Thermal transport in crystals as a kinetic theory of relaxons

Thermal conductivity in dielectric crystals is the result of the relaxation of lattice vibrations described by the phonon Boltzmann transport equation. Remarkably, an exact microscopic definition of the heat carriers and their relaxation times is still missing: phonons, typically regarded as the relevant excitations for thermal transport, cannot be identified as the heat carriers when most scattering events conserve momentum and do not dissipate heat flux. This is the case for two-dimensional or layered materials at room temperature, or three-dimensional crystals at cryogenic temperatures. In this work we show that the eigenvectors of the scattering matrix in the Boltzmann equation define collective phonon excitations, termed here relaxons. These excitations have well defined relaxation times, directly related to heat flux dissipation, and provide an exact description of thermal transport as a kinetic theory of the relaxon gas. We show why Matthiessen's rule is violated, and construct a procedure for obtaining the mean free paths and relaxation times of the relaxons. These considerations are general, and would apply also to other semiclassical transport models, such as the electronic Boltzmann equation. For heat transport, they remain relevant even in conventional crystals like silicon, but are of the utmost importance in the case of two-dimensional materials, where they can revise by several orders of magnitude the relevant time- and length-scales for thermal transport in the hydrodynamic regime.

[1]  Nicola Marzari,et al.  Acoustic phonon lifetimes and thermal transport in free-standing and strained graphene. , 2012, Nano letters.

[2]  J. Kirkwood.  The statistical mechanical theory of irreversible processes , 1949 .

[3]  L. Colombo,et al.  Intrinsic thermal conductivity in monolayer graphene is ultimately upper limited: A direct estimation by atomistic simulations , 2015 .

[4]  G. P. Srivastava.  Derivation and calculation of complementary variational principles for the lattice thermal conductivity , 1976 .

[5]  Laurent Chaput,et al.  Direct solution to the linearized phonon Boltzmann equation. , 2013, Physical review letters.

[6]  Gang Chen,et al.  Heat transport in silicon from first-principles calculations , 2011, 1107.5288.

[7]  Testa,et al.  Green's-function approach to linear response in solids. , 1987, Physical review letters.

[8]  J. Krumhansl.  Thermal conductivity of insulating crystals in the presence of normal processes , 1965 .

[9]  Natalio Mingo,et al.  Lattice thermal conductivity of silicon from empirical interatomic potentials , 2005 .

[10]  R. Hardy.  ENERGY-FLUX OPERATOR FOR A LATTICE , 1963 .

[11]  Francesco Mauri,et al.  Ab initio variational approach for evaluating lattice thermal conductivity , 2012, 1212.0470.

[12]  D. Broido,et al.  Enhanced thermal conductivity and isotope effect in single-layer hexagonal boron nitride , 2011 .

[13]  R. Peierls,et al.  Zur kinetischen Theorie der Wärmeleitung in Kristallen , 1929 .

[14]  Nicola Marzari,et al.  Thermal conductivity of graphene and graphite: collective excitations and mean free paths. , 2014, Nano letters.

[15]  John Ziman,et al.  Electrons and Phonons: The Theory of Transport Phenomena in Solids , 2001 .

[16]  K. Nelson,et al.  Reconstructing phonon mean-free-path contributions to thermal conductivity using nanoscale membranes , 2014, 1408.6747.

[17]  Baoling Huang,et al.  Unusual Enhancement in Intrinsic Thermal Conductivity of Multilayer Graphene by Tensile Strains. , 2015, Nano letters.

[18]  X. Gonze,et al.  Density-functional approach to nonlinear-response coefficients of solids. , 1989, Physical review. B, Condensed matter.

[19]  Boris Kozinsky,et al.  Role of disorder and anharmonicity in the thermal conductivity of silicon-germanium alloys: a first-principles study. , 2011, Physical review letters.

[20]  Stefano de Gironcoli,et al.  Phonons and related crystal properties from density-functional perturbation theory , 2000, cond-mat/0012092.

[21]  Baroni,et al.  Anharmonic Phonon Lifetimes in Semiconductors from Density-Functional Perturbation Theory. , 1995, Physical review letters.

[22]  C. de Tomas,et al.  From kinetic to collective behavior in thermal transport on semiconductors and semiconductor nanostructures , 2013, 1310.7127.

[23]  Boris Kozinsky,et al.  AiiDA: Automated Interactive Infrastructure and Database for Computational Science , 2015, ArXiv.

[24]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[25]  L. Paulatto,et al.  Anharmonic properties from a generalized third order ab~initio approach: theory and applications to graphite and graphene , 2013, 1304.2626.

[26]  Thermal conductivity of nanotubes revisited: effects of chirality, isotope impurity, tube length, and temperature. , 2005, The Journal of chemical physics.

[27]  J. Parrott,et al.  Variational Calculation of the Thermal Conductivity of Germanium , 1969 .

[28]  N. Mingo,et al.  Intrinsic lattice thermal conductivity of semiconductors from first principles , 2007 .

[29]  Natalio Mingo,et al.  Flexural phonons and thermal transport in graphene , 2010 .

[30]  Melville S. Green,et al.  Markoff Random Processes and the Statistical Mechanics of Time-Dependent Phenomena , 1952 .

[31]  R. Kubo.  Statistical-Mechanical Theory of Irreversible Processes : I. General Theory and Simple Applications to Magnetic and Conduction Problems , 1957 .

[32]  R. Guyer,et al.  Solution of the Linearized Phonon Boltzmann Equation , 1966 .

[33]  A. Majumdar,et al.  Nanoscale thermal transport , 2003, Journal of Applied Physics.

[34]  First-principles study of the thermal expansion of Be(1010) , 2002 .

[35]  Stefano de Gironcoli,et al.  Ab initio calculation of phonon dispersions in semiconductors. , 1991, Physical review. B, Condensed matter.

[36]  G. Verma,et al.  Lattice Thermal Conductivity at Low Temperatures , 1962 .

[37]  Giulia Galli,et al.  Atomistic simulations of heat transport in silicon nanowires. , 2009, Physical review letters.

[38]  R. Hardy.  Lowest‐Order Contribution to the Lattice Thermal Conductivity , 1965 .

[39]  Stefano de Gironcoli,et al.  QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials , 2009, Journal of physics. Condensed matter : an Institute of Physics journal.

[40]  Alan J. H. McGaughey,et al.  Quantitative validation of the Boltzmann transport equation phonon thermal conductivity model under the single-mode relaxation time approximation , 2004 .

[41]  M. Dresselhaus,et al.  Thermal conductivity spectroscopy technique to measure phonon mean free paths. , 2011, Physical review letters.

[42]  Sebastian Volz,et al.  Molecular dynamics simulation of thermal conductivity of silicon nanowires , 1999 .

[43]  L. J. Lewis,et al.  On the importance of collective excitations for thermal transport in graphene , 2015 .

[44]  Cristina H Amon,et al.  Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreflectance , 2013, Nature Communications.

[45]  Zhifeng Ren,et al.  Coherent Phonon Heat Conduction in Superlattices , 2012, Science.

[46]  Gang Chen,et al.  Hydrodynamic phonon transport in suspended graphene , 2015, Nature Communications.

[47]  Amelia Carolina Sparavigna,et al.  Heat transport in dielectric solids with diamond structure , 1997 .

[48]  Gang Chen,et al.  Applied Physics Reviews Nanoscale Thermal Transport. Ii. 2003–2012 , 2022 .

[49]  J. Callaway.  Model for Lattice Thermal Conductivity at Low Temperatures , 1959 .

[50]  R. Hardy.  Phonon Boltzmann Equation and Second Sound in Solids , 1970 .

[51]  N. Marzari,et al.  First-Principles Determination of Phonon Lifetimes, Mean Free Paths, and Thermal Conductivities in Crystalline Materials: Pure Silicon and Germanium , 2014 .

[52]  Amelia Carolina Sparavigna,et al.  Beyond the isotropic-model approximation in the theory of thermal conductivity. , 1996, Physical review. B, Condensed matter.

[53]  Gernot Deinzer,et al.  Ab initio theory of the lattice thermal conductivity in diamond , 2009 .

[54]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[55]  Alan J. H. McGaughey,et al.  Strongly anisotropic in-plane thermal transport in single-layer black phosphorene , 2015, Scientific Reports.

[56]  L. Paulatto,et al.  Phonon hydrodynamics in two-dimensional materials , 2015, Nature Communications.

[57]  Y. Koh,et al.  The role of low-energy phonons with mean-free-paths >0.8 um in heat conduction in silicon , 2015, 1507.03422.

[58]  C. N. Lau,et al.  Superior thermal conductivity of single-layer graphene. , 2008, Nano letters.