Thermal transport in crystals: from the quantum Dyson equation to mesoscopic phonon hydrodynamics

Abstract

Thermal transport in dielectric, non-magnetic crystals is mediated by quantized lattice vibrations, which drift and interact when driven out of equilibrium by a temperature gradient. This phenomenon can be described at multiple theoretical levels, ranging from fully quantum descriptions to semiclassical and mesoscopic continuum approaches. This review rigorously discusses the theoretical steps and approximations connecting these levels, bridging quantum phonon Dyson and Kadanoff-Baym equations and semiclassical Boltzmann transport formalism, and discussing the coarse-graining procedures that yield mesoscopic viscous heat equations for non-diffusive, hydrodynamic heat transport in devices. We show how the Guyer-Krumhansl and dual-phase-lag equations emerge as special linear-isotropic-band and inviscid limits of the viscous heat equations, respectively; most importantly, we demonstrate that these equations predict not only Poiseuille flow and second sound, but also more exotic effects such as negative thermal resistance, steady-state thermal backflow and vortices. We highlight how combining these frameworks with first-principles simulations connects microscopic phonon physics to observable non-diffusive heat-transport phenomena and guides their detection, amplification, and control. We recast the viscous heat equations in terms of Helmholtz and biharmonic equations solved analytically, and use this to discuss similarities and differences between the macroscopic behavior of the phonon fluid and other hydrodynamic systems, such as classical and electron fluids, focusing on compressibility, vorticity, and their influence on phonon hydrodynamics. We conclude with a roadmap to generalize the tools used to describe phonon hydrodynamics to other quasiparticles, motivating future advances in collective quantum transport phenomena in solids.

arXiv comment165 pages, 34 figures, 3 tables

References in corpus (62)

• Thermal Properties of Graphene, Carbon Nanotubes and Nanostructured Carbon Materials
• Distribution of phonon lifetime in Brillouin zone
• Negative local resistance caused by viscous electron backflow in graphene
• Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene
• Unified theory of thermal transport in crystals and disordered solids
• Self-consistent phonon calculations of lattice dynamical properties in cubic SrTiO${}_3$ with first-principles anharmonic force constants
• Schrieffer-Wolff transformation for quantum many-body systems
• Evidence for hydrodynamic electron flow in PdCoO${}_2$
• Anharmonic free energies and phonon dispersions from the stochastic self-consistent harmonic approximation: application to platinum and palladium hydrides
• Entropy driven stabilization of energetically unstable crystal structures explained from first principles theory
• Electron Viscosity, Current Vortices and Negative Nonlocal Resistance in Graphene
• First-principles calculations of charge carrier mobility and conductivity in bulk semiconductors and two-dimensional materials
• Significant reduction of lattice thermal conductivity by electron-phonon interaction in silicon with high carrier concentrations: a first-principles study
• Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations
• almaBTE: a solver of the space-time dependent Boltzmann transport equation for phonons in structured materials
• Observation of second sound in graphite at temperatures above 100 K
• First-Principles Theory of Anharmonicity and the Inverse Isotope Effect in Superconducting Palladium-Hydride Compounds
• Ab initio variational approach for evaluating lattice thermal conductivity
• Non-local transport and the hydrodynamic shear viscosity in graphene
• The Stochastic Self-Consistent Harmonic Approximation: Calculating Vibrational Properties of Materials with Full Quantum and Anharmonic Effects
• Second order structural phase transitions, free energy curvature, and temperature-dependent anharmonic phonons in the self-consistent harmonic approximation: theory and stochastic implementation
• Phonon hydrodynamics and ultrahigh-room-temperature thermal conductivity in thin graphite
• Imaging phonon-mediated hydrodynamic flow in WTe2
• Thermal transport and phonon hydrodynamics in strontium titanate
• Phonon Hydrodynamic Heat Conduction and Knudsen Minimum in Graphite
• The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics
• Onset of non-diffusive phonon transport in transient thermal grating decay
• Improved Callaway model for Lattice Thermal Conductivity
• Nonequilibrium Green's function method for phonon-phonon interaction and ballistic-diffusive thermal transport
• Anharmonic phonon frequency shift in MgB2
• First-Principles Phonon Quasiparticle Theory Applied to a Strongly Anharmonic Halide Perovskite
• Superdiffusive heat conduction in semiconductor alloys -- I. Theoretical foundations
• First-principles calculations of phonon frequencies, lifetimes and spectral functions from weak to strong anharmonicity: the example of palladium hydrides
• Kinetic equation for strongly interacting dense Fermi systems
• Transport properties of strongly coupled electron-phonon liquids
• Observation of current whirlpools in graphene at room temperature
• Electron-phonon hydrodynamics
• Crossover from Boltzmann to Wigner thermal transport in thermoelectric skutterudites
• Boltzmann Transport in Nanostructures as a Friction Effect
• Quantum mechanical modeling of anharmonic phonon-phonon scattering in nanostructures
• Experimental demonstration of a generalized Fourier's Law for non-diffusive thermal transport
• Accessing temperature waves: a dispersion relation perspective
• Extending the range of validity of Fourier's law into the kinetic transport regime via asymptotic solution of the phonon Boltzmann transport equation
• Space-time dependent thermal conductivity in nonlocal thermal transport
• Transport Waves as Crystal Excitations
• Skyrmion fluid and bimeron glass protected by a chiral spin liquid on a kagome lattice
• Temperature in a Peierls-Boltzmann Treatment of Nonlocal Phonon Heat Transport
• Large variation in the boundary-condition slippage for a rarefied gas flowing between two surfaces
• Phonon hydrodynamics in crystalline GeTe at low temperature
• A unified theory of second sound in two dimensional materials
• Thermal dynamics and electronic temperature waves in layered correlated materials
• Using oxides to compute with heat
• Hydrodynamic signatures in thermal transport in devices based on 2D materials: an ab initio study
• BTE-Barna: An extension of almaBTE for thermal simulation of devices based on 2D materials
• Time Reversal Invariance of quantum kinetic equations: Nonequilibrium Green Functions Formalism
• Analysis of nonlocal phonon thermal conductivity simulations showing the ballistic to diffusive crossover
• Analytical Green's Function of Multidimensional Boltzmann Transport Equation for Modeling Hydrodynamic Second Sound
• Phonon hydrodynamic regimes in sapphire
• Effect of characteristic size on the collective phonon transport in crystalline GeTe
• Imaging heat transport in suspended diamond nanostructures with integrated spin defect thermometers
• Vortices and backflow in hydrodynamic heat transport
• The thermodynamics of CaSiO3 in Earth's lower mantle