A brief review of thermal transport in mesoscopic systems from nonequilibrium Green’s function approach
Zhi-Zhou Yu, Guo-Huan Xiong, Li-Fa Zhang
A brief review of thermal transport in mesoscopic systems from nonequilibrium Green’s function approach
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.
thermal transport / nonequilibrium Green’s function
[1] |
E. Pop, S. Sinha, and K. E. Goodson, Heat generation and transport in nanometer-scale transistors, Proc. IEEE 94(8), 1587 (2006)
CrossRef
ADS
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)
CrossRef
ADS
Google scholar
|
[3] |
G. Zhang and Y. W. Zhang, Thermal properties of two-dimensional materials, Chin. Phys. B 26(3), 034401 (2017)
CrossRef
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)
CrossRef
ADS
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)
CrossRef
ADS
Google scholar
|
[6] |
B. Li, L. Wang, and G. Casati, Thermal diode: Rectification of heat flux, Phys. Rev. Lett. 93(18), 184301 (2004)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[9] |
L. Wang and B. Li, Thermal logic gates: Computation with phonons, Phys. Rev. Lett. 99(17), 177208 (2007)
CrossRef
ADS
Google scholar
|
[10] |
L. Wang and B. Li, Thermal memory: A storage of phononic information, Phys. Rev. Lett. 101(26), 267203 (2008)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
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)
CrossRef
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)
CrossRef
ADS
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 twodimensional thermoelectric materials, Nano-Micro Lett. 12(1), 36 (2020)
CrossRef
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)
CrossRef
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)
CrossRef
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 ZTin n-type SnSe crystals, Science 360(6390), 778 (2018)
CrossRef
ADS
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[23] |
A. P. Jauho, N. S. Wingreen, and Y. Meir, Timedependent transport in interacting and noninteracting resonant-tunneling systems, Phys. Rev. B 50(8), 5528 (1994)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[26] |
N. Sergueev, D. Roubtsov, and H. Guo, Ab initioanalysis of electron–phonon coupling in molecular devices, Phys. Rev. Lett. 95(14), 146803 (2005)
CrossRef
ADS
Google scholar
|
[27] |
T. Shimazaki and Y. Asai, Bias voltage dependence on the vibronic electric current, Phys. Rev. B 77(7), 075110 (2008)
CrossRef
ADS
Google scholar
|
[28] |
M. Paulsson, T. Frederiksen, and M. Brandbyge, Modeling inelastic phonon scattering in atomic- and molecularwire junctions, Phys. Rev. B 72(20), 201101 (2005)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[35] |
G. Zhang and H. Zhang, Thermal conduction and rectification in few-layer graphene Y junctions, Nanoscale 3(11), 4604 (2011)
CrossRef
ADS
Google scholar
|
[36] |
R. Yang and G. Chen, Thermal conductivity modeling of periodic two-dimensional nanocomposites, Phys. Rev. B 69(19), 195316 (2004)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[41] |
H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer-Verlag, Berlin, 1998
|
[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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[44] |
L. Zhang, J. Thingna, D. He, J. S. Wang, and B. Li, Nonlinearity enhanced interfacial thermal conductance and rectification, EPL (Europhys. Lett.) 103(6), 64002 (2013)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
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)
CrossRef
ADS
Google scholar
|
[48] |
J. Chen, J. H. Walther, and P. Koumoutsakos, Covalently bonded graphene–carbon nanotube hybrid for high-performance thermal interfaces, Adv. Funct. Mater. 25(48), 7539 (2015)
CrossRef
ADS
Google scholar
|
[49] |
W. A. Little, The transport of heat between dissimilar solids at low temperatures, Can. J. Phys. 37(3), 334 (1959)
CrossRef
ADS
Google scholar
|
[50] |
E. T. Swartz and R. O. Pohl, Thermal boundary resistance, Rev. Mod. Phys. 61(3), 605 (1989)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[53] |
G. Xiong, J. S. Wang, D. Ma, and L. Zhang, Dramatic enhancement of interfacial thermal transport by massgraded and coupling-graded materials, EPL (Europhys. Lett.) 128(5), 54007 (2020)
CrossRef
ADS
Google scholar
|
[54] |
B. Chen and L. Zhang, Optimized couplers for interfacial thermal transport, J. Phys.: Condens. Matter 27(12), 125401 (2015)
CrossRef
ADS
Google scholar
|
[55] |
D. He, J. Thingna, J. S. Wang, and B. Li, Quantum thermal transport through anharmonic systems: A selfconsistent approach, Phys. Rev. B 94(15), 155411 (2016)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[57] |
J. C. Klöckner, M. Bürkle, J. C. Cuevas, and F. Pauly, Length dependence of the thermal conductance of alkanebased single-molecule junctions: An ab initio study, Phys. Rev. B 94(20), 205425 (2016)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
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 selfassembled monolayers with different terminal groups, J. Phys. Chem. C 124(31), 16748 (2020)
CrossRef
ADS
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[70] |
Y. Blanter and M. Büttiker, Shot noise in mesoscopic conductors, Phys. Rep. 336(1–2), 1 (2000)
CrossRef
ADS
Google scholar
|
[71] |
M. Büttiker, Four-terminal phase-coherent conductance, Phys. Rev. Lett. 57(14), 1761 (1986)
CrossRef
ADS
Google scholar
|
[72] |
M. Büttiker, Absence of backscattering in the quantum Hall effect in multiprobe conductors, Phys. Rev. B 38(14), 9375 (1988)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[77] |
Y. Gu, Mode-dependent phonon transmission in a Tshaped three-terminal graphene nanojunction, Carbon 158, 818 (2020)
CrossRef
ADS
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[81] |
M. Wei, M. Zhou, B. Wang, and Y. Xing, Thermoelectric transport properties of ferromagnetic graphene with CTinvariant quantum spin Hall effect, Phys. Rev. B 102(7), 075432 (2020)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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 (2011)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[89] |
K. Saito and A. Dhar, Fluctuation theorem in quantum heat conduction, Phys. Rev. Lett. 99(18), 180601 (2007)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[93] |
P. Reddy, S. Y. Jang, R. A. Segalman, and A. Majumdar, Thermoelectricity in molecular junctions, Science 315(5818), 1568 (2007)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[100] |
J. Li, B. Wang, F. Xu, Y. Wei, and J. Wang, Spindependent Seebeck effects in graphene-based molecular junctions, Phys. Rev. B 93(19), 195426 (2016)
CrossRef
ADS
Google scholar
|
[101] |
B. Zhou, B. Zhou, Y. Yao, G. Zhou, and M. Hu, Spindependent Seebeck effects in a graphene superlattice p–n junction with different shapes, J. Phys.: Condens. Matter 29(40), 405303 (2017)
CrossRef
ADS
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[104] |
G. D. Mahan, Many-Particle Physics, Springer, New York, 2000
CrossRef
ADS
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[107] |
G. E. Bauer, A. H. MacDonald, and S. Maekawa, Spin caloritronics, Solid State Commun. 150(11–12), 459 (2010)
CrossRef
ADS
Google scholar
|
[108] |
G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Spin caloritronics, Nat. Mater. 11(5), 391 (2012)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[111] |
M. Zeng, Y. Feng, and G. Liang, Graphene-based spin caloritronics, Nano Lett. 11(3), 1369 (2011)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
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)
CrossRef
ADS
Google scholar
|
[116] |
C. E. Nebel, Electrons dance in diamond, Nat. Mater. 12(8), 690 (2013)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[118] |
D. Gunlycke and C. T. White, Graphene valley filter using a line defect, Phys. Rev. Lett. 106(13), 136806 (2011)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[120] |
Z. Yu, F. Xu, and J. Wang, Valley Seebeck effect in gate tunable zigzag graphene nanoribbons, Carbon 99, 451 (2016)
CrossRef
ADS
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)
CrossRef
ADS
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[134] |
F. G. Eich, M. Di Ventra, and G. Vignale, Temperaturedriven transient charge and heat currents in nanoscale conductors, Phys. Rev. B 93(13), 134309 (2016)
CrossRef
ADS
Google scholar
|
[135] |
Č. Lozej and T. Rejec, Time-dependent thermoelectric transport in nanosystems: Reflectionless Luttinger field approach, Phys. Rev. B 98(7), 075427 (2018)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
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)
CrossRef
ADS
Google scholar
|
[146] |
G. Tang, X. Chen, J. Ren, and J. Wang, Rectifying fullcounting statistics in a spin Seebeck engine, Phys. Rev. B 97(8), 081407 (2018)
CrossRef
ADS
Google scholar
|
/
〈 | 〉 |