Lattice thermal conductivity

Since we know that Cv approaches 0 with a 7 dependence at low temperatures, one would expect the thermal conductivity to do likewise and this accounts for the low thermal conductivity of nonmetals at low temperature. At higher temperatures, thermal conductivity is limited by the mean free path A. Phonons are scattered inelastically from defects, impurities, and grain boimdaries. Phonons may also interact with other phonons through the anharmonic terms in the lattice potential. For example, two phonons may combine to produce a third phonon. The conservation of momentum requires 

If both fci and 2 Tr/2fl, the resultant vector will always remain in the first Brillouin zone and the collision will be an N-process. However, as they become larger than ir/2fl, a li-process becomes more likely. Using the Debye model, the frequency associated with a phonon whose fc = Tr/2fl is w = Vo u/la. The ratio of this frequency to the Debye frequency is 

The probability of exciting a mode with frequency (o is given by 

Illustration of an N-process in which the collision is elastic and a U-process in which the collision is inelastic. If the resulting vector from a phonon-phonon collision falls outside the first Brillouin zone (the square box), the momentum is Bragg reflected back into the first Brillouin zone with a transfer oi G1r momentum to the lattice in the form of crystal momentum. 

Thus the li-processes can be expected to become important when the temperature approaches 0.4 d and will limit the conductivity as the temperature increases. 

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ZT Y r A A A A A AC dimensionless thermoelectric figure of merit electronic coefficient of heat capacity (1+ZT)F2 crystal field singlet non-Kramers doublet (crystal field state) crystal field triplet crystal field triplet hybridization gap jump in heat capacity at Tc K KL -min P 6>d X JCO total thermal conductivity of solid thermal conductivity of electrons or holes thermal conductivity of lattice minimum lattice thermal conductivity electrical resistivity Debye temperature magnetic susceptibility magnetic susceptibility at T = 0... 

Balandin, A., and Wang, K.L., Significant Decrease of the Lattice Thermal Conductivity due to Phonon Confinement in a Free-standing Semiconductor Quantum Well. Physical Review B, 1998. 58(3) p. 1544-1549. 

Ladd, A., B. Moran, and W.G. Hoover, Lattice Thermal Conductivity A Comparison of Molecular Dynamics andAnharmonic Lattice Dynamics. Physical Review B, 1986. 34 p. 5058-5064. McGaughey, A.J. and M. Kaviany, Quantitative Validation of the Boltzmann Transport Equation Phonon Thermal Conductivity Model Under the Single-Mode Relaxation Time Approximation. Physical Review B, 2004. 69(9) p. 094303(1)-094303(11). 

For the lattice thermal conductivity, the model due to Steigmeier and Abels is adopted, here [3]. The lattice thermal conductivity is given by... 

Here o is electrical conductivity, u is thermopower, k is thermal conductivity, t is energy of carrier, p is chemical potential, e is bare charge of electron, and f (e) is Fermi-Dirac distribution function. In deriving eq.(2) we treat the lattice thermal conductivity as a constant. Following we consider the n-type semiconductors, then the change of differential conductivity can be given by ... 

Origin of the notable difference can be easily understood, if one examine grain size dependence of electric conductivity and lattice thermal conductivity show in fig. 4. Lattice... 

Figure 4. Grain size dependence of electric conductivity a and lattice thermal conductivity Kl for (a) SiGe and (b) PbTe.

The best material system available at this time to meet the above criteria is the bismuth-antimony crystal system [ ]. In pure bismuth, the conduction and valence bands are slightly overlapped and, in the proper orientation, both holes and electrons have high and reasonably equal mobilities. The addition of a small amount of antimony has the effect of greatly reducing the lattice thermal conductivity. However, the addition of antimony also has the effect of decreasing the overlap of the principal conduction and valence bands in fact, for antimony concentrations greater than 5 atomic %, a gap exists and the system becomes a semiconductor, which is undesirable for these purposes [ ]. At present it appears that the optimum antimony content is about 3 atomic %. 

Two-electron covalent bonds are formed between the metal (Me) and silicon atoms in monosilicides. The overall electron-valence nature of these bonds determines the uniformity of the crystal structure of silieides. The high value of their lattice thermal conductivity (Table 2) is an indirect confirmation that stable covalent bonds are present in monosllicide crystals. The uniformity of the structures amounts not only... 

Therefore, the anisotropy of the carrier mobility (Up) and the lattice thermal conductivity ( Kiatt) Is due to the characteristic features of the distribution of chemical bonds in the crystal according to the scheme in [8],... 

By analyzing the temperature dependence of the electrical properties, using our results (Fig. 5) and published data [9,10], another characteristic feature of the structure of CrSi2 crystals becomes apparent, which makes the energy spectrum of valence electrons in this compound more precise. A calculation of the lattice thermal conductivity of single crystals, taken as the difference between the total and electronic thermal conductivities (Fig. 5), as a function of temperature, shows that it decreases continuously up to the maximum measurement temperature. This Indicates the absence of an additional heat transfer component due to ambipolar diffusion of carriers [18] in the intrinsic conduction range. 

Fig. 5. Temperature dependence of some electrical properties of CrSi2 a) thermoelectric power b) electrical conductivity c) total thermal conductivity d) lattice thermal conductivity H) parallel to the c axis i.) perpendicular to the c axis.

While thermal energy can be transported through a solid via a variety of different mechanisms [24], the two most important for TE applications are diffusive transport of energy by the mobile charge carriers (electronic thermal conductivity, iCg) and phonons (lattice thermal conductivity, Kl). Since it is a relatively good approximation to treat and Kl as independent for many solids, significant emphasis has been placed on the development of TE materials with low icl values [6]. Amorphous solids are typically characterized by low carrier mobilities and thus low a values, however they also exhibit some of the lowest known thermal conductivities (excluding porous materials) and serve as useful benchmark materials for TE materials research. 

Lattice thermal conductivity data below 300 K for representative clathrates are summarized in Fig. 6.4 [35, 40 4], These data, for a representative number of compositions collected from both polycrystalline and single crystalline specimens, allow for a comparison of the effect of guest and framework composition on Kl. While some compositions have lattice thermal conductivities that are characteristic of glasses, others have Kl values that more closely resemble the typical temperature dependence of defect-free crystalline solids in which Umklapp processes [24] produce a monotonically decreasing Kl with increasing temperature above 10 K. 

The transport properties of the clathrate EugGaigGego were first investigated by Cohn et al. [6]. The lattice thermal conductivity was shown to be extremely low and glass-like [6], as was the case for SrgGaigGego [7], much like the type-1... 

The lattice thermal conductivity of EugGaieGeso was smdied by several groups below room temperature [6,13,15, 18, 20,24,25]. In all cases it was estimated via Eqs. (9.2) and (9.3) using p T) data. This naturally puts a larger error bar on Ki than on the electronic transport quantities p and S. In addition, in the generally employed steady-state heat flow technique, measurements of k above about 100 K are encumbered by radiation losses. Thus, differences in absolute values between results from different groups should be discussed with caution. Here we focus on the overall trends and relative changes within the sample series measured and analyzed in the same way. 

Fig. 9.4 Lattice thermal conductivity K versus temperature of type-I (top) and type-VIII EugGaie. cGe3o+ , (bottom). Plot adapted from Bentien et al. [18] and completed with data on type-I EugGaigGe3o single crystals [6, 13, 15]...

Fig. 9.5 Lattice thermal conductivity K at 150 K (top) and 300 K (bottom) versus guest free space Figure adapted from [30] and completed with data from [15, 31-34]. The K data at 300 K are, except for CsgZn4Sn42, all for single crystalline samples...

Fig. 9.6 Lattice thermal conductivity k versus charge carrier concentration n isotherm at 193 K of type-I (JS, left) and type-VIII EugGaigGeso (a, right). Plot based on data from Bentien et al. [18]...

Figures 9.5 and 9.6 show that indeed both structural and electronic properties influence the lattice thermal conductivity of intermetaUic clathrates. In Fig. 9.5, /C/ is plotted for different type-I clathrates versus the guest free space, an analysis done for many type-I clathrates by Suekuni et al. [29,30]. For calculating the guest free space...

Partial substitution of Sr for Eu has been observed in the pioneering work of Cohn et al. [6], where the type-I clathrate Sr4Eu4Gai6Ge3o was investigated. Below 40 K, the lattice thermal conductivity of Sr4Eu4Gai6Ge3o was shown to be distinctly reduced with respect to EugGaieGeso, being two times smaller at 6 K, the lowest temperature of that experiment. Synchrotron X-ray powder diffraction showed that in spite of the small difference of the covalent radii of Eu (185 pm)... 

K. Suekuni, S. Yamamoto, M.A. Avila, T. Takabatake, Relation between guest free space, U. lattice thermal conductivity reduction by anharmonic rattling in type 1 clathrates. J. Phys. Soc. Jpn. 77SA, 61 (2008)... 

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