Abstract
The Green-Kubo theory of thermal transport has long be considered incompatible with modern simulation methods based on electronic-structure theory, because it is based on such concepts as energy density and current, which are ill-defined at the quantum-mechanical level. Besides, experience with classical simulations indicates that the estimate of heat-transport coefficients requires analysing molecular trajectories that are more than one order of magnitude longer than deemed feasible using ab initio molecular dynamics. In this paper we report on recent theoretical advances that are allowing one to overcome these two obstacles. First, a general gauge invariance principle has been established, stating that thermal conductivity is insensitive to many details of the microscopic expression for the energy density and current from which it is derived, thus permitting to establish a rigorous expression for the energy flux from Density-Functional Theory, from which the conductivity can be computed in practice. Second, a novel data analysis method based on the statistical theory of time series has been proposed, which allows one to considerably reduce the simulation time required to achieve a target accuracy on the computed conductivity. These concepts are illustrated in detail, starting from a pedagogical introduction to the Green-Kubo theory of linear response and transport, and demonstrated with a few applications done with both classical and quantum-mechanical simulation methods.
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References
Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19:716–723. https://doi.org/10.1109/TAC.1974.1100705
Anderson TW (1994) The statistical analysis of time series. Wiley-Interscience, New York
Baroni S, Giannozzi P, Testa A (1987) Green’s-function approach to linear response in solids. Phys Rev Lett 58:1861–1864. https://doi.org/10.1103/PhysRevLett.58.1861
Bertossa R, Ercole L, Baroni S (2018) Transport coefficients in multi-component fluids from equilibrium molecular dynamics, arXiv:1808.03341 [cond-mat.stat-mech]
Baroni S, de Gironcoli S, Dal Corso A, Giannozzi P (2001) Phonons and related crystal properties from density-functional perturbation theory. Rev Mod Phys 73:515–562. https://doi.org/10.1103/RevModPhys.73.515
Bouzid A, Zaoui H, Palla PL, Ori G, Boero M, Massobrio C, Cleri F, Lampin E (2017) Thermal conductivity of glassy GeTe4 by first-principles molecular dynamics. Phys Chem Chem Phys 19:9729–9732. https://doi.org/10.1039/C7CP01063J
Broido DA, Malorny M, Birner G, Mingo N, Stewart DA (2007) Intrinsic lattice thermal conductivity of semiconductors from first principles. Appl Phys Lett 91:231922. https://doi.org/10.1063/1.2822891
Car R, Parrinello M (1985) Unified approach for molecular dynamics and density-functional theory. Phys Rev Lett 55:2471–2474. https://doi.org/10.1103/PhysRevLett.55.2471
Carbogno C, Ramprasad R, Scheffler M (2017) Ab initio Green-Kubo approach for the thermal conductivity of solids. Phys Rev Lett 118:175901. https://doi.org/10.1103/PhysRevLett.118.175901
Casimir HBG (1945) On Onsager’s principle of microscopic reversibility. Rev Mod Phys 17:343–350. https://doi.org/10.1103/RevModPhys.17.343
Chetty N, Martin R (1992) First-principles energy density and its applications to selected polar surfaces. Phys Rev B 45:6074–6088. https://doi.org/10.1103/PhysRevB.45.6074
Childers DG, Skinner DP, Kemerait RC (1977) The cepstrum: a guide to processing. Proc IEEE 65:1428–1443. https://doi.org/10.1109/PROC.1977.10747
Claeskens G, Hjort NL (2008) Model selection and model averaging. Cambridge University Press, Cambridge
Debernardi A, Baroni S, Molinari E (1995) Anharmonic phonon lifetimes in semiconductors from density-functional perturbation theory. Phys Rev Lett 75:1819–1822. https://doi.org/10.1103/PhysRevLett.75.1819
Ercole L, Marcolongo A, Umari P, Baroni S (2016) Gauge invariance of thermal transport coefficients. J Low Temp Phys 185:79–86. https://doi.org/10.1007/s10909-016-1617-6
Ercole L, Marcolongo A, Baroni S (2017) Accurate thermal conductivities from optimally short molecular dynamics simulations. Sci Rep 7:15835. https://doi.org/10.1038/s41598-017-15843-2
Foster D (1975) Hydrodynamic fluctuations, broken symmetry, and correlation functions. Benjamin, Reading
Giannozzi P, De Gironcoli S, Pavone P, Baroni S (1991) Ab initio calculation of phonon dispersions in semiconductors. Phys Rev B 43:7231–7242. https://doi.org/10.1103/PhysRevB.43.7231
Giannozzi P, Andreussi O, Brumme T, Bunau O, Nardelli MB, Calandra M, Car R, Cavazzoni C, Ceresoli D, Cococcioni M, Colonna N, Carnimeo I, Corso AD, de Gironcoli S, Delugas P, Jr RAD, Ferretti A, Floris A, Fratesi G, Fugallo G, Gebauer R, Gerstmann U, Giustino F, Gorni T, Jia J, Kawamura M, Ko HY, Kokalj A, Küçükbenli E, Lazzeri M, Marsili M, Marzari N, Mauri F, Nguyen NL, Nguyen HV, de-la Roza AO, Paulatto L, Poncé S, Rocca D, Sabatini R, Santra B, Schlipf M, Seitsonen AP, Smogunov A, Timrov I, Thonhauser T, Umari P, Vast N, Wu X, Baroni S (2017) Advanced capabilities for materials modelling with quantum espresso. J Phys Condens Matter 29:465901. https://doi.org/10.1088/1361-648X/aa8f79
Goodman NR (1963a) The distribution of the determinant of a complex Wishart distributed matrix. Ann Math Stat 34:178–180. https://doi.org/10.1214/aoms/1177704251
Goodman NR (1963b) Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction). Ann Math Stat 34:152–177
Green MS (1952) Markoff random processes and the statistical mechanics of time-dependent phenomena. J Chem Phys 20:1281–1295. https://doi.org/10.1063/1.1700722
Green MS (1954) Markoff random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids. J Chem Phys 22:398–413. https://doi.org/10.1063/1.1740082
Helfand E (1960) Transport coefficients from dissipation in a canonical ensemble. Phys Rev 119: 1–9. https://doi.org/10.1103/PhysRev.119.1
Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136:B864–B871. https://doi.org/10.1103/PhysRev.136.B864
Irving JH, Kirkwood JG (1950) The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J Chem Phys 18:817. https://doi.org/10.1063/1.1747782
Jones RE, Mandadapu KK (2012) Adaptive Green-Kubo estimates of transport coefficients from molecular dynamics based on robust error analysis. J Chem Phys 136:154102. https://doi.org/10.1063/1.3700344
Kadanoff LP, Martin PC (1963) Hydrodynamic equations and correlation functions. Ann Phys 24:419–469. https://doi.org/10.1016/0003-4916(63)90078-2
Kang J, Wang LW (2017) First-principles Green-Kubo method for thermal conductivity calculations. Phys Rev B 96:20302. https://doi.org/10.1103/PhysRevB.96.020302
Khintchine A (1934) Korrelationstheorie der stationären stochastischen Prozesse. Math Ann 109:604–615. https://doi.org/10.1007/BF01449156
Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140:A1133–A1138. https://doi.org/10.1103/PhysRev.140.A1133
Kshirsagar AM (1959) Bartlett decomposition and wishart distribution. Ann Math Statist 30:239–241. https://doi.org/10.1214/aoms/1177706379
Kubo R (1957) Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J Phys Soc Jpn 12:570–586. https://doi.org/10.1143/JPSJ.12.570
Kubo R, Yokota M, Nakajima S (1957) Statistical-mechanical theory of irreversible processes. II. Response to thermal disturbance. J Phys Soc Jpn 12:1203–1211. https://doi.org/10.1143/JPSJ.12.1203
Lampin E, Palla PL, Francioso PA, Cleri F (2013) Thermal conductivity from approach-to-equilibrium molecular dynamics. J Appl Phys 114:033525. https://doi.org/10.1063/1.4815945
Marcolongo A (2014) Theory and ab initio simulation of atomic heat transport. PhD thesis, Scuola Internazionale Superiore di Studi Avanzati, Trieste, an optional note
Marcolongo A, Umari P, Baroni S (2016) Microscopic theory and ab initio simulation of atomic heat transport. Nat Phys 12:80–84. https://doi.org/10.1038/nphys3509
Martin RM (2008) Electronic structure: basic theory and practical methods. Cambridge University Press, Cambridge
Marx D, Hutter J (2009) Ab initio molecular dynamics: basic theory and advanced methods. Cambridge University Press, Cambridge
Matsunaga N, Nagashima A (1983) Transport properties of liquid and gaseous D2O over a wide range of temperature and pressure. J Phys Chem Ref Data 12:933–966. https://doi.org/10.1063/1.555694
Müller-Plathe F (1997) A simple nonequilibrium molecular dynamics method for calculating the thermal conductivity. J Chem Phys 106:6082–6085. https://doi.org/10.1063/1.473271
Nagar DK, Gupta AK (2011) Expectations of functions of complex wishart matrix. Acta Appl Math 113:265–288. https://doi.org/10.1007/s10440-010-9599-x
Oliveira LDS, Greaney PA (2017) Method to manage integration error in the Green-Kubo method. Phys Rev E 95:023308. https://doi.org/10.1103/PhysRevE.95.023308
Onsager L (1931a) Reciprocal relations in irreversible processes. I. Phys Rev 37:405–426. https://doi.org/10.1103/PhysRev.37.405
Onsager L (1931b) Reciprocal relations in irreversible processes. II. Phys Rev 38:2265. https://doi.org/10.1103/PhysRev.38.2265
Peierls R (1929) Zur kinetischen theorie der wärmeleitung in kristallen. Ann Phys (Berlin) 395:1055–1101. https://doi.org/10.1002/andp.19293950803
Peligrad M, Wu WB (2010) Central limit theorem for Fourier transforms of stationary processes. Ann Prob 38:2009–2022. https://doi.org/10.1214/10-AOP530
Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77:3865–3868. https://doi.org/10.1103/PhysRevLett.77.3865
Puligheddu M, Gygi F, Galli G (2017) First-principles simulations of heat transport. Phys Rev Mater 1:060802. https://doi.org/10.1103/PhysRevMaterials.1.060802
Ramires MLV, de Castro CAN, Nagasaka Y, Nagashima A, Assael MJ, Wakeham WA (1995) Standard reference data for the thermal conductivity of water. J Phys Chem Ref Data 24:1377–1381. https://doi.org/10.1063/1.555963
Schelling PK, Phillpot SR, Keblinski P (2002) Comparison of atomic-level simulation methods for computing thermal conductivity. Phys Rev B 65:144306. https://doi.org/10.1103/PhysRevB.65.144306
Sindzingre P, Gillan MJ (1990) A computer simulation study of transport coefficients in alkali halides. J Phys Condens Matter 2:7033
Sit PHL, Marzari N (2005) Static and dynamical properties of heavy water at ambient conditions from first-principles molecular dynamics. J Chem Phys 122:204510. https://doi.org/10.1063/1.1908913
Stackhouse S, Stixrude L, Karki BB (2010) Thermal conductivity of periclase (MgO) from first principles. Phys Rev Lett 104:208501. https://doi.org/10.1103/PhysRevLett.104.208501
Turney JE, Landry ES, McGaughey AJH, Amon CH (2009) Predicting phonon properties and thermal conductivity from anharmonic lattice dynamics calculations and molecular dynamics simulations. Phys Rev B 79:064301. https://doi.org/10.1103/PhysRevB.79.064301
Volz SG, Chen G (2000) Molecular-dynamics simulation of thermal conductivity of silicon crystals. Phys Rev B 61:2651–2656. https://doi.org/10.1103/PhysRevB.61.2651
Wang Z, Safarkhani S, Lin G, Ruan X (2017) Uncertainty quantification of thermal conductivities from equilibrium molecular dynamics simulations. Int J Heat Mass Trans 112:267–278. https://doi.org/10.1016/j.ijheatmasstransfer.2017.04.077
Weisstein EW (MovingAverage) Moving average. From MathWorld – a Wolfram Web Resource. http://mathworld.wolfram.com/MovingAverage.html
Weisstein EW (PolyGamma) Polygamma functions. From MathWorld – a Wolfram Web Resource. http://mathworld.wolfram.com/PolygammaFunction.html
Wiener N (1930) Generalized harmonic analysis. Acta Math 55:117–258. https://doi.org/10.1007/BF02546511
Zhang Y, Otani A, Maginn EJ (2015) Reliable viscosity calculation from equilibrium molecular dynamics simulations: a time decomposition method. J Chem Theory Comput 11:3537–3546. https://doi.org/10.1021/acs.jctc.5b00351
Zhou J, Liao B, Chen G (2016) First-principles calculations of thermal, electrical, and thermoelectric transport properties of semiconductors. Semicond Sci Technol 31:043001. https://doi.org/10.1088/0268-1242/31/4/043001
Acknowledgements
This work was supported in part by the MaX EU Centre of Excellence, grant no 676598. SB, LE, and FG are grateful to Davide Donadio for insightful discussions all over the Summer of 2017 and beyond.
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Baroni, S., Bertossa, R., Ercole, L., Grasselli, F., Marcolongo, A. (2018). Heat Transport in Insulators from Ab Initio Green-Kubo Theory. In: Andreoni, W., Yip, S. (eds) Handbook of Materials Modeling. Springer, Cham. https://doi.org/10.1007/978-3-319-50257-1_12-1
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