Abstract
This review deals with the nonequilibrium Green’s function (NEGF) method applied to the problems of energy transport due to atomic vibrations (phonons), primarily for small junction systems. We present a pedagogical introduction to the subject, deriving some of the well-known results such as the Laudauer-like formula for heat current in ballistic systems. The main aim of the review is to build the machinery of the method so that it can be applied to other situations, which are not directly treated here. In addition to the above, we consider a number of applications of NEGF, not in routine model system calculations, but in a few new aspects showing the power and usefulness of the formalism. In particular, we discuss the problems of multiple leads, coupled left-right-lead system, and system without a center. We also apply the method to the problem of full counting statistics. In the case of nonlinear systems, we make general comments on the thermal expansion effect, phonon relaxation time, and a certain class of mean-field approximations. Lastly, we examine the relationship between NEGF, reduced density matrix, and master equation approaches to thermal transport.
Similar content being viewed by others
References and notes
J. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys., 1961, 2(3): 407
L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics, Benjamin/Cummings, 1962
L. V. Keldysh, Diagram technique for nonequilibrium processes, Sov. Phys. JETP, 1965, 20: 1018
K. C. Chou, Z. B. Su, B. L. Hao, and L. Yu, Equilibrium and nonequilibrium formalisms made unified, Phys. Rep., 1985, 118(1–2): 1
P. Danielewicz, Quantum theory of nonequilibrium processes (I), Ann. Phys., 1984, 152(2): 239
J. Rammer and H. Smith, Quantum field-theoretical methods in transport theory of metals, Rev. Mod. Phys., 1986, 58(2): 323
M. Bonitz (Ed.), Progress in Nonequilibrium Green’s Functions, Singapore: World Scientific, 2000
M. Bonitz and D. Semkat (Eds.), Progress in Nonequilibrium Green’s Functions (II), Singapore: World Scientific, 2003
C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, Direct calculation of the tunneling current, J. Phys. C, 1971, 4(8): 916
Y. Meir and N. S. Wingreen, Landauer formula for the current through an interacting electron region, Phys. Rev. Lett., 1992, 68(16): 2512
A. Prociuk, H. Phillips, and B. D. Dunietz, Modeling transient aspects of coherence-driven electron transport, J. Phys.: Conf. Ser., 2010, 220: 012008
U. Aeberhard, Theory and simulation of quantum photovoltaic devices based on the non-equilibrium Green’s function formalism, J. Comput. Electron., 2011, 10(4): 394
N. A. Zimbovskaya and M. R. Pederson, Electron transport through molecular junctions, Phys. Rep., 2011, 509(1): 1
B. K. Nikolić, K. K. Saha, T. Markussen, and K. S. Thygesen, First-principles quantum transport modeling of thermoelectricity in single-molecule nanojunctions with graphene nanoribbon electrodes, J. Comput. Electron., 2012, 11(1): 78
J. S. Wang, J. Wang, and J. T. Lü, Quantum thermal transport in nanostructures, Eur. Phys. J. B, 2008, 62(4): 381
J. Lan, J. S. Wang, C. K. Gan, and S. K. Chin, Edge effects on quantum thermal transport in graphene nanoribbons: Tight-binding calculations, Phys. Rev. B, 2009, 79(11): 115401
P. E. Hopkins, P. M. Norris, M. S. Tsegaye, and A. W. Ghosh, Extracting phonon thermal conductance across atomic junctions: Nonequilibrium Green’s function approach compared to semiclassical methods, J. Appl. Phys., 2009, 106(6): 063503
Z. X. Xie, K. Q. Chen, and W. Duan, Thermal transport by phonons in zigzag graphene nanoribbons with structural defects, J. Phys.: Condens. Matter, 2011, 23(31): 315302
Z. Tian, K. Esfarjani, and G. Chen, Enhancing phonon transmission across a Si/Ge interface by atomic roughness: First-principles study with the Green’s function method, Phys. Rev. B, 2012, 86(23): 235304
M. Bachmann, M. Czerner, S. Edalati-Boostan, and C. Heiliger, Ab initio calculations of phonon transport in ZnO and ZnS, Eur. Phys. J. B, 2012, 85(5): 146
P. S. E. Yeo, K. P. Loh, and C. K. Gan, Strain dependence of the heat transport properties of graphene nanoribbons, Nanotechnology, 2012, 23(49): 495702
P. Brouwer, 2005, http://www.physics.udel.edu/~bnikolic/QTTG/shared/reviews/brouwer_notes.pdf
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th Ed., Singapore: World Scientific, 2009
J. W. Jiang, J. S. Wang, and B. Li, Thermal expansion in single-walled carbon nanotubes and graphene: Nonequilibrium Green’s function approach, Phys. Rev. B, 2009, 80(20): 205429
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, 2012, 85(5 Pt 1): 051142
A. Böhm, Quantum Mechanics, Heidelberg: Springer-Verlag, 1979
K. Huang, Statistical Mechanics, 2nd Ed., New York: John Wiley & Sons, 1987
R. Kubo, Statistical-mechanical theory of irreversible processes (I): General theory and simple applications to magnetic and Conduction Problems, J. Phys. Soc. Jpn., 1957, 12(6): 570
P. C. Martin and J. Schwinger, Theory of many-particle systems (I), Phys. Rev., 1959, 115(6): 1342
A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971
R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II, Nonequilibrium Statistical Mechanics, Springer, 1992
A. Altland and B. Simons, Condsensed Matter Field Theory, 2nd Ed., Cambridge: Cambridge University Press, 2010
H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer, 1996
A. M. Zagoskin, Quantum Theory of Many-Body Systems, Springer, 1998
J. Rammer, Quantum Field Theory of Non-Equilibrium States, Cambridge: Cambridge University Press, 2007
M. Di Ventra, Electrical Transport in Nanoscale Systems, Cambridge: Cambridge University Press, 2008
A. Kamenev, Field Theory of Non-Equilibrium Systems, Cambridge: Cambridge University Press, 2011
D. C. Langreth, in: Linear and Nonlinear Electron Transport in Solids, edited by J. T. Devreese and E. van Doren, Plenum, 1976: 3–32
C. Niu, D. L. Lin, and T. H. Lin, Equation of motion for nonequilibrium Green functions, J. Phys.: Condens. Matter, 1999, 11(6): 1511
A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover Publ., 1963
S. Doniach and E. H. Sondheimer, Green’s Functions for Solid State Physicists, W. A. Benjamin, 1974
G. D. Mahan, Many-Particle Physics, 3rd Ed., Kluwer Academic, 2000
H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics: An introduction, Oxford: Oxford University Press, 2004
H. L. Friedman, A Course in Statistical Mechanics, Prentice-Hall, 1985
B. K. Agarwalla, Study of full-counting statistics in heat transport in transient and steady state and quantum fuctuation theorems, Ph.D. thesis, National University Singapore, 2013
P. C. K. Kwok, Green’s function method in lattice dynamics, Solid State Phys., 1968, 20: 213
M. L. Leek, Mathematical details in the application of nonequilibrium Green’s functions (NEGF) and quantum kinetic equations (QKE) to thermal transport, arXiv: 1207.6204, 2012
H. Kleinert, A. Pelster, B. Kastening, and M. Bachmann, Recursive graphical construction of Feynman diagrams and their multiplicities in φ4 and φ2A theory, Phys. Rev. E, 2000, 62(2): 1537
S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge: Cambridge University Press, 1995
R. Landauer, Spatial variation of currents and fields due to localized scatterers in metallic conduction, IBM J. Res. Develop., 1957, 1(3): 223
R. Landauer, Electrical resistance of disordered onedimensional lattices, Philos. Mag., 1970, 21(172): 863
A. Ozpineci and S. Ciraci, Quantum effects of thermal conductance through atomic chains, Phys. Rev. B, 2001, 63(12): 125415
D. Segal, A. Nitzan, and P. Hänggi, Thermal conductance through molecular wires, J. Chem. Phys., 2003, 119(13): 6840
N. Mingo and L. Yang, Phonon transport in nanowires coated with an amorphous material: An atomistic Green’s function approach, Phys. Rev. B, 2003, 68(24): 245406
A. Dhar and D. Roy, Heat transport in harmonic lattices, J. Stat. Phys., 2006, 125(4): 805
A. Dhar and D. Sen, Nonequilibrium Green’s function formalism and the problem of bound states, Phys. Rev. B, 2006, 73(8): 085119
J. S. Wang, J. Wang, and N. Zeng, Nonequilibrium Green’s function approach to mesoscopic thermal transport, Phys. Rev. B, 2006, 74(3): 033408
T. Yamamoto and K. Watanabe, Nonequilibrium Green’s function approach to phonon transport in defective carbon nanotubes, Phys. Rev. Lett., 2006, 96(25): 255503
W. Zhang, T. S. Fisher, and N. Mingo, The atomistic Green’s function method: An efficient simulation approach for nanoscale phonon transport, Numer. Heat Transf. B, 2007, 51(4): 333
S. G. Das and A. Dhar, Landauer formula for phonon heat conduction: Relation between energy transmittance and transmission coefficient, Eur. Phys. J. B, 2012, 85(11): 372
M. P. L. Sancho, J. M. L. Sancho, and J. Rubio, Quick iterative scheme for the calculation of transfer matrices: Application to Mo(100), J. Phys. F, 1984, 14(5): 1205
M. P. L. Sancho, J. M. L. Sancho, and J. Rubio, Highly convergent schemes for the calculation of bulk and surface Green functions, J. Phys. F, 1985, 15(4): 851
A. P. Arya, Introduction to Classical Mechanics, Allyn and Bacon, 1990, Chap. 15.
E. C. Cuansing, H. Li, and J. S. Wang, Role of the on-site pinning potential in establishing quasi-steady-state conditions of heat transport in finite quantum systems, Phys. Rev. E, 2012, 86(3): 031132
J. S. Wang, N. Zeng, J. Wang, and C. K. Gan, Nonequilibrium Green’s function method for thermal transport in junctions, Phys. Rev. E, 2007, 75(6): 061128
J. Wang and J. S. Wang, Mode-dependent energy transmission across nanotube junctions calculated with a lattice dynamics approach, Phys. Rev. B, 2006, 74(5): 054303
L. Zhang, P. Keblinski, J. S. Wang, and B. Li, Interfacial thermal transport in atomic junctions, Phys. Rev. B, 2011, 83(6): 064303
M. Büttiker, Four-terminal phase-coherent conductance, Phys. Rev. Lett., 1986, 57(14): 1761
M. Büttiker, Symmetry of electrical conduction, IBM J. Res. Develop., 1988, 32(3): 317
M. Büttiker, Absence of backscattering in the quantum Hall effect in multiprobe conductors, Phys. Rev. B, 1988, 38(14): 9375
Y. M. Blanter and M. Büttiker, Shot noise in mesoscopic conductors, Phys. Rep., 2000, 336(1–2): 1
L. Zhang, J.-S. Wang, and B. Li, Ballistic thermal rectification in nanoscale three-terminal junctions, Phys. Rev. B, 2010, 81(10): 100301(R)
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., 2012, 100(18): 183110
A. Dhar and D. Roy, Heat transport in harmonic lattices, J. Stat. Phys., 2006, 125(4): 801
D. Roy, Crossover from ballistic to diffusive thermal transport in quantum Langevin dynamics study of a harmonic chain connected to self-consistent reservoirs, Phys. Rev. E, 2008, 77(6): 062102
M. Bandyopadhyay and D. Segal, Quantum heat transfer in harmonic chains with self-consistent reservoirs: exact numerical simulations, Phys. Rev. E, 2011, 84(1): 011151
L. Zhang, J. S. Wang, and B. Li, Phonon Hall effect in fourterminal nano-junctions, New J. Phys., 2009, 11(11): 113038
H. Li, B. K. Agarwalla, and J. S. Wang, Generalized Caroli formula for the transmission coefficient with lead-lead coupling, Phys. Rev. E, 2012, 86(1): 011141
M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys., 2009, 81(4): 1665
M. Campisi, P. Hänggi, and P. Talkner, Quantum fluctuation relations: Foundations and applications, Rev. Mod. Phys., 2011, 83(3): 771
U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys., 2012, 75(12): 126001
M. L. Roukes, Yoctocalorimetry: Phonon counting in nanostructures, Physica B, 1999, 263: 1
H. Touchette, The large deviation approach to statistical mechanics, Phys. Rep., 2009, 478(1–3): 1
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, 2012, 86(16): 165425
J. S. Wang, B. K. Agarwalla, and H. Li, Transient behavior of full counting statistics in thermal transport, Phys. Rev. B, 2011, 84(15): 153412
A. O. Gogolin and A. Komnik, Towards full counting statistics for the Anderson impurity model, Phys. Rev. B, 2006, 73(19): 195301
H. Li, B. K. Agarwalla, B. Li, and J. S. Wang, Cumulants of heat transfer in nonlinear quantum systems, arXiv: 1210.2798, 2012
L. S. Levitov and G. B. Lesovik, Charge distribution in quantum shot noise, JETP Lett., 1993, 58: 230
L. S. Levitov, H. Lee, and G. B. Lesovik, Electron counting statistics and coherent states of electric current, J. Math. Phys., 1996, 37(10): 4845
K. Saito and A. Dhar, Fluctuation theorem in quantum heat conduction, Phys. Rev. Lett., 2007, 99(18): 180601
G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Lett., 1995, 74(14): 2694
A. Kundu, S. Sabhapandit, and A. Dhar, Large deviations of heat flow in harmonic chains, J. Stat. Mech., 2011, 2011(03): P03007
K. Saito and A. Dhar, Generating function formula of heat transfer in harmonic networks, Phys. Rev. E, 2011, 83(4 Pt 1): 041121
E. C. Cuansing and J. S. Wang, Transient behavior of heat transport in a thermal switch, Phys. Rev. B, 2010, 81(5): 052302
N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders College, 1976
J. W. Jiang, J. S. Wang, and B. Li, Thermal contraction in silicon nanowires at low temperatures, Nanoscale, 2010, 2(12): 2864
J. W. Jiang and J. S. Wang, Thermal expansion in multiple layers of graphene, arXiv: 1108.5820, 2011
A. A. Maradudin and A. E. Fein, Scattering of neutrons by an anharmonic crystal, Phys. Rev., 1962, 128(6): 2589
Y. Xu, J. S. Wang, W. Duan, B. L. Gu, and B. Li, Nonequilibrium Green’s function method for phonon-phonon interactions and ballistic-diffusive thermal transport, Phys. Rev. B, 2008, 78(22): 224303
N. Mingo, Anharmonic phonon flow through molecular-sized junctions, Phys. Rev. B, 2006, 74(12): 125402
M. Luisier, Atomistic modeling of anharmonic phononphonon scattering in nanowires, Phys. Rev. B, 2012, 86(24): 245407
J. T. Lü and J. S. Wang, Coupled electron and phonon transport in one-dimensional atomic junctions, Phys. Rev. B, 2007, 76(16): 165418
P. Myöhänen, A. Stan, G. Stefanucci, and R. van Leeuwen, Kadanoff-Baym approach to quantum transport through interacting nanoscale systems: From the transient to the steady-state regime, Phys. Rev. B, 2009, 80(11): 115107
L. A. Wu and D. Segal, Quantum heat transfer: A Born-Oppenheimer method, Phys. Rev. E, 2011, 83(5): 051114
L. Lindsay, D. A. Broido, and N. Mingo, Lattice thermal conductivity of single-walled carbon nanotubes: Beyond the relaxation time approximation and phonon-phonon scattering selection rules, Phys. Rev. B, 2009, 80(12): 125407
L. Zhang, J. Thingna, D. He, J.-S. Wang, and B. Li, Nonlinearity enchanced interfacial thermal conducntance and rectification, 2013 (in preparation)
D. He, S. Buyukdagli, and B. Hu, Thermal conductivity of anharmonic lattices: effective phonons and quantum corrections, Phys. Rev. E, 2008, 78(6): 061103
J. Thingna, Steady-state transport properties of anharmonic systems, Ph.D. thesis, National University Singapore, 2013
A. Dhar, K. Saito, and P. Hänggi, Nonequilibrium densitymatrix description of steady-state quantum transport, Phys. Rev. E, 2012, 85(1): 011126
H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford: Oxford University Press, 2002
K. Saito, Strong evidence of normal heat conduction in a one-dimensional quantum system, Europhys. Lett., 2003, 61(1): 34
D. Segal and A. Nitzan, Spin-boson thermal rectifier, Phys. Rev. Lett., 2005, 94(3): 034301
D. Segal, Heat flow in nonlinear molecular junctions: Master equation analysis, Phys. Rev. B, 2006, 73(20): 205415
W. Pauli, in: Festschrift zum 60. Geburtstage A. Sommerfeld, Hirzel, Leipzig, 1928
A. O. Caldeira and A. J. Leggett, Path integral approach to quantum Brownian motion, Physica A, 1983, 121(3): 587
A. G. Redfield, On the theory of relaxation processes, IBM J. Res. Develop., 1957, 1(1): 19
G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys., 1976, 48(2): 119
T. Mori and S. Miyashita, Dynamics of the density matrix in contact with a thermal bath and the quantum master equation, J. Phys. Soc. Jpn., 2008, 77(12): 124005
C. H. Fleming and N. I. Cummings, Accuracy of perturbative master equations, Phys. Rev. E, 2011, 83(3): 031117
J. Thingna, J. S. Wang, and P. Hänggi, Generalized Gibbs state with modified Redfield solution: exact agreement up to second order, J. Chem. Phys., 2012, 136(19): 194110
B. B. Laird, J. Budimir, and J. L. Skinner, Quantummechanical derivation of the Bloch equations: Beyond the weak-coupling limit, J. Chem. Phys., 1991, 94(6): 4391
S. Jang, J. Cao, and R. J. Silbey, Fourth-order quantum master equation and its Markovian bath limit, J. Chem. Phys., 2002, 116(7): 2705
S. Nakajima, On quantum theory of transport phenomena, Prog. Theor. Phys., 1958, 20(6): 948
R. Zwanzig, Ensemble method in the theory of irreversibility, J. Chem. Phys., 1960, 33(5): 1338
F. Shibata, Y. Takahashi, and N. Hashitsume, A generalized stochastic Liouville equation, non-Markovian versus memoryless master equations, J. Stat. Phys., 1977, 17(4): 171
G. Nan, Q. Shi, and Z. Shuai, Nonperturbative time-convolutionless quantum master equation from the path integral approach, J. Chem. Phys., 2009, 130(13): 134106
J. Thingna, J. L. Garcá-Palacios, and J. S. Wang, Steadystate thermal transport in anharmonic systems: Application to molecular junctions, Phys. Rev. B, 2012, 85(19): 195452
L. A. Wu, C. X. Yu, and D. Segal, Nonlinear quantum heat transfer in hybrid structures: Sufficient conditions for thermal rectification, Phys. Rev. E, 2009, 80(4): 041103
J. Thingna and J.-S. Wang, 2013 (in preparation)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, JS., Agarwalla, B.K., Li, H. et al. Nonequilibrium Green’s function method for quantum thermal transport. Front. Phys. 9, 673–697 (2014). https://doi.org/10.1007/s11467-013-0340-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11467-013-0340-x