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The impact of energy spectrum width in the energy selective electron low-temperature thermionic heat engine at maximum power
Physics Letters A 377 (2013) 1566–1570
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
The impact of energy spectrum width in the energy selective electron low-temperature thermionic heat engine at maximum power Xiaoguang Luo a , Cong Li b , Nian Liu c , Ruiwen Li b , Jizhou He b , Teng Qiu a,∗ a b c
Department of Physics, Southeast University, Nanjing 211189, China Department of Physics, Nanchang University, Nanchang 330031, China Department of Physical and Electronics, Anhui Science and Technology University, Bengbu 233100, China
a r t i c l e
i n f o
Article history: Received 11 December 2012 Received in revised form 22 April 2013 Accepted 24 April 2013 Available online 29 April 2013 Communicated by C.R. Doering Keywords: Energy spectrum width Energy selective electron Thermionic Maximum power
a b s t r a c t A model of thermionic heat engine with the energy selective electron mechanism is studied. Analytical expressions of the power output and efficiency of this device are derived at low temperature, where the chemical potentials of the reservoirs are assumed to be constant. After discussing the impact of the energy spectrum width of the energy selective electron mechanism, we find two bounds (η± ) of efficiency at maximum power exist naturally. When the energy spectrum width increases gradually from zero and then to the semi-infinite case with the infinite upper limit, the efficiency at maximum power decreases monotonously from the upper bound η+ to the lower bound η− at a given temperature ratio of the cold and hot reservoirs. The two bound are given by numerical simulation and by an analytical expression respectively. These results may provide some guidance for the application of the practical energy selective electron heat engines. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Nowadays, the performance of conventional heat engines can be expressed well by the Carnot theorem and Curzon–Ahlborn (CA) efficiency limitation at maximum power [1]. To these energy conversion devices, the efficiency is always smaller than the Carnot value (ηC = 1 − T c / T h , where T h and T c are the temperatures of the hot and cold reservoirs respectively). Moreover, the efficiency at maximum √ power (EMP) should be smaller than the CA efficiency (ηCA = 1 − T c / T h ) which can be obtained from, in addition to traditional Carnot heat engine, linear irreversible thermodynamics for perfectly coupled systems [2–4] and symmetric dissipation systems [5,6]. It is known that Carnot efficiency is extended to be a universal limitation without any exceptions. However, the CA efficiency is not the unique limitation any more for EMP of some systems, such as Feynman ratchet and pawl model [7], stochastic procedure of a Brownian particle undergoing a Carnot cycle through the modulation of a harmonic potential [8] etc. Consequently, to find out the properties of some different heat engines working at maximum power becomes meaningful for fundamental thermodynamics and engineering applications. Here we focus on a thermionic device which consists of two different electron heat reservoirs and an electronic conductor between them. Generally, the chemical potentials of the electron
*
Corresponding author. E-mail addresses: [email protected] (X.G. Luo), [email protected] (T. Qiu).
0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.04.045
reservoirs can be assumed to be constant approximatively at relatively low temperature, such as several or few dozens of kelvin [9,10]. If the chemical potential of the cold reservoir is higher than that of the hot one (i.e. μc > μh ), the electrons in the two reservoirs could be exchanged because of the temperature gradient (hot electrons transmitted from the hot to cold reservoir) and chemical potential gradient (cold electrons transmitted from the cold to hot reservoir). The heat flux, coupled with the electronic flux, will flow from one reservoir to the other. According to the direction of the heat flux, the thermionic device can work as a heat engine or a refrigerator that controlled by the energy selective electron (ESE) mechanism [9]. With the development of the nanotechnology, many nanostructures exhibit good ESE properties for electron transmission, such as nanowires, quantum ratchets, quantum wells and quantum dots [10–12] and so on. In theory, these transmission properties can be simulated well by rectangular model [13], Lorentzian model [9,14], Gaussian model [15], multi-barrier model [16], single-level quantum dot [17,18] and so on. After investigated by experiments and numerical simulations, the efficiency of the heat engine can definitely be improved effectively and even reach Carnot value by adjusting the selection of the electronic conductor. For instance, a single-level channel between the two reservoirs can help to realize the Carnot efficiency, however, the power output vanishes in the meanwhile [9,17]. In practice, it is hard to find a material of electronic conductor behaving like the singlelevel quantum dot. And the resonant electron transmission with a resonant energy spectrum width may be an ideal potentially ESE mechanism in some way [10–12].
X.G. Luo et al. / Physics Letters A 377 (2013) 1566–1570
Fig. 1. The diagram of the ideal ESE thermionic heat engine where where F = f h − f c .
μc − μh = −eV (a), and the working area (gray area above zero) with respect to the electron energy (b)
Since high efficiency can be obtained at the cost of lowering the power output, the optimization between them will become more meaningful. And some strategies, focused on the efficiency and the power of the ESE thermionic heat engine, have been carried out [19,20]. However, the practical factors, such as multi-resonance channels [21], heat leakage [22], load [23] etc., will reduce the efficiency of heat engine. In fact, these factors can be prevented as far as possible by microcosmic science and engineering technology. Anyway, the natural lower and upper bounds without heat loss, similar to the temperature-ratio-dependent EMP bounds of Carnot heat engine [6], have not been obtained theoretically for the ESE thermionic heat engine. In this Letter, a thermionic model with an ideal rectangular model ESE mechanism is discussed. The position and the width of the energy spectrum range of the ESE mechanism will dramatically affect the performance of this thermionic heat engine, and we have found the natural lower and upper bounds for ESE thermionic heat engine with respect to the width of ESE mechanism, where the upper bound is just the result from the single-level quantum dot model [17] and the lower bound can be realized in the quantum well thermionic heat engine. 2. Thermionic heat engine with ESE mechanism The electrons in the ideal rectangular ESE thermionic model can be exchanged between the two electron reservoirs only when their energy E satisfy E E E + E, as shown in Fig. 1(a). The model can work as a heat engine if the energy of exchanged electrons E E 0 ≡ (μc T h − μh T c )/( T h − T c ), where the electrons with energy E 0 can be transmitted spontaneously without entropy increase [9]. The transmission probability of the electrons from one to the other reservoir in this ESE mechanism can be written as
ζ (E) =
1, E < E < E + E , 0, elsewhere.
(1)
factor of f h − f c (shown in Fig. 1(b)) being positive or negative depends on the electron energy E. However, the net electric current will vanish as both positive and negative gray areas in Fig. 1(b), equal to each other, locate in the energy spectrum range of the ESE mechanism E. Moreover, if E → 0, the electric current will also vanish, and the device will work reversibly when the energy of exchanged electrons E = E 0 . The energy change of the hot/cold reservoir is E − μh/c when an electron is transmitted out. Therefore, without heat loss, the heat flux out of the hot/cold reservoir per unit area per unit time can be given by
Q˙ h/c = ±
I=
h
( f h − f c ) dE ,
(2)
where e is the charge of an electron, h is the Planck constant, the factor of 2 accounts for degeneracy due to electron spin, and f h/c = 1/{1 + exp[( E − μh/c )/k B T h/c ]} with Boltzmann constant k B is the Fermi distribution function of the electrons in the hot/cold reservoir. From Eqs. (1) and (2), the direction of the net electric current between the two reservoirs can be controlled by adjusting the parameters E and E of the ESE mechanism, because the
+ E E
( E − μh/c )( f h − f c ) dE .
h
(3)
E
P = Q˙ h + Q˙ c =
2 h
+ E E
(μc − μh )( f h − f c ) dE ,
(4)
E
and the efficiency can be calculated by η ≡ P / Q˙ h . For nondimensionalization, Γ = E /[k B ( T c + T h )/2] is used to represent the energy spectrum width, where Γ → 0 as E → 0 donates nearly no electron can be exchanged and Γ → ∞ as E → ∞ implies plenty of electrons with different energy can be exchanged freely. The performance parameters in this model can be expressed analytically after using two integral functions of
b a
1/(1 + x) dx = ln(1 + x)|ab and
b a
ln(x)/(1 + x) dx = [ln(x) ln(1 +
x) + PolyLog(2, −x)]|ab , where PolyLog(2, −x) is the Nielsen function (e.g. PolyLog(2, 0) = 0 and PolyLog(2, −1) = −π 2 /12). Hence, the heat flux out of the hot reservoir per unit area per unit time can be expressed by
Q˙ h =
+ E E
E
2
Then, one can get the power output of the heat engine from
Combined with Landauer formula [24], the electrical current density flowing from the cold to the hot reservoir can be calculated by
2e
1567
2 h
k2B T h2
f e −Γh e −rh + g e −Γh e −rh − f e −rh − g e −rh
− (1 − ηC )2 f e −Γc e −rc + g e −Γc e −rc − f e −rc − g e −rc + (1 − ηC )rh − (1 − ηC )2 rc
× ln 1 + e −Γc e −rc − ln 1 + e −rc ,
(5)
where f (x) = ln(x) ln(1 + x) and g (x) = PolyLog(2, −x) for simplification, and Γh = Γ (1 − ηC /2), Γc = Γ (1 − ηC /2)/(1 − ηC ), rh = ( E − μh )/k B T h and rc = ( E − μc )/k B T c are the dimensionless scaled energies. Thus, the expression of the power of this heat engine, i.e. Eq. (4), can be rewritten as
P=
2 h
k2B T h2 rh − (1 − ηC )rc
rc
rh e +1 e +1 − ( , 1 − η ) ln × ln r C e h + e −Γh erc + e −Γc
(6)
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X.G. Luo et al. / Physics Letters A 377 (2013) 1566–1570
from which one can see that the power output can be directly regarded as the function of temperature T h , energy spectrum width Γ and dimensionless scaled energies. Consequently, at the arbitrarily given T h and Γ , the maximum power with respect to ηC can be obtained by optimizing the dimensionless scaled energies. So,
2
= rh − (1 − ηC )rc
rc
rh e +1 e +1 − ( × ln r 1 − η ) ln C e h + e −Γh erc + e −Γc
P∗ = P
h
k2B T h2
(7)
can be used to represent the power. Then the optimization will focus on the dimensionless variable P ∗ which will still be stand for power in the following discussions for convenience. The factor of f h − f c < 0 in Eq. (4) denotes that the net electron flux flows from cold to hot reservoir and then the power output will be reduced. In order to guarantee maximum power, the net electron flux should flow from the hot to cold reservoir as much as possible. Thus, the energy of all transmitted electrons should satisfy E E 0 which results in E E 0 , as shown in Fig. 1. To this system, the entropy increase caused by one exchanged electron from hot to cold reservoir can be expressed by
S = −
E − μh Th
+
E − μc Tc
(8)
.
It is clearly shown that S 0 owing to E E 0 . As a result,
μh −E − + T h
E −μc Tc
= k B (rc − rh ) 0. So we have rc rh where the
equality holds as E = E 0 .
3. Optimization with maximum power From Eq. (7), the values of rh and rc at maximum power can be found by optimizing the power function through ∂ P ∗ /∂ rc = ∂ P ∗ /∂ rh = 0. After that, it is shown the efficiency has nothing to do with the actual temperatures of the reservoirs directly but their ratio at the fixed energy spectrum width Γ . Anyway, these values are determined by the following two equations:
ln
e rh + 1 erh + e −Γh
− (1 − ηC ) ln
+ rh − (1 − ηC )rc
erc + 1
erc + e −Γc
e (e −Γc − 1) rc
= 0, (erc + 1)(erc + e −Γc ) e rh + 1 erc + 1 ln r − ( 1 − η ) ln C e h + e −Γh erc + e −Γc r h e (e −Γh − 1) + rh − (1 − ηC )rc r = 0. (e h + 1)(erh + e −Γh )
(9a)
(9b)
According to our calculation (see Fig. 2): rc rh and they will reach the maximum values simultaneously when the energy spectrum width Γ → 0; the values of both rh and rc will decrease monotonously if Γ increases, and rc = rh = 1.14455 as Γ → ∞. The case of Γ → 0 accords the situation of E → E P and E → 0 where F reaches the maximum value (the thick line shown in Fig. 1(b)), while the case of Γ → ∞ accords the semi-infinite situation of E → E 0 and E → ∞ (the gray area above zero shown in Fig. 1(b)). It is noted that the cases of Γ → 0 and Γ → ∞ are, actually, equivalent to the single-level quantum dot model and the semi-infinite quantum well model respectively. During the process, the intrinsic parameters of E , E, T c /h and μc /h are related to the optimization indirectly by the variables of Γ , ηC , rc and rh . We can always find the values of intrinsic parameters corresponding to the optimized results. For instance, if T c = T h /2 = 10 K and the real energy width E = 15k B K (i.e. Γ = 1 and
Fig. 2. Scaled energy rc (upper) and rh (lower) at maximum power when the energy spectrum width Γ → 0 (dotted lines), Γ = 1 (dashed lines) and Γ → ∞ (solid lines where rc = rh ).
ηC = 0.5 where k B = 1.38 × 10−23 J/K), then, rc = 2.21511 and
rh = 1.70695 can be obtained from the optimized results, as shown in Fig. 2. After assuming μc = 1000k B K, we find that the heat engine works at maximum power when E = 1022.1511k B K and μh = 988.0121k B K. Related to Carnot efficiency and energy spectrum width, we get 3D figures of EMP and the maximum power, as shown in Figs. 3(a) and 3(b), where Γ vary from 0 to 10 because only the electrons with the energy locating in several k B T around the chemical potential of the reservoir can be exchanged. It is found that EMP will be increased by enlarging ηC and lessening Γ while the corresponding maximum power will be increased by enlarging both ηC and Γ . ηC = 0 implies that the temperatures of the two electron reservoirs are equal T h = T c , where all the transmitted electrons are from one reservoir to the other one due to the chemical potential gradient without power output, so EMP is always zero. At some fixed nonzero values of ηC , as also shown in Figs. 3(c) and 3(d), EMP will decrease monotonously if Γ increases, meanwhile, the corresponding maximum power will increase monotonously. However, they both become constant when Γ is large enough. When ηC → 1 and Γ → 0, EMP will tend to ηC theoretically while the corresponding maximum power vanishes. In a word, two bounds of EMP exist naturally in our model if Γ varies from 0 to ∞, and it is meaningful to find the upper/lower bound η+ /η− . When the energy spectrum width is infinitesimal, i.e. Γ → 0, Eq. (3) can be approximated by Q˙ h/c = ± h2 ( E − μh/c )( f h − f c )δ E, where the infinitesimal variable δ E → 0. Then, the heat flux out of the hot reservoir can be written as
Q˙ h =
2 h
k B T h rh
1 e rh + 1
−
1 erc + 1
δ E.
(10)
And the power output can also be expressed in the same way:
P=
2 h
k B T h rh − (1 − ηC )rc
1 e rh + 1
−
1 erc + 1
δ E.
(11)
Therefore, the efficiency of this thermionic heat engine can be represented with respect to the dimensionless scaled energy and Carnot efficiency as η = 1 − (1 − ηC )rc /rh . By using the numerical data of Fig. 2, η+ can be obtained at the fixed ηC . Seen from Fig. 4, the upper bound of EMP of our model is distinctly bigger than CA efficiency when Carnot efficiency of the heat engine is big enough. And at small relative temperature differences, η+ can be numerically fitted up to the third-order term of ηC which is given by [17]
η+ |ηC →0 =
ηC 2
+
ηC2 8
+
[7 + csc h2 (a0 /2)]ηC3 96
+ O ηC4 ,
(12)
X.G. Luo et al. / Physics Letters A 377 (2013) 1566–1570
Fig. 3. The efficiency at maximum power (a) and the corresponding maximum power (b) versus Carnot efficiency contour lines of figures (a) and (b) respectively after fixing the ηC from 0.1 to 0.9 successively (from bottom to top).
1569
ηC and energy spectrum width Γ . (c) and (d) are the
Furthermore, the efficiency of this thermionic heat engine now can be obtained by
η=
ηC , 1 − 2α + αηC Polylog(2,−e −r0 ) . r0 ln(1+e −r0 )
It is clearly shown that EMP at Γ → ∞ is the function of r0 and the Carnot efficiency. And from ∂ P /∂ r0 = 0, one can get the unique solution of r0 = 1.14455. Interestingly, combining with Fig. 2, we find that r0 = rc = rh when Γ → ∞ which is because r0 = T T−hT c rh − T T−c T c rc . For simplicity, where
α=
(15)
h
h
the lower bound of EMP can be expressed by
η− = Fig. 4. The efficiency at maximum power as a function of ηC . Black solid lines denote the upper and lower bounds (η± ) obtained in this Letter, the dashed line is the CA efficiency, and the dotted lines indicate the bounds in Ref. [6].
where a0 = 2.39936. Compared with the expansion of ηCA at the same condition ηCA |ηC →0 = ηC /2 + ηC2 /8 + 6ηC3 /96 + O (ηC4 ), one can find the upper bound of EMP is still bigger than CA efficiency even when Carnot efficiency is small enough. The other extreme situation of Γ → ∞ can be analyzed in the same way, however, Eq. (3) should be rewritten as Q˙ h/c =
∞
( E − μh/c )( f h − f c ) dE. And the range of integration ( E , ∞) E should be replaced by ( E 0 , ∞) after the power output is optimized. Then the heat flux out of the hot reservoir can be represented by
± h2
Q˙ h =
2 h
k2B T h2
ηC r0 ln 1 + e−r0 + ηC2 − 2ηC Polylog 2, −e−r0 , (13)
where r0 =
P=
2 h
μc −μh
k B (T h −T c )
. And the power output can be expressed as
k2B T h2 ηC2 r0 ln 1 + e −r0 .
(14)
ηC b 0 − c 0 ηC
,
(16)
where b0 = 2.87186 and c 0 = 0.93593. So far, we obtain an efficiency area, in which all the EMPs at arbitrary Γ and arbitrary ηC should locate, seen from the gray area in Fig. 4. Moreover, compared with the bounds obtained from Carnot heat engine [6], our bounds are always smaller except the lower bound is a little larger at very high Carnot efficiency. Consequently, the bounds in the two heat engine model are very different from each other. 4. Conclusions The thermionic heat engine with an ideal rectangular ESE mechanism has been studied. After introducing the Nielsen function, the analytic expressions of power output and the efficiency are obtained. Then, some optimizations focusing on power output are made to get the EMP. From the numerical results, the behavior that EMP can reach Carnot value as the Carnot value ηC → 1 is recovered if the energy spectrum width Γ → 0, which is equivalent to the single-level quantum dot model [17]. Moreover, at the fixed ηC , EMP decreases from the upper bound η+ to the lower bound η− monotonously if Γ varies from 0 to ∞, and meanwhile, the corresponding maximum power increases monotonously from
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X.G. Luo et al. / Physics Letters A 377 (2013) 1566–1570
0 to an upper bound. To the EMP, the upper bound is a little bigger than the CA efficiency, and the lower and upper bounds are very different from those obtained in Carnot heat engine [6]. Consequently, no matter Γ is, the EMP η∗ of a thermionic heat engine with an ideal ESE mechanism should converge to the efficiency area limited by the two bounds: η− η∗ η+ . These results can represent some features of the actual thermionic heat engine with nanowires, quantum dots, quantum wells etc. Acknowledgements This work is supported by the National Natural Science Foundations of China (Grant No. 11065008) and the College Student Innovation Training Projects of Nanchang University (Grant No. 2012038). References [1] [2] [3] [4] [5]
F. Curzon, B. Ahlborn, Am. J. Phys. 43 (1975) 22. C. van den Broeck, Phys. Rev. Lett. 95 (2005) 190602. B.J. de Cisneros, A. Calvo Hernández, Phys. Rev. Lett. 98 (2007) 130602. A. Gomez-Marin, J.M. Sancho, Phys. Rev. E 74 (2006) 062102. C. de Tomás, A. Calvo Hernández, J.M.M. Roco, Phys. Rev. E 85 (2012) 010104(R). [6] M. Esposito, R. Kawai, K. Lindenberg, C. Van den Broeck, Phys. Rev. Lett. 105 (2010) 150603. [7] Z.C. Tu, J. Phys. A: Math. Theor. 41 (2008) 312003.
[8] T. Schmiedl, U. Seifert, Europhys. Lett. 81 (2008) 20003. [9] T.E. Humphrey, R. Newbury, R.P. Taylor, H. Linke, Phys. Rev. Lett. 89 (2002) 116801; M.F. O’Dwyer, R.A. Lewis, C. Zhang, T.E. Humphrey, Phys. Rev. B 72 (2005) 205330. [10] T.E. Humphrey, H. Linke, R. Newbury, Physica E 11 (2001) 281; M.F. O’Dwyer, T.E. Humphrey, H. Linke, Nanotechnology 17 (2006) S338. [11] G.B. Akguc, O. Gülseren, Phys. Rev. B 85 (2012) 075432. [12] N. Nakpathomkun, H.Q. Xu, H. Linke, Phys. Rev. B 82 (2010) 235428; O. Karlström, H. Linke, G. Karlström, A. Wacker, Phys. Rev. B 84 (2012) 113415. [13] Z. Ding, L. Chen, F. Sun, Int. J. Energy Environ. 2 (2011) 627; Z. Ding, L. Chen, F. Sun, Rev. Mex. Fis. 56 (2010) 125. [14] J. He, B. He, Acta Phys. Sin. 59 (2010) 2345; H. Wang, G. Wu, H. Lu, Phys. Scr. 83 (2011) 055801. [15] L. Chen, Z. Ding, F. Sun, Energy 36 (2011) 4011; Z. Ding, L. Chen, F. Sun, Math. Comput. Model. 54 (2011) 2064; X.G. Luo, J.Z. He, Chin. Phys. B 20 (2011) 030509. [16] B.X. He, J.Z. He, Acta Phys. Sin. 59 (2010) 3846; X.G. Luo, J.Z. He, Acta Phys. Sin. 60 (2011) 090506. [17] M. Esposito, K. Lindenberg, C. Van den Broeck, Europhys. Lett. 85 (2009) 60010. [18] F. Chi, J. Zheng, Y.S. Liu, Y. Guo, Appl. Phys. Lett. 100 (2012) 233106. [19] T.E. Humphrey, M.F. O’Dwyer, H. Linke, J. Phys. D: Appl. Phys. 38 (2005) 2051. [20] Z. Ding, L. Chen, F. Sun, Appl. Math. Model. 35 (2011) 276. [21] X. Wang, J. He, W. Tang, Chin. Phys. B 18 (2009) 984; X.G. Luo, J.Z. He, Chin. Phys. B 20 (2011) 030509. [22] Z. Ding, L. Chen, F. Sun, Sci. China: Phys. Mech. Astron. 54 (2011) 1925; Z. Ding, L. Chen, F. Sun, J. Energy Inst. 85 (2012) 227; J. He, X. Wang, H. Liang, Phys. Scr. 80 (2009) 035701. [23] S. Su, J. Guo, G. Su, J. Chen, Energy 44 (2012) 570. [24] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
The impact of energy spectrum width in the energy selective electron low-temperature thermionic heat engine at maximum power Xiaoguang Luo a , Cong Li b , Nian Liu c , Ruiwen Li b , Jizhou He b , Teng Qiu a,∗ a b c
Department of Physics, Southeast University, Nanjing 211189, China Department of Physics, Nanchang University, Nanchang 330031, China Department of Physical and Electronics, Anhui Science and Technology University, Bengbu 233100, China
a r t i c l e
i n f o
Article history: Received 11 December 2012 Received in revised form 22 April 2013 Accepted 24 April 2013 Available online 29 April 2013 Communicated by C.R. Doering Keywords: Energy spectrum width Energy selective electron Thermionic Maximum power
a b s t r a c t A model of thermionic heat engine with the energy selective electron mechanism is studied. Analytical expressions of the power output and efficiency of this device are derived at low temperature, where the chemical potentials of the reservoirs are assumed to be constant. After discussing the impact of the energy spectrum width of the energy selective electron mechanism, we find two bounds (η± ) of efficiency at maximum power exist naturally. When the energy spectrum width increases gradually from zero and then to the semi-infinite case with the infinite upper limit, the efficiency at maximum power decreases monotonously from the upper bound η+ to the lower bound η− at a given temperature ratio of the cold and hot reservoirs. The two bound are given by numerical simulation and by an analytical expression respectively. These results may provide some guidance for the application of the practical energy selective electron heat engines. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Nowadays, the performance of conventional heat engines can be expressed well by the Carnot theorem and Curzon–Ahlborn (CA) efficiency limitation at maximum power [1]. To these energy conversion devices, the efficiency is always smaller than the Carnot value (ηC = 1 − T c / T h , where T h and T c are the temperatures of the hot and cold reservoirs respectively). Moreover, the efficiency at maximum √ power (EMP) should be smaller than the CA efficiency (ηCA = 1 − T c / T h ) which can be obtained from, in addition to traditional Carnot heat engine, linear irreversible thermodynamics for perfectly coupled systems [2–4] and symmetric dissipation systems [5,6]. It is known that Carnot efficiency is extended to be a universal limitation without any exceptions. However, the CA efficiency is not the unique limitation any more for EMP of some systems, such as Feynman ratchet and pawl model [7], stochastic procedure of a Brownian particle undergoing a Carnot cycle through the modulation of a harmonic potential [8] etc. Consequently, to find out the properties of some different heat engines working at maximum power becomes meaningful for fundamental thermodynamics and engineering applications. Here we focus on a thermionic device which consists of two different electron heat reservoirs and an electronic conductor between them. Generally, the chemical potentials of the electron
*
Corresponding author. E-mail addresses: [email protected] (X.G. Luo), [email protected] (T. Qiu).
0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.04.045
reservoirs can be assumed to be constant approximatively at relatively low temperature, such as several or few dozens of kelvin [9,10]. If the chemical potential of the cold reservoir is higher than that of the hot one (i.e. μc > μh ), the electrons in the two reservoirs could be exchanged because of the temperature gradient (hot electrons transmitted from the hot to cold reservoir) and chemical potential gradient (cold electrons transmitted from the cold to hot reservoir). The heat flux, coupled with the electronic flux, will flow from one reservoir to the other. According to the direction of the heat flux, the thermionic device can work as a heat engine or a refrigerator that controlled by the energy selective electron (ESE) mechanism [9]. With the development of the nanotechnology, many nanostructures exhibit good ESE properties for electron transmission, such as nanowires, quantum ratchets, quantum wells and quantum dots [10–12] and so on. In theory, these transmission properties can be simulated well by rectangular model [13], Lorentzian model [9,14], Gaussian model [15], multi-barrier model [16], single-level quantum dot [17,18] and so on. After investigated by experiments and numerical simulations, the efficiency of the heat engine can definitely be improved effectively and even reach Carnot value by adjusting the selection of the electronic conductor. For instance, a single-level channel between the two reservoirs can help to realize the Carnot efficiency, however, the power output vanishes in the meanwhile [9,17]. In practice, it is hard to find a material of electronic conductor behaving like the singlelevel quantum dot. And the resonant electron transmission with a resonant energy spectrum width may be an ideal potentially ESE mechanism in some way [10–12].
X.G. Luo et al. / Physics Letters A 377 (2013) 1566–1570
Fig. 1. The diagram of the ideal ESE thermionic heat engine where where F = f h − f c .
μc − μh = −eV (a), and the working area (gray area above zero) with respect to the electron energy (b)
Since high efficiency can be obtained at the cost of lowering the power output, the optimization between them will become more meaningful. And some strategies, focused on the efficiency and the power of the ESE thermionic heat engine, have been carried out [19,20]. However, the practical factors, such as multi-resonance channels [21], heat leakage [22], load [23] etc., will reduce the efficiency of heat engine. In fact, these factors can be prevented as far as possible by microcosmic science and engineering technology. Anyway, the natural lower and upper bounds without heat loss, similar to the temperature-ratio-dependent EMP bounds of Carnot heat engine [6], have not been obtained theoretically for the ESE thermionic heat engine. In this Letter, a thermionic model with an ideal rectangular model ESE mechanism is discussed. The position and the width of the energy spectrum range of the ESE mechanism will dramatically affect the performance of this thermionic heat engine, and we have found the natural lower and upper bounds for ESE thermionic heat engine with respect to the width of ESE mechanism, where the upper bound is just the result from the single-level quantum dot model [17] and the lower bound can be realized in the quantum well thermionic heat engine. 2. Thermionic heat engine with ESE mechanism The electrons in the ideal rectangular ESE thermionic model can be exchanged between the two electron reservoirs only when their energy E satisfy E E E + E, as shown in Fig. 1(a). The model can work as a heat engine if the energy of exchanged electrons E E 0 ≡ (μc T h − μh T c )/( T h − T c ), where the electrons with energy E 0 can be transmitted spontaneously without entropy increase [9]. The transmission probability of the electrons from one to the other reservoir in this ESE mechanism can be written as
ζ (E) =
1, E < E < E + E , 0, elsewhere.
(1)
factor of f h − f c (shown in Fig. 1(b)) being positive or negative depends on the electron energy E. However, the net electric current will vanish as both positive and negative gray areas in Fig. 1(b), equal to each other, locate in the energy spectrum range of the ESE mechanism E. Moreover, if E → 0, the electric current will also vanish, and the device will work reversibly when the energy of exchanged electrons E = E 0 . The energy change of the hot/cold reservoir is E − μh/c when an electron is transmitted out. Therefore, without heat loss, the heat flux out of the hot/cold reservoir per unit area per unit time can be given by
Q˙ h/c = ±
I=
h
( f h − f c ) dE ,
(2)
where e is the charge of an electron, h is the Planck constant, the factor of 2 accounts for degeneracy due to electron spin, and f h/c = 1/{1 + exp[( E − μh/c )/k B T h/c ]} with Boltzmann constant k B is the Fermi distribution function of the electrons in the hot/cold reservoir. From Eqs. (1) and (2), the direction of the net electric current between the two reservoirs can be controlled by adjusting the parameters E and E of the ESE mechanism, because the
+ E E
( E − μh/c )( f h − f c ) dE .
h
(3)
E
P = Q˙ h + Q˙ c =
2 h
+ E E
(μc − μh )( f h − f c ) dE ,
(4)
E
and the efficiency can be calculated by η ≡ P / Q˙ h . For nondimensionalization, Γ = E /[k B ( T c + T h )/2] is used to represent the energy spectrum width, where Γ → 0 as E → 0 donates nearly no electron can be exchanged and Γ → ∞ as E → ∞ implies plenty of electrons with different energy can be exchanged freely. The performance parameters in this model can be expressed analytically after using two integral functions of
b a
1/(1 + x) dx = ln(1 + x)|ab and
b a
ln(x)/(1 + x) dx = [ln(x) ln(1 +
x) + PolyLog(2, −x)]|ab , where PolyLog(2, −x) is the Nielsen function (e.g. PolyLog(2, 0) = 0 and PolyLog(2, −1) = −π 2 /12). Hence, the heat flux out of the hot reservoir per unit area per unit time can be expressed by
Q˙ h =
+ E E
E
2
Then, one can get the power output of the heat engine from
Combined with Landauer formula [24], the electrical current density flowing from the cold to the hot reservoir can be calculated by
2e
1567
2 h
k2B T h2
f e −Γh e −rh + g e −Γh e −rh − f e −rh − g e −rh
− (1 − ηC )2 f e −Γc e −rc + g e −Γc e −rc − f e −rc − g e −rc + (1 − ηC )rh − (1 − ηC )2 rc
× ln 1 + e −Γc e −rc − ln 1 + e −rc ,
(5)
where f (x) = ln(x) ln(1 + x) and g (x) = PolyLog(2, −x) for simplification, and Γh = Γ (1 − ηC /2), Γc = Γ (1 − ηC /2)/(1 − ηC ), rh = ( E − μh )/k B T h and rc = ( E − μc )/k B T c are the dimensionless scaled energies. Thus, the expression of the power of this heat engine, i.e. Eq. (4), can be rewritten as
P=
2 h
k2B T h2 rh − (1 − ηC )rc
rc
rh e +1 e +1 − ( , 1 − η ) ln × ln r C e h + e −Γh erc + e −Γc
(6)
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X.G. Luo et al. / Physics Letters A 377 (2013) 1566–1570
from which one can see that the power output can be directly regarded as the function of temperature T h , energy spectrum width Γ and dimensionless scaled energies. Consequently, at the arbitrarily given T h and Γ , the maximum power with respect to ηC can be obtained by optimizing the dimensionless scaled energies. So,
2
= rh − (1 − ηC )rc
rc
rh e +1 e +1 − ( × ln r 1 − η ) ln C e h + e −Γh erc + e −Γc
P∗ = P
h
k2B T h2
(7)
can be used to represent the power. Then the optimization will focus on the dimensionless variable P ∗ which will still be stand for power in the following discussions for convenience. The factor of f h − f c < 0 in Eq. (4) denotes that the net electron flux flows from cold to hot reservoir and then the power output will be reduced. In order to guarantee maximum power, the net electron flux should flow from the hot to cold reservoir as much as possible. Thus, the energy of all transmitted electrons should satisfy E E 0 which results in E E 0 , as shown in Fig. 1. To this system, the entropy increase caused by one exchanged electron from hot to cold reservoir can be expressed by
S = −
E − μh Th
+
E − μc Tc
(8)
.
It is clearly shown that S 0 owing to E E 0 . As a result,
μh −E − + T h
E −μc Tc
= k B (rc − rh ) 0. So we have rc rh where the
equality holds as E = E 0 .
3. Optimization with maximum power From Eq. (7), the values of rh and rc at maximum power can be found by optimizing the power function through ∂ P ∗ /∂ rc = ∂ P ∗ /∂ rh = 0. After that, it is shown the efficiency has nothing to do with the actual temperatures of the reservoirs directly but their ratio at the fixed energy spectrum width Γ . Anyway, these values are determined by the following two equations:
ln
e rh + 1 erh + e −Γh
− (1 − ηC ) ln
+ rh − (1 − ηC )rc
erc + 1
erc + e −Γc
e (e −Γc − 1) rc
= 0, (erc + 1)(erc + e −Γc ) e rh + 1 erc + 1 ln r − ( 1 − η ) ln C e h + e −Γh erc + e −Γc r h e (e −Γh − 1) + rh − (1 − ηC )rc r = 0. (e h + 1)(erh + e −Γh )
(9a)
(9b)
According to our calculation (see Fig. 2): rc rh and they will reach the maximum values simultaneously when the energy spectrum width Γ → 0; the values of both rh and rc will decrease monotonously if Γ increases, and rc = rh = 1.14455 as Γ → ∞. The case of Γ → 0 accords the situation of E → E P and E → 0 where F reaches the maximum value (the thick line shown in Fig. 1(b)), while the case of Γ → ∞ accords the semi-infinite situation of E → E 0 and E → ∞ (the gray area above zero shown in Fig. 1(b)). It is noted that the cases of Γ → 0 and Γ → ∞ are, actually, equivalent to the single-level quantum dot model and the semi-infinite quantum well model respectively. During the process, the intrinsic parameters of E , E, T c /h and μc /h are related to the optimization indirectly by the variables of Γ , ηC , rc and rh . We can always find the values of intrinsic parameters corresponding to the optimized results. For instance, if T c = T h /2 = 10 K and the real energy width E = 15k B K (i.e. Γ = 1 and
Fig. 2. Scaled energy rc (upper) and rh (lower) at maximum power when the energy spectrum width Γ → 0 (dotted lines), Γ = 1 (dashed lines) and Γ → ∞ (solid lines where rc = rh ).
ηC = 0.5 where k B = 1.38 × 10−23 J/K), then, rc = 2.21511 and
rh = 1.70695 can be obtained from the optimized results, as shown in Fig. 2. After assuming μc = 1000k B K, we find that the heat engine works at maximum power when E = 1022.1511k B K and μh = 988.0121k B K. Related to Carnot efficiency and energy spectrum width, we get 3D figures of EMP and the maximum power, as shown in Figs. 3(a) and 3(b), where Γ vary from 0 to 10 because only the electrons with the energy locating in several k B T around the chemical potential of the reservoir can be exchanged. It is found that EMP will be increased by enlarging ηC and lessening Γ while the corresponding maximum power will be increased by enlarging both ηC and Γ . ηC = 0 implies that the temperatures of the two electron reservoirs are equal T h = T c , where all the transmitted electrons are from one reservoir to the other one due to the chemical potential gradient without power output, so EMP is always zero. At some fixed nonzero values of ηC , as also shown in Figs. 3(c) and 3(d), EMP will decrease monotonously if Γ increases, meanwhile, the corresponding maximum power will increase monotonously. However, they both become constant when Γ is large enough. When ηC → 1 and Γ → 0, EMP will tend to ηC theoretically while the corresponding maximum power vanishes. In a word, two bounds of EMP exist naturally in our model if Γ varies from 0 to ∞, and it is meaningful to find the upper/lower bound η+ /η− . When the energy spectrum width is infinitesimal, i.e. Γ → 0, Eq. (3) can be approximated by Q˙ h/c = ± h2 ( E − μh/c )( f h − f c )δ E, where the infinitesimal variable δ E → 0. Then, the heat flux out of the hot reservoir can be written as
Q˙ h =
2 h
k B T h rh
1 e rh + 1
−
1 erc + 1
δ E.
(10)
And the power output can also be expressed in the same way:
P=
2 h
k B T h rh − (1 − ηC )rc
1 e rh + 1
−
1 erc + 1
δ E.
(11)
Therefore, the efficiency of this thermionic heat engine can be represented with respect to the dimensionless scaled energy and Carnot efficiency as η = 1 − (1 − ηC )rc /rh . By using the numerical data of Fig. 2, η+ can be obtained at the fixed ηC . Seen from Fig. 4, the upper bound of EMP of our model is distinctly bigger than CA efficiency when Carnot efficiency of the heat engine is big enough. And at small relative temperature differences, η+ can be numerically fitted up to the third-order term of ηC which is given by [17]
η+ |ηC →0 =
ηC 2
+
ηC2 8
+
[7 + csc h2 (a0 /2)]ηC3 96
+ O ηC4 ,
(12)
X.G. Luo et al. / Physics Letters A 377 (2013) 1566–1570
Fig. 3. The efficiency at maximum power (a) and the corresponding maximum power (b) versus Carnot efficiency contour lines of figures (a) and (b) respectively after fixing the ηC from 0.1 to 0.9 successively (from bottom to top).
1569
ηC and energy spectrum width Γ . (c) and (d) are the
Furthermore, the efficiency of this thermionic heat engine now can be obtained by
η=
ηC , 1 − 2α + αηC Polylog(2,−e −r0 ) . r0 ln(1+e −r0 )
It is clearly shown that EMP at Γ → ∞ is the function of r0 and the Carnot efficiency. And from ∂ P /∂ r0 = 0, one can get the unique solution of r0 = 1.14455. Interestingly, combining with Fig. 2, we find that r0 = rc = rh when Γ → ∞ which is because r0 = T T−hT c rh − T T−c T c rc . For simplicity, where
α=
(15)
h
h
the lower bound of EMP can be expressed by
η− = Fig. 4. The efficiency at maximum power as a function of ηC . Black solid lines denote the upper and lower bounds (η± ) obtained in this Letter, the dashed line is the CA efficiency, and the dotted lines indicate the bounds in Ref. [6].
where a0 = 2.39936. Compared with the expansion of ηCA at the same condition ηCA |ηC →0 = ηC /2 + ηC2 /8 + 6ηC3 /96 + O (ηC4 ), one can find the upper bound of EMP is still bigger than CA efficiency even when Carnot efficiency is small enough. The other extreme situation of Γ → ∞ can be analyzed in the same way, however, Eq. (3) should be rewritten as Q˙ h/c =
∞
( E − μh/c )( f h − f c ) dE. And the range of integration ( E , ∞) E should be replaced by ( E 0 , ∞) after the power output is optimized. Then the heat flux out of the hot reservoir can be represented by
± h2
Q˙ h =
2 h
k2B T h2
ηC r0 ln 1 + e−r0 + ηC2 − 2ηC Polylog 2, −e−r0 , (13)
where r0 =
P=
2 h
μc −μh
k B (T h −T c )
. And the power output can be expressed as
k2B T h2 ηC2 r0 ln 1 + e −r0 .
(14)
ηC b 0 − c 0 ηC
,
(16)
where b0 = 2.87186 and c 0 = 0.93593. So far, we obtain an efficiency area, in which all the EMPs at arbitrary Γ and arbitrary ηC should locate, seen from the gray area in Fig. 4. Moreover, compared with the bounds obtained from Carnot heat engine [6], our bounds are always smaller except the lower bound is a little larger at very high Carnot efficiency. Consequently, the bounds in the two heat engine model are very different from each other. 4. Conclusions The thermionic heat engine with an ideal rectangular ESE mechanism has been studied. After introducing the Nielsen function, the analytic expressions of power output and the efficiency are obtained. Then, some optimizations focusing on power output are made to get the EMP. From the numerical results, the behavior that EMP can reach Carnot value as the Carnot value ηC → 1 is recovered if the energy spectrum width Γ → 0, which is equivalent to the single-level quantum dot model [17]. Moreover, at the fixed ηC , EMP decreases from the upper bound η+ to the lower bound η− monotonously if Γ varies from 0 to ∞, and meanwhile, the corresponding maximum power increases monotonously from
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X.G. Luo et al. / Physics Letters A 377 (2013) 1566–1570
0 to an upper bound. To the EMP, the upper bound is a little bigger than the CA efficiency, and the lower and upper bounds are very different from those obtained in Carnot heat engine [6]. Consequently, no matter Γ is, the EMP η∗ of a thermionic heat engine with an ideal ESE mechanism should converge to the efficiency area limited by the two bounds: η− η∗ η+ . These results can represent some features of the actual thermionic heat engine with nanowires, quantum dots, quantum wells etc. Acknowledgements This work is supported by the National Natural Science Foundations of China (Grant No. 11065008) and the College Student Innovation Training Projects of Nanchang University (Grant No. 2012038). References [1] [2] [3] [4] [5]
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