PHONONS (original) (raw)
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Journal of Statistical Physics, 2006
For low density gases the validity of the Boltzmann transport equation is well established. The central object is the one-particle distribution function, f , which in the Boltzmann-Grad limit satisfies the Boltzmann equation. Grad and, much refined, Cercignani argue for the existence of this limit on the basis of the BBGKY hierarchy for hard spheres. At least for a short kinetic time span, the argument can be made mathematically precise following the seminal work of Lanford. In this article a corresponding program is undertaken for weakly nonlinear, both discrete and continuum, wave equations. Our working example is the harmonic lattice with a weakly nonquadratic on-site potential. We argue that the role of the Boltzmann f -function is taken over by the Wigner function, which is a very convenient device to filter the slow degrees of freedom. The Wigner function, so to speak, labels locally the covariances of dynamically almost stationary measures. One route to the phonon Boltzmann equation is a Gaussian decoupling, which is based on the fact that the purely harmonic dynamics has very good mixing properties. As a further approach the expansion in terms of Feynman diagrams is outlined. Both methods are extended to the quantized version of the weakly nonlinear wave equation.
The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics
Journal of Statistical Physics, 2006
For low density gases the validity of the Boltzmann transport equation is well established. The central object is the one-particle distribution function, f , which in the Boltzmann-Grad limit satisfies the Boltzmann equation. Grad and, much refined, Cercignani argue for the existence of this limit on the basis of the BBGKY hierarchy for hard spheres. At least for a short kinetic time span, the argument can be made mathematically precise following the seminal work of Lanford. In this article a corresponding program is undertaken for weakly nonlinear, both discrete and continuum, wave equations. Our working example is the harmonic lattice with a weakly nonquadratic on-site potential. We argue that the role of the Boltzmann f -function is taken over by the Wigner function, which is a very convenient device to filter the slow degrees of freedom. The Wigner function, so to speak, labels locally the covariances of dynamically almost stationary measures. One route to the phonon Boltzmann equation is a Gaussian decoupling, which is based on the fact that the purely harmonic dynamics has very good mixing properties. As a further approach the expansion in terms of Feynman diagrams is outlined. Both methods are extended to the quantized version of the weakly nonlinear wave equation.
Lattice vibrations in the Frenkel-Kontorova Model. II. Thermal conductivity
Physical Review B, 2015
We studied the lattice vibrations of two inter-penetrating atomic sublattices via the Frenkel-Kontorova (FK) model of a linear chain of harmonically interacting atoms subjected to an on-site potential, using the technique of thermodynamic Green's functions based on quantum field-theoretical methods. General expressions were deduced for the phonon frequency-wave-vector dispersion relations, number density, and energy of the FK model system. As the application of the theory, we investigated in detail cases of linear chains with various periods of the on-site potential of the FK model. Some unusual but interesting features for different amplitudes of the on-site potential of the FK model are discussed. In the commensurate structure, the phonon spectrum always starts at a finite frequency, and the gaps of the spectrum are true ones with a zero density of modes. In the incommensurate structure, the phonon spectrum starts from zero frequency but at a nonzero wave vector; there are some modes inside these gap regions, but their density is very low. In our approximation, the energy of a higher-order commensurate state of the onedimensional system at finite temperature may become indefinitely close to the energy of an incommensurate state. This finding implies that the higher-order incommensuratecommensurate transitions are continuous ones, and that the phase transition may exhibit a "devil's staircase" behavior at a finite temperature.
Energy Current Correlations for Weakly Anharmonic Lattices
New Trends in Mathematical Physics, 2009
A solid transports energy. Besides the mobile electrons, one important mechanism for energy transport are the vibrations of the crystal lattice. There is no difficulty in writing down the appropriate lattice dynamics. To extract from it the thermal conductivity remains a fairly untractable problem. The most successful approach exploits that even rather close to the melting temperature the typical deviations of the crystal atoms from their equilibrium position are small as compared to the lattice constant. This observation then leads to the phonon kinetic equation, which goes back to the seminal paper by Peierls [1]. (For electron transport a corresponding idea was put forward by Nordheim [2].) Phonon kinetic theory flourished in the 50ies, an excellent account of the 1960 status being the book by Ziman . Of course, transport of heat and thermal conductivity remain an important experimental research area, in particular since novel materials become available and since more extreme properties are in demand. On the other hand, if the very recent collection of articles by Tritt [4] is taken to be representative, it is obvious that after 1960 hardly any new elements have been added to the theory. The real innovation are fast and efficient molecular dynamics algorithms. The currently available techniques allow the simulation of 6 × 6 × 6 periodized lattices with two atoms per unit cell .
Physical Review B, 2010
We use classical molecular dynamics to evaluate the thermal conductivity ͑T͒ from the heat-flux correlation ͗j͑0͒j͑t͒͘ for a two-dimensional Lennard-Jones triangular lattice. Our work, which follows Ladd, Moran, and Hoover ͓Phys. Rev. B 34, 5058 ͑1986͔͒, finds large deviations from the Eucken-Debye result ͑T͒ = A / T predicted by the phonon-gas model, even though phonon quasiparticles are fairly well defined. The main source of deviations comes from higher order ͑anharmonic͒ terms in the heat-flux operator j. By separating different orders of terms j = j ͑2͒ + j ͑3͒ +¯, we examine various separate contributions to ͑T͒Ϸ 22 + 23 +¯, both from the harmonic and the anharmonic heat fluxes. We find that 22 ͑T͒ϷA / T follows quasiparticle theory fairly well but important terms from 23 and 24 are independent of T in the classical ͑high T͒ limit. We use diagrammatic perturbation theory applied to the quantum Kubo formula, to check and explain the T dependence found numerically from anharmonic heat fluxes. We also demonstrate the importance of vertex correction in obtaining the correct quasiparticle coefficient of 1 / T.
Hot electrons in low-dimensional phonon systems
Physical Review B, 2005
A simple bulk model of electron-phonon coupling in metals has been surprisingly successful in explaining experiments on metal films that actually involve surface-or other low-dimensional phonons. However, by an exact application of this standard model to a semi-infinite substrate with a free surface, making use of the actual vibrational modes of the substrate, we show that such agreement is fortuitous, and that the model actually predicts a low-temperature crossover from the familiar T 5 temperature dependence to a stronger T 6 log T scaling. Comparison with existing experiments suggests a widespread breakdown of the standard model of electron-phonon thermalization in metals. PACS numbers: 63.22.+m, 85.85.+j The coupling between electrons and phonons plays a crucial role in determining the thermal properties of nanostructures. The widely used "standard" model of low temperature electron-phonon thermal coupling and hot-electron effects in bulk metals [1, 2] assumes (i) a clean three-dimensional free-electron gas with a spherical Fermi surface, rapidly equilibrated to a temperature T el ; (ii) a continuum description of the acoustic phonons, which have a temperature T ph ; (iii) a negligible Kapitzalike thermal boundary resistance [3] between the metal and any surrounding dielectric, an assumption that is often well justified experimentally; and (iv), a deformationpotential electron-phonon coupling, expected to be the dominant interaction at long-wavelengths. In a bulk metal, the net rate P of thermal energy transfer between the electron and phonon subsystems is [2] P = ΣV el T 5 el − T 5 ph ,
Thermal conduction in classical low-dimensional lattices
2003
Deriving macroscopic phenomenological laws of irreversible thermodynamics from simple microscopic models is one of the tasks of non-equilibrium statistical mechanics. We consider stationary energy transport in crystals with reference to simple mathematical models consisting of coupled oscillators on a lattice. The role of lattice dimensionality on the breakdown of the Fourier's law is discussed and some universal quantitative aspects are emphasized: the divergence of the finite-size thermal conductivity is characterized by universal laws in one and two dimensions. Equilibrium and non-equilibrium molecular dynamics methods are presented along with a critical survey of previous numerical results. Analytical results for the non-equilibrium dynamics can be obtained in the harmonic chain where the role of disorder and localization can be also understood. The traditional kinetic approach, based on the Boltzmann-Peierls equation is also briefly sketched with reference to one-dimensional chains. Simple toy models can be defined in which the conductivity is finite. Anomalous transport in integrable nonlinear systems is briefly discussed. Finally, possible future research themes are outlined.
Non-local phonon thermal conductivity and the ballistic to diffusive crossover
arXiv: Materials Science, 2016
The local Fourier relation between heat current to temperature gradient, J = -kdT/dr, does not hold on length scales shorter than carrier mean free paths. In insulating crystals, long phonon mean free paths enhance nonlocality of the kernel k(r,r') that relates a steady state heat current J(r) to remote temperature gradients dT(r')/dr'. If the system is spatially homogeneous, k(r,r') = k(r-r'), and in Fourier space, J(q) = -k(q)dT(q)/dr. A local relation has k(q) independent of q, or k(r-r')~delta(r-r'). In nanoscale systems, nonlocality, or equivalently, mixed ballistic/diffusive behavior, complicates heat transfer. Non-local information is starting to be measurable by modern sub-micron imaging methods. This paper derives the formula for k(q) from the Peierls-Boltzmann equation (PBE). There is a given applied thermal power P(k)=ikJ(k), which acts as a source term in the PBE. A new specific form of this source term is presented. Closed form results are ob...