Theory of Thermal Conduction in Nonmetals
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Theory General Expression for
From the kinetic theory of gases, the expression for phonon (lattice) thermal conductivity is
1 Cv , 3
(1)
where C is the specific heat (i.e., heat capacity per unit volume), v is the average
MRS BULLETIN/JUNE 2001
particle (or group) speed, and is the phonon mean free path. This expression can be written in its full form by taking into consideration contributions from phonons in all possible modes, as follows:1
2 3VkBT 2
c q qs qs nqs 2 s
2
qs
nqs 1 ,
(2)
where is Planck’s constant divided by 2, V represents crystal volume, T is temperature, kB is Boltzmann’s constant, (qs) is the frequency, cs(q) (qs) is the group velocity, n(qs) is the Bose–Einstein distribution function, and (qs) (qs)/cs(qs) is an effective relaxation time for phonons with wave vector q and polarization index s. It is clear from Equation 2 that a proper calculation of thermal conductivity requires three essential ingredients: (1) the phonon dispersion relation (qs), (2) the effective relaxation time for phonons in all possible modes and at different temperatures, and (3) a reliable scheme for performing the summations in Equation 2. While phonon dispersion relations can be calculated by using several reliable methods, Brillouin zone summations can be very time-consuming. In addition, there is a lack of knowledge about the effective anharmonic phonon relaxation time. With these difficulties in mind, most calculations of have been made by employing the isotropic continuum dispersion relation (qs) qcs and the Debye radius cutoff qD to simplify the Brillouin zone integration. This reduces Equation 2 to a much simpler form,
2qD5 6 2kBT 2
c dxx nn 1 , 1
4 s
s
4
0
(3)
where x q/qD , and and n are functions of x and the polarization index s. The justification for employing the Debye model is that the majority of heat carriers are low-frequency acoustic phonons, whose dispersion relation can satisfactorily be described within the continuum model. Figure 1a shows the phonon dispersion curve and density of states (DOS) for AlN in the zinc-blende structure obtained using realistic methods,3 and Figure 1b shows the results obtained within the Debye model. Clearly, the Debye picture is a good representation up to approximately 60% of the full acoustic phonon spectrum. While the longitudinal optical branch shows a reasonable dispersive behavior, transverse optical modes can be reasonably well represented within Einstein’s flat dispersion model.
Effective Phonon Relaxation Time
In order to calculate properly, it is important to obtain the magnitude as well as the frequency and temperaturedependence of the effective phonon relaxation time . This is the most difficult parameter to deal with, as in a general situation there is no exact method available to solve the phonon Boltzmann equation for it (see, e.g., Srivastava1 for a good discussion). Both acoustic and optical phonons can carry heat. However, as can be appreciated from Figure
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