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A model of quantum dot refrigerator driven by photon, which consists of two two-level quantum dots, a photon reservoir and two leads, is proposed in this paper. Comparing with previous studies, we consider the transitions of electrons between different energy levels in a single quantum dot, which is more practical.Based on the theory of master equation and the assumption of weak coupling, we derive the expression of the cooling rate and the coefficient of performance of the refrigerator and obtain the condition of the tight coupling of the refrigerator operation. Next, we plot numerically the performance characteristic curves between the cooling rate and the coefficient of performance in the case of the tight coupling and in the general case. We find that the curves between the cooling rate and the coefficient of performance are opened loops for tight coupling, but they are closed loops in the general case. And we gain the conclusions that the refrigerator can be reversible under the condition of the tight coupling, while it can be irreversible in the general case. Then the optimally operating range of the refrigerator is determined. Finally, the effect of the temperature of the photon reservoir, transition coefficient, and temperature ratio on the performance of refrigerator under the conditions of the maximum cooling rate are studied, and also the coefficient of performance under the maximum cooling rate, the maximum coefficient of performanceand the cooling rate under the maximum coefficient of performanceare analyzed in detail.
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Keywords:
- irreversible thermodynamics /
- quantum dot /
- thermoelectric refrigerator /
- performance optimization
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图 2 紧耦合条件下制冷率与制冷系数在不同温度
${T_{\rm{S}}}$ 下的关系Fig. 2. The relation curves of the cooling rate and the coefficient of performance at different temperature
${T_{\rm{S}}}$ under the condition of tight coupling.
图 3 紧耦合条件下制冷率与制冷系数在不同跃迁系数
${\varGamma _{n{\rm{r}}}}$ 下的关系Fig. 3. The relation curves of the cooling rate and the coefficient of performance at different transition coefficient
${\varGamma _{n{\rm{r}}}}$ under the condition of tight coupling.
图 4 一般情况下制冷率与制冷系数在不同温度
${T_{\rm{S}}}$ 下的关系Fig. 4. The relation curves of the cooling rate and the coefficient of performance at different temperature
${T_{\rm{S}}}$ in the general case.
图 5 一般情况下制冷率与制冷系数在不同跃迁系数
${\varGamma _{n{\rm{r}}}}$ 下的关系Fig. 5. The relation curves of the cooling rate and the coefficient of performanceat different transition coefficient
${\varGamma _{n{\rm{r}}}}$ in the general case.
图 6 在不同温度
${T_{\rm{S}}}$ 下, 两个优化性能参数${\dot Q}_{\rm{R}}^{\max }$ 和${\eta ^{{Q_{\rm{R}}}}}$ 随温比的变化Fig. 6. The curves of two optimal performance parameters
${\dot Q}_{\rm{R}}^{\max }$ and${\eta ^{{Q_{\rm{R}}}}}$ changing with the temperature ratio at different temperature${T_{\rm{S}}}$ 
图 7 在不同跃迁系数
${\varGamma _{n{\rm{r}}}}$ 下, 两个优化性能参数${\dot Q}_{\rm{R}}^{\max }$ 和${\eta ^{{Q_{\rm{R}}}}}$ 随着温比的变化Fig. 7. The curves of two optimal performance parameters
${\dot Q}_{\rm{R}}^{\max }$ and${\eta ^{{Q_{\rm{R}}}}}$ changing with the temperature ratio at different transition coefficient${\varGamma _{n{\rm{r}}}}$ 
图 8 在不同温度
${T_{\rm{S}}}$ 下, 两个优化性能参数${\dot Q}_{\rm{R}}^{\max }$ 和${\eta ^{{Q_{\rm{R}}}}}$ 随温比的变化Fig. 8. The curves of two optimalperformance parameters
${\dot Q}_{\rm{R}}^{\max }$ and${\eta ^{{Q_{\rm{R}}}}}$ changing with the temperature ratio at different temperature${T_{\rm{S}}}$ .
图 10 在不同温度
${T_{\rm{S}}}$ 下, 两个优化性能参数${\eta ^{\max }}$ 和${\dot Q}_{\rm{R}}^\eta $ 随温比的变化Fig. 10. The curves of two optimal performance parameters
${\eta ^{\max }}$ and${\dot Q}_{\rm{R}}^\eta $ changing with the temperature ratio at different temperature${T_{\rm{S}}}$ .
图 9 在不同跃迁系数
${\varGamma _{n{\rm{r}}}}$ 下, 两个优化性能参数${\dot Q}_{\rm{R}}^{\max }$ 和${\eta ^{{Q_{\rm{R}}}}}$ 随着温比的变化Fig. 9. The curves of two optimal performance parameters
${\dot Q}_{\rm{R}}^{\max }$ and${\eta ^{{Q_{\rm{R}}}}}$ changing with the temperature ratio at different transition coefficient${\varGamma _{n{\rm{r}}}}$ . -
[1] Giazotto F, Heikkila T T, Luukanen A, Savin A M, Pekola J P 2006 Rev. Mod. Phys. 78 217
Google Scholar
[2] Koumoto K, Mori T 2013 Thermoelectric Nanomaterials: Materials Design and Applications (Vol. 182) (Berlin: Springer Press) pp255－285
[3] Maciá E 2015 Thermoelectric Materials: Advances and Applications (Jenny: Stanford Publishing)
[4] Pichanusakorn P, Bandaru P 2010 Mater. Sci. Eng. R 67 19
Google Scholar
[5] Benenti G, Casati G, Saito K, Whitney R S 2017 Phys. Reports 694 1
Google Scholar
[6] Sothmann B, Sánchez R, Jordan A N 2014 Nanotechnology 26 032001
Google Scholar
[7] Sánchez R, Büttiker M 2011 Phys. Rev. B 83 085428
Google Scholar
[8] Dare A M, Lombardo P 2017 Phys. Rev. B 96 115414
Google Scholar
[9] Zhang Y C, Gin G X, Chen J C 2015 Phys. Rev. E 91 052118
Google Scholar
[10] Sothmann B, Sánchez R, Jordan A N, Büttiker M 2012 Phys. Rev. B 85 205301
Google Scholar
[11] Zhang Y C, Zhang X, Ye Z L, Gin G X, Chen J C 2017 Appl. Phys. Lett. 110 153501
Google Scholar
[12] Jordan A N, Sothmann B, Sánchez R, Buttiker M 2013 Phys. Rev. B 87 075312
Google Scholar
[13] Prance J R, Smith C G, Griffiths J P, Chorley S J, Anderson D, Jones G A C, Farrer I, Ritchie D A 2009 Phys. Rev. Lett. 102 146602
Google Scholar
[14] Sothmann B, Sánchez R, Jordan A N, Büttiker M 2013 N. J. Phys. 15 095021
Google Scholar
[15] Choi Y J, Jordan A N 2015 Physica E 74 465
Google Scholar
[16] Su S H, Zhang Y C, Chen J C, Shih T M 2016 Sci. Rep. 6 21425
Google Scholar
[17] Wohlman O E, Imry Y, Aharony A 2015 Phys. Rev. B 91 054302
Google Scholar
[18] Szukiewicz B, Eckern U, Wysokinski K I 2016 New J. Phys. 18 023050
Google Scholar
[19] Whitney R S, Sánchez R, Haupt F, Splettstoesser J 2016 Physica E 75 257
Google Scholar
[20] Lim J S, Sanchez D, Lopez R 2018 New J. Phys. 20 023038
Google Scholar
[21] Walldorf N, Jauho A P, Kaasbjerg K 2017 Phys. Rev. B 96 115415
[22] Jiang J H, Entin-Wohlman O, Imry Y 2013 New J. Phys. 15 075021
Google Scholar
[23] Su H, Shi Z C, He J Z 2015 Chin. Phys. Lett. 32 100501
Google Scholar
[24] Lin Z B, Li W, Fu J, Yang Y Y, He J Z 2019 Chin. Phys. Lett. 36 060501
Google Scholar
[25] Josefsson M, Svilans A, Burke A M, Hoffmann E A, Fahlvik S, Thelander C, Leijnse M, Linke H 2018 Nature Nanotechnol. 13 920
Google Scholar
[26] Thierschmann H, Sanchez R, Sothmann B, Arnold F, Heyn C, Hansen W, Buhmann H, Molenkamp L W 2015 Nature Nanotechnol. 10 854
Google Scholar
[27] Roche B, Roulleau P, Ulien T J, Jompol Y, Farrer I, Ritchie D A, Glattli D C 2015 Nature Commun. 6 6738
Google Scholar
[28] Hartmann F, Pfeffer P, Hofling S, Kamp M, Worschech L 2015 Phys. Rev. Lett. 114 146805
Google Scholar
[29] Cleuren B, Rutten B, van den Broeck C 2012 Phys. Rev. Lett. 108 120603
Google Scholar
[30] Levy A, Alicki R, Kosloff R 2012 Phys. Rev. Lett. 109 248901
Google Scholar
[31] Wang J H, Lai Y M, Ye Z L, He J Z, Ma Y L, Liao Q H 2015 Phys. Rev. E 91 050102
Google Scholar
[32] van den Broeck C 2005 Phys. Rev. Lett. 95 190602
Google Scholar
[33] Yuan Y, Wang R, He J Z, Ma Y L, Wang J H 2014 Phys. Rev. E 90 052151
Google Scholar
[34] Sheng S Q, Tu Z C 2014 Phys. Rev. E 89 012129
Google Scholar
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