Phonons, heat conduction

The motion of the exciton wavepacket causes the transport of energy. In order to find the appropriate energy diffusion coefficient we must estimate the mean free path and the mean free time of the wavepackets. This situation is quite similar to that of phonon heat conductivity (see, for example, (12)). 

Yang, B. Chen, G. (2003). Partially coherent phonon heat conduction in supierlattices, Phys. Rev. B 67 195311. 

The heat conductivity in solids occurs via phonons. This conductivity is ideal in single crystals and is considerably reduced in porous solids, by one to two orders of magnitude. Therefore thermal insulation materials are built up of small particles which should touch each other at only a few points. This effect is of course enhanced by a low density of the material. 

In order to estimate the phonon scattering strength and thus the heat conductivity, we need to know the effective scattering density of states, the transition amplitudes, and the coupling of these transitions to the phonons. 

An accurate calculation of the heat conductivity requires solving a kinetic equation for the phonons coupled with the multilevel systems, which would account for thermal saturation effects and so on. We encountered one example of such saturation in the expression (21) for the scattering strength by a two-level system, where the factor of tanh((3co/2) reflected the difference between thermal populations of the two states. Neglecting these effects should lead to an error on the order of unity for the thermal frequencies. Within this single relaxation time approximation for each phonon frequency, the Fermi golden rule yields, for the scattering rate of a phonon with Ha kgT,... 

Figure 19. The predicted low T heat conductivity. The no coupling case neglects phonon coupling effects on the ripplon spectrum. The (scaled) experimental data are taken from Smith [112] for a-Si02 (AsTj/ScOd 4.4) and from Freeman and Anderson [19] for polybutadiene (ksTg/Hcao — 2.5). The empirical universal lower T ratio l /l 150 [19], used explicitly here to superimpose our results on the experiment, was predicted by the present theory earlier within a factor of order unity, as explained in Section lllB. The effects of nonuniversaUty due to the phonon coupling are explained in Section IVF.

If the two solids are of the same (or similar) materials and the depth of surface impurities (e.g. oxides) is thin in comparison with the heat carrier wavelength, the expected contact thermal resistance is Rc oc T 1 (see eq. 3.36) for metals, and Rc oc 7 3 (see eq. 3.33) for dielectric material. For a dirty contact between metals (heat conduction by phonons only) Rc oc T 2 (see eq. 3.35). These dependences of Rc have been observed experimentally. 

Low-temperature (T < 1K) heat conduction of a pure metal, like copper of our experiment (Cu Debye temperature 0D 340K), is mostly electronic [27] and the phonon contribution should be negligible. With the latter hypothesis, in the 30-150 mK temperature range ... 

Low-temperature thermometers are obtained by cutting a metallized wafer of NTD Ge into chips of small size (typically few mm3) and bonding the electrical contacts onto the metallized sides of the chip. These chips are seldom used as calibrated thermometers for temperatures below 30 mK, but find precious application as sensors for low-temperature bolometers [42,56], When the chips are used as thermometers, i.e. in quasi-steady applications, their heat capacity does not represent a problem. In dynamic applications and at very low temperatures T < 30 mK, the chip heat capacity, together with the carrier-to-phonon thermal conductance gc d, (Section 15.2.1.3), determines the rise time of the pulses of the bolometer. 

Film conductances are also often defined for the impedance to thermal conduction when two solid conductors are placed in mechanical contact. A significant contact resistance is often observed when, on a microscopic scale, heat transfer involves an air-gap between the materials. Under such conditions, phonon propagation must be replaced by the kinetic interaction amongst gaseous atoms and then back to phonon heat transfer in the next solid. Fibrous and foam insulation axe effective thermal insulators because of the numerous contact resistances involved in the transfer of heat. 

Most theories have the common feature that they explain the phenomenon of heat conductivity (in melts and amorphous solids) on the basis of the so-called "phonon" model. The process is supposed to occur in such a way that energy is passed quantumwise from layer to layer with sonic velocity and the amount of energy transferred is assumed to be proportional to density and heat capacity. No large-scale transfer of molecules takes place. 

Chen, G., Phonon Wave Heat Conduction in Thin Films and Superlattices. Journal of Heat Transfer, 1999.121 p. 945-953. 

Narumanchi, S.V.J., J.Y. Murthy, and C.H. Amon. Simulations of Heat Conduction in Sub-micron Silicon-on-insulator Transistors Accounting for Phonon Dispersion and Polarization, in ASME Internatirjnal Mechanical Engineering Congress and Exposition, IMECE 2003-42447. 2003. Washington, DC. 

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