Point Defect Scattering of Phonons

Chapter 7 Solid Solutions between Lead Chalcogenides

7.4 Point Defect Scattering of Phonons

137 Table 7.3. Mobility for undoped p-type PbSe1-xSx compared with calculated mobility for p-type

PbSe, and PbSe1-xSx without alloy scattering contribution, assuming the same carrier density.

Sample n_H (10¹⁸ cm^-3) µ_H (cm²/Vs) µ_H PbSe µ_H PbSe_1-xS_x

p-50 2.4 487 784 520

p-60 0.8 593 694 467

p-80 1.2 780 739 606

In general, the value of alloy scattering potential U is of key importance to the magnitude of alloy scattering and the mobility reduction in solid solutions. Values of U, however, are reported only for a few systems220, 221, 224-227. These are shown in Table 7.4 along with some other physical property differences of the alloy components^{228, 229}. The typical magnitude of U is found between 0.6 to 2 eV, which is much smaller than a typical effective deformation potential coefficient Ξ (8 to 35 eV).

Table 7.4. A comparison of alloy scattering potential U in a few solid solutions together with the difference between two constituents in electron affinity ΔX, band gap ΔEg, molar mass ΔM, and

lattice parameter Δα.

Alloy system ΔX (eV)

ΔE_g

(eV) ΔM Δα (Å) U (eV) note

n-Al1-xGaxN 3.5 2.69 42.7 0.21 1.5 – 2.0 Δa compare c direction n-Al1-xGaxAs 0.43 1.72 42.7 0.01 1.1

n-Cd1-xZnxTe 0.8 0.88 88.2 0.37 0.8

n-InAs_1-xP_x 0.5 0.99 44 0.19 0.6

n-Si_1-xGe_x 0.05 0.46 44.5 0.23 0.6 - 1.0 n-PbSe_1-xTe_x 0.1 0.02 48.6 0.33 1.1

n-PbSe1-xSx 0.1 0.12 47.0 0.19 1.0

Similar as modeling the charge carrier mobility and thus conductivity, the problem of heat conduction is depicted using the concept of phonon scattering and relaxation time approximation.

The total thermal conductivity is usually calculated using the Debye-Callaway model^{230, 231}:

!L= k_B⁴

2"²!³vT³ !total

x⁴e^x (e^x!1)²dx

0

"/T

#

Equation 7.8

Here v is the averaged speed of sound, Θ the Debye temperature, x stands for ħω/kBT, and τtotal is the combined relaxation time of different scattering mechanisms. At high temperatures, the last term in the integrant can be approximated by x², as shown in Figure 7.18. In fact, as long as T is above Debye temperature, which for the majority of thermoelectric compounds is below or around 300 K, this approximation will lead to error that does not exceeding 8%.

Figure 7.18. The integrant in Debye model can be approximated at high temperature with x². So that Equation 7.8 at high temperatures becomes:

!_L= k_B⁴

2"²!³vT³ #_totalx²dx

0

"

!/T Equation 7.9

For bulk materials, there are three different scattering mechanisms that are usually considered: the Umklapp phonon scattering, the point defect scattering, and the boundary scattering.

Umklapp process refers to phonon-phonon collision where the final wave vector q is out of the Brillion zone boundary resulting in an energy loss and thermal resistance. The relaxation time under Umklapp scattering is formulated by Slack²³² as:

!_U^!1(")=!!²"²

Mv²"Texp(!"/ 3T)= k_B²!²

!Mv²"x²T³exp(!"/ 3T) Equation 7.10

0.0 0.2 0.4 0.6 0.8 1.0

0.90 0.92 0.94 0.96 0.98 1.00

y

x= Θ/T

y = x²e^x (e^x -1)²

139 ω is phonon frequency and γ is the Grüneisen parameter that characterizes the anharmonicity of lattice vibration. The other type of phonon-phonon interaction is the Normal process where the final wave vector q is in the same Brillion zone, Normal process is generally considered having no contribution to thermal resistance because there is no energy loss for the phonons. Some researchers, for example Zaitsev et al. argued²³³ that the Normal process has its contribution to thermal resistance by redistributing phonons so that more will participate the Umklapp scattering.

The treatment of Normal process, is essentially adding a phenomenological factor to the relaxation time of Umklapp scattering.

The point defect scattering of phonons has a Rayleigh scattering characteristic that scattering rate changes with λ^-­‐4, so the short wavelength phonons will get scattering most.

!D

!1(")= "#

4#v³"⁴= k_B⁴"#

4#!⁴v³x⁴T⁴ Equation 7.11

Γ is the point defect scattering parameter.

The boundary scattering is simply a function of the distance between the boundaries, usually grain size.

!B

!1=v L

Equation 7.12 Some researchers would also add the transmissivity²³⁴ of phonons into this equation:

!_B^!1=v L

4(1!t)

3t Equation 7.13

This version does not necessarily lead to more accurate estimate because the transmissivity t is not an easily observable and well-determined parameter, so essentially the second term would become a fitting pre-factor.

In nano-structured systems, researchers also tend to introduce other scattering mechanisms such as scattering from precipitates or dislocations around the precipitates. Theories on such mechanisms are rather complicated and the expressions of relaxation time include one or more parameters that can’t be well determined experimentally²³⁵. Also because the samples are always annealed for a long time and are single phase, these mechanisms are not considered.

Adding all three mechanisms and calculating κL using Equation 7.8 has been successfully adopted for studies of other material systems, but for Pb chalcogenide, the results even for pure binary compound at room temperature are found overestimated significantly. The difficulty of accurately calculating κL of binary compounds at any given temperature has made us use another treatment^235-

237 suggested by Callaway and Klemens for κL of solid solutions, which allows us to use experimentally determined κL for the binary compounds as an input. The assumption behind his derivation is, in addition to the high temperature approximation, that a combined scattering mechanism with Umklapp-type relaxation time for the unalloyed “pure” compound (this would allow the contribution from Normal process), and the only added scattering mechanism is the point defect scattering in the alloys or, solid solutions. Klemens’ formula can be written as:

!L,alloy

!L,pure

=arctan(u)

u Equation 7.14

where u is given by:

u²=!"D!

2!v² !_L,pure" Equation 7.15

For the calculation of scattering parameter Γ there has been some discrepancy. In the early works on the thermal conductivity affected by point defects, researchers primarily studied the influence from the isotopes and their expression of Γ only took into account the mass difference. Many researchers continued to use such expression because the impurity is very close in atomic size to the atom it replaced. Actually, Klemens’ full analysis from continuum theorem took into account influences from mass difference, binding force difference and strain field induced by a point defect. The full expression of Γ contains 3 terms:

!= !i= x_i{("M

M )²+2[("K

K )#2Q("R R)]²}

i

$

i

$

Equation 7.16

M, K and R are the atomic mass, bulk modulus and bonding length of the pure compound and ΔM and ΔK are the difference of each quantity between matrix and introduced compounds. ΔR is the difference of local bonding length caused by a point defect. Q accounts for the accumulative influences on surrounding bonds from the point defect for which Klemens²³⁵ used 4.2 for substitutional impurities and 3.2 for vacancies. Abeles¹⁹⁴ replaced the term ΔR/R with Δd/R where Δd is the difference of bonding length of matrix and alloying atom in their own lattices so that for binary (A1-xBx type) or pseudo-binary ((AB)1-x(AC)x type) systems:

141

!=x(1"x)[(#M

M )²+!(#a

a )²] Equation 7.17

ΔM and Δα are the difference in mass and lattice constants between two constituents, M and α are the molar mass and lattice constant of the alloy. The parameter ε is related to the Grüneisen parameter γ and elastic properties. Abeles wrote ε as¹⁹⁴ (for some reason Abeles used 6.4 instead of 8.4 as would expected from Klemens’ value for the factor Q):

! =2

9[(G+6.4")1+r

1!r]² Equation 7.18

G is a ratio between the contrast in bulk modulus (ΔK/K) and that in the bonding length (ΔR/R). The ratio G is relatively constant within similar materials systems. Scientists have been studying the relationship between bulk modulus and the volume of unit cell for a lot of compounds. An example can be seen from Anderson’s study^{238, 239} on lots of crystals of different types. From the slopes seen in his plot it is found that for covalent IV and III-V structures G = 4, for more ionic II-VI and I-VII structures G = 3, for complex oxide structures G = 9. So for Pb chalcogenides we suggest the use of G =3. r is the Poisson ratio, which can be calculated from the longitudinal and transverse speed of sound via:

v_l

v_t = 2(1!r) 1!2r

Equation 7.19 the Poisson ratio r for most semiconducting compounds is found²⁴⁰ between 0.15 and 0.3 (for most solids it is between 0 to 0.5). With all the parameters in Equation 7.18 determined, the value of ε could be calculated for any binary compounds, given adequate information. ε calculated for PbTe, PbSe and PbS are 100, 110 and 150, respectively. ε for PbTe based systems was previously suggested to be 65 by Alekseeva²⁴¹ by fitting experimental results, this value has been used in lots of studies on relevant materials including ours. We now believe the use of 100 instead of 65 is of more physical origin and it does not lead to significant inconsistency between model and experiments. In studies of other compounds in general, ε is mostly taken as an empirically adjustable fitting parameter, which is understandable since the elastic parameters needed to calculate ε for a compound are not all necessarily studied.

When there is more than one type of substitution, how to determine the scattering parameter Γ in the point defect model is of debate. Abeles has suggested to generalize the expression in Equation 7.17 into:

!= !i i

"

^{= xi $}_%^&#M_M^i'₍⁾

2

+! ^#"_i

"

$

%& ' ()

* 2

+ ,,

- . //

i

"

Equation 7.20

Yang et al.’s treatment²⁴² is another widely adopted method, which considered different substitutions on each different site all together. Detail about such treatment was described in Appl Phys Letter 85, 1140 2004. Expressions from these two treatments are not consistent and the difference (which is not simply proportional) diverges as the number of atoms in a unit cell increases. It is worth noticing that Yang’s derivation was based on Slack’s expression²⁴³ of Γ but arguably generalized it to cases where both mass and size differences are considered, while Slack’s work only considered the mass difference. So without careful derivation, replacing the equation (7) in Yang’s paper with

!_s=

c_i ri

r

"

#$ %

&

'

i=1 n

(

2

f_i¹f_i²!_i ^ri

1)ri 2

r_i

"

#$ %

&

'

2

c_i

i)1 n

"

(

#$ %

&

'

Equation 7.21

seems to be more justifiable.

No matter which treatment is used, there hasn’t been a systematic study on the thermal conductivity of a multi-substituted solid solution to compare the theory with. The Gurieva et al. suggested²⁴⁴ that the effect from multiple substitutional atoms are not additive, which is reasonable considering they both primarily affect phonons with the same wavelength range, and Matthiessen’s rule works best only when different mechanisms acts on different parts of the phonon spectrum.

Si_1-xGe_x or Mg₂Si_1-xSn_x are examples where the lattice thermal conductivity in solid solutions is greatly reduced (up to 94% reduction in Si1-xGex245 and 75% in Mg2Si1-xSnx233) and hence a significant improvement in zT has resulted. The reduction in (PbTe)_1-x(PbSe)_x was found only about 45%, which is considerably less than such examples because the mean free path is already small due to efficient Umklapp scattering in lead chalcogenides. In Table 7.5 the κL reduction is compared among different material systems with the same degree of alloying (30%), together with the magnitude of the contribution from mass (ΔM/M) and strain field (Δα/α and ε) contrast. It is seen that the strain field contrast does not change much for different material systems and none of them is large due to the required similarity in lattice constant for the solid solution to form. The PbTe based system has one of the largest ε, which actually leads to the largest total strain contribution among these systems. The mass contrast in PbTe0.7Se0.3 on the other hand is small compared to the

143 other systems. The factor with the highest correlation with the thermal conductivity reduction (Δκ_L/κ_L,pure) is the magnitude of κ_L,pure.

Table 7.5. Relative lattice thermal conductivity reduction in alloy systems with the same degree of alloying (30%) together with the contribution from mass (ΔM/M) and strain field (Δα/α and ε)

contrast. ε stands for values been used as fitting parameter, and εcalc are calculated values. κ_L,pure for the major constituent compound.

ΔκL/ κL,pure (300 K) ΔM/M Δα/α ε εcalc κL,pure (300 K)

PbTe_0.7Se_0.3 45% 0.15 0.05 65 100 1.9

PbSe0.7S0.3 30% 0.17 0.03 110 1.7

Si_0.7Ge_0.3 94% 1.1 0.04 39 47 150

Mg₂Si_0.7Sn_0.3 75% 0.87 0.06 23 67 7.9

Ga_0.7In_0.3As 86% 0.29 0.07 45 51 45