Most materials can be synthesized in either glassy or crystalline form. Amorphous materials are characterized only by the presence of short-range atomic order, whereas crystals are characterized
by the presence of long-range atomic order (either periodic or quasiperiodic). Though glasses typically show very small thermal conductivity values, they are very poor electrical conductors as well, which leads to very smallZ Tvalues in turn. This intrinsic drawback of glasses cannot be alleviated by doping them, as it is usually done when dealing with semiconducting materials, or even for certain types of insulators, such as the so-calledconducting polymers (see Section 6.3.1). This feature then precludes the possible use of amorphous materials in competitive TE devices.a Accordingly hereafter, we will focus our attention on the study of materials grown in crystalline form (Exercise 3.1). Broadly speaking, the structural complexityof a crystalline solid can be measured in terms of several parameters, including:
• the number of different chemical species, nS, forming the compound. Thus,nS =1 for elemental solids,nS =2 for binary compounds, and so on,
• the number of lattice parameters,nL, required to fully describe the unit cell geometry. For the different crystallographic systems we have:nL=1 (cubic),nL=2 (rhombohedral, tetragonal, and hexagonal),nL = 3 (orthorhombic),nL = 4 (monoclinic), and nL=6 (triclinic),
• the unit cell volume,VU,
• the number of atoms present in the primitive unit cell,nU,
• the number, nE, and nature of stablechemical entities which can be present as inclusions (e.g., interstitial atoms, guest atoms in voids, and small solvent molecules) or building blocks (e.g., chains of bonded atoms, small molecules, or molecular cluster frameworks) in the unit cell,
• the coordination index of different atoms within the unit cell,nC
• the number of stable isotopes,nI, for a given element as well as the atomic mass ratio between the atoms composing binary and ternary compounds, respectively accounting for isotopic and alloying dynamical effects.
aSome years ago, Nolas and Goldsmid proposed a possible scenario under which the FOM may be higher in the amorphous state than in the crystalline state for a given solid. Essentially, this may occur when the mean free path of phonons is greater than that of charge carriers [71].
In order to illustrate the role of atomic structure on the transport properties of TE interest, let us consider the effect of (a) increasing the unit cell size and (b) the inclusion of a second alloying element in a simple elemental solid having a cubic lattice with lattice constant a. In the former caseVU =a3, so that to increase theavalue leads to a smaller reciprocal lattice volume (2π/a)3, resulting in a smaller Brillouin zone. This Brillouin zone reduction, in turn, gives rise to the presence of a large number of phonon bands in the frequency spectrum of the solid, due to an enhanced band-folding effect.
A similar effect is obtained when an alloy is formed from a solid solution process, keeping the original unit cell size almost constant,a but increasing the number of atomsnU present in it. In this case, a larger number of optical phonons, 3(nU −1) in number, must be accommodated in the frequency spectrum whennUis increased (see Fig. 3.1).
In both cases, one gets quite fragmented frequency spectra, characterized by the presence of many low dispersion, flat bands, for energies, say, higher than∼10 meV. Since the group velocity of the related phonons is determined by the slope of their dispersion relations, according to the expression vph ∼ dω/dq (where ω is the phonon frequency and q is its momentum), one realizes that the presence of flat bands results in a significant reduction of the phonons’ group velocities, thereby degrading the thermal conductivity. In addition, in the case of the lattice size increase, the Brillouin size reduction in reciprocal space also enhances the role of phonon–phonon Umklapp processes in the solid (see Section 1.2.2.3), further reducing the thermal conductivity. Thus, from general principles, we conclude that solids characterized by large unit cells containing many atoms therein must exhibit remarkably low lattice thermal conductivities. We will further discus on this important result throughout this chapter.
aAccording to the empirical Vegard’s law, a linear relation exists between the crystal lattice parameter of an alloy and the concentration of its constituent elements at constant temperature. Therefore, the lattice volume usually changes upon alloying (seeTables 3.5and3.6). Notwithstanding this, for the sake of clarity we are now ignoring this fact in order to conceptually distinguish the effects arising from an increase in thenumberof atoms in the unit cell from those related to itsvolume change.
Figure 3.1 Extended zone scheme dispersion relation for two binary chains (nS = 2) of atoms Aand Bwhose one-dimensional unit cells are given by A B (nU =2, left) and A B A A B A B A A B A A B (nU = 13, right), respectively. For the sake of comparison, the dispersion relation of the simple binary chain with unit cellA Bis shown by black solid curves on the right panel [23]. Adapted from T. Janssen, G. Chapuis, and M. de Boissieu, Aperiodic Crystals: From Modulated Phases to Quasicrystals, Oxford (2009).
By permission of Oxford University Press. The significant shortening of the Brillouin zone boundaries (ZB, shown by a vertical dashed line) can be clearly appreciated. The phonon wave vectorqis measured in relative lattice units.