Phonons mean free path

The discussion of the previous section would also lead us to believe that since most ceramics are poor electrical conductors (with a few notable exceptions) due to a lack of free electrons, electronic conduction would be negligible compared to lattice, or phonon, conduction. This is indeed the case, and we will see that structural effects such as complexity, defects, and impurity atoms have a profound effect on thermal conductivity due to phonon mean free path, even if heat capacity is relatively unchanged. 

Amorphous materials have no long-range structural order, so there is no continuous lattice in which atoms can vibrate in concert in order for phonons to propagate. As a result, phonon mean free paths are restricted to distances corresponding to interatomic spacing, and the (effective) thermal conductivity of (oxide) glasses remains low and increases only with photon conduction (Figure 8.2). 

Figure 6 shows the MD predicted in-plane and out-of-plane thermal eonduetivities at 376K (Fig. 6a) and lOOOK (Fig. 6b) as a function of film thickness. It is seen that both the in-plane and out-of-plane thermal conductivities are affeeted by the thiekness of the film. For thiekness smaller than the phonon mean free path (approximately 300 nm and 30 nm at 300K and lOOOK, respeetively), both the in-plane and out-of-plane thermal eonduetivities deerease with deereasing thiekness, an effeet attributed to the scattering of phonons with the boundaries of the thin film. This effeet is more pronounced in the out-of-plane direction, where the dimensions of the thin film make the phonon transport ballistic. At large thicknesses, the thermal conductivities approach the bulk value (shown as dashed lines in Fig. 6). The bulk value is reached at smaller thicknesses at lOOOK due to the smaller phonon mean free path at this temperature.

A Phonon mean free path [m]. with itself. 

As the temperature is increased beyond 0D/1O, C becomes a constant (=3r kB). Also, the phonon density at high frequency and large wave vectors becomes sufficiently high that phonon-phonon Umklapp scattering dominates and determines the phonon mean free path. 

Above about 0.5 GHz, Brillouin scattering provides an elegant technique of measuring phonon mean free paths (which are inversely proportional to the sound attenuation coefficient) by evaluating the width of the Brillouin lines [53]. However, as a result of finite spectrometer resolution and contrast, this method is limited to aerogel densities above 180 kg/m. ... 

What is the thermal conductivity of silicon nanowires, n-alkane single molecules, carbon nanotubes, or thin films How does the conductivity depend on the nanowiie dimension, nanotube chirality, molecular length and temperature, or the film thickness and disorder More profoundly, what are the mechanisms of heat transfer at the nanoscale, in constrictions, at low tanperatures Recent experiments and theoretical studies have dononstrated that the thermal conductivity of nanolevel systems significantly differ from their macroscale analogs [1]. In macroscopic-continuum objects, heat flows diffusively, obeying the Fourier s law (1808) of heat conduction, J = -KVT, J is the current, K is the thermal conductivity and VT is the temperature gradient across the structure. It is however obvious that at small scales, when the phonon mean free path is of the order of the device dimension, distinct transport mechanisms dominate the dynamics. In this context, one would like to understand the violation of the Fourier s... 

The phonon mean free path greatly increases with decreasing temperature it is 10 i.m at 20 K. 

Debye (in 1914) applied Eq, 34.15 to phonon conduction to describe thermal conductivity in dielectric solids. Then C is the heat capacity of the phonons, v is the phonon velocity, and / is the phonon mean free path. 

FIGURE 34.4 Inverse phonon mean free paths for several crystalline oxides and for vitreous silica. 

Slack [25] and Cahill et al. [26] explored the theoretical limits on k for solids within a phonon model of heat transport. Their work utilized the concept of the minimum thermal conductivity, Kj n- At this minimum value the mean free path for all heat carrying phonons in a material approaches the phonon wavelengths [25]. In this limit, the material behaves as an Einstein solid in which energy transport occurs via a random walk of energy transfer between localized vibrations in the solid. Experimentally, K an is often comparable to the value in the amorphous state of the same composition. In principle jc in can be achieved by the introduction of one or more phonon scattering mechanisms that reduce the phonon mean free path to its minimum value over a broad range of frequencies, and therefore reduces Kl over a broad range of temperatures. In practice, there are relatively few crystalline compounds for which this limit is approached. 

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