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2.4.2 Boltzmann Transport Equation

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)

 (2.61)

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

 (2.62)

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

 (2.63)

and therefore

 (2.64)

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:

 (2.65)

and

 (2.66)

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:

 (2.70)

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

 (2.72)

Next: 3. Thermoelectric Properties of Graphene-Based Nanostructures Up: 2.4 Phonon Transport Previous: 2.4.1 Landauer Formula   Contents
H. Karamitaheri: Thermal and Thermoelectric Properties of Nanostructures