Thermal transport calculation .1 Minimum thermal conductivity .1 Minimum thermal conductivity

Role of filler in thermal transport

4.3 Thermal transport calculation .1 Minimum thermal conductivity .1 Minimum thermal conductivity

It is important for us to know the minimum thermal conductivity of skutterudites when trying to decrease the lattice thermal conductivity. There are two frequently used formulas for minimum thermal conductivity calculation. The one proposed by Cahill¹⁰⁵ was initially for amorphous materials and based on a physical picture of a modified Einstein model, which consists of a random walk of energy between uncorrelated localized oscillators of varying sizes and frequencies. The dominant energy transport is between nearest neighbors. The minimum phonon mean free path is one half of its wavelength (relaxation time equals to one half of one period of vibration). The Cahill minimum thermal conductivity can be expressed as:

(Eq. 4.27)

where the summation is over the three sound modes (one longitudinal and two transverse modes), and Ω represents the average volume per atom,𝑣𝑣_𝑡𝑡 is the sound velocity for the longitudinal and transverse modes, and Θ_𝑡𝑡 =𝑣𝑣_𝑡𝑡(ℏ 𝑘𝑘⁄ _𝐵𝐵)(6𝜋𝜋²⁄Ω)^1/3 is the Debye temperature for each polarization in degrees K. Equation 4.27 contains no free parameters, since both Ω and 𝑣𝑣_𝑡𝑡 are known.

In the high temperature limit,

,

so the equation becomes:

(Eq. 4.28) Slack⁸⁹ proposed a different formula to calculate minimum thermal conductivity based on the notion that heat is carried by waves, and for this they must live longer than one period of vibration. The minimum mean free path is thus one wavelength instead of one half according to Cahill.

According to Slack, the minimum thermal conductivity at high temperatures can also be estimated using a Debye approximation as:

(Eq. 4.29)

(Eq. 4.30)

(Eq. 4.31) where 𝑁𝑁 is the number of atoms per primitive cell. The optic contribution is calculated from a sum of 3(𝑁𝑁 −1) equally spaced optic modes.

For CoSb3, 𝑣𝑣_𝑙𝑙 = 4743𝑚𝑚/𝑠𝑠, 𝑣𝑣_𝑡𝑡 = 2830𝑚𝑚/𝑠𝑠 according to speed of sound measurement, see Section 4.4.2.

The average velocity can be calculated from:

(Eq. 4.32)

we get

3132m/s.

The Debye temperature

Θ

can be calculated from:

Θ=𝑣𝑣(ℏ 𝑘𝑘⁄ _𝐵𝐵)(6𝜋𝜋²⁄Ω)^1/3 (Eq. 4.33)

With Ω=_2𝑁𝑁^𝑚𝑚3 =^9.0348_2∗16³∗10⁻³⁰𝑚𝑚⁻³= 23.05∗10⁻³⁰𝑚𝑚⁻³, Θ= 327.8K. Both the speed of sound measurement data and the Debye temperature match well with literature data (𝑣𝑣_𝑙𝑙 = 4590𝑚𝑚/𝑠𝑠, 𝑣𝑣𝑡𝑡 = 2643𝑚𝑚/𝑠𝑠,Θ= 307K)⁶.

With the values obtained we get the minimum thermal conductivity for CoSb3 skutterudites:

=0.204+0.232 (W/m/K)=0.436 W/m/K (Eq. 4.34)

W/m/K (Eq. 4.35)

These values (Eq. 4.34 and Eq. 4.35) should serve as a guide in search of lower lattice thermal conductivity. Notice there is some discrepancy between the values obtained from both methods.

The Cahill method surprisingly gives a higher thermal conductivity with only half of the wavelength used in the Slack model.

4.3.2 Electronic contribution to thermal conductivity

The data from thermal diffusivity measurements 𝐷𝐷_𝑇𝑇gives us experimental estimations of thermal conductivity 𝜅𝜅 (𝜅𝜅=𝑑𝑑𝐷𝐷_𝑇𝑇𝐶𝐶_𝑃𝑃,𝐶𝐶_𝑃𝑃= 3𝑘𝑘_𝐵𝐵,𝑑𝑑 𝑖𝑖𝑠𝑠 𝑑𝑑𝑒𝑒𝑛𝑛𝑠𝑠𝑖𝑖𝑡𝑡𝑑𝑑). 𝜅𝜅 is the sum of electronic contribution to the thermal conductivity 𝜅𝜅_𝑒𝑒 and lattice thermal conductivity 𝜅𝜅_𝑙𝑙. In order to probe the role of fillers in reducing lattice thermal conductivity, we have to separate 𝜅𝜅_𝑒𝑒 from 𝜅𝜅 first. 𝜅𝜅_𝑒𝑒 can be expressed as in Eq. 4.36:

𝜅𝜅_𝑒𝑒=𝜅𝜅_{𝑒𝑒,𝑒𝑒}+𝜅𝜅_𝑒𝑒,ℎ+𝜅𝜅_𝑏𝑏 (Eq. 4.36)

where 𝜅𝜅_{𝑒𝑒,𝑡𝑡} denotes the electronic thermal conductivity from the carrier type i. 𝜅𝜅_𝑏𝑏 denotes the electronic thermal conductivity from bipolar conduction, which usually occurs when the band gap 𝐸𝐸_𝑔𝑔 is small and temperature 𝑇𝑇 is high enough that intrinsic conduction can happen (𝑘𝑘_𝐵𝐵𝑇𝑇>𝐸𝐸_𝑔𝑔) with both electrons and holes excited in pairs.

The values of 𝜅𝜅_{𝑒𝑒,𝑡𝑡} is large in heavily doped semiconductor and can be easily calculated using the Wiedemann-Franz law. The Wiedemann-Franz relationship states that the electronic contribution from species i is related to the electrical conductivity of species i (𝜎𝜎𝑡𝑡), temperature T, and the corresponding Lorenz number 𝐿𝐿𝑡𝑡 via:

(Eq. 4.37) In a single parabolic band model with carrier scattering time of the form :

(Eq. 4.38) For acoustic phonon scattering, 𝑟𝑟=−1/2, so the Lorenz number can be expressed as:

(Eq. 4.39) For electronic thermal conductivity due to bipolar conduction,

^(Eq.

4.40)

For more details, please refer to Appendix A-D.

4.3.3 Callaway model

Once the electronic thermal conductivity is exacted from the total thermal conductivity, the lattice contribution can be estimated experimentally. A Callaway model can be used to calculate theoretical lattice thermal conductivity, taking into account of various phonon scattering mechanisms. By comparing results from Callaway model to experimental values, underlying phonon scattering mechanisms can be revealed. This is quite useful in accessing the role of fillers in reducing the lattice thermal conductivity.

The Callaway model is based on a Debye approximation, with which the lattice thermal conductivity can be calculated as:

(Eq. 4.41)

(Eq. 4.42)

is the boundary scattering relaxation time, which is determined by the speed of sound in the material 𝑣𝑣_𝑠𝑠 and average grain size 𝐿𝐿.

^{(Eq. 4.43)} is the Umklapp scattering relaxation time. Empirically it can be expressed as:

(Eq. 4.44) where

,

ω is phonon frequency, 𝑀𝑀 is the average mass of a single atom, and γ is the Grüneisen parameter that characterizes the anharmonicity of lattice vibration.

There are two types of phonon-phonon scattering mechanisms: Umklapp scattering and Normal scattering. Both processes involve three phonons. The difference is that in Normal scattering process, the total phonon momentum is conserved and is generally considered as having no contribution to the thermal resistance, whereas in Umklapp scattering process, with increasing phonon momentum of the incoming wave vectors (

),

the outgoing wave vector can point out of the Brillouin zone, which can be equivalently transferred into a phonon wave vector inside the Brillouin zone but with a smaller magnitude (

,

is a reciprocal lattice vector).

This leads to a change in the phonon momentum, and results in energy loss and adds thermal resistivity. Umklapp scattering is the main mechanism for thermal resistance at high temperatures for materials with low defect concentration.

As discussed previously, denotes empirically the resonant scattering time: 𝜏𝜏_𝑟𝑟⁻¹=𝐶𝐶_0(𝑤𝑤2^𝑤𝑤_−𝑤𝑤²

02)²

(section 4.2.1); denotes the point defect scattering time 𝜏𝜏𝑃𝑃𝐷𝐷−1=_{4𝜋𝜋𝜈𝜈}^Ω₃𝜔𝜔⁴Γ (section 4.2.2);

denotes the electron phonon scattering time 𝜏𝜏𝑒𝑒𝑒𝑒−1=𝐶𝐶𝑒𝑒𝑒𝑒𝜔𝜔²(section 4.2.3). Note that the nature of rattling remains unclear, given that both the relaxation time reduction (due to resonant scattering) and the group velocity reduction (because of the avoided crossing in the phonon spectra) can lead to similar reduction in lattice thermal conductivity (Figure 4.1). However here we are using the resonant scattering for simplicity.