413 597
[9] Edwards H L, Niu Q and De L A L 1993 Appl. Phys. Lett. 63 1815
[10] Edwards H L, Niu Q, Georgakis G A and De L A L 1995 Phys. Rev. B 52 5714
[11] Jordan A N, Sothmann B, Sánchez R and Büttiker M 2013 Phys. Rev. B 87 075312
[12] Sothmann B, Sánchez R, Jordan A N and Büttiker M 2013 New J. Phys. 15 095021
[13] Lin Z B, Li W, Fu J, Yang Y Y and He J Z 2019 Chin. Phys. Lett. 36 060501
[14] Lin Z B, Yang Y Y, Li W, Wang J H and He J Z 2020 Phys. Rev. E 101 022117
[15] Choi Y and Jordan A N 2015 Physica E 74 465
[16] Jiang J H, Entin-Wohlman O and Imry Y 2012 Phys. Rev. B 85 075412
[17] Jiang J H, Entin-Wohlman O and Imry Y 2013 New J. Phys. 15 075021
[18] Su G Z, Pan Y Z, Zhang Y C, Shih T M and Chen J C 2016 Energy 113 723
[19] Peng W L, Ye Z L, Zhang X and Chen J C 2018 Energy Convers. Manage. 166 74
[20] Qiu S S, Ding Z M, Chen L, Meng F K and Sun F R 2019 Eur. Phys. J. Plus 134 273
[21] Ding Z M, Chen L G, Ge Y L and Xie Z H 2019 Mach. Learn.: Sci. Technol. 62 397
[22] Shi Z C, Fu J, Qin W F and He J Z 2017 Chin. Phys. Lett. 34 110501
[23] Li W, Yang Y Y, Fu J and He J Z 2020 ES Energy & Environ. 7 40
[24] Shakouri A and Bowers J E 1997 Appl. Phys. Lett. 71 1234
[25] Mahan G D, Sofo J O and Bartkowiak M 1998 J. Appl. Phys. 83 4683
[26] Mahan G D and Woods L M 1998 Phys. Rev. Lett. 80 4016
[27] Vining C B and Mahan G D 1999 J. Appl. Phys. 86 6852
[28] Ulrich M D, Barnes P A and Vining C B 2001 J. Appl. Phys. 90 1625
[29] Zebarjadi M 2017 Phys. Rev. Appl. 8 014008
[30] Chen C C, Li Z, Shi L and Cronin S B 2015 Nano Res. 8 666
[31] Wang X, Zebarjadi M and Esfarjani K 2016 Nanoscale 8 14695
[32] Liang S J, Liu B, Hu W, Zhou K and Ang L K 2017 Sci. Rep. 7 46211
[33] Wang X, Zebarjadi M and Esfarjani K 2018 Sci. Rep. 8 9303
[34] Humphrey T E, O'Dwyer M F and Linke H 2005 J. Phys. D 38 2051
[35] Vashaee D and Shakouri A 2004 J. Appl. Phys. 95 1233
[36] Luo X G, He J Z, Long K L, Wang J, Liu N and Qiu T 2014 J. Appl. Phys. 115 244306
[37] Nakpathomkun N, Xu H Q and Linke H 2010 Phys. Rev. B 82 235428
[38]Ferry D and Goodnick S M 1999 Transport in Nanostructures (Cambridge: Cambridge University Press)
[39] Yuan Y, Wang R, He J Z, Ma Y L and Wang J H 2014 Phys. Rev. E 90 052151
[40] De T C, Hernández A C and Roco J M M 2012 Phys. Rev. E 85 010104
[41] De T C, Roco J M M, Hernández A C, Wang Y and Tu Z C 2013 Phys. Rev. E 87 012105
[42] Wang Y, Li M, Tu Z C, Hernández A C and Roco J M M 2012 Phys. Rev. E 86 011127