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Optimum energy conversion strategies of a nano-scaled three-terminal quantum dot thermoelectric device
Energy 85 (2015) 200e207
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Optimum energy conversion strategies of a nano-scaled threeterminal quantum dot thermoelectric device Yanchao Zhang, Chuankun Huang, Junyi Wang, Guoxing Lin, Jincan Chen* Department of Physics, Xiamen University, Xiamen 361005, People's Republic of China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 November 2014 Received in revised form 10 March 2015 Accepted 14 March 2015 Available online 23 April 2015
A model of the three-terminal nano-scaled energy conversion system as a heat engine based on two capacitively coupled quantum dots in the Coulomb-blockade regime is established within four quantum states that include the essential physical features. The dynamical properties of the model are calculated by master equation approach account for the quantitative behavior of such a system. Expressions for the power output and efficiency of the three-terminal quantum dot heat engine are derived. The characteristic curves between the power output and the efficiency are plotted. Moreover, the optimal values of main performance parameters are determined by the numerical calculation. The influence of dissipative tunnel processes on the optimal performance is discussed in detail. The results obtained here can provide some theoretical guidelines for the design and operation of practical three-terminal quantum dot heat engines. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Quantum dot heat engine Performance characteristics Dissipative tunnel Optimum design
1. Introduction In recent years, investigations on the thermoelectric effects of nano-thermoelectric devices have attracted considerable interest due to their importance in developing miniaturized devices which help to utilize energy resources at the microscopic scale [1e5]. In particular, there is broad, current interest in developing high performance thermoelectric materials for the generation of electric power from heat sources such as waste heat. It was pointed out that structures of reduced dimension can give rise to an increased thermoelectric figure of merit ZT as compared to bulk structures made from the same materials [6,7] and that sharp spectral features can improve thermoelectric performances characterized by a high value of ZT in materials with a delta-like density of states [8]. Furthermore, it was reported that the Carnot efficiency can be reached for reversible electron transport between two reservoirs at different temperatures and chemical potentials by using a sharply tuned energy filter for which the electron density is the same in both reservoirs [9e11]. Quantum dots naturally provide these sharp spectral features. Hence, there are many investigations of the quantum dot thermoelectric devices because quantum dots can be applied to the high-
* Corresponding author. Tel.: þ86 592 2180922; fax: þ86 592 2189426. E-mail address: [email protected] (J. Chen). http://dx.doi.org/10.1016/j.energy.2015.03.087 0360-5442/© 2015 Elsevier Ltd. All rights reserved.
potential solid-state energy conversion devices [12e15]. Hence, the study of nano-thermoelectric heat engines and refrigerators using the quantum dot system as the working substance embedded between two reservoirs at different temperatures and chemical potentials has attracted considerable interest [16e21]. In twoterminal nano-thermoelectric device, when both the gradients of temperature and chemical potential are applied to the device and the thermoelectric response as heat engines and refrigerators is investigated, it is found that the heat flow will be accompanied by the electron current. But, for the purpose of energy harvesting, it suffers from the fact that the different parts of the thermoelectric response must be at different temperatures which make thermal isolation difficult [22]. Hence, some theoretical works on the thermoelectric effects in three-terminal quantum dot devices have attracted considerable interest [23e33]. This is because that such a three-terminal thermoelectric device offers the advantage of spatially separating the hot and cold reservoirs and exhibits a crossed flow of heat and electric currents. This is useful for energy harvesting applications because such a three-terminal thermoelectric device can be driven by electronic sources [23e25], phonon sources [26e28], or photon sources [29e31]. The maximum power output and efficiency of a mesoscopic heat engine based on a hot chaotic cavity capacitively coupled to a cold cavity were investigated [24]. A resonant-tunneling quantum dot heat engine was researched. Such a heat engine converts a part of the heat into electrical current in a three-terminal geometry which permits one
Y. Zhang et al. / Energy 85 (2015) 200e207
Nomenclature T U ZT C ~ C
U G p I Q
hC h hmax hmP P Pmax Pmh DV DVopen DVP
DV h kB q
optimum voltage output at maximum efficiency (V) Boltzmann's constant (J K1) elementary charge (C)
temperature (K) Coulomb interaction (J) figure of merit capacitance (F)
m
effective capacitance (F) normalization factor transition rate occupation probability charge currents (W) heat flow (W) Carnot efficiency efficiency maximum efficiency efficiency at maximum power output power output (W) maximum power output (W) power output at maximum efficiency (W) voltage output (V) open circuit voltage (V) optimum voltage output at maximum power output (V)
Subscripts s conductor dot g gate dot n electron occupation number l left side reservoir r right side reservoir max maximum mh maximum efficiency mP maximum power opt optimal
to separate current and heat flows. This provides an option to highly efficient solid-state energy harvesting [25]. Soon after, Sothmann et al. investigated a heat engine consisting of a singlelevel quantum dot coupled to two ferromagnetic metals and one ferromagnetic insulator held at different temperatures and demonstrated that in the tight-coupling limit the device can reach the Carnot efficiency [28]. Photon-driven heat engines or refrigerators were also studied [29e33]. Rutten et al. used the quantum dot structure to design a photoelectric device as a heat engine and studied the thermodynamic efficiency of the heat engine [29]. Ruokola et al. theoretically researched single-electron heat engines coupled to electromagnetic environments and predicted that high efficiencies are possible with the quantum dot system [30]. Only recently, a possibly realized hybrid microwave cavity heat engine consisting of two macroscopically separated quantum dot conductors coupled capacitively to the fundamental mode of a microwave cavity was proposed. Such a heat engine can reach the Carnot efficiency when the optimal conversion is achieved [31]. The thermoelectric properties of two capacitively coupled quantum dots in the Coulomb-blockade regime in a three-terminal nano-sized structure thermoelectric system were firstly analyzed nchez et al. [23]. They showed that such a system can be used by Sa to transform a part of the heat flowing from a hot reservoir into electric current. Furthermore, they demonstrated that a threeterminal heat engine can act as an ideal thermal-to-electric energy converter that can reach the Carnot efficiency. But, these authors only considered the ideal situation and ignored many nonideal factors. Thus, it is lack of practical guidance significance for actual engine devices. On the basis of the previous works, the main focus in this paper is to analyze the performance characteristics of a thermoelectric quantum dot heat engine, to discuss the influence of dissipative tunnel processes on the performance in detail, and to optimally design the main parameters of the heat engine. This paper is organized as follows. In Section. 2, we briefly describe the model and basic physical theory of a three-terminal quantum dot thermoelectric heat engine. In Section. 3, we
201
Greek letters chemical potential (J) ε single energy level (J) Z reduced Planck constant (J s) l dissipation factor g bare tunneling rate D difference a quantum dot
investigate the performance characteristics of the heat engine in different dissipative tunnel processes. The influence of dissipative tunnel processes on the optimal performance of the heat engine is discussed in Section. 4. Finally, the main results are summarized in Section. 5. 2. Model and theory The model of a three-terminal quantum dot heat engine is illustrated in Fig. 1, where the system consists of three independent electron reservoirs and two quantum dots. The conductor dot s is coupled to two reservoirs via two tunnel contacts which permit particle and energy exchange between the left reservoir at temperature Tl and potential ml and the right reservoir at temperature Tr and potential mr. The gate dot g is coupled to a single gate reservoir with temperature Tg and potential mg. U is the long-range Coulomb interaction between the electrons of the conductor dot s and the
Fig. 1. The schematic diagram of a three-terminal quantum dot thermoelectric device.
202
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gate dot g. The transition rates Gl, Gr, and Gg describe the tunneling of the electrons between the reservoirs and two quantum dots. If the two quantum dots are far from each other, they can be bridged to obtain a strong coupling and at the same time ensure good thermal isolation between the conductor reservoir and the gate reservoir [34,35]. Thus, the quantum dots s and g are capacitively coupled to each other and interact only though the long-range Coulomb force such that they can only exchange energy but no particles. Quantum dots s and g have their respective single energy levels εs and εg. Because Coulomb interactions prevent two electrons to be present at one energy level at the same time, the single energy level εs or εg can be occupied only by zero or one electron. The dynamical evolution of such a system is characterized by four quantum states ns ng i with respective probabilities pns ng , where the electron occupation numbers {ns,ng} of quantum dots s and g are equal to {0,0}, {0,1} {1,0}, and {1,1}, respectively, as shown in Fig. 2, where transition rates G± an describe the tunneling of electrons into (þ) or out () one quantum dot through barrier a (a ¼ s,g) when the other quantum dot has n (n ¼ 0,1) electrons. In sequential tunneling approximation, the broadening of energy levels can be neglected and the transmission through tunnel barriers is defined by sequential tunneling of a single electron. Thus, the evolution of the occupation probabilities of quantum states is described by master equation [36e38].
0 1 þ 1 Gþ G G 0 s0 g0 s0 Gg0 p_ 00 B C þ C B p_ 10 C B G 0 G Gþ s0 Gg1 g1 s0 C B C¼B þ þ C @ p_ 01 A B G 0 G G G @ A g0 s1 g0 s1 þ p_ 11 0 Gþ G G G s1 g1 g1 s1 0 1 p00 B p10 C C B @ p01 A; p11 0
where f ðxÞ ¼ fexp½ðx ma Þ=ðkB Ta Þ þ 1g1 is the Fermi distribution, kB is the Boltzmann constant, and gan are the bare tunneling rate between the quantum dots and each of the reservoirs. The capacitances associated with each tunnel junction are defined by the charging energies Uan of quantum dot a (a ¼ s,g), depending on whether the other quantum dot is empty (n ¼ 0) or occupied (n ¼ 1). The charging energies are, respectively, given by Ref. [23].
Us0
0 1 X q @q C þ CSg ¼ Cs ms þ CCg mg A; ~ 2 Sg CC
Ug0
0 1 X q @q C þ CSs Cg mg þ C ¼ Cs ms A; ~ 2 Ss CC
Gþ an
¼ gan f ðεa þ Uan Þ;
Us1 ¼ Us0 þ U;
Ug1 ¼ Ug0 þ U;
(7)
~ is the exchanged energy between where U ¼ Ua1 Ua0 ¼ 2q2 =C the two systems when an electron tunnels into the empty system but leaves it only after a second electron has occupied the other quantum dot, where q is the elementary charge. The total capacitance of each quantum dot is defined by CSs ¼ ClþCrþC and ~ ¼ ðCSs CSg C 2 Þ=C. CSg ¼ CgþC, and the effective capacitance C In the steady-state, i.e., p_ ns ng ¼ 0, the solutions of the occupation probability are as follows
p00 ¼
1 þ þ Gs1 Gg1 Gs0 þ G g0 Gg1 Gs0 þ Gg0 Gg1 Gs1 þ Gg0 Gs0 Gs1 ; U (8)
p10 ¼
1 þ þ þ þ Gg0 Gs1 Gg1 þ Gþ Gþ G g1 þ Gs0 Gg0 Gg1 þ Gs0 Gg0 Gs1 ; s0 s1 U (9)
p01 ¼
G an ¼ gan ½1 f ðεa þ Uan Þ;
(6)
and
(2)
and
(5)
s¼l;r
(1) ± ± ± where G± sn ¼ Gln þ Grn , (s ¼ l,r). The transition rates Gan are, respectively, given by
(4)
s¼l;r
1 þ þ þ þ þ G G G þ Gþ g0 Gg1 Gs0 þ Gg0 Gg1 Gs1 þ Gg0 Gs0 Gs1 ; U g1 s0 s1 (10)
(3) and
p11 ¼
1 þ þ þ þ þ þ Gg0 Gs1 Gg1 þ Gþ Gþ Gþ þ Gþ G g0 Gg1 þ Gs0 Gg0 Gs1 ; s0 s1 g1 s0 U (11)
where U is the normalization factor that ensures the sum of probabilities to be equal to unity. According to Eq. (8)e(11), the charge currents from the quantum dot system to the left (l) and right (r) side reservoirs are, respectively, given by
þ þ Il ¼ q G l0 p10 Gl0 p00 þ Gl1 p11 Gl1 p01
(12)
and
þ þ Ir ¼ q G r0 p10 Gr0 p00 þ Gr1 p11 Gr1 p01 :
Fig. 2. Available transitions for the situation described in Fig. 1.
(13)
In the steady state, the magnitude of charge currents Ir and Il is the same but their directions are opposite, i.e. Ir¼Il. The heat flow from the gate dot g into the reservoir g is given by
Y. Zhang et al. / Energy 85 (2015) 200e207
203
þ Qg ¼ εg þ Ug0 mg G g0 p01 Gg0 p00 þ εg þ Ug1 mg þ G g1 p11 Gg1 p10 : (14) Such a system can constitute a heat engine to transform a part of the heat flowing from a hot reservoir into electric [23]. In what follows, it is assumed that Tl ¼ Tr≡Ts and Tg>Ts. In this case, one can obtain an electric current I (¼Ir) and a voltage output DV ¼ (mrml)/q. Thus, the power output of the heat engine is simply given by
P ¼ IDV
(15)
and the efficiency of the heat engine is defined as
h¼
IDV Qg
(16)
3. Performance characteristics To illustrate thermodynamic performances resulting from these transition processes, a set of physically reasonable parameters are adopted in following calculations [23], i.e., kB Tl ¼ kB Tr ≡kB Ts ¼ 5Zg, Tg ¼ 2Ts, q2 =Ca ¼ 20Zg, q2 =C ¼ 50Zg, εa¼εg¼0, and gan ¼ g except gl1¼gr0¼lg, where Z is the reduced Planck constant and l is defined as the dissipation factor. Using Eq. (13), one can generate the three-dimension projection graph of the electric current I varying with the dissipation factor l and voltage output DV, as shown in Fig. 3. It is seen from Fig. 3 that the electric current I is a monotonically decreasing function of the voltage output DV and dissipation factor l and attains its maximum value when DV ¼ 0 and l ¼ 0. In order to make the electric current Ir0, the dimensionless dissipation factor must be 0l1. In practical applications, one can choose the dissipation factor by adjusting the gate voltage for different materials [39,40]. Fig. 3 also implies the fact that the smaller two parameters gl1 and gr0, the larger the electric current I, because gl1 and gr0 proportional to l. In order to obtain a large electric current, gl1 and gr0 should be as small as possible. Thus, the tunneling processes gl1 and gr0 are undesired. According to Eqs. (15), (16), the curves of the power output and efficiency as a function of the voltage output are plotted, as shown in Figs. 4 and 5, respectively. It is seen from Fig. 4 that the power
Fig. 3. Three-dimension projection graph of the electric current I varying with the dissipation factor l and voltage output DV.
Fig. 4. Power output as a function of the voltage output for different values of the dissipation factor l.
output increases firstly and then decreases as the voltage output increases for a given l. There exists an optimum voltage output DVP leading to a maximum power output Pmax. Fig. 5 shows that the efficiency is a non-monotonous function of the voltage output for given values of ls0. There exists an optimum voltage output DVh leading to a maximum efficiency hmax. In the region of DVDVh, the efficiency decreases as the voltage output increases. It is seen that the maximum power output and efficiency gradually decreases with the increase of l. For zero voltage output, the power output and efficiency are equal to zero. In addition, for the case of l¼0, the power output vanishes when the voltage output arrives at the so called the open circuit voltage DVopen, because the electric current is equal to zero. At this point, the efficiency of the heat engine can achieve its maximum value, i.e. the Carnot efficiency hC, which means that the heat engine works at an ideal situation. But, for the case of ls0, both the power output and efficiency vanish at the open circuit voltage DVopen, This shows that for the case of ls0, the heat engine cannot be operated at the ideal situation.
Fig. 5. Efficiency as a function of the voltage output for different values of the dissipation factor l.
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The variation curve of the open circuit voltage as a function of the dissipation factor l is plotted, as shown in Fig. 6. It is found that the open circuit voltage rapidly increases with the decrease of the dissipation factor and reaches its maximum value qDVopen =ðZgÞ ¼ 2:5 at the ideal situation of l¼0. This means that the tunneling processes gl1 and gr0 are equivalent to the actual internal resistance of the circuit. Therefore, the parameter l is defined as the dissipation factor. By using the data in Figs. 4, 5, the characteristic curves of the power output versus efficiency can be plotted, as shown in Fig. 7. Due to the contribution of these undesired processes gl1 and gr0 can be negligible for the case of l ¼ 0, the electronic transport is ideal and of no dissipation. Therefore, it is seen that the characteristic curve between the power output and the efficiency is open shaped. In this case, no power output can be extracted from a heat engine working at the Carnot efficiency. But, for the case of ls0, the electronic transport of these undesired processes will gradually increase with the increase of l. Therefore, the characteristic curves are closed loop-shaped. This means that the power output at the maximum efficiency does not vanish. For actual heat engines, engineers always want to get an efficiency as high as possible and at the same time obtain one large power output. Therefore, the optimally operating regions of the quantum dot heat engine should be located in those of the h~P curves with negative slopes, as shown in Fig. 7. Thus, the optimal regions of the three-terminal quantum dot heat engine should be
Pmh P Pmax ;
(17)
and
Fig. 7. Characteristic curves of the power output versus efficiency for different values of the dissipation factor l.
voltage output. In this optimal voltage output region, the quantum dot heat engine is located in the range of the characteristic curves with negative slopes. The optimal voltage output regions gradually decreases with the increase of l. These results obtained here indicate a way to operate heat engines at the optimum ranges.
4. Maximum power output and efficiency
hmP h hmax ;
(18)
where Pmh,Pmax, hmP, and hmax are four important parameters which determine the lower and upper bounds of the optimized power output and efficiency of a quantum dot heat engine. Moreover, the variations of the optimal voltage output at the maximum power output and maximum efficiency with the dissipation factor l are plotted in Fig. 8. It is found that the optimal region of the voltage output should be
DVp DV DVh :
(19)
Obviously, DVp and DVh are also two important parameters which determine the lower and upper bounds of the optimized
Fig. 6. Open voltage output as a function of the dissipation factor l.
For the actual application, it is more useful to discuss the efficiency at the maximum power output [13,16,17,41,42]. Hence, the curves of the efficiency hmp at the maximum power output varying with the Carnot efficiency hC ¼ 1Ts/Tg are plotted for different values of l, as shown in Fig. 9(a). It is clearly shown that the efficiency hmp ¼ hC/2 at the maximum power output for the ideal case (l¼0). The result was revealed by Sanchez et al. [23]. The efficiency at the maximum power output in the dissipation case (ls0) is smaller than that in the ideal case (l¼0) and obviously decreases as the dissipation factor l increases. The curves of the maximum power output Pmax/g varying with the Carnot efficiency hC are plotted, as shown in Fig. 9(b). It is seen that the maximum power output is a
Fig. 8. Optimal voltage outputs at the maximum power output and efficiency as a function of the dissipation factor l.
Y. Zhang et al. / Energy 85 (2015) 200e207
205
Fig. 9. (a) Efficiency at the maximum power output and (b) maximum power output as a function of the Carnot efficiency for different values of the dissipation factor l.
monotonically increasing function of hC and gradually decreases with the increase of the dissipation factor l. Fig. 10 shows (a) the maximum efficiency hmax and (b) the power output at the maximum efficiency Pmh/g as a function the Carnot efficiency for different values of the dissipation factor l. It is seen that the maximum efficiency hmax decreases as the dissipation factor l increases. For the case of l ¼ 0, the maximum efficiency is equal to the Carnot efficiency i.e. hmax ¼ hC, but the corresponding power output is zero. While the power output Pmh/g at the maximum efficiency increases firstly and then decreases as the dissipation factor l increases. Therefore, the dissipation factor l has an optimal value at which the power output Pmh/g at the maximum efficiency will attains its maximum. This will be discussed hereinafter. Fig. 11 shows the maximum power output Pmax/g and the corresponding efficiency hmp at the maximum power output as a function of the dissipation factor l. It is found that for the case of
ls0, both of the maximum power output Pmax/g and the corresponding efficiency hmp obviously decrease as the dissipation factor l increases, because undesired processes gl1 and gr0 cannot be negligible. Hence, these undesired processes gl1 and gr0 should be minimized as largely as possible in the actual maximum power output optimization design. The variation curves of the maximum efficiency hmax and corresponding power output Pmh/g at the maximum efficiency with the dissipation factor l are plotted, as shown in Fig. 12. It is seen that the maximum efficiency hmax decreases as the dissipation factor l increases, while the corresponding power output at the maximum efficiency increases firstly and then decreases as the dissipation factor l increases and it reaches its maximum value Pmh/ g ¼ 3.366 at lopt ¼ 0.017. So the maximum efficiency optimization is different from the maximum power output optimization. In order to achieve the maximum efficiency and at the same time obtain a large power output, the quantum dot heat engine should work in
Fig. 10. (a) Maximum efficiency and (b) power output at maximum efficiency as a function of the Carnot efficiency for different values of the dissipation factor l.
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Y. Zhang et al. / Energy 85 (2015) 200e207
maximum efficiency and at the same time obtain a large power output, the dissipation factor l should be chosen in the range of 0
Fig. 11. Maximum power output and efficiency at the maximum power output as a function of the dissipation factor l.
the area of llopt, as shown in Fig. 12. Thus, in the actual design, the optimal regions of the dissipation factor l should be 0
Fig. 12. Power output at the maximum efficiency and maximum efficiency as a function of the dissipation factor l.
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Energy journal homepage: www.elsevier.com/locate/energy
Optimum energy conversion strategies of a nano-scaled threeterminal quantum dot thermoelectric device Yanchao Zhang, Chuankun Huang, Junyi Wang, Guoxing Lin, Jincan Chen* Department of Physics, Xiamen University, Xiamen 361005, People's Republic of China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 November 2014 Received in revised form 10 March 2015 Accepted 14 March 2015 Available online 23 April 2015
A model of the three-terminal nano-scaled energy conversion system as a heat engine based on two capacitively coupled quantum dots in the Coulomb-blockade regime is established within four quantum states that include the essential physical features. The dynamical properties of the model are calculated by master equation approach account for the quantitative behavior of such a system. Expressions for the power output and efficiency of the three-terminal quantum dot heat engine are derived. The characteristic curves between the power output and the efficiency are plotted. Moreover, the optimal values of main performance parameters are determined by the numerical calculation. The influence of dissipative tunnel processes on the optimal performance is discussed in detail. The results obtained here can provide some theoretical guidelines for the design and operation of practical three-terminal quantum dot heat engines. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Quantum dot heat engine Performance characteristics Dissipative tunnel Optimum design
1. Introduction In recent years, investigations on the thermoelectric effects of nano-thermoelectric devices have attracted considerable interest due to their importance in developing miniaturized devices which help to utilize energy resources at the microscopic scale [1e5]. In particular, there is broad, current interest in developing high performance thermoelectric materials for the generation of electric power from heat sources such as waste heat. It was pointed out that structures of reduced dimension can give rise to an increased thermoelectric figure of merit ZT as compared to bulk structures made from the same materials [6,7] and that sharp spectral features can improve thermoelectric performances characterized by a high value of ZT in materials with a delta-like density of states [8]. Furthermore, it was reported that the Carnot efficiency can be reached for reversible electron transport between two reservoirs at different temperatures and chemical potentials by using a sharply tuned energy filter for which the electron density is the same in both reservoirs [9e11]. Quantum dots naturally provide these sharp spectral features. Hence, there are many investigations of the quantum dot thermoelectric devices because quantum dots can be applied to the high-
* Corresponding author. Tel.: þ86 592 2180922; fax: þ86 592 2189426. E-mail address: [email protected] (J. Chen). http://dx.doi.org/10.1016/j.energy.2015.03.087 0360-5442/© 2015 Elsevier Ltd. All rights reserved.
potential solid-state energy conversion devices [12e15]. Hence, the study of nano-thermoelectric heat engines and refrigerators using the quantum dot system as the working substance embedded between two reservoirs at different temperatures and chemical potentials has attracted considerable interest [16e21]. In twoterminal nano-thermoelectric device, when both the gradients of temperature and chemical potential are applied to the device and the thermoelectric response as heat engines and refrigerators is investigated, it is found that the heat flow will be accompanied by the electron current. But, for the purpose of energy harvesting, it suffers from the fact that the different parts of the thermoelectric response must be at different temperatures which make thermal isolation difficult [22]. Hence, some theoretical works on the thermoelectric effects in three-terminal quantum dot devices have attracted considerable interest [23e33]. This is because that such a three-terminal thermoelectric device offers the advantage of spatially separating the hot and cold reservoirs and exhibits a crossed flow of heat and electric currents. This is useful for energy harvesting applications because such a three-terminal thermoelectric device can be driven by electronic sources [23e25], phonon sources [26e28], or photon sources [29e31]. The maximum power output and efficiency of a mesoscopic heat engine based on a hot chaotic cavity capacitively coupled to a cold cavity were investigated [24]. A resonant-tunneling quantum dot heat engine was researched. Such a heat engine converts a part of the heat into electrical current in a three-terminal geometry which permits one
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Nomenclature T U ZT C ~ C
U G p I Q
hC h hmax hmP P Pmax Pmh DV DVopen DVP
DV h kB q
optimum voltage output at maximum efficiency (V) Boltzmann's constant (J K1) elementary charge (C)
temperature (K) Coulomb interaction (J) figure of merit capacitance (F)
m
effective capacitance (F) normalization factor transition rate occupation probability charge currents (W) heat flow (W) Carnot efficiency efficiency maximum efficiency efficiency at maximum power output power output (W) maximum power output (W) power output at maximum efficiency (W) voltage output (V) open circuit voltage (V) optimum voltage output at maximum power output (V)
Subscripts s conductor dot g gate dot n electron occupation number l left side reservoir r right side reservoir max maximum mh maximum efficiency mP maximum power opt optimal
to separate current and heat flows. This provides an option to highly efficient solid-state energy harvesting [25]. Soon after, Sothmann et al. investigated a heat engine consisting of a singlelevel quantum dot coupled to two ferromagnetic metals and one ferromagnetic insulator held at different temperatures and demonstrated that in the tight-coupling limit the device can reach the Carnot efficiency [28]. Photon-driven heat engines or refrigerators were also studied [29e33]. Rutten et al. used the quantum dot structure to design a photoelectric device as a heat engine and studied the thermodynamic efficiency of the heat engine [29]. Ruokola et al. theoretically researched single-electron heat engines coupled to electromagnetic environments and predicted that high efficiencies are possible with the quantum dot system [30]. Only recently, a possibly realized hybrid microwave cavity heat engine consisting of two macroscopically separated quantum dot conductors coupled capacitively to the fundamental mode of a microwave cavity was proposed. Such a heat engine can reach the Carnot efficiency when the optimal conversion is achieved [31]. The thermoelectric properties of two capacitively coupled quantum dots in the Coulomb-blockade regime in a three-terminal nano-sized structure thermoelectric system were firstly analyzed nchez et al. [23]. They showed that such a system can be used by Sa to transform a part of the heat flowing from a hot reservoir into electric current. Furthermore, they demonstrated that a threeterminal heat engine can act as an ideal thermal-to-electric energy converter that can reach the Carnot efficiency. But, these authors only considered the ideal situation and ignored many nonideal factors. Thus, it is lack of practical guidance significance for actual engine devices. On the basis of the previous works, the main focus in this paper is to analyze the performance characteristics of a thermoelectric quantum dot heat engine, to discuss the influence of dissipative tunnel processes on the performance in detail, and to optimally design the main parameters of the heat engine. This paper is organized as follows. In Section. 2, we briefly describe the model and basic physical theory of a three-terminal quantum dot thermoelectric heat engine. In Section. 3, we
201
Greek letters chemical potential (J) ε single energy level (J) Z reduced Planck constant (J s) l dissipation factor g bare tunneling rate D difference a quantum dot
investigate the performance characteristics of the heat engine in different dissipative tunnel processes. The influence of dissipative tunnel processes on the optimal performance of the heat engine is discussed in Section. 4. Finally, the main results are summarized in Section. 5. 2. Model and theory The model of a three-terminal quantum dot heat engine is illustrated in Fig. 1, where the system consists of three independent electron reservoirs and two quantum dots. The conductor dot s is coupled to two reservoirs via two tunnel contacts which permit particle and energy exchange between the left reservoir at temperature Tl and potential ml and the right reservoir at temperature Tr and potential mr. The gate dot g is coupled to a single gate reservoir with temperature Tg and potential mg. U is the long-range Coulomb interaction between the electrons of the conductor dot s and the
Fig. 1. The schematic diagram of a three-terminal quantum dot thermoelectric device.
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gate dot g. The transition rates Gl, Gr, and Gg describe the tunneling of the electrons between the reservoirs and two quantum dots. If the two quantum dots are far from each other, they can be bridged to obtain a strong coupling and at the same time ensure good thermal isolation between the conductor reservoir and the gate reservoir [34,35]. Thus, the quantum dots s and g are capacitively coupled to each other and interact only though the long-range Coulomb force such that they can only exchange energy but no particles. Quantum dots s and g have their respective single energy levels εs and εg. Because Coulomb interactions prevent two electrons to be present at one energy level at the same time, the single energy level εs or εg can be occupied only by zero or one electron. The dynamical evolution of such a system is characterized by four quantum states ns ng i with respective probabilities pns ng , where the electron occupation numbers {ns,ng} of quantum dots s and g are equal to {0,0}, {0,1} {1,0}, and {1,1}, respectively, as shown in Fig. 2, where transition rates G± an describe the tunneling of electrons into (þ) or out () one quantum dot through barrier a (a ¼ s,g) when the other quantum dot has n (n ¼ 0,1) electrons. In sequential tunneling approximation, the broadening of energy levels can be neglected and the transmission through tunnel barriers is defined by sequential tunneling of a single electron. Thus, the evolution of the occupation probabilities of quantum states is described by master equation [36e38].
0 1 þ 1 Gþ G G 0 s0 g0 s0 Gg0 p_ 00 B C þ C B p_ 10 C B G 0 G Gþ s0 Gg1 g1 s0 C B C¼B þ þ C @ p_ 01 A B G 0 G G G @ A g0 s1 g0 s1 þ p_ 11 0 Gþ G G G s1 g1 g1 s1 0 1 p00 B p10 C C B @ p01 A; p11 0
where f ðxÞ ¼ fexp½ðx ma Þ=ðkB Ta Þ þ 1g1 is the Fermi distribution, kB is the Boltzmann constant, and gan are the bare tunneling rate between the quantum dots and each of the reservoirs. The capacitances associated with each tunnel junction are defined by the charging energies Uan of quantum dot a (a ¼ s,g), depending on whether the other quantum dot is empty (n ¼ 0) or occupied (n ¼ 1). The charging energies are, respectively, given by Ref. [23].
Us0
0 1 X q @q C þ CSg ¼ Cs ms þ CCg mg A; ~ 2 Sg CC
Ug0
0 1 X q @q C þ CSs Cg mg þ C ¼ Cs ms A; ~ 2 Ss CC
Gþ an
¼ gan f ðεa þ Uan Þ;
Us1 ¼ Us0 þ U;
Ug1 ¼ Ug0 þ U;
(7)
~ is the exchanged energy between where U ¼ Ua1 Ua0 ¼ 2q2 =C the two systems when an electron tunnels into the empty system but leaves it only after a second electron has occupied the other quantum dot, where q is the elementary charge. The total capacitance of each quantum dot is defined by CSs ¼ ClþCrþC and ~ ¼ ðCSs CSg C 2 Þ=C. CSg ¼ CgþC, and the effective capacitance C In the steady-state, i.e., p_ ns ng ¼ 0, the solutions of the occupation probability are as follows
p00 ¼
1 þ þ Gs1 Gg1 Gs0 þ G g0 Gg1 Gs0 þ Gg0 Gg1 Gs1 þ Gg0 Gs0 Gs1 ; U (8)
p10 ¼
1 þ þ þ þ Gg0 Gs1 Gg1 þ Gþ Gþ G g1 þ Gs0 Gg0 Gg1 þ Gs0 Gg0 Gs1 ; s0 s1 U (9)
p01 ¼
G an ¼ gan ½1 f ðεa þ Uan Þ;
(6)
and
(2)
and
(5)
s¼l;r
(1) ± ± ± where G± sn ¼ Gln þ Grn , (s ¼ l,r). The transition rates Gan are, respectively, given by
(4)
s¼l;r
1 þ þ þ þ þ G G G þ Gþ g0 Gg1 Gs0 þ Gg0 Gg1 Gs1 þ Gg0 Gs0 Gs1 ; U g1 s0 s1 (10)
(3) and
p11 ¼
1 þ þ þ þ þ þ Gg0 Gs1 Gg1 þ Gþ Gþ Gþ þ Gþ G g0 Gg1 þ Gs0 Gg0 Gs1 ; s0 s1 g1 s0 U (11)
where U is the normalization factor that ensures the sum of probabilities to be equal to unity. According to Eq. (8)e(11), the charge currents from the quantum dot system to the left (l) and right (r) side reservoirs are, respectively, given by
þ þ Il ¼ q G l0 p10 Gl0 p00 þ Gl1 p11 Gl1 p01
(12)
and
þ þ Ir ¼ q G r0 p10 Gr0 p00 þ Gr1 p11 Gr1 p01 :
Fig. 2. Available transitions for the situation described in Fig. 1.
(13)
In the steady state, the magnitude of charge currents Ir and Il is the same but their directions are opposite, i.e. Ir¼Il. The heat flow from the gate dot g into the reservoir g is given by
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203
þ Qg ¼ εg þ Ug0 mg G g0 p01 Gg0 p00 þ εg þ Ug1 mg þ G g1 p11 Gg1 p10 : (14) Such a system can constitute a heat engine to transform a part of the heat flowing from a hot reservoir into electric [23]. In what follows, it is assumed that Tl ¼ Tr≡Ts and Tg>Ts. In this case, one can obtain an electric current I (¼Ir) and a voltage output DV ¼ (mrml)/q. Thus, the power output of the heat engine is simply given by
P ¼ IDV
(15)
and the efficiency of the heat engine is defined as
h¼
IDV Qg
(16)
3. Performance characteristics To illustrate thermodynamic performances resulting from these transition processes, a set of physically reasonable parameters are adopted in following calculations [23], i.e., kB Tl ¼ kB Tr ≡kB Ts ¼ 5Zg, Tg ¼ 2Ts, q2 =Ca ¼ 20Zg, q2 =C ¼ 50Zg, εa¼εg¼0, and gan ¼ g except gl1¼gr0¼lg, where Z is the reduced Planck constant and l is defined as the dissipation factor. Using Eq. (13), one can generate the three-dimension projection graph of the electric current I varying with the dissipation factor l and voltage output DV, as shown in Fig. 3. It is seen from Fig. 3 that the electric current I is a monotonically decreasing function of the voltage output DV and dissipation factor l and attains its maximum value when DV ¼ 0 and l ¼ 0. In order to make the electric current Ir0, the dimensionless dissipation factor must be 0l1. In practical applications, one can choose the dissipation factor by adjusting the gate voltage for different materials [39,40]. Fig. 3 also implies the fact that the smaller two parameters gl1 and gr0, the larger the electric current I, because gl1 and gr0 proportional to l. In order to obtain a large electric current, gl1 and gr0 should be as small as possible. Thus, the tunneling processes gl1 and gr0 are undesired. According to Eqs. (15), (16), the curves of the power output and efficiency as a function of the voltage output are plotted, as shown in Figs. 4 and 5, respectively. It is seen from Fig. 4 that the power
Fig. 3. Three-dimension projection graph of the electric current I varying with the dissipation factor l and voltage output DV.
Fig. 4. Power output as a function of the voltage output for different values of the dissipation factor l.
output increases firstly and then decreases as the voltage output increases for a given l. There exists an optimum voltage output DVP leading to a maximum power output Pmax. Fig. 5 shows that the efficiency is a non-monotonous function of the voltage output for given values of ls0. There exists an optimum voltage output DVh leading to a maximum efficiency hmax. In the region of DV
Fig. 5. Efficiency as a function of the voltage output for different values of the dissipation factor l.
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The variation curve of the open circuit voltage as a function of the dissipation factor l is plotted, as shown in Fig. 6. It is found that the open circuit voltage rapidly increases with the decrease of the dissipation factor and reaches its maximum value qDVopen =ðZgÞ ¼ 2:5 at the ideal situation of l¼0. This means that the tunneling processes gl1 and gr0 are equivalent to the actual internal resistance of the circuit. Therefore, the parameter l is defined as the dissipation factor. By using the data in Figs. 4, 5, the characteristic curves of the power output versus efficiency can be plotted, as shown in Fig. 7. Due to the contribution of these undesired processes gl1 and gr0 can be negligible for the case of l ¼ 0, the electronic transport is ideal and of no dissipation. Therefore, it is seen that the characteristic curve between the power output and the efficiency is open shaped. In this case, no power output can be extracted from a heat engine working at the Carnot efficiency. But, for the case of ls0, the electronic transport of these undesired processes will gradually increase with the increase of l. Therefore, the characteristic curves are closed loop-shaped. This means that the power output at the maximum efficiency does not vanish. For actual heat engines, engineers always want to get an efficiency as high as possible and at the same time obtain one large power output. Therefore, the optimally operating regions of the quantum dot heat engine should be located in those of the h~P curves with negative slopes, as shown in Fig. 7. Thus, the optimal regions of the three-terminal quantum dot heat engine should be
Pmh P Pmax ;
(17)
and
Fig. 7. Characteristic curves of the power output versus efficiency for different values of the dissipation factor l.
voltage output. In this optimal voltage output region, the quantum dot heat engine is located in the range of the characteristic curves with negative slopes. The optimal voltage output regions gradually decreases with the increase of l. These results obtained here indicate a way to operate heat engines at the optimum ranges.
4. Maximum power output and efficiency
hmP h hmax ;
(18)
where Pmh,Pmax, hmP, and hmax are four important parameters which determine the lower and upper bounds of the optimized power output and efficiency of a quantum dot heat engine. Moreover, the variations of the optimal voltage output at the maximum power output and maximum efficiency with the dissipation factor l are plotted in Fig. 8. It is found that the optimal region of the voltage output should be
DVp DV DVh :
(19)
Obviously, DVp and DVh are also two important parameters which determine the lower and upper bounds of the optimized
Fig. 6. Open voltage output as a function of the dissipation factor l.
For the actual application, it is more useful to discuss the efficiency at the maximum power output [13,16,17,41,42]. Hence, the curves of the efficiency hmp at the maximum power output varying with the Carnot efficiency hC ¼ 1Ts/Tg are plotted for different values of l, as shown in Fig. 9(a). It is clearly shown that the efficiency hmp ¼ hC/2 at the maximum power output for the ideal case (l¼0). The result was revealed by Sanchez et al. [23]. The efficiency at the maximum power output in the dissipation case (ls0) is smaller than that in the ideal case (l¼0) and obviously decreases as the dissipation factor l increases. The curves of the maximum power output Pmax/g varying with the Carnot efficiency hC are plotted, as shown in Fig. 9(b). It is seen that the maximum power output is a
Fig. 8. Optimal voltage outputs at the maximum power output and efficiency as a function of the dissipation factor l.
Y. Zhang et al. / Energy 85 (2015) 200e207
205
Fig. 9. (a) Efficiency at the maximum power output and (b) maximum power output as a function of the Carnot efficiency for different values of the dissipation factor l.
monotonically increasing function of hC and gradually decreases with the increase of the dissipation factor l. Fig. 10 shows (a) the maximum efficiency hmax and (b) the power output at the maximum efficiency Pmh/g as a function the Carnot efficiency for different values of the dissipation factor l. It is seen that the maximum efficiency hmax decreases as the dissipation factor l increases. For the case of l ¼ 0, the maximum efficiency is equal to the Carnot efficiency i.e. hmax ¼ hC, but the corresponding power output is zero. While the power output Pmh/g at the maximum efficiency increases firstly and then decreases as the dissipation factor l increases. Therefore, the dissipation factor l has an optimal value at which the power output Pmh/g at the maximum efficiency will attains its maximum. This will be discussed hereinafter. Fig. 11 shows the maximum power output Pmax/g and the corresponding efficiency hmp at the maximum power output as a function of the dissipation factor l. It is found that for the case of
ls0, both of the maximum power output Pmax/g and the corresponding efficiency hmp obviously decrease as the dissipation factor l increases, because undesired processes gl1 and gr0 cannot be negligible. Hence, these undesired processes gl1 and gr0 should be minimized as largely as possible in the actual maximum power output optimization design. The variation curves of the maximum efficiency hmax and corresponding power output Pmh/g at the maximum efficiency with the dissipation factor l are plotted, as shown in Fig. 12. It is seen that the maximum efficiency hmax decreases as the dissipation factor l increases, while the corresponding power output at the maximum efficiency increases firstly and then decreases as the dissipation factor l increases and it reaches its maximum value Pmh/ g ¼ 3.366 at lopt ¼ 0.017. So the maximum efficiency optimization is different from the maximum power output optimization. In order to achieve the maximum efficiency and at the same time obtain a large power output, the quantum dot heat engine should work in
Fig. 10. (a) Maximum efficiency and (b) power output at maximum efficiency as a function of the Carnot efficiency for different values of the dissipation factor l.
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maximum efficiency and at the same time obtain a large power output, the dissipation factor l should be chosen in the range of 0
Fig. 11. Maximum power output and efficiency at the maximum power output as a function of the dissipation factor l.
the area of llopt, as shown in Fig. 12. Thus, in the actual design, the optimal regions of the dissipation factor l should be 0
Fig. 12. Power output at the maximum efficiency and maximum efficiency as a function of the dissipation factor l.
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