Point defect scattering

Role of filler in thermal transport

4.2 Complex phonon modes

4.2.2 Point defect scattering

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 𝑉𝑉_𝑔𝑔²(𝑤𝑤) in the vicinity of 𝑤𝑤₀. d) 𝜅𝜅_𝑠𝑠(𝑤𝑤)for an empty BvK framework, using Umklapp and boundary scattering terms (curve A). Including resonant scattering reduces 𝜅𝜅_𝑠𝑠(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 𝑉𝑉_𝑔𝑔²(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 𝑉𝑉_𝑔𝑔²(𝑤𝑤) and 𝜏𝜏(𝑤𝑤) are required to unravel the intertwined effects on 𝜅𝜅𝐿𝐿.

lattice thermal conductivity values (as shown below from a Klemens model): the larger the scattering parameter, the lower the lattice thermal conductivity.

According to Callaway and Klemens^94-96, if we consider a combined scattering mechanism including Umklapp scattering for the pure crystal without disorder, and extra point defect scattering process for the crystal with disorder, then the ratio of 𝜅𝜅_𝐿𝐿 of the crystal with disorder to that without disorder, 𝜅𝜅_𝐿𝐿^𝑃𝑃 is:

𝜅𝜅_𝐿𝐿

𝜅𝜅_𝐿𝐿^𝑃𝑃=arctan (𝑢𝑢)

𝑢𝑢 (Eq. 4.4) 𝑢𝑢²= ^𝜋𝜋_ℎ𝜈𝜈^{2𝜃𝜃𝐷𝐷𝛺𝛺}

𝑠𝑠2 𝜅𝜅_𝐿𝐿^𝑃𝑃𝛤𝛤𝑒𝑒𝑚𝑚𝑒𝑒𝑡𝑡 (Eq. 4.5) where 𝑢𝑢, ℎ, and 𝛤𝛤_{𝑒𝑒𝑚𝑚𝑒𝑒𝑡𝑡} are the disorder scaling parameter, the Planck constant, and the experimental disorder scattering parameter, respectively. Thus, 𝛤𝛤_{𝑒𝑒𝑚𝑚𝑒𝑒𝑡𝑡} can be derived from 𝜅𝜅_𝐿𝐿 measurements using Eq. 4.4 and 4.5. It can then be compared to calculated values, as detailed in the following.

Γ can be separated into two components:

𝛤𝛤=𝛤𝛤_𝑀𝑀+ 𝛤𝛤_𝑆𝑆 (Eq. 4.6) where the scattering parameters 𝛤𝛤_𝑀𝑀 and 𝛤𝛤_𝑆𝑆 represent mass and strain field fluctuations due to the introduction of point defects (e.g., R into the void), respectively.

(Eq. 4.7) where 𝜀𝜀 is an empirical fitting parameter related to Gruneisen parameter and elastic properties. 𝑎𝑎 is the lattice constant of pure alloy whereas Δ𝑎𝑎 is the difference in lattice constant between the alloy with point defects and the pure alloy.

When there is more than one constituent element in the compound (as in the case of skutterudites RxCo4Sb12), there is some discrepancy regarding the definition of 𝛤𝛤𝑀𝑀.

According to Klemens:

(Eq. 4.8) where ∆Mis the mass difference between two species on the same site and Mis the molar mass of the compound.

In the case of skutterudite compound, with Yb filling, we are comparing YbxCo4Sb12 to unfilled Co4Sb12, ∆ =M M_Yb=173.04 and

4 12 (1 ) 4 12 173.04* 4*58.93 12*121.76

YbCo Sb Co Sb

M=xM + −x M = x+ +

So

2 2

173.04

(1 ) (1 )

173.04 * 4 *58.93 12 *121.76

M

x x M x x

M x

∆   

Γ = −   = −  + +  (Eq. 4.9)

Later, Yang⁹⁷ proposed a different way of calculating 𝛤𝛤_𝑀𝑀which weighs the influence of mass fluctuation on each site by its degeneracy. To be more specific, it takes into account not only the number of atoms on a specific site but also the mass contrast between the specific site and the rest of the sites. For example, the chemical composition of a material can be expressed as A1c1A2c2A3c3A4c4…Ancn, where the Ai are crystallographic sublattices in the structure and the ci are the relative degeneracies of the respective sites. In this context, the skutterudite compound YbxCo4Sb12 has n=3, A1=Yb, A2=Co, A3=Sb, and c1=1, c2=4, c3=12. In general there will be several different types of atoms that occupy each sublattice, and the k^th atom of the i^th sublattice has mass 𝑀𝑀_𝑡𝑡^𝑘𝑘, radius 𝑟𝑟_𝑡𝑡^𝑘𝑘, and fractional occupation 𝑓𝑓_𝑡𝑡^𝑘𝑘. The average mass and radius of atoms on the i^th sublattice are:

k k

i i i

k

M =

∑

f M (Eq. 4.10)

k k

i i i

k

r =

∑

f r (Eq. 4.11) The mass fluctuation scattering parameter is then given by:

2

, 1

1 n

i

i M i

i

M n

i i

c M M

c

=

=

 

  Γ

 

Γ =  

 

 

∑

∑

(Eq. 4.12)

where the mass fluctuation scattering parameter for the i^th sublattice is:

2

, 1

k

k i

M i i

k i

f M M

 

Γ =  − 

 

∑

(Eq. 4.13) and the average atomic mass of the compound is:

1

1 n

i i

i n

i i

c M M

c

=

=

=  

 

 

∑

∑

(Eq. 4.14)

In the case of skutterudite compound YbxCo4Sb12, there is no mass fluctuation on sites A2 (Co) and A3 (Sb).

2 ( ) 58.93

M =M Co = ,

3 ( ) 121.76 M =M Sb = .

Consequently Γ_M,2 = Γ_M,3=

0

For site A1 (Yb/void), M₁= f M₁^Yb ₁^Yb+ f₁^voidM₁^void =173.04*x

2 2

1 1

,1 1 1

1 1

2 2

2

1 1

173.04 0

1 (1 ) 1

173.04 * 173.04 *

( 1) 1 1

(1 ) (1 )(1 )

Yb void

Yb void

M

M M

f f

M M

x x

x x

x x x

x x

x x x

   

Γ =  −  +  − 

   

   

=  −  + −  − 

− − −

= + − = − + =

1 1 2 3

1

4 12 173.04 * 4 *58.93 12 *121.76

1 4 12 17

n

i i

i n

i i

c M M M M x

M

c

=

=

+ + + +

= = =

  + +

 

 

∑

∑

2 2

1

, ,1

1

1

2

2

17

173.04 *

173.04 * 4 *58.93 12 *121.76 17 1

17 173.04

17 (1 )

173.04 * 4 *58.93 12 *121.76

n i

i M i M

i

M n

i i

M M

c

M M

c

x x

x x x x x

=

=

   

Γ Γ

   

   

Γ = =

 

 

 

 

 

 + + 

 

  −

=

 

=  + +  −

∑

∑

(Eq. 4.15)

Now if we compare Eq. 4.15 to Klemens’s result in Eq. 4.9, we can see that there is clearly a difference of 17 times which equals to the total number of atoms per compound formula.

If we modify Yang’ formula such that the degeneracy of each site is still taken into account but with a less complicated form, we get

𝛤𝛤_𝑀𝑀 =𝑥𝑥(1− 𝑥𝑥)𝑓𝑓 �^{𝛥𝛥𝑀𝑀}_𝑀𝑀��² (Eq. 4.16) This formula is quite similar to Klemens’s formula (Eq. 4.8) except with an extra term f and a different definition for 𝑀𝑀�, which stand for the fraction of substituted lattice sites and the average atomic mass per number of atoms in the compound (rather than the previous molar mass for the compound) respectively. ΔM still holds for the mass difference between the host and impurity atom.

In the case of skutterudite compound, ∆ =M M_Yb=173.04 and

[

_{YbCo Sb}4 12 (1 ) _{Co Sb}4 12 173.04* 4*58.93 12*121.76 /17

]

M= xM + −x M = x+ +

So

2 2

2

1 173.04

(1 ) (1 )

173.04 * 4 *58.93 12 *121.76 17

17 173.04

17 (1 )

173.04 * 4 *58.93 12 *121.76

M

x x f M x x

M x

x x

x

 

 

∆ 

Γ = −   = −  + + 

 

 

= −  + + 

(Eq. 4.17)

Compare Eq. 4.9, Eq. 4.15 and Eq.17, we get

(Eq. 4.18) Let us take Cu2ZnGeSexS4-x for another example. Using Klemens model, we have:

A = Cu2ZnGeSe4

B = Cu2ZnGeS4

Cu2ZnGeSexS4-x = (x/4)*Cu2ZnGeSe4 + (1-x/4)*Cu2ZnGeS4

( )

( )

2 2

,

2

(1 ) (1 ) 4

4 4 4 4 * (4 ) * 2 *

(1 ) 1

4 4 4

Se S

M Klemens

Se S Zn Ge Cu

Se S

M M

x x M x x

M x M x M M M M

M M

x x

M

−

 

∆ 

Γ = −   = −  + − + + + 

−

 

= −  

 

(Eq. 4.19)

(Eq. 4.20)

(Eq. 4.21)

Compare Eq. 4.19, Eq. 4.20, and Eq. 4.21, and we get

(Eq. 4.22)

Thus the modified equation for calculating gives the same result as Yang’s formula, yet it is easier to implement. From now on, we will only compare Yang’s formula and Klemens’s formula.

In a more general case, for compound A1c1A2c2A3c3A4c4…Ancn, the relationship between the calculated scattering parameters from Klemens’s formula (Eq. 4.8) and Yang’s formula (Eq. 4.12) will be:

(Eq. 4.23)

with the i^th element being substituted and Ai position having a Ci degeneracy.

The difference between Yang’s formula and Klemens’ formula is that Yang’s formula takes the degeneracy of each site as the weighing factor in calculating scattering parameter, which makes more sense because doping on the Co site (CoxNi1-xSb3) should result in a smaller scattering parameter from doping on the Sb site (Co(SbxTe1-x)3) even when the doping content x is the same.

Since is always larger than

,

the scattering parameter of Yang’s formula is always larger than Klemens’, which leads to a smaller lattice thermal conductivity.