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A brief review of thermal transport in mesoscopic systems from nonequilibrium Green’s function approach

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Abstract

With the rapidly increasing integration density and power density in nanoscale electronic devices, the thermal management concerning heat generation and energy harvesting becomes quite crucial. Since phonon is the major heat carrier in semiconductors, thermal transport due to phonons in mesoscopic systems has attracted much attention. In quantum transport studies, the nonequilibrium Green’s function (NEGF) method is a versatile and powerful tool that has been developed for several decades. In this review, we will discuss theoretical investigations of thermal transport using the NEGF approach from two aspects. For the aspect of phonon transport, the phonon NEGF method is briefly introduced and its applications on thermal transport in mesoscopic systems including one-dimensional atomic chains, multi-terminal systems, and transient phonon transport are discussed. For the aspect of thermoelectric transport, the caloritronic effects in which the charge, spin, and valley degrees of freedom are manipulated by the temperature gradient are discussed. The time-dependent thermoelectric behavior is also presented in the transient regime within the partitioned scheme based on the NEGF method.

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References

  1. E. Pop, S. Sinha, and K. E. Goodson, Heat generation and transport in nanometer-scale transistors, Proc. IEEE 94(8), 1587 (2006)

    Article  Google Scholar 

  2. N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, and B. Li, Phononics: Manipulating heat flow with electronic analogs and beyond, Rev. Mod. Phys. 84(3), 1045 (2012)

    Article  ADS  Google Scholar 

  3. G. Zhang and Y. W. Zhang, Thermal properties of two-dimensional materials, Chin. Phys. B 26(3), 034401 (2017)

    Article  ADS  Google Scholar 

  4. X. Chen, Y. Liu, and W. Duan, Thermal engineering in low-dimensional quantum devices: A tutorial review of nonequilibrium Green’s function methods, Small Methods 2(6), 1700343 (2018)

    Article  Google Scholar 

  5. D. G. Cahill, P. V. Braun, G. Chen, D. R. Clarke, S. Fan, K. E. Goodson, P. Keblinski, W. P. King, G. D. Mahan, A. Majumdar, H. J. Maris, S. R. Phillpot, E. Pop, and L. Shi, Nanoscale thermal transport (II): 2003–2012, Appl. Phys. Rev. 1(1), 011305 (2014)

    Article  ADS  Google Scholar 

  6. B. Li, L. Wang, and G. Casati, Thermal diode: Rectification of heat flux, Phys. Rev. Lett. 93(18), 184301 (2004)

    Article  ADS  Google Scholar 

  7. B. Li, L. Wang, and G. Casati, Negative differential thermal resistance and thermal transistor, Appl. Phys. Lett. 88(14), 143501 (2006)

    Article  ADS  Google Scholar 

  8. W. Chung Lo, L. Wang, and B. Li, Thermal Transistor: Heat Flux Switching and Modulating, J. Phys. Soc. Jpn. 77(5), 054402 (2008)

    Article  ADS  Google Scholar 

  9. L. Wang and B. Li, Thermal logic gates: Computation with phonons, Phys. Rev. Lett. 99(17), 177208 (2007)

    Article  ADS  Google Scholar 

  10. L. Wang and B. Li, Thermal memory: A storage of phononic information, Phys. Rev. Lett. 101(26), 267203 (2008)

    Article  ADS  Google Scholar 

  11. H. Zhu, J. Yi, M. Y. Li, J. Xiao, L. Zhang, C. W. Yang, R. A. Kaindl, L. J. Li, Y. Wang, and X. Zhang, Observation of chiral phonons, Science 359(6375), 579 (2018)

    Article  MathSciNet  ADS  Google Scholar 

  12. J. Lu, C. Qiu, M. Ke, and Z. Liu, Valley vortex states in sonic crystals, Phys. Rev. Lett. 116(9), 093901 (2016)

    Article  ADS  Google Scholar 

  13. J. Lu, C. Qiu, L. Ye, X. Fan, M. Ke, F. Zhang, and Z. Liu, Observation of topological valley transport of sound in sonic crystals, Nat. Phys. 13(4), 369 (2017)

    Article  Google Scholar 

  14. Y. Liu, Y. Xu, S. C. Zhang, and W. Duan, Model for topological phononics and phonon diode, Phys. Rev. B 96(6), 064106 (2017)

    Article  ADS  Google Scholar 

  15. S. Twaha, J. Zhu, Y. Yan, and B. Li, A comprehensive review of thermoelectric technology: Materials, applications, modelling and performance improvement, Renew. Sustain. Energy Rev. 65, 698 (2016)

    Article  Google Scholar 

  16. D. Li, Y. Gong, Y. Chen, J. Lin, Q. Khan, Y. Zhang, Y. Li, H. Zhang, and H. Xie, Recent progress of two-dimensional thermoelectric materials, Nano-Micro Lett. 12(1), 36 (2020)

    Article  ADS  Google Scholar 

  17. L. D. Zhao, S. H. Lo, Y. Zhang, H. Sun, G. Tan, C. Uher, C. Wolverton, V. P. Dravid, and M. G. Kanatzidis, Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals, Nature 508(7496), 373 (2014)

    Article  ADS  Google Scholar 

  18. M. J. Lee, J. H. Ahn, J. H. Sung, H. Heo, S. G. Jeon, W. Lee, J. Y. Song, K. H. Hong, B. Choi, S. H. Lee, and M. H. Jo, Thermoelectric materials by using two-dimensional materials with negative correlation between electrical and thermal conductivity, Nat. Commun. 7(1), 12011 (2016)

    Article  ADS  Google Scholar 

  19. C. Chang, M. Wu, D. He, Y. Pei, C. F. Wu, X. Wu, H. Yu, F. Zhu, K. Wang, Y. Chen, L. Huang, J. F. Li, J. He, and L. D. Zhao, 3D charge and 2D phonon transports leading to high out-of-plane ZT in n-type SnSe crystals, Science 360(6390), 778 (2018)

    Article  Google Scholar 

  20. H. Babaei, J. M. Khodadadi, and S. Sinha, Large theoretical thermoelectric power factor of suspended single-layer MoS2, Appl. Phys. Lett. 105(19), 193901 (2014)

    Article  ADS  Google Scholar 

  21. C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, Direct calculation of the tunneling current, J. Phys. C 4(8), 916 (1971)

    Article  ADS  Google Scholar 

  22. Y. Meir, and N. S. Wingreen, Landauer formula for the current through an interacting electron region, Phys. Rev. Lett. 68(16), 2512 (1992)

    Article  ADS  Google Scholar 

  23. A. P. Jauho, N. S. Wingreen, and Y. Meir, Time-dependent transport in interacting and noninteracting resonant-tunneling systems, Phys. Rev. B 50(8), 5528 (1994)

    Article  ADS  Google Scholar 

  24. J. S. Wang, J. Wang, and N. Zeng, Nonequilibrium Green’s function approach to mesoscopic thermal transport, Phys. Rev. B 74(3), 033408 (2006)

    Article  ADS  Google Scholar 

  25. J. S. Wang, N. Zeng, J. Wang, and C. K. Gan, Nonequilibrium Green’s function method for thermal transport in junctions, Phys. Rev. E 75(6), 061128 (2007)

    Article  ADS  Google Scholar 

  26. N. Sergueev, D. Roubtsov, and H. Guo, Ab initio analysis of electron-phonon coupling in molecular devices, Phys. Rev. Lett. 95(14), 146803 (2005)

    Article  ADS  Google Scholar 

  27. T. Shimazaki and Y. Asai, Bias voltage dependence on the vibronic electric current, Phys. Rev. B 77(7), 075110 (2008)

    Article  ADS  Google Scholar 

  28. M. Paulsson, T. Frederiksen, and M. Brandbyge, Modeling inelastic phonon scattering in atomic- and molecular-wire junctions, Phys. Rev. B 72(20), 201101 (2005)

    Article  ADS  Google Scholar 

  29. A. Ferretti, A. Calzolari, R. Di Felice, F. Manghi, M. J. Caldas, M. B. Nardelli, and E. Molinari, First-principles theory of correlated transport through nanojunctions, Phys. Rev. Lett. 94(11), 116802 (2005)

    Article  ADS  Google Scholar 

  30. K. S. Thygesen and A. Rubio, Conserving GW scheme for nonequilibrium quantum transport in molecular contacts, Phys. Rev. B 77(11), 115333 (2008)

    Article  ADS  Google Scholar 

  31. J. Taylor, H. Guo, and J. Wang, Ab initio modeling of quantum transport properties of molecular electronic devices, Phys. Rev. B 63(24), 245407 (2001)

    Article  ADS  Google Scholar 

  32. M. Brandbyge, J. L. Mozos, P. Ordejón, J. Taylor, and K. Stokbro, Density-functional method for nonequilibrium electron transport, Phys. Rev. B 65(16), 165401 (2002)

    Article  ADS  Google Scholar 

  33. Z. Y. Ong and E. Pop, Effect of substrate modes on thermal transport in supported graphene, Phys. Rev. B 84(7), 075471 (2011)

    Article  ADS  Google Scholar 

  34. G. Zhang and B. Li, Thermal conductivity of nanotubes revisited: Effects of chirality, isotope impurity, tube length, and temperature, J. Chem. Phys. 123(11), 114714 (2005)

    Article  ADS  Google Scholar 

  35. G. Zhang and H. Zhang, Thermal conduction and rectification in few-layer graphene Y junctions, Nanoscale 3(11), 4604 (2011)

    Article  ADS  Google Scholar 

  36. R. Yang and G. Chen, Thermal conductivity modeling of periodic two-dimensional nanocomposites, Phys. Rev. B 69(19), 195316 (2004)

    Article  ADS  Google Scholar 

  37. W. Li, N. Mingo, L. Lindsay, D. A. Broido, D. A. Stewart, and N. A. Katcho, Thermal conductivity of diamond nanowires from first principles, Phys. Rev. B 85(19), 195436 (2012)

    Article  ADS  Google Scholar 

  38. W. Li, J. Carrete, N. A. Katcho, and N. Mingo, Sheng-BTE: A solver of the Boltzmann transport equation for phonons, Comput. Phys. Commun. 185(6), 1747 (2014)

    Article  MATH  ADS  Google Scholar 

  39. J. S. Wang, J. Wang, and J. T. Lü, Quantum thermal transport in nanostructures, Eur. Phys. J. B 62(4), 381 (2008)

    Article  ADS  Google Scholar 

  40. J.-S. Wang, B. K. Agarwalla, H. Li, and J. Thingna, Nonequilibrium Green’s function method for quantum thermal transport, Front. Phys. 9(6), 673 (2014)

    Article  ADS  Google Scholar 

  41. H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer-Verlag, Berlin, 1998

    Google Scholar 

  42. N. Mingo and L. Yang, Phonon transport in nanowires coated with an amorphous material: An atomistic Green’s function approach, Phys. Rev. B 68(24), 245406 (2003)

    Article  ADS  Google Scholar 

  43. T. Yamamoto and K. Watanabe, Nonequilibrium Green’s function approach to phonon transport in defective carbon nanotubes, Phys. Rev. Lett. 96(25), 255503 (2006)

    Article  ADS  Google Scholar 

  44. L. Zhang, J. Thingna, D. He, J. S. Wang, and B. Li, Non-linearity enhanced interfacial thermal conductance and rectification, EPL (Europhys. Lett.) 103(6), 64002 (2013)

    Article  ADS  Google Scholar 

  45. J. T. Lü and J. S. Wang, Coupled electron and phonon transport in one-dimensional atomic junctions, Phys. Rev. B 76(16), 165418 (2007)

    Article  ADS  Google Scholar 

  46. L. Zhang, J. T. Lü, J. S. Wang, and B. Li, Thermal transport across metal-insulator interface via electron-phonon interaction, J. Phys.: Condens. Matter 25(44), 445801 (2013)

    Google Scholar 

  47. K. Gordiz and A. Henry, Examining the effects of stiffness and mass difference on the thermal interface conductance between Lennard-Jones solids, Sci. Rep. 5(1), 18361 (2015)

    Article  ADS  Google Scholar 

  48. J. Chen, J. H. Walther, and P. Koumoutsakos, Co-valently bonded graphene-carbon nanotube hybrid for high-performance thermal interfaces, Adv. Funct. Mater. 25(48), 7539 (2015)

    Article  Google Scholar 

  49. W. A. Little, The transport of heat between dissimilar solids at low temperatures, Can. J. Phys. 37(3), 334 (1959)

    Article  ADS  Google Scholar 

  50. E. T. Swartz and R. O. Pohl, Thermal boundary resistance, Rev. Mod. Phys. 61(3), 605 (1989)

    Article  ADS  Google Scholar 

  51. L. Zhang, P. Keblinski, J. S. Wang, and B. Li, Interfacial thermal transport in atomic junctions, Phys. Rev. B Condens. Matter Mater. Phys. 83(6), 064303 (2011)

    Article  ADS  Google Scholar 

  52. C. B. Saltonstall, C. A. Polanco, J. C. Duda, A. W. Ghosh, P. M. Norris, and P. E. Hopkins, Effect of interface adhesion and impurity mass on phonon transport at atomic junctions, J. Appl. Phys. 113(1), 013516 (2013)

    Article  ADS  Google Scholar 

  53. G. Xiong, J. S. Wang, D. Ma, and L. Zhang, Dramatic enhancement of interfacial thermal transport by mass-graded and coupling-graded materials, EPL (Europhys. Lett.) 128(5), 54007 (2020)

    Article  Google Scholar 

  54. B. Chen and L. Zhang, Optimized couplers for interfacial thermal transport, J. Phys.: Condens. Matter 27(12), 125401 (2015)

    ADS  Google Scholar 

  55. D. He, J. Thingna, J. S. Wang, and B. Li, Quantum thermal transport through anharmonic systems: A self-consistent approach, Phys. Rev. B 94(15), 155411 (2016)

    Article  ADS  Google Scholar 

  56. J. Fang, X. Qian, C. Y. Zhao, B. Li, and X. Gu, Monitoring anharmonic phonon transport across interfaces in one-dimensional lattice chains, Phys. Rev. E 101(2), 022133 (2020)

    Article  ADS  Google Scholar 

  57. J. C. Klöckner, M. Bürkle, J. C. Cuevas, and F. Pauly, Length dependence of the thermal conductance of alkane-based single-molecule junctions: An ab initio study, Phys. Rev. B 94(20), 205425 (2016)

    Article  ADS  Google Scholar 

  58. J. C. Klöckner, R. Siebler, J. C. Cuevas, and F. Pauly, Thermal conductance and thermoelectric figure of merit of C60-based single-molecule junctions: Electrons, phonons, and photons, Phys. Rev. B 95(24), 245404 (2017)

    Article  ADS  Google Scholar 

  59. L. Cui, R. Miao, C. Jiang, E. Meyhofer, and P. Reddy, Perspective: Thermal and thermoelectric transport in molecular junctions, J. Chem. Phys. 146(9), 092201 (2017)

    Article  ADS  Google Scholar 

  60. L. Hu, L. Zhang, M. Hu, J. S. Wang, B. Li, and P. Keblinski, Phonon interference at self-assembled monolayer interfaces: Molecular dynamics simulations, Phys. Rev. B 81(23), 235427 (2010)

    Article  ADS  Google Scholar 

  61. J. Lu, K. Yuan, F. Sun, K. Zheng, Z. Zhang, J. Zhu, X. Wang, X. Zhang, Y. Zhuang, Y. Ma, X. Cao, J. Zhang, and D. Tang, Self-assembled monolayers for the polymer/semiconductor interface with improved interfacial thermal management, ACS Appl. Mater. Interfaces 11(45), 42708 (2019)

    Article  Google Scholar 

  62. H. Fan, M. Wang, D. Han, J. Zhang, J. Zhang, and X. Wang, Enhancement of interfacial thermal transport between metal and organic semiconductor using self-assembled monolayers with different terminal groups, J. Phys. Chem. C 124(31), 16748 (2020)

    Article  Google Scholar 

  63. X. Chen, Y. Xu, X. Zou, B. L. Gu, and W. Duan, Interfacial thermal conductance of partially unzipped carbon nanotubes: Linear scaling and exponential decay, Phys. Rev. B 87(15), 155438 (2013)

    Article  ADS  Google Scholar 

  64. W. Zhang, N. Mingo, and T. S. Fisher, Simulation of phonon transport across a non-polar nanowire junction using an atomistic Green’s function method, Phys. Rev. B 76(19), 195429 (2007)

    Article  ADS  Google Scholar 

  65. Y. Xu, X. Chen, B. L. Gu, and W. Duan, Intrinsic anisotropy of thermal conductance in graphene nanoribbons, Appl. Phys. Lett. 95(23), 233116 (2009)

    Article  ADS  Google Scholar 

  66. Y. Xu, X. Chen, J. S. Wang, B. L. Gu, and W. Duan, Thermal transport in graphene junctions and quantum dots, Phys. Rev. B 81(19), 195425 (2010)

    Article  ADS  Google Scholar 

  67. Z. Ding, Q. X. Pei, J. W. Jiang, W. Huang, and Y. W. Zhang, Interfacial thermal conductance in graphene/MoS2 heterostructures, Carbon 96, 888 (2016)

    Article  Google Scholar 

  68. S. Sadasivam, N. Ye, J. P. Feser, J. Charles, K. Miao, T. Kubis, and T. S. Fisher, Thermal transport across metal silicide-silicon interfaces: First-principles calculations and Green’s function transport simulations, Phys. Rev. B 95(8), 085310 (2017)

    Article  ADS  Google Scholar 

  69. Z. Zhang, Y. Xie, Q. Peng, and Y. Chen, Phonon transport in single-layer boron nanoribbons, Nanotechnology 27(44), 445703 (2016)

    Article  Google Scholar 

  70. Y. Blanter and M. Büttiker, Shot noise in mesoscopic conductors, Phys. Rep. 336(1–2), 1 (2000)

    Article  ADS  Google Scholar 

  71. M. Büttiker, Four-terminal phase-coherent conductance, Phys. Rev. Lett. 57(14), 1761 (1986)

    Article  ADS  Google Scholar 

  72. M. Büttiker, Absence of backscattering in the quantum Hall effect in multiprobe conductors, Phys. Rev. B 38(14), 9375 (1988)

    Article  ADS  Google Scholar 

  73. L. Zhang, J. S. Wang, and B. Li, Ballistic thermal rectification in nanoscale three-terminal junctions, Phys. Rev. B 81(10), 100301 (2010)

    Article  ADS  Google Scholar 

  74. Y. Ming, Z. X. Wang, Z. J. Ding, and H. M. Li, Ballistic thermal rectification in asymmetric three-terminal mesoscopic dielectric systems, New J. Phys. 12(10), 103041 (2010)

    Article  ADS  Google Scholar 

  75. T. Ouyang, Y. Chen, Y. Xie, X. L. Wei, K. Yang, P. Yang, and J. Zhong, Ballistic thermal rectification in asymmetric three-terminal graphene nanojunctions, Phys. Rev. B 82(24), 245403 (2010)

    Article  ADS  Google Scholar 

  76. Z. X. Xie, K. M. Li, L. M. Tang, C. N. Pan, and K. Q. Chen, Nonlinear phonon transport and ballistic thermal rectification in asymmetric graphene-based three terminal junctions, Appl. Phys. Lett. 100(18), 183110 (2012)

    Article  ADS  Google Scholar 

  77. Y. Gu, Mode-dependent phonon transmission in a T-shaped three-terminal graphene nanojunction, Carbon 158, 818 (2020)

    Article  Google Scholar 

  78. L. Zhang, J. S. Wang, and B. Li, Phonon Hall effect in four-terminal nano-junctions, New J. Phys. 11(11), 113038 (2009)

    Article  ADS  Google Scholar 

  79. Y. Xing, Q. F. Sun, and J. Wang, Nature of spin Hall effect in a finite ballistic two-dimensional system with Rashba and Dresselhaus spin-orbit interaction, Phys. Rev. B 73(20), 205339 (2006)

    Article  ADS  Google Scholar 

  80. Y. Xing, Q. F. Sun, and J. Wang, Symmetry and transport property of spin current induced spin-Hall effect, Phys. Rev. B 75(7), 075324 (2007)

    Article  ADS  Google Scholar 

  81. M. Wei, M. Zhou, B. Wang, and Y. Xing, Thermoelectric transport properties of ferromagnetic graphene with CT-invariant quantum spin Hall effect, Phys. Rev. B 102(7), 075432 (2020)

    Article  ADS  Google Scholar 

  82. C. Strohm, G. L. J. A. Rikken, and P. Wyder, Phenomenological evidence for the phonon Hall effect, Phys. Rev. Lett. 95(15), 155901 (2005)

    Article  ADS  Google Scholar 

  83. E. C. Cuansing and J. S. Wang, Transient behavior of heat transport in a thermal switch, Phys. Rev. B 81(5), 052302 (2010)

    Article  ADS  Google Scholar 

  84. R. Tuovinen, N. Säkkinen, D. Karlsson, G. Stefanucci, and R. van Leeuwen, Phononic heat transport in the transient regime: An analytic solution, Phys. Rev. B 93(21), 214301 (2016)

    Article  ADS  Google Scholar 

  85. E. C. Cuansing and J. S. Wang, Erratum: Transient behavior of heat transport in a thermal switch [Phys. Rev. B 81, 052302 (2010)], Phys. Rev. B 83(1), 019902 (201

    Article  ADS  Google Scholar 

  86. J. S. Wang, B. K. Agarwalla, and H. Li, Transient behavior of full counting statistics in thermal transport, Phys. Rev. B 84(15), 153412 (2011)

    Article  ADS  Google Scholar 

  87. B. K. Agarwalla, B. Li, and J. S. Wang, Full-counting statistics of heat transport in harmonic junctions: Transient, steady states, and fluctuation theorems, Phys. Rev. E. 85(5), 051142 (2012)

    Article  ADS  Google Scholar 

  88. B. K. Agarwalla, J. H. Jiang, and D. Segal, Full counting statistics of vibrationally assisted electronic conduction: Transport and fluctuations of thermoelectric efficiency, Phys. Rev. B 92(24), 245418 (2015)

    Article  ADS  Google Scholar 

  89. K. Saito and A. Dhar, Fluctuation theorem in quantum heat conduction, Phys. Rev. Lett. 99(18), 180601 (2007)

    Article  ADS  Google Scholar 

  90. K. Saito and A. Dhar, Generating function formula of heat transfer in harmonic networks, Phys. Rev. E 83(4), 041121 (2011)

    Article  ADS  Google Scholar 

  91. Y. Dubi and M. Di Ventra, Heat flow and thermoelectricity in atomic and molecular junctions, Rev. Mod. Phys. 83(1), 131 (2011)

    Article  ADS  Google Scholar 

  92. A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang, Enhanced thermoelectric performance of rough silicon nanowires, Nature 451(7175), 163 (2008)

    Article  ADS  Google Scholar 

  93. P. Reddy, S. Y. Jang, R. A. Segalman, and A. Majumdar, Thermoelectricity in molecular junctions, Science 315(5818), 1568 (2007)

    Article  ADS  Google Scholar 

  94. T. Gunst, T. Markussen, A. P. Jauho, and M. Brandbyge, Thermoelectric properties of finite graphene antidot lattices, Phys. Rev. B 84(15), 155449 (2011)

    Article  ADS  Google Scholar 

  95. Y. Chen, T. Jayasekera, A. Calzolari, K. W. Kim, and M. B. Nardelli, Thermoelectric properties of graphene nanoribbons, junctions and superlattices, J. Phys.: Condens. Matter 22(37), 372202 (2010)

    Google Scholar 

  96. K. Yang, Y. Chen, R. D’Agosta, Y. Xie, J. Zhong, and A. Rubio, Enhanced thermoelectric properties in hybrid graphene/boron nitride nanoribbons, Phys. Rev. B 86(4), 045425 (2012)

    Article  ADS  Google Scholar 

  97. Y. Xing, Q. F. Sun, and J. Wang, Nernst and Seebeck effects in a graphene nanoribbon, Phys. Rev. B 80(23), 235411 (2009)

    Article  ADS  Google Scholar 

  98. M. M. Wei, Y. T. Zhang, A. M. Guo, J. J. Liu, Y. Xing, and Q. F. Sun, Magnetothermoelectric transport properties of multiterminal graphene nanoribbons, Phys. Rev. B 93(24), 245432 (2016)

    Article  ADS  Google Scholar 

  99. B. Wang, J. Zhou, R. Yang, and B. Li, Ballistic thermoelectric transport in structured nanowires, New J. Phys. 16(6), 065018 (2014)

    Article  ADS  Google Scholar 

  100. J. Li, B. Wang, F. Xu, Y. Wei, and J. Wang, Spin-dependent Seebeck effects in graphene-based molecular junctions, Phys. Rev. B 93(19), 195426 (2016)

    Article  ADS  Google Scholar 

  101. B. Zhou, B. Zhou, Y. Yao, G. Zhou, and M. Hu, Spin-dependent Seebeck effects in a graphene superlattice p-n junction with different shapes, J. Phys.: Condens. Matter 29(40), 405303 (2017)

    Google Scholar 

  102. P. N. Butcher, Thermal and electrical transport formalism for electronic microstructures with many terminals, J. Phys.: Condens. Matter 2(22), 4869 (1990)

    ADS  Google Scholar 

  103. U. Sivan and Y. Imry, Multichannel Landauer formula for thermoelectric transport with application to thermopower near the mobility edge, Phys. Rev. B 33(1), 551 (1986)

    Article  ADS  Google Scholar 

  104. G. D. Mahan, Many-Particle Physics, Springer, New York, 2000

    Book  Google Scholar 

  105. J. Ren, J. X. Zhu, J. E. Gubernatis, C. Wang, and B. Li, Thermoelectric transport with electron-phonon coupling and electron-electron interaction in molecular junctions, Phys. Rev. B85(15), 155443 (2012)

    Article  ADS  Google Scholar 

  106. K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Observation of the spin Seebeck effect, Nature 455(7214), 778 (2008)

    Article  ADS  Google Scholar 

  107. G. E. Bauer, A. H. MacDonald, and S. Maekawa, Spin caloritronics, Solid State Commun. 150(11–12), 459 (2010)

    Article  ADS  Google Scholar 

  108. G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Spin caloritronics, Nat. Mater. 11(5), 391 (2012)

    Article  ADS  Google Scholar 

  109. M. Hatami, G. E. W. Bauer, Q. Zhang, and P. J. Kelly, Thermal spin-transfer torque in magnetoelectronic devices, Phys. Rev. Lett. 99(6), 066603 (2007)

    Article  ADS  Google Scholar 

  110. Z. Zhang, L. Bai, X. Chen, H. Guo, X. L. Fan, D. S. Xue, D. Houssameddine, and C. M. Hu, Observation of thermal spin-transfer torque via ferromagnetic resonance in magnetic tunnel junctions, Phys. Rev. B 94(6), 064414 (2016)

    Article  ADS  Google Scholar 

  111. M. Zeng, Y. Feng, and G. Liang, Graphene-based spin caloritronics, Nano Lett. 11(3), 1369 (2011)

    Article  ADS  Google Scholar 

  112. X. Q. Yu, Z. G. Zhu, G. Su, and A. P. Jauho, Thermally driven pure spin and valley currents via the anomalous nernst effect in monolayer Group-VI dichalcogenides, Phys. Rev. Lett. 115(24), 246601 (2015)

    Article  ADS  Google Scholar 

  113. S. G. Cheng, Y. Xing, Q. F. Sun, and X. C. Xie, Spin Nernst effect and Nernst effect in two-dimensional electron systems, Phys. Rev. B78(4), 045302 (2008)

    Article  ADS  Google Scholar 

  114. Q. Wang, J. Li, Y. Nie, F. Xu, Y. Yu, and B. Wang, Pure spin current and phonon thermoelectric transport in a triangulene-based molecular junction, Phys. Chem. Chem. Phys. 20(23), 15736 (2018)

    Article  Google Scholar 

  115. D. Xiao, W. Yao, and Q. Niu, Valley-contrasting physics in graphene: Magnetic moment and topological transport, Phys. Rev. Lett. 99(23), 236809 (2007)

    Article  ADS  Google Scholar 

  116. C. E. Nebel, Electrons dance in diamond, Nat. Mater. 12(8), 690 (2013)

    Article  ADS  Google Scholar 

  117. A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, Valley filter and valley valve in graphene, Nat. Phys. 3(3), 172 (2007)

    Article  Google Scholar 

  118. D. Gunlycke and C. T. White, Graphene valley filter using a line defect, Phys. Rev. Lett. 106(13), 136806 (2011)

    Article  ADS  Google Scholar 

  119. Y. Jiang, T. Low, K. Chang, M. I. Katsnelson, and F. Guinea, Generation of pure bulk valley current in graphene, Phys. Rev. Lett. 110(4), 046601 (2013)

    Article  ADS  Google Scholar 

  120. Z. Yu, F. Xu, and J. Wang, Valley Seebeck effect in gate tunable zigzag graphene nanoribbons, Carbon 99, 451 (2016)

    Article  Google Scholar 

  121. L. Zhang, Z. Yu, F. Xu, and J. Wang, Influence of dephasing and B/N doping on valley Seebeck effect in zigzag graphene nanoribbons, Carbon 126, 183 (2018)

    Article  Google Scholar 

  122. X. Chen, L. Zhang, and H. Guo, Valley caloritronics and its realization by graphene nanoribbons, Phys. Rev. B 92(15), 155427 (2015)

    Article  ADS  Google Scholar 

  123. X. Zhai, W. Gao, X. Cai, D. Fan, Z. Yang, and L. Meng, Spin-valley caloritronics in silicene near room temperature, Phys. Rev. B 94(24), 245405 (2016)

    Article  ADS  Google Scholar 

  124. Z. P. Niu and S. Dong, Valley and spin thermoelectric transport in ferromagnetic silicene junctions, Appl. Phys. Lett. 104(20), 202401 (2014)

    Article  ADS  Google Scholar 

  125. X. Zhai, S. Wang, and Y. Zhang, Valley-spin Seebeck effect in heavy group-IV monolayers, New J. Phys. 19(6), 063007 (2017)

    Article  ADS  Google Scholar 

  126. G. Stefanucci and C. O. Almbladh, Time-dependent partition-free approach in resonant tunneling systems, Phys. Rev. B 69(19), 195318 (2004)

    Article  ADS  Google Scholar 

  127. M. Cini, Time-dependent approach to electron transport through junctions: General theory and simple applications, Phys. Rev. B 22(12), 5887 (1980)

    Article  ADS  Google Scholar 

  128. C. Caroli, R. Combescot, D. Lederer, P. Nozieres, and D. Saint-James, A direct calculation of the tunnelling current (II): Free electron description, J. Phys. C 4(16), 2598 (1971)

    Article  ADS  Google Scholar 

  129. Z. Yu, J. Yuan, and J. Wang, Time-dependent thermoelectric transport in mesoscopic systems under a quantum quench, Phys. Rev. B 101(23), 235433 (2020)

    Article  ADS  Google Scholar 

  130. B. Wang, J. Wang, and H. Guo, Current partition: A nonequilibrium Green’s function approach, Phys. Rev. Lett. 82(2), 398 (1999)

    Article  ADS  Google Scholar 

  131. J. Chen, M. ShangGuan, and J. Wang, A gauge invariant theory for time dependent heat current, New J. Phys. 17(5), 053034 (2015)

    Article  ADS  Google Scholar 

  132. X. Chen, J. Yuan, G. Tang, J. Wang, Z. Zhang, C. M. Hu, and H. Guo, Transient spin current under a thermal switch, J. Phys. D 51(27), 274004 (2018)

    Article  ADS  Google Scholar 

  133. F. G. Eich, A. Principi, M. Di Ventra, and G. Vignale, Luttinger-field approach to thermoelectric transport in nanoscale conductors, Phys. Rev. B 90(11), 115116 (2014)

    Article  ADS  Google Scholar 

  134. F. G. Eich, M. Di Ventra, and G. Vignale, Temperature-driven transient charge and heat currents in nanoscale conductors, Phys. Rev. B 93(13), 134309 (2016)

    Article  ADS  Google Scholar 

  135. C. Lozej and T. Rejec, Time-dependent thermoelectric transport in nanosystems: Reflectionless Luttinger field approach, Phys. Rev. B 98(7), 075427 (2018)

    Article  ADS  Google Scholar 

  136. A. Crépieux, F. Šimkovic, B. Cambon, and F. Michelini, Enhanced thermopower under a time-dependent gate voltage, Phys. Rev. B 83(15), 153417 (2011)

    Article  ADS  Google Scholar 

  137. A. Kara Slimane, P. Reck, and G. Fleury, Simulating time-dependent thermoelectric transport in quantum systems, Phys. Rev. B 101(23), 235413 (2020)

    Article  ADS  Google Scholar 

  138. M. M. Odashima and C. H. Lewenkopf, Time-dependent resonant tunneling transport: Keldysh and Kadanoff-Baym nonequilibrium Green’s functions in an analytically soluble problem, Phys. Rev. B 95(10), 104301 (2017)

    Article  ADS  Google Scholar 

  139. M. Ridley, and R. Tuovinen, Formal equivalence between partitioned and partition-free quenches in quantum transport, J. Low Temp. Phys. 191(5–6), 380 (2018)

    Article  ADS  Google Scholar 

  140. A. M. Daré and P. Lombardo, Time-dependent thermoelectric transport for nanoscale thermal machines, Phys. Rev. B 93(3), 035303 (2016)

    Article  ADS  Google Scholar 

  141. Z. Yu, L. Zhang, Y. Xing, and J. Wang, Investigation of transient heat current from first principles using complex absorbing potential, Phys. Rev. B 90(11), 115428 (2014)

    Article  ADS  Google Scholar 

  142. Z. Yu, G. M. Tang, and J. Wang, Full-counting statistics of transient energy current in mesoscopic systems, Phys. Rev. B 93(19), 195419 (2016)

    Article  ADS  Google Scholar 

  143. H. Li, B. K. Agarwalla, and J. S. Wang, Cumulant generating function formula of heat transfer in ballistic systems with lead-lead coupling, Phys. Rev. B 86(16), 165425 (2012)

    Article  ADS  Google Scholar 

  144. M. Ridley, M. Galperin, E. Gull, and G. Cohen, Numerically exact full counting statistics of the energy current in the Kondo regime, Phys. Rev. B 100(16), 165127 (2019)

    Article  ADS  Google Scholar 

  145. G. Tang, J. Thingna, and J. Wang, Thermodynamics of energy, charge, and spin currents in a thermoelectric quantum-dot spin valve, Phys. Rev. B 97(15), 155430 (2018)

    Article  ADS  Google Scholar 

  146. G. Tang, X. Chen, J. Ren, and J. Wang, Rectifying full-counting statistics in a spin Seebeck engine, Phys. Rev. B 97(8), 081407 (2018)

    Article  ADS  Google Scholar 

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Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 12074190, 11975125, 11890703, and 11874221).

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  1. NNU-SULI Thermal Energy Research Center (NSTER) & Center for Quantum Transport and Thermal Energy Science (CQTES), School of Physics and Technology, Nanjing Normal University, Nanjing, 210023, China

    Zhi-Zhou Yu, Guo-Huan Xiong & Li-Fa Zhang

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  1. Zhi-Zhou Yu
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Correspondence to Li-Fa Zhang.

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arXiv: 2102.13332. Special Topic: Thermodynamics and Thermal Metamaterials (Editor: Ji-Ping Huang). This article can also be found at http://journal.hep.com.cn/fop/EN/10.1007/s11467-021-1051-3.

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Yu, ZZ., Xiong, GH. & Zhang, LF. A brief review of thermal transport in mesoscopic systems from nonequilibrium Green’s function approach. Front. Phys. 16, 43201 (2021). https://doi.org/10.1007/s11467-021-1051-3

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