Skip to main content
SpringerLink
Log in

Study of phonon transport across Si/Ge interfaces using Full-Band phonon Monte Carlo simulation

  • Published:
Journal of Computational Electronics Aims and scope Submit manuscript

Abstract

A Full Band Monte Carlo simulator has been developed to consider phonon transmission across interfaces disposed perpendicularly to the heat flux. This solver of the Boltzmann transport equation does not require any assumption on the shape the phonon distribution and can naturally consider all phonon transport regimes from the diffusive to the fully ballistic regime. This simulator is used to study single and double Si/Ge heterostructures from the micrometer scale down to the nanometer scale, i.e. in all phonon transport regime from fully diffusive to ballistic. A methodology to determine the thermal conductivity at thermal interfaces is presented. It is also shown that the different transport regimes are correlated to different spectral contributions of the phonon modes to the heat flux along the devices. This local indicator of the transport regime gives new insights into the out-of-equilibrium phonon transport near the interfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Liu, W., Jie, Q., Kim, H.S., Ren, Z.: Current progress and future challenges in thermoelectric power generation: from materials to devices. Acta Mater. 87, 357–376 (2015). https://doi.org/10.1016/j.actamat.2014.12.042

    Article  Google Scholar 

  2. The international roadmap for devices and systems: more Moore report 2017. IRDS (2017) [Online]. Available: https://irds.ieee.org/images/files/pdf/2017/2017IRDS_MM.pdf

  3. Fourier, J. B. J., & Darboux, G.: Théorie analytique de la chaleur, Vol. 504, Didot, Paris (1822)

  4. Cahill, D.G., et al.: Nanoscale thermal transport. II. 2003–2012. Appl. Phys. Rev. 1(1), 011305 (2014). https://doi.org/10.1063/1.4832615

    Article  MathSciNet  Google Scholar 

  5. Kaiser, J., Feng, T., Maassen, J., Wang, X., Ruan, X., Lundstrom, M.: Thermal transport at the nanoscale: a Fourier’s law vs. phonon Boltzmann equation study. J. Appl. Phys. 121(4), 044302 (2017). https://doi.org/10.1063/1.4974872

    Article  Google Scholar 

  6. Péraud, J.-P.M., Hadjiconstantinou, N.G.: Extending the range of validity of Fourier’s law into the kinetic transport regime via asymptotic solution of the phonon Boltzmann transport equation. Phys. Rev. B 93(4), 045424 (2016). https://doi.org/10.1103/PhysRevB.93.045424

    Article  Google Scholar 

  7. Ramu, A.T., Bowers, J.E.: A generalized enhanced Fourier law. J. Heat Transf. 139(3), 034501 (2017). https://doi.org/10.1115/1.4034796

    Article  Google Scholar 

  8. Ordonez-Miranda, J., Yang, R., Alvarado-Gil, J.J.: A constitutive equation for nano-to-macro-scale heat conduction based on the Boltzmann transport equation. J. Appl. Phys. 109(8), 084319 (2011). https://doi.org/10.1063/1.3573512

    Article  Google Scholar 

  9. Kapitza, P.: The study of heat transfer in helium II. J. Phys. (USSR) 4(1–6), 181–210 (1941)

    Google Scholar 

  10. Little, W.A.: The transport of heat between dissimilar solids at low temperatures. Can. J. Phys. 37(3), 334–349 (1959). https://doi.org/10.1139/p59-037

    Article  Google Scholar 

  11. Swartz, E.T., Pohl, R.O.: Thermal boundary resistance. Rev. Mod. Phys. 61(3), 605 (1989). https://doi.org/10.1103/RevModPhys.61.605

    Article  Google Scholar 

  12. Reddy, P., Castelino, K., Majumdar, A.: Diffuse mismatch model of thermal boundary conductance using exact phonon dispersion. Appl. Phys. Lett. 87(21), 211908 (2005). https://doi.org/10.1063/1.2133890

    Article  Google Scholar 

  13. Larroque, J., Dollfus, P., Saint-Martin, J.: Phonon transmission at Si/Ge and polytypic Ge interfaces using full-band mismatch based models. J. Appl. Phys. 123(2), 025702 (2018). https://doi.org/10.1063/1.5007034

    Article  Google Scholar 

  14. Merabia, S., Termentzidis, K.: Thermal conductance at the interface between crystals using equilibrium and nonequilibrium molecular dynamics. Phys. Rev. B (2012). https://doi.org/10.1103/PhysRevB.86.094303

    Article  Google Scholar 

  15. Chen, G.: Thermal conductivity and ballistic-phonon transport in the cross-plane direction of superlattices. Phys. Rev. B 57(23), 14958–14973 (1998). https://doi.org/10.1103/PhysRevB.57.14958

    Article  Google Scholar 

  16. Hua, C., Minnich, A.J.: Semi-analytical solution to the frequency-dependent Boltzmann transport equation for cross-plane heat conduction in thin films. J. Appl. Phys. 117(17), 175306 (2015). https://doi.org/10.1063/1.4919432

    Article  Google Scholar 

  17. Davier, B., Dollfus, P., Le, N.D., Volz, S., Shiomi, J., Saint-Martin, J.: Revisiting thermal conductivity and interface conductance at the nanoscale. Int. J. Heat Mass Transf. 183, 122056 (2022). https://doi.org/10.1016/j.ijheatmasstransfer.2021.122056

    Article  Google Scholar 

  18. Chalopin, Y., Esfarjani, K., Henry, A., Volz, S., Chen, G.: Thermal interface conductance in Si/Ge superlattices by equilibrium molecular dynamics. Phys. Rev. B 85(19), 195302 (2012). https://doi.org/10.1103/PhysRevB.85.195302

    Article  Google Scholar 

  19. Sellan, D.P., Landry, E.S., Turney, J.E., McGaughey, A.J.H., Amon, C.H.: Size effects in molecular dynamics thermal conductivity predictions. Phys. Rev. B 81(21), 214305 (2010). https://doi.org/10.1103/PhysRevB.81.214305

    Article  Google Scholar 

  20. Landry, E.S., McGaughey, A.J.H.: Thermal boundary resistance predictions from molecular dynamics simulations and theoretical calculations. Phys. Rev. B 80(16), 165301–165304 (2009). https://doi.org/10.1103/PhysRevB.80.165304

    Article  Google Scholar 

  21. Qiu, B., Ruan, X.: Molecular dynamics simulations of lattice thermal conductivity of bismuth telluride using two-body interatomic potentials. Phys. Rev. B (2009). https://doi.org/10.1103/PhysRevB.80.165203

    Article  Google Scholar 

  22. Duong, T.-Q., Massobrio, C., Ori, G., Boero, M., Martin, E.: Thermal resistance of an interfacial molecular layer by first-principles molecular dynamics. J. Chem. Phys 6, (2020). https://doi.org/10.1063/5.0014232

  23. Mingo, N., Yang, L., Li, D., Majumdar, A.: Predicting the thermal conductivity of Si and Ge nanowires. Nano Lett. 3(12), 1713–1716 (2003). https://doi.org/10.1021/nl034721i

    Article  Google Scholar 

  24. Mazzamuto, F., et al.: Enhanced thermoelectric properties in graphene nanoribbons by resonant tunneling of electrons. Phys. Rev. B 83(23), 235426 (2011). https://doi.org/10.1103/PhysRevB.83.235426

    Article  Google Scholar 

  25. Alkurdi, A., Pailhès, S., Merabia, S.: Critical angle for interfacial phonon scattering: results from ab initio lattice dynamics calculations. Appl. Phys. Lett. 111(9), 093101 (2017). https://doi.org/10.1063/1.4997912

    Article  Google Scholar 

  26. Carrete, J., et al.: Phonon transport across crystal-phase interfaces and twin boundaries in semiconducting nanowires. Nanoscale 11(34), 16007–16016 (2019). https://doi.org/10.1039/C9NR05274G

    Article  Google Scholar 

  27. Lee, Y., et al.: Quantum treatment of phonon scattering for modeling of three-dimensional atomistic transport. Phys. Rev. B 6 (2017). https://doi.org/10.1103/PhysRevB.95.201412

  28. Mazumder, S., Majumdar, A.: Monte Carlo study of phonon transport in solid thin films including dispersion and polarization. J. Heat Transf. 123(4), 749 (2001). https://doi.org/10.1115/1.1377018

    Article  Google Scholar 

  29. Lacroix, D., Joulain, K., Lemonnier, D.: Monte Carlo transient phonon transport in silicon and germanium at nanoscales. Phys. Rev. B (2005). https://doi.org/10.1103/PhysRevB.72.064305

    Article  Google Scholar 

  30. Péraud, J.-P.M., Landon, C.D., Hadjiconstantinou, N.G.: Monte Carlo methods for solving the Boltzmann transport equation. Annu. Rev. Heat Transf. 17, 205–265 (2014). https://doi.org/10.1615/AnnualRevHeatTransfer.2014007381

    Article  Google Scholar 

  31. Ran, X., Guo, Y., Hu, Z., Wang, M.: Interfacial phonon transport through Si/Ge multilayer film using Monte Carlo scheme with spectral transmissivity. Front. Energy Res. 6, 28 (2018). https://doi.org/10.3389/fenrg.2018.00028

    Article  Google Scholar 

  32. Jeng, M.-S., Yang, R., Song, D., Chen, G.: Modeling the thermal conductivity and phonon transport in nanoparticle composites using Monte Carlo simulation. J. Heat Transf. 130(4), 042410 (2008). https://doi.org/10.1115/1.2818765

    Article  Google Scholar 

  33. Hori, T., Shiomi, J., Dames, C.: Effective phonon mean free path in polycrystalline nanostructures. Appl. Phys. Lett. 106(17), 171901 (2015). https://doi.org/10.1063/1.4918703

    Article  Google Scholar 

  34. Davier, B., et al.: Heat transfer in rough nanofilms and nanowires using full band ab initio Monte Carlo simulation. J. Phys. Condens. Matter 30(49), 495902 (2018). https://doi.org/10.1088/1361-648X/aaea4f

    Article  Google Scholar 

  35. Yang, L., Jiang, Y., Zhou, Y.: Quantitatively predicting modal thermal conductivity of nanocrystalline Si by full-band Monte Carlo simulations. Phys. Rev. B 104(19), 195303 (2021). https://doi.org/10.1103/PhysRevB.104.195303

    Article  Google Scholar 

  36. Yang, L., Minnich, A.J.: Thermal transport in nanocrystalline Si and SiGe by ab initio based Monte Carlo simulation. Sci. Rep. 7, 44254 (2017). https://doi.org/10.1038/srep44254

    Article  Google Scholar 

  37. Chaput, L., Larroque, J., Dollfus, P., Saint-Martin, J., Lacroix, D.: Ab initio based calculations of the thermal conductivity at the micron scale. Appl. Phys. Lett. 112(3), 033104 (2018). https://doi.org/10.1063/1.5010959

    Article  Google Scholar 

  38. Weber, W.: Adiabatic bond charge model for the phonons in diamond, Si, Ge, and α-Sn. Phys. Rev. B 15(10), 4789–4803 (1977). https://doi.org/10.1103/PhysRevB.15.4789

    Article  Google Scholar 

  39. Valentin, A., Sée, J., Galdin-Retailleau, S., Dollfus, P.: Study of phonon modes in silicon nanocrystals using the adiabatic bond charge model. J. Phys. Condens. Matter 20(14), 145213 (2008). https://doi.org/10.1088/0953-8984/20/14/145213

    Article  Google Scholar 

  40. Larroque, J., Dollfus, P., Saint-Martin, J.: Full-band modelling of phonons in polytype Ge and Si. J. Phys. Conf. Ser. 906, 012007 (2017). https://doi.org/10.1088/1742-6596/906/1/012007

    Article  Google Scholar 

  41. Togo, A., Chaput, L., Tanaka, I.: Distributions of phonon lifetimes in Brillouin zones. Phys. Rev. B 91(9), 094306 (2015). https://doi.org/10.1103/PhysRevB.91.094306

    Article  Google Scholar 

  42. Soffer, S.B.: Statistical model for the size effect in electrical conduction. J. Appl. Phys. 38(4), 1710–1715 (1967). https://doi.org/10.1063/1.1709746

    Article  Google Scholar 

  43. Sadasivam, S., et al.: Thermal transport across metal silicide–silicon interfaces: first-principles calculations and Green’s function transport simulations. Phys. Rev. B 95(8), 085310 (2017). https://doi.org/10.1103/PhysRevB.95.085310

    Article  Google Scholar 

  44. Pernot, G., et al.: Precise control of thermal conductivity at the nanoscale through individual phonon-scattering barriers. Nat. Mater. 9(6), 491–495 (2010). https://doi.org/10.1038/nmat2752

    Article  Google Scholar 

  45. Glassbrenner, C.J., Slack, G.A.: Thermal conductivity of silicon and germanium from 3°K to the melting point. Phys. Rev. 134(4A), A1058–A1069 (1964). https://doi.org/10.1103/PhysRev.134.A1058

    Article  Google Scholar 

  46. Vermeersch, B., Carrete, J., Mingo, N.: Cross-plane heat conduction in thin films with ab-initio phonon dispersions and scattering rates. Appl. Phys. Lett. 108(19), 193104 (2016). https://doi.org/10.1063/1.4948968

    Article  Google Scholar 

  47. Johnson, J.A., et al.: Direct measurement of room-temperature nondiffusive thermal transport over micron distances in a silicon membrane. Phys. Rev. Lett. 110(2), 025901 (2013). https://doi.org/10.1103/PhysRevLett.110.025901

    Article  Google Scholar 

  48. Goto, M., et al.: Ultra-low thermal conductivity of high-interface density Si/Ge amorphous multilayers. Appl. Phys. Express 11(4), 045202 (2018). https://doi.org/10.7567/APEX.11.045202

    Article  Google Scholar 

  49. Rurali, R., Cartoixà, X., Colombo, L.: Heat transport across a SiGe nanowire axial junction. Interface thermal resistance and thermal rectification. Phys. Rev. B (2014). https://doi.org/10.1103/PhysRevB.90.041408

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by a public Grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (Labex NanoSaclay, reference: ANR-10-LABX-0035).

Author information

Authors and Affiliations

  1. CNRS, Centre de Nanosciences et de Nanotechnologies, Université Paris-Saclay, 91120, Palaiseau, France

    N. D. Le, B. Davier, N. Izitounene, P. Dollfus & J. Saint-Martin

  2. Department of Mechanical Engineering, The University of Tokyo, Tokyo, 113-856, Japan

    B. Davier

Authors
  1. N. D. Le
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Davier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. Izitounene
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. P. Dollfus
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. J. Saint-Martin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Saint-Martin.

Ethics declarations

Conflict of interest

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Le, N.D., Davier, B., Izitounene, N. et al. Study of phonon transport across Si/Ge interfaces using Full-Band phonon Monte Carlo simulation. J Comput Electron 21, 744–755 (2022). https://doi.org/10.1007/s10825-022-01885-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10825-022-01885-x

Keywords

Search

Navigation