Prior empirical models

Chapter V: Origin of acoustic excitations in amorphous silicon

5.3 Prior empirical models

71

Figure 5.4: The internal friction versus the temperature for several glassy materials in Ref. [184]. The internal friction (𝑄⁻¹) is defined by𝑣 𝛼/𝜔, where𝑣is the sound velocity,𝜔is the acoustic frequency, and the𝛼is the attenuation coefficient [178]. At a high temperature up to 100 - 300 K, the internal friction exhibits a clear temperature dependence, which varies depending on the chemical composition. Below 100 - 300 K, as the temperature decreases, hence the internal friction decreases with weaker temperature dependence. The values of the internal friction in this temperature range for amorphous materials are reported to fall into similar values, separated within a factor of 20 as indicated by the glassy range (dashed line) [178] . For temperatures less than 1 K, the temperature dependence of the internal friction again shows marked temperature dependence.

Several empirical models have been proposed to explain the anomalies mentioned above. The minimum thermal conductivity model was employed to explain the thermal transport amorphous materials [185,186]. The two level system (TLS) was developed to explain both the anomalies of the excess heat capacity and the damping mechanism in amorphous materials [146,149,183].

previous descriptions.

Minimum thermal conductivity

The minimum thermal conductivity is one of the empirical approaches to explain the thermal transport in glassy substances. The proof of concept was proposed by Einstein [187], and the theory was applied to glasses by Cahill [185, 186]. The minimum thermal conductivity assumes the heat transfer process as a random walk of the heat energy. The minimum thermal conductivity assumes that the atomic vibrations oscillate near the equilibrium point and that they have an incoherent phase. Corresponding distance resulting from the random walk isΛ = 𝜋/𝜔which gives the expression for the lower limit to the thermal conductivity

𝜅_𝑚𝑖𝑛 = 𝜋

6 1/3

𝑘_𝐵𝑛^2/3 Õ

𝑖

𝑣_𝑖 𝑇

𝜃_𝑖

2∫ 𝜃𝑖/𝑇

0

𝑥³𝑒^𝑥

(𝑒^𝑥−1)² (5.1) where 𝜃_𝑖 is the Debye temperature, and 𝑛is the number density. It was discussed that this theory is not applicable to the crystals, since periodic arrangements of the atoms are known to produce a coherence between the vibrations [186]. On the other hand, Ref. [186] demonstrated that the prediction from the minimum thermal conductivity agrees with experimental observations in amorphous materials.

However, the applicability of this theory should be re-examined. In fact, the min- imum thermal conductivity is not favorable for explaining the transport of the low energy excitations. First, analogous to acoustic phonons in crystals, the wave- length of the low energy excitations is relatively long, indicating that the excitations propagate collectively rather than randomly diffuse within disordered length scales.

Second, the frequency dependent transport properties are missing in the minimum thermal conductivity theory. Vibrations in solids are dispersive and correspond- ing transport properties are different depending on the frequency. Therefore, if the mechanism of the damping of the excitations would be different at higher fre- quencies, the corresponding frequency dependence of the MFP should be different, which the minimum thermal conductivity model cannot precisely capture.

Two-level system (TLS)

Earlier works have described the linear temperature dependence of the heat capac- ity by the two-level system (TLS) damping of the low energy excitations, initially proposed by Anderson, Halperin, and Varma [149]. According to this model, the damping of the acoustic excitation is due to resonant absorption at very low temper- atures (typically less than 5 K), above which is due to thermally activated relaxation

73 [146, 183]. In the following, we briefly describe the frequency dependence of the MFP induced by resonant absorption.

Figure 5.5: Assymetric potential well versus the position proposed by Anderson, Halperin, and Varma [149]. The figure was taken from Ref. [149], and modified.

Briefly, this theory assumes that the impurities induce an asymmetric double-well potential system with a constant density as follows

𝐶_{𝑇 𝐿 𝑆} ∝

∫

(ℏ𝜔)𝑛(𝜔)𝑑 𝜔 ∝ 𝑘_𝐵𝑇

ℏ (5.2)

The theory of the TLS assumes that the elastic waves near TLS deform the double- well potential, which causes the tunneling of acoustic excitation through the poten- tials. Here, we describe an earlier mathematical framework proposed by Anderson, Halperin, and Varma [149]. The interaction between local TLS and the phonon can be described by the first-order perturbation theory where the Hamiltonian can be described by

𝐻 =𝐻₀+𝐻₁= 1 2

𝜖₁ Δ Δ 𝜖₂

!

+ 𝛿𝜖 0 0 𝛿𝜖

!

(5.3) where the𝜖₁(𝜖₂) is the local minima of the asymmetric potential,𝛿𝜖is the deforma- tion potential, andΔis the zero-point energy due to the overlap of the wavefunction at the local minima. The mean free path of the acoustic excitation forℏ𝜔 > 𝑘_𝐵𝑇

Λ(𝜔) = (𝜎h𝜏⁻¹i)⁻¹ (5.4)

where𝜎 =4𝜋𝑣²𝜔²is the scattering cross section, andh𝜏⁻¹iis the ensemble averaged transition rate between the eigenstates of the matrix above. Applying a Fermi-golden rule under an approximation ofℏ𝜔 = 𝜖²

1 +𝜖²

2 (resonant absorption), the transition rate is given by

𝜏⁻¹ = 3𝜋 𝐵² 𝑚 𝑣²

𝜔 𝜔³

𝐷

! Δ²

ℏ (5.5)

h𝜏⁻¹i =

∫

𝑝(𝜖₁−𝜖₂;Δ;𝐵)𝜏⁻¹

𝑑𝐵 𝑑Δ𝑑(𝜖₁−𝜖₂) (5.6) where 𝑝 is the probability distribution over ensemble of levels, and𝐵is the differ- ences in the constants for the deformation potential. Corresponding MFP can be simplified to

Λ∝ 1 𝑣²

𝜔⁻¹ (5.7)

Finally, considering the temperature dependence, an empirical factor was added such that

Λ𝑟 𝑒 𝑠 ∝ 1 𝑣²

𝜔⁻¹coth ℏ𝜔

2𝑘_𝐵𝑇

(5.8) whereΛ𝑟 𝑒 𝑠is the MFP due to resonant absorption. For𝜔 𝑘_𝐵𝑇/ℏ, Eq.5.8can be simplified to𝑙 ≈ 𝜔²/𝑇 for𝜔 𝑘_𝐵𝑇/ℏ. The theory of the TLS has shown excellent agreement with the low-temperature thermal conductivity as well as the attenuation of the low frequency excitations below a few K [148].

Density fluctuation

The density fluctuation is based on the structural inhomogeniety as a primary mech- anism for the scattering of the acoustic excitations [182]. The model considers free excess sub-volumes in an amorphous structure [188]. The re-distribution of the sub-volumes would occur, resulting in the local density deviating from the average density. The local density deviation becomes a scattering cross-section for the ex- citations, similar to point defect phonon scattering. Here, we briefly describe the frequency dependence shown in Ref. [182].

We consider a sub-volume𝑣with a different density (𝜌) in an amorphous structure.

The corresponding scattering cross-section for an acoustic excitation is given by 𝜎₁≈ 1

12𝜋

𝑞⁴𝑣²(Δ𝜌 𝜌

)² (𝑞 𝑣^1/3 < 1) (5.9)

𝜎₂≈ 𝑣^−1/3(Δ𝜌 𝜌

)² (𝑞 𝑣^1/3 >1) (5.10)

75 where𝑞is the wave vector of the acoustic excitation, and 𝜌is the density deviation.

The above expression is analogous to the scattering cross-section for Rayleigh scattering and geometric scattering, respectively.

First, considering a short wave vector excitations, the corresponding mean free path of the excitation from Eq.5.9is

Λ⁻¹₁ =Õ

Δ𝜌

𝑃(Δ𝜌) 1 12𝜋

𝑞⁴𝑣²(Δ𝜌 𝜌

)² = 1 12𝜋

𝑞⁴𝑣² Δ𝜌²

𝜌²

(5.11)

Δ𝜌²

𝜌²

= 𝑉 𝑝

𝑣 𝑁(1−𝑝) (5.12)

where𝑃is the probability of the given density fluctuation,Δ𝜌²is the mean square deviation of the density, 𝜌² is the average density, N is the number of the sub- volumes, and (𝑝) is the probability for density deviation. For the volumes from𝑣 to𝑣+𝑑 𝑣, using Eq.5.12and Eq. 5.11, the corresponding MFP of the excitation is given by

Λ⁻¹₁ = 𝑞⁴ 12𝜋

𝑝 1− 𝑝

𝑑 𝑣 (5.13)

which yieldsΛ⁻¹ ∝𝑞⁴for short wave vector limit.

Similarly, for𝑞 𝑎 >1,

Λ⁻¹₂ =Õ

Δ𝜌

𝑃(Δ𝜌)𝜎₂ ≈𝑣^−4/3 𝑝 1− 𝑝

𝑑 𝑣 (5.14)

The integration over𝑑 𝑣 gives an expression for the MFP

Λ⁻¹ =

∫

𝑣

Λ⁻¹₁ (𝑣 , 𝑞) +Λ⁻¹₂ (𝑣 , 𝑞)

𝑑 𝑣∝ 𝑞 (5.15)

The above expression for the MFP can be re-written as the MFP versus the frequency at different frequency range

Λ∝ 𝜔⁻⁴ (5.16)

for high frequency, whereas for low frequency,

Λ∝ 𝜔⁻¹ (5.17)

The density fluctuation theory has shown a good agreement with the thermal con- ductivity data [18, 182], provided that the information of the density deviation is well-defined. However, the theory may not be applicable to the damping of the excitation below 1 K, since the extremely large length scale of the fluctuation is required for explaining the damping at very low temperatures.