<figure><img src="image_1.png" /></figure>

<figure><img src="image_2.png" /></figure>

Desmarchelier Paul¹², Giordano Valentina⁴, Beltukov Yaroslav³, Termendzidis Konstantinos², Tanguy Anne¹

¹LaMCoS, INSA Lyon, Villeurbanne Cedex, France. ²Department of Solid State Physics, Ioffe Physical Technical Institute, Saint Petersburg, Russian Federation. ³CETHIL, INSA Lyon, Villeurbanne, France. ⁴ILM, University Lyon, Villeurbanne, France.

# 1 Introduction and Configurations

Thermal management becomes more and more important in decreasing the sizes of devices and increasing their density. The evolution of fabrication allows to avoid hot spots using isolating and heat dissipative materials.

* Effect of crystalline inclusion rigidity and comparison with bulk amorphous silicon

* On propagation/diffusion of excitation

* Varying rigidity

* Variation of the 3 body interaction

* $\chi = \frac{E_{\text{Incl}}}{E_{\text{Mat}}} = 1$ or 4.6

<figure><img src="image_3.png" /><figcaption>Crystalline
nano-inclusion⁶</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Unit Cell⁵</figcaption></figure>

# 4 Propagation

<figure><img src="image_5.png" /><figcaption>Maximum value of $E_k$ along the propagation direction: nano-composite</figcaption></figure>

* Nano-composite

* Increasing acoustic mismatch intensifies attenuation

* Most important at low frequencies

* Local maximum of $E_k$ between inclusions

* Stronger attenuation in the matrix

* Ballistic at low frequencies/diffusive at higher frequencies

* Amorphous

* Stronger propagation at low frequencies

* Higher attenuation at higher frequencies

<figure><img src="image_6.png" /><figcaption>Bulk amorphous</figcaption></figure>

# 6 Consequences on the thermal conductivity

* For phonons below the Ioffe-Regel criterium the thermal conductivity is defined by

* $\kappa_{\text{prop}} = \int_{0}^{\omega_{\text{IR}}} C(\omega, T)\nu(\omega)l(\omega)g(\omega)d\omega$

* For phonons above the Ioffe-Regel we have

* $\kappa_{\text{diff}} = \int_{\omega_{\text{IR}}}^{\infty} D(\omega)C(\omega, T)g(\omega)d\omega$

* The higher the mismatch, the stronger the reduction of $\kappa$

* Inclusion effect mostly sensible on $\kappa_{\text{prop}}^7$

* Small effect on the conductivity at 300K

<figure><img src="image_7.png" /></figure>

<table><thead><tr><th>At 300K</th><th>χ = 1</th><th>χ = 4.6</th><th>Am.</th></tr></thead><tbody><tr><td>κ<sub>tot</sub> W/mK</td><td>1.43</td><td>1.23</td><td>1.48</td></tr></tbody></table>

# 2 Simulation Methods

* MD simulation

* SW⁴ interaction potential for Si

* LAMMPS³

* Wave-Packet excitation²

* Displacement imposed on central atoms between inclusions

* $u_i = \frac{A}{\omega}\sin(\omega(t - 3\tau))e^{-\frac{(t-3\tau)^2}{2\tau^2}}$

* Polarization

* Longitudinal

* Propagation analysis

* Random

* Diffusivity computation

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Evolution of a wave-packet with $\omega = 3.65$ THz and $\chi = 4.6$</figcaption></figure>

# 3 Kinetic Energy Distribution

* Kinetic energy maps as a function of time and position

* For frequencies between 0.5 and 11 THz

* lower limit due to resolution

* higher limit due to energy propagation limitation

* Propagation/Diffusion

* Pinning between inclusions

<figure><img src="image_10.png" /><figcaption>Time evolution of kinetic energy distribution for different frequencies</figcaption></figure>

# 5 Diffusivity

* Randomly polarized excitations allows the computation of the diffusivity¹

* From the MSD of the energy

$$\text{front } R^2(t) = \frac{1}{E_{\text{tot}}} \sum_{i=1}^{N} x_i^2 E_i(t)$$

* Diffusivity defined by the slope $R^2(t) = 2D(\omega)t$

* Small discrepancies between $\chi = 1,4.6$ or bulk amorphous

* Highly dependent on the glass structure

# 7 Conclusion

Through wave packets excitation simulations we have separately computed the propagative and diffusive parts of heat conduction. Though propagation is lowered by the rigid inclusions, the diffusion tends to be increased. This work can be continued with the study of different nano-composites.

# References

[1] Y. M. Beltukov, V. I. Kozub, and D. A. Parshin. "Ioffe-Regel criterion and diffusion of vibrations in random lattices". In: *Physical Review B - Condensed Matter and Materials Physics* 87.13 (2013). [5]

[2] Y. M. Beltukov, D. A. Parshin, V. Giordano, and A. Tanguy. "Propagative and diffusive regimes of acoustic damping in bulk amorphous material". In: *Physical Review E* 98.2 (2018). [6]

[3] S. Plimpton. "Fast Parallel Algorithms for Short-Range Molecular Dynamics". In: *Journal of Computational Physics* 117.6 (1995). [7]

[4] F. H. Stillinger and T. A. Weber. "Computer simulation of local order in condensed phases of silicon". In: *Physical Review B* 31.8 (1985).

A. Stukowski. "Visualization and analysis of atomistic simulation data with OVITO-the Open Visualization Tool". In: *Modelling and Simulation in Materials Science and Engineering* 18.1 (2010).

A. Tili, S. Pailhès, R. Debord, B. Ruta, S. Gravier, J.-J. Blandin, D. Blanchard, S. Gomès, A. Assy, A. Tanguy, and V. M. Giordano. "Thermal transport properties in amorphous / nanocrystalline metallic composites : A microscopic insight". In: *Acta Materialia* 136 (2017).

A. Tili, V. Giordano, Y. Beltukov, P. Desmarchelier, S. Merabia, and A. Tanguy. "Enhancement and anticipation of the Ioffe-Regel crossover in amorphous/nanocrystalline composites". In: Submitted (2019).