At equilibrium, the distribution of phonons in branch and wavevector is given by the Bose-Einstein distribution function :

(2.59) |

Under non-equilibrium conditions, the distribution of phonons deviates from its equilibrium distribution, and transport of phonons is computed using the Boltzmann transport formalism. The non-equilibrium distribution function , in general, is a function of time and position . The BTE can be written as:

(2.60) |

and for the steady state:

(2.61) |

Under a temperature gradient, the BTE can be written as [60]:

In the relaxation time approximation, the change of the distribution function due to the scattering events can be given by:

(2.63) |

and therefore

where is the relaxation time of phonons of frequency . In this work we use a linearized form of Eq. 2.64, which assumes that the temperature gradient causes only a small deviation from Bose-Einstein distribution function [61,62], so that:

and

where shows the deviation from the equilibrium distribution. Then, one may eliminate the temperature gradient using and write:

(2.67) |

Since the equilibrium distribution does not carry any heat flux, the heat flux equals to [62]:

(2.68) |

On the other hand, it holds the differential form of Fourier's law:

(2.69) |

Therefore, one can obtain the lattice thermal conductivity as:

Under the single-mode relaxation time (SMRT) approximation [62], follows from the linearized BTE (Eqs. 2.64-2.66) as:

(2.71) |

Here, is the scattering time in SMRT approximation. Therefore, Eq. 2.70 becomes