Phonon boltzmann equation abstract the effective thermal conductivity of nano. Derivation of the linearized boltzmann equation from a. Later on we will discuss electronphonon scattering, which is inelastic. A phonon boltzmann transport equation approach is employed to describe anharmonic phonon. As a consequence the usual derivation of hydrodynamic equations from a boltzmann equation is not valid in the case of a phonon system. Transport properties boltzmann equation current density.
The boltzmann equation of phonon thermal transport solved in. In chapter 2, we study the theoretical properties of the linearized and nonlinear pbe using standard. Solution of boltzmanns equation for semiconductors using. The boltzmann equation written in abstract form as df dt cf 2. Distribution function f t contains all the information about the transport in the system. Iterative solution method for the linearized poissonboltzmann equation. Theory of phonon thermal transport in singlewalled carbon nanotubes and graphene lucas lindsay advisor. Efficient methods for solving the boltzmann equation for. Simultaneous diagonalization of the collision kernel of that equation and a symmetry crystal class operator allow us to obtain a spectral representation of the lattice thermal conductivity valid at finite frequency. The boltzmann equation of phonon thermal transport solved in the relaxation time approximation i theory amelia carolina sparavigna1 1 department of applied science and technology, politecnico di torino, italy doi 10.
Variancereduced particle simulation of the boltzmann. Novel feature in the formalism is that arbitrary finite concentration of electrons is admitted. The frequency dependent phonon boltzmann equation is transformed to an integral equation over the irreducible part of the brillouin zone. Evaluating the thermoelectric properties of batis3 by. Ballisticdiffusive phonon transport and size induced. Surfaces et spectroscopiesinternational audiencethe frequency dependent phonon boltzmann equation is transformed to an integral equation over the irreducible part of the brillouin zone. In contrast, phonon boltzmann equation modeling is a mesoscopic approach avoiding or relieving the aforementioned three drawbacks 16,17. Phonon transport conclusions e cient methods for solving the boltzmann equation for nanoscale transport applications nicolas g. Boltzmann equation an overview sciencedirect topics. Lecture notes microscopic theory of transport nuclear. A predictive model for the size dependent thermal conductivity is. A theory is presented for describing the lattice thermal conductivities of graphene and singlewalled carbon nanotubes.
We exhibit an intertwining operator for the two semigroups involved. The perturbed distribution function shows slope discontinuities at all multiples of the polar phonon energy. C, s mechanical engineering, purdue university, west lafayette, in, u. Phonon properties and thermal conductivity from first. The second approach, on the contrary, corrects the formula for thermal conductivity by solving the linearized boltzmann equation with given boundary conditions but uses bulk acoustic phonon dispersion and phonon mfp. In the second order in the scattering strength and after a short period of forgetting details of initial conditions. Heat transport in twodimensional materials by directly. Direct solution to the linearized phonon boltzmann equation. To have the boltzmann linearized equation in the general case, let us.
Derivation of the linearized boltzmann equation from a deterministic dynamics of hard spheres thierry bodineau joint works with isabelle gallagher, laure saintraymond. We have developed a formalism of the exact solution to the linearized phonon boltzmann transport equation bte for thermal conductivity calculations including three and fourphonon scattering. Insert this expression into the linearized boltzmann equation, and remove the term with a. This representation is particularly convenient as a basis for solutions, since in the isotropic dispersionless case the temperature deviation. Thermal networks form phonon boltzmanns transport equation. Boltzmann transport equations for nanoscience applications. Linear alternative to the boltzmann equation without. The approach given may be used for any type of particle. Here, we evaluate the prospect of batis3 as a thermoelectric material by using the linearized electron and. Boltzmanns equation describes the evolution of the oneparticle distribution function f fx, u, t, where the vector x, with components x 1, x 2, x 3, is the position vector, u, with components u 1, u 2, u 3, is the velocity vector, and t is the time. In this paper, we investigate the crossplane phonon transport from the viewpoint of the phonon boltzmann equation. In this paper we compare the nonlinear boltzmann equation appearing in the kinetic theory of gases, with its linearized version. Introduction one must distinguish between the linear boltzmann equation and the linearized boltzmann equation.
Other collisional invariants include energy when all sources are accounted for, spin total spin. Firstly the gray phonon boltzmanns transport equation is introduced. Solution of boltzmanns equation for semiconductors using a spherical harmonic expansion. We note that the time coordinate, formally used for the purpose of the derivation, is not.
The equivalence of the new linearized transport equations with the holstein equation is. An exact solution of the linearized boltzmann equation. The linearized boltzmann equation for the pure phonon field may be solved formally in terms of the eigenvectors of the normalprocess collision operator. In this paper, we develop an adjoint formulation of the linearized boltzmann transport equation for phonon transport. The collision terms in the boltzmann equation have several.
Hadjiconstantinou, department of mechanical engineering, massachusetts institute of technology cambridge, ma 029, usa may 3, 2019 abstract we derive the continuum equations and. Phonon dispersion relations and transition rates are obtained from density functional theory calculations. Get a printable copy pdf file of the complete article 253k. Pdf on the boltzmann equation of thermal transport for. We do not assume from the reader any familiarity with boltzmanns equation. Solve linearized bte selfconsistently to determine occupations. This can be easily verified by considering the ballistic and linearized. The phonon boltzmann equation, properties and link to. Adjointbased deviational monte carlo methods for phonon. We present a deviational monte carlo method for solving the boltzmann equation for phonon transport subject to the linearized ab initio 3phonon scattering operator. A deviational monte carlo formulation of ab initio phonon. Pdf solution of the linearized phonon boltzmann equation.
Hadjiconstantinou massachusetts institute of technology department of mechanical engineering 8 november 2011 acknowledgements. In appendix b we show how conservation laws result for the linearized functions in the new method. The former has no self interaction, just scattering with the medium, whereas the latter is the linearization of the fully nonlinear boltzmann equation. One important approach to study phonon transport in solids is the boltzmann transport equation bte. It is proved that the eigenvalue spectrum of the linearized collision operator of peierls phononboltzmann equation has no gap. In the case of a gas of elastic sphere and in the absence of external forces, this equation takes the form. This paper revisits the problem of the linearized boltzmann transport equation bte, or, equivalently, of the momentum relaxation time, momentum relaxation time mrt, for the calculation of low field mobility, which in previous works has been almost universally solved in approximated forms. T and the heat current q are related to zeroeigenvalue eigenfunctions of this operator. The boltzmann equation of phonon thermal transport solved. The phonon boltzmann equation, properties and link to weakly. Transport properties boltzmann equation relaxation time approximation equilibrium distribution relaxation time electrons in oscillating electric field small deviations linearized.
An exact method is given for solving the linearized boltzmann equation for arbitrary magnetic field strengths in the presence of polar mode scattering and elastic scattering mechanisms. The phonon boltzmann equation approach includes the exact ab initio solution of the linearized phonon boltzmann equation corresponding author. The boltzmann equation of phonon thermal transport solved in the relaxation time approximation ii data analysis amelia carolina sparavigna1 1 department of applied science and technology, politecnico di torino, torino, italy doi 10. A new form of projector is suggested leading to timeconvolutionless generalized master equations tclgme for the singleelectron density matrix or the wigner distribution function. To motivate the work, we provide a thorough discussion of the poissonboltzmann equation, including derivation from a few basic assumptions, discussions of special case solutions, as well as common analytical approximation techniques. The phononcontributed part of the total thermal conductivity is the lattice thermal conductivity. Phonon properties including frequencies, velocities and scattering rates are largely determined by interatomic force constants ifcs.
In this section some numerical results are presented, which might be of importance for mesoscopics. Solution of the linearized phonon boltzmann equation physical. When this condition is satisfied, the bte may be linearized, resulting. Graphene singlewalled carbon nanotubes and theory of. An exact solution of the linearized boltzmann equation with applications to the hall mobility and hall factor of ngaas k fletcher and p n butcherl 1c. Spectral gap and coercivity estimates for linearized. The variance reduction is achieved by simulating only the deviation from equilibrium. We call this approach a1 in the rest of the article. Pdf the linearized boltzmann equation for the pure phonon field may be solved formally in terms of the eigenvectors of the normalprocess. Simultaneous diagonalization of the collision kernel of that equation and a symmetry crystal class operator allow us to obtain a spectral. The phonon boltzmann equation, properties and link to weakly anharmonic lattice dynamics. An exact solution of the linearized boltzmann transport. Solution of the linearized phonon boltzmann equation.
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