Role of filler in thermal transport
4.2 Complex phonon modes
4.2.1 Rattling – resonant scattering or avoided crossing
The decrease of lattice thermal conductivity in RxCo4Sb12 skutterudites has been largely attributed to the “rattling” effect of filler atom R. Due to the size difference between the void and filler atoms (see Table 4.1)9,87,88, fillers are under-constrained and weakly bound, which allows them to “rattle”
in the void. The concept of fillers as “rattlers” was first proposed by Slack89. Rattlers are supposed to scatter phonon propagation effectively such that the material behaves as a glass whereas the electron conduction is mostly unaffected as it is in a crystal. Materials with such characteristic features are known as PGEC (phonon glass electron crystal).
Even though the concept of rattling has been widely accepted in the skutterudite community, the microscopic mechanism of rattling, as to be more precise, the interaction between guest atom phonon modes and host atom phonon modes, is still unclear and often debated.
The first and most common explanation is resonant scattering by guest atom vibrations. In this mechanism, a localized mode of the guest atom (Einstein or so-called rattling mode) within the acoustic frequency range is introduced which is uncorrelated to the phonon modes of the framework described by the usual Debye model. The rattling mode can be approximated as a quantized harmonic oscillator, with a single-frequency 𝑤𝑤0=�𝐾𝐾 𝑀𝑀� , where K is the force constant which depends on the bonding strength between R and Sb atoms and M is the rattler mass. A few Einstein frequencies calculated along [100] direction for different fillers with comparison to experimental results are listed in Table 4.1 9,87,88. As we can see from this equation, the heavier the rattler and the smaller its ionic radius, the lower the rattling frequency would be. The scattering mechanism due to the guest atom vibrations is called resonant scattering, in which only the lattice phonons with the
energy similar to that of the local mode introduced by guest atoms would get scattered. The resonant scattering relaxation time can be expressed as:
𝜏𝜏𝑟𝑟−1=𝐶𝐶0 𝑤𝑤2
(𝑤𝑤2−𝑤𝑤02)2 (Eq. 4.2)
where 𝐶𝐶0 is a constant that is proportional to the concentration of rattlers, and 𝑤𝑤0 is the frequency of Einstein mode of rattlers. Since the majority of heat is transferred by acoustic phonons with relatively low frequencies, the lower the rattling frequency, the more acoustic phonons are scattered. This can explain why Yb is such good filler in decreasing lattice thermal conductivity.
Evidence of resonant scattering has been found in many experiments. Large atomic displacement parameter (ADP) of fillers was observed from Rietveld refinements of X-ray diffraction data37. The existence of single-frequency Einstein modes was also found either from specific heat data, or from peaked response close to the supposed Einstein mode frequency in the partial densities of phonon states (PDOS) from either nuclear inelastic spectroscopy (NIS)90 or time-of-flight inelastic neutron scattering (INS). Moreover, as the void size increases, the decrease in the lattice thermal conductivity and Einstein temperature (vibration frequency) also supports the resonant scattering mechanism37,91.
Table 4.1 Ionic radii, atomic mass, and rattling frequency of different filler atoms. Coordination for all fillers is chosen to be eight except for Ga (coordination = 6).
Filler type Charge Ionic radius (Å)
Atomic Mass
𝒘𝒘𝟎𝟎_𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
(cm-1)
𝒘𝒘𝟎𝟎_𝒆𝒆𝒆𝒆𝒆𝒆 (cm-1)
Na +1 1.18 22.99 113 -
K +1 1.51 39.10 142 -
Ca +2 1.12 40.08 - -
Sr +2 1.26 87.62 91 -
Ba +2 1.42 137.33 94 -
Eu +2 1.25 151.96 59 -
Yb +2 1.14 173.05 43 40
Ga +3 0.62 69.72 - -
In +3 0.92 114.82 - -
Ce +3 1.14 140.12 55 55
La +3 1.16 138.91 68 55
Void - 1.89 - - -
The resonant scattering mechanism has been challenged when evidence of avoided crossing mechanism was discovered in recent years. In the resonant scattering picture, the rattling mode is considered to be a vibration motion lacking phase coherence compared to the host motion. However recent studies from both experimental work (the lower reduced mass compared to rattler mass from Extended X-ray fine absorption measurements (EXAFS)91, inelastic neutron29, and nuclear inelastic scattering90) and theoretical calculations92 cast doubt on this assumption. From inelastic neutron scattering data, Koza et al.29 showed coherency of coupling between the localized guest mode and host lattice mode in Ce and La-filled Fe4Sb12 skutterudites. This interaction between the localized rattling mode and the host acoustic modes could lead to avoided crossing on the phonon spectra and thus effectively lower group velocity, largely scattering the acoustic phonons and reducing thermal conductivity. In addition, this coupling also leads to an increased anharmonicity for the rattling mode manifested as a higher Gruneisen parameter, which will contribute to more Umklapp scattering, as we will discuss later. Recently it was also discovered that the anharmonicity associated with phonon hybridization can decrease as the phonons modes decouple at high pressure, meaning the rattling mode can be switched off at high pressure90.
The different origin in decreasing lattice thermal conductivity between resonant scattering and avoided crossing can be understood as follows. From Eq. 4.1 we know that the lattice thermal conductivity can be expressed as 𝜅𝜅𝑙𝑙𝑚𝑚𝑡𝑡𝑡𝑡𝑡𝑡𝑐𝑐𝑒𝑒 =13∫0𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝐶𝐶𝑐𝑐(𝑤𝑤)𝑉𝑉𝑔𝑔2(𝑤𝑤)𝜏𝜏(𝑤𝑤)𝑑𝑑𝑤𝑤; while resonant scattering reduces the scattering time near the rattling mode frequency 𝜏𝜏(𝑤𝑤) (Figure 4.1a and blue curve in 4.1d), the avoided crossing due to coupling of localized guest atom and host lattice modes leads to a large reduction in the group phonon velocity 𝑉𝑉𝑔𝑔2(𝑤𝑤) near the guest vibration mode frequency, as illustrated in Figure 4.1b, 4.1c and orange curve in 4.1d 93.
Figure 4.1 Different mechanisms in reducing lattice thermal conductivity between resonant scattering and avoided crossing on the phonon spectra. a) The resonant scattering model targets phonons near 𝑤𝑤0. b) BvK phonon dispersions for a stiff framework (m1, k1) and loosely bound guest atoms (m2, k2). Increased k2 stiffness results in increased coupling (extent of avoided crossing) between the framework and guest modes. c) The avoided crossing reduces 𝑉𝑉𝑔𝑔2(𝑤𝑤) in the vicinity of 𝑤𝑤0. d) 𝜅𝜅𝑠𝑠(𝑤𝑤)for an empty BvK framework, using Umklapp and boundary scattering terms (curve A). Including resonant scattering reduces 𝜅𝜅𝑠𝑠(𝑤𝑤) near 𝑤𝑤0 (curve B). If instead the effect of coupling on 𝑉𝑉𝑔𝑔2(𝑤𝑤) is accounted for, a similar reduction is observed (curve C). Reproduced with copyright
permission obtained © Royal Society of Chemistry [2011]
(http://pubs.rsc.org/en/Content/ArticleLanding/2011/JM/C1JM11754H#!divAbstract).
From Figure 4.1 we can see that both the resonant scattering effect (reduction of scattering time 𝜏𝜏(𝑤𝑤)) and the effect of avoided crossing on the phonon spectra (reduction of group velocity 𝑉𝑉𝑔𝑔2(𝑤𝑤)) will result in the similar reduction of lattice thermal conductivity. It is thus difficult to distinguish between the two effects. Temperature dependent thermal conductivity measurements are not enough but rather frequency-dependent measurements of 𝑉𝑉𝑔𝑔2(𝑤𝑤) and 𝜏𝜏(𝑤𝑤) are required to unravel the intertwined effects on 𝜅𝜅𝐿𝐿.