The Phonon Contribution

The thermal conductivity of a pure metal is lowered by alloying, whether the alloy formed is a single phase (solid solution) or multiphase mixture. There are several reasons for this. First, electrons are scattered by crystal imperfections and solute atoms (electron-defect scattering). Second, a substantial portion of the thermal conductivity in alloys, in contrast to that of pure metals, is by phonons, Kph (phonons are the sole contribution in electrically insulating solids) and phonons are also scattered by defects. Finally, electron-phonon interactions limit both Kei and Kp.  

For impurity atoms with masses lighter than those of the other atoms in the lattice, a spatially localized vibrational mode called an impurity exciton is generated, which is of a frequency above that of the maximum allowed for propagating waves. The amplitude of the vibration decays to zero, not far from the impurity. For heavier atoms, the amplitude of 

With regards to the second feature of real crystals mentioned earlier, there are different types of anharmonicity-induced phonon-phonon scattering events that may occur. However, only those events that result in a total momentum change can produce resistance to the flow of heat. A special type, in which there is a net phonon momentum change (reversal), is the three-phonon scattering event called the Umklapp process. In this process, two phonons combine to give a third phonon propagating in the reverse direction. 

In this equation, h is Planck s constant divided by 2tt, V is the crystal volume, T is temperature, fej, is Boltzmann s constant, u is the phonon frequency, is the wave packet, or phonon group velocity, t is the effective relaxation time, n is the Bose-Einstein distribution function, and q and s are the phonon wave vector and polarization index, respectively. 

Equation 6.20 is a rather formidable expression. From the experimentalist s standpoint, it will be beneficial to point out some simple, yet useful, criteria when toiling with thermal conductivity. First, it is noted that thermal conductivity is not an additive property. It is generally not possible to predict the thermal conductivity of an alloy or compound from the known thermal conductivities of the substituent pure elements. For example, the thermal conductivity of polycrystaUine silver and bismuth are. 

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Figure 11. Displayed are the TLS heat capacities as computed from Eq. (29) appropriate to the experiment time scales on the order of a few microseconds, seconds, and hours. A value of c = 0.1 was used here. If one makes an assumption on the specific value of Aq, it is possible to superimpose the Debye contribution on this graph, which would serve as the lowest bound on the total heat capacity. As checked for Aq = cod. the phonon contribution is negligible at these temperatures.

At high temperatures (T>100K) cv is due to the phonon contribution cph which approaches the classical Dulong and Petit value of 3 nR = 24.94n [J/mol K], where n is the number of atoms in the molecule. 

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 ... 

Since in our temperature range, the Debye temperature of Ge is 370K [47], the phonon contribution to the heat capacity can be neglected. Hence, the heat capacity of our samples is expected to follow the equation ... 

Note that the full curve drawn in the figure corresponds to the specific heat calculated with no free parameters. While the agreement of this model to the specific heat is rather impressive, it is in fact the 3 inter-cluster degrees of freedom, i.e. motion of the whole cluster, which govern the phonon contribution to the specific heat at temperatures below about 15-20 K [99]. 

To elucidate the effect of temperature, we performed calculations of the rate of multiphonon non-radiative transitions. We considered a case when l and l1 belong to different rows of the same representation. The phonons, contributing to a nondiagonal vibronic interaction are considered in an Einstein-like model with the parabolic distribution function (14) (note that the results are not sensitive to the actual shape of the phonon bands) interaction is arbitrary. In this model the Green function is described by simple expression (16). In the case of a strong linear diagonal vibronic interaction one can expand the gr(f)-function into a series and take into account the terms up to the quadratic terms with respect to t gT(—t) iSjt — Ojt2/2. Here = Oq/wq, cD0 is the mean frequency of totally symmetric... 

The phonon contribution to the thermal conductivity is strongly suppressed by bond-length fluctuations. 

Figure 61 shows the ZFC and FC M(T) curves for the FMM phase II, which show a first-order transition at 7c where the bond-length fluctuations become frozen out to restore the phonon contribution to the thermal conductivity, fig. 62. In the I + II two-phase samples t = 0.973 and t = 0.974, phase II is suppressed and phase fluctuations inhibit phonon formation below Tco or 7c. Tokura et al. (1996) applied hydrostatic pressure to the FM phase II of (Ndo.i25Smo.875)o.5Sro.5Mn03 having / /c they reported the appearance of a CE AFI second phase having a volume fraction that increased with pressure below Too = 7n- Since pressure... 

Zimmermann and Konig211) introduce the phonon contribution of the lattice by a Debye model with an interpolated Debye temperature... 

The onset of electron-phonon interaction in the superconducting state is unusual in term of conventional electron-phonon interaction where one would expect that the phonon contribution is weakly dependent on the temperature [19], and increase at high T. Indeed, based on this naive expectation, this type of unconventional T dependence has been often used to rule out phonons. Here, however, we see clearly that this reasoning is not justified. Moreover, this type of unconventional enhancement of the electron phonon interaction below a characteristic temperature scale is actually expected for other systems such as spin-Peierls systems or charge density wave (CDW) systems. Thus, our results put an important constraint on the nature of the electron phonon interaction in these systems. 

Use the modihed Wiedemann-Franz-Lorenz law to estimate the phonon contribution to the thermal conductivity of a semiconductor at 298 K whose total thermal conductivity is 2.2 W m if the electrical conductivity is 0.4 x 10 S m ... 

The experimental techniques most commonly used to measure the phonon distributions are IR absorption, Raman scattering and neutron scattering. The IR and Raman spectra of crystalline silicon reflect the selection rules for optical transitions and are very different from the phonon density of states. The momentum selection rules are relaxed in the amorphous material so that all the phonons contribute to the spectrum. 

However, very soon it became clear that the situation is more complex (e.g. [9, 10]). The obvious problem arises with the fact that the red-yellow PL from PS is relatively slow with a decay time in the range of tens of microseconds, which, together with some further experimental observations [11] and theoretical calculations [9,10,12], is considered as strong evidence for an indirect band gap. However, as pointed out by Hybertsen [13], the electron and hole wave functions in small crystallites are spread in k space so that it is no longer meaningful to debate whether the gap is direct or indirect. Detailed calculations show that the phonon assisted transitions dominate in crystallites larger than about 1.5 nm, where an important part of the phonon contribution comes from scattering at the surface of the crystallites and a part from the bulk phonons. 

To reiterate the statements made above, this model shares two important features with reality First g (w) a> as (a -> 0, and second, the existence of a characteristic cutoff associated with the total number of normal modes. The fact that the model accounts for the low-frequency spectrum of lattice vibrations enables it to describe correctly the low-temperature behavior of the phonon contribution to the heat capacity. Indeed, using (4.51) in (4.35) leads to... 

This electronic contribution to the heat capacity is linear in the temperature T. This should be contrasted with the cubic form of the low temperature dependence of the phonon contribution, Eqs (4.54) and (4.57). 

Here Q ps, I2pv are the results of integration in (6.44) with respect to atom momenta of the surface layer and the cluster body, respectively Qqs and Qqv are the integration results with respect to atom coordinate deviations from equilibrium values they contain, in particular, the phonon contribution. 

Retardation effects. If the phonon contribution dominates the bare vertex (6), the retardation effects associated with heavy ions can play an important role in the many-body theory. In order to develop this point in greater detail let us, for reasons of clarity, ignore the Coulomb contribution to the bare vertex (6). Some simple vertex corrections are shown in Fig. 1. These particular diagrams are chosen because in the one-dimensional case (ij = r/ — 0) they all yield the same > g6log2T contribution to the vertex, provided that the retardation effects are neglected. Such a degenerate situation is usually named parquet. 

M. Sunjic and A. Lucas, "On the Phonon Contribution to the X-ray Photoemission Linewidths in Polar Crystal Films", /to be published/. 

The phonon contribution to the electronic self-energy is obtained perturbatively in the phonon-electron coupling. We recast the phonon contribution (first term on the RHS of Eq. (C25) for S lp) in the interaction picture by writing... 

Here the superscript 0 represents the trace with respect to the non-interacting density matrix. The zeroth order Green functions are given in Eq. (55). The terms coming from the lead-molecule coupling (V. ) vanish because they are odd in creation and annihilation operators. Substituting Eq. (C34) in Eq. (C25) gives for the phonon contribution... 

The phonon contribution to the heat capacity is the most important one, but others also occur. As noted in Section 2.3.7, the heat capacity due to free electrons in metals is small but significant heat capacity changes accompany phase changes, such as order-disorder changes of the type noted with respect to ferroelectrics (Sections 11.3.5 and 11.3.6), or when a ferromagnetic solid becomes paramagnetic (Sections 12.1.2 and 12.3.1). 

The first two terms in eq. (43) are replaced by SJ" of eq. (42). This is equivalent now to the assumption that to the first approximation the impurity and the phonon contribution of corresponding magnetic and non-magnetic compounds are equal. These considerations lead to the following expression for the spin-disorder contribution for the thermopower of GdX (X = AI2, Cua, Ni)... 

Thermal energy is transported by two mechanisms in solids—electronic conduction and lattice or phonon conduction. An electrical analog for thermal conduction is shown in Fig. 2 [% The total thermal conductivity. A, is the sum of the electronic term and the lattice term. For pure metals and dilute alloys, thermal conduction is dominated by the electronic term, while for heavily alloyed metals, the phonon contribution is appreciable. 

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