# Heat Transport Driven by the Coupling of Polaritons and Phonons in a Polar Nanowire

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Models

#### 2.1. Numerical Approach

#### 2.2. Analytical Approach

## 3. Results and Discussions

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Rego, L.G.C.; Kirczenow, G. Quantized Thermal Conductance of Dielectric Quantum Wires. Phys. Rev. Lett.
**1998**, 81, 232. [Google Scholar] [CrossRef][Green Version] - Yamamoto, T.; Watanabe, S.; Watanabe, K. Universal Features of Quantized Thermal Conductance of Carbon Nanotubes. Phys. Rev. Lett.
**2004**, 92, 075502. [Google Scholar] [CrossRef][Green Version] - Glavin, B.A. Low-Temperature Heat Transfer in Nanowires. Phys. Rev. Lett.
**2001**, 86, 4318. [Google Scholar] [CrossRef] [PubMed][Green Version] - Chalopin, Y.; Gillet, J.-N.; Volz, S. Predominance of thermal contact resistance in a silicon nanowire on a planar substrate. Phys. Rev. B
**2008**, 77, 233309. [Google Scholar] [CrossRef] - Venkatesh, R.; Amrit, J.; Chalopin, Y.; Volz, S. Thermal resistance of metal nanowire junctions in the ballistic régime. Phys. Rev. B
**2011**, 83, 115425. [Google Scholar] [CrossRef] - Büttiker, M. Four-Terminal Phase-Coherent Conductance. Phys. Rev. Lett.
**1986**, 57, 1761. [Google Scholar] [CrossRef] - Rego, L.G.C.; Kirczenow, G. Fractional exclusion statistics and the universal quantum of thermal conductance: A unifying approach. Phys. Rev. B
**1999**, 59, 13080. [Google Scholar] [CrossRef][Green Version] - Chiatti, O.; Nicholls, J.T.; Proskuryakov, Y.Y.; Lumpkin, N.; Farrer, I.; Ritchie, D.A. Quantum Thermal Conductance of Electrons in a One-Dimensional Wire. Phys. Rev. Lett.
**2006**, 97, 056601. [Google Scholar] [CrossRef][Green Version] - Schwab, K.; Henriksen, E.A.; Worlock, J.M.; Roukes, M.L. Measurement of the quantum of thermal conductance. Nature
**2000**, 404, 974. [Google Scholar] [CrossRef] [PubMed] - Biehs, S.A.; Messina, R.; Venkataram, P.S.; Rodriguez, A.W.; Cuevas, J.C.; Ben-Abdallah, P. Near-field radiative heat transfer in many-body systems. Rev. Mod. Phys.
**2021**, 93, 025009. [Google Scholar] [CrossRef] - Dong, J.; Zhao, J.; Liu, L. Long-distance near-field energy transport via propagating surface waves. Phys. Rev. B
**2018**, 97, 075422. [Google Scholar] [CrossRef][Green Version] - Agranovich, V.M. Surface Polaritons, 3rd ed.; Elsevier: Amsterdam, The Netherlands, 2012. [Google Scholar]
- Zayats, A.V.; Smolyaninov, I.I.; Maradudin, A.A. Nano-optics of surface plasmon polaritons. Phys. Rep.
**2005**, 408, 0370. [Google Scholar] [CrossRef] - Jiunn-Woei, L.; Szu-Yao, M.; Jia-Yun, L.; Yun-Cheng, K.; Mao-Kuen, K. Nano-optics of surface plasmon polaritons. Opt. Express
**2021**, 29, 18876. [Google Scholar] - Ordonez-Miranda, J.; Tranchant, L.; Antoni, T.; Chalopin, Y.; Volz, S. Thermal conductivity of nano-layered systems due to surface phonon-polaritons. J. Appl. Phys.
**2014**, 115, 054311–054315. [Google Scholar] [CrossRef][Green Version] - Gluchko, S.; Palpant, B.; Volz, S.; Braive, R.; Antoni, T. Thermal excitation of broadband and long-range surface waves on SiO
_{2}submicron films. Appl. Phys. Lett.**2017**, 110, 263108. [Google Scholar] [CrossRef][Green Version] - Tranchant, L.; Hamamura, S.; Orrdonez-Miranda, J.; Yabuki, T.; Vega-Flick, A.; Cervantes-Alvarez, A.; Alvarado-Gil, J.J.; Volz, S.; Miyazaki, K. Two-Dimensional Phonon Polariton Heat Transport. Nano Lett.
**2019**, 19, 6924–6930. [Google Scholar] [CrossRef] - Ordonez-Miranda, J.; Tranchant, L.; Kim, B.; Chalopin, Y.; Antoni, T.; Volz, S. Effects of anisotropy and size of polar nano thin films on their thermal conductivity due to surface phonon-polaritons. Appl. Phys. Express
**2014**, 7, 035201–035204. [Google Scholar] [CrossRef][Green Version] - Ordonez-Miranda, J.; Tranchant, L.; Joulain, K.; Ezzahri, Y.; Drevillon, J.; Volz, S. Thermal energy transport in a surface phonon-polariton crystal. Phys. Rev. B
**2016**, 93, 035428. [Google Scholar] [CrossRef] - Ordonez-Miranda, J.; Tranchant, L.; Tokunaga, T.; Kim, B.; Palpant, B.; Chalopin, Y.; Antoni, T.; Volz, S. Anomalous thermal conductivity by surface phonon-polaritons of polar nano thin films due to their asymmetric surrounding media. J. Appl. Phys.
**2013**, 113, 084311–084318. [Google Scholar] [CrossRef] - Ordonez-Miranda, J.; Tranchant, L.; Kim, B.; Chalopin, Y.; Antoni, T.; Volz, S. Quantized thermal conductance of nanowires at room temperature due to Zenneck surface-phonon polaritons. Phys. Rev. Lett.
**2014**, 112, 055901–055905. [Google Scholar] [CrossRef][Green Version] - Ordonez-Miranda, J.; Tranchant, L.; Gluchko, S.; Volz, S. Energy transport of surface phonon polaritons propagating along a chain of spheroidal nanoparticles. Phys. Rev. B
**2015**, 92, 115409–115413. [Google Scholar] [CrossRef][Green Version] - Wu, Y.; Ordonez-Miranda, J.; Gluchko, S.; Anufriev, R.; Meneses, D.D.S.; Del Campo, L.; Volz, S.; Nomura, M. Enhanced thermal conduction by surface phonon-polaritons. Sci. Adv.
**2020**, 6, eabb4461. [Google Scholar] [CrossRef] - Li, X.; Lee, S. Crossover of ballistic, hydrodynamic, and diffusive phonon transport in suspended graphene. Phys. Rev. B
**2019**, 99, 085202. [Google Scholar] [CrossRef][Green Version] - Chen, D.-Z.A.; Narayanaswamy, A.; Chen, G. Surface phonon-polariton mediated thermal conductivity enhancement of amorphous thin films. Phys. Rev. B
**2005**, 72, 155435. [Google Scholar] [CrossRef][Green Version] - Majumdar, A. Microscale heat conduction in dielectric thin films. ASME J. Heat Transf.
**1993**, 115, 7–16. [Google Scholar] [CrossRef] - Minnich, A.J.; Chen, G.; Mansoor, S.; Yilbas, B.S. Quasiballistic heat transfer studied using the frequency-dependent Boltzmann transport equation. Phys. Rev. B
**2011**, 84, 235207. [Google Scholar] [CrossRef][Green Version] - Chen, D.Z.A.; Chen, G. Heat flow in thin films via surface phonon-polaritons. Front. Heat Mass Transf.
**2010**, 1, 023005. [Google Scholar] [CrossRef] - Kathmann, C.; Messina, R.; Ben-Abdallah, P.; Biehs, S.A. Limitations of kinetic theory to describe near-field heat exchanges in many-body systems. Phys. Rev. B
**2018**, 98, 115434. [Google Scholar] [CrossRef][Green Version] - Kittel, C. Introduction to Solid State Physics, 8th ed.; John Wiley & Sons, Inc.: New York, NY, USA, 2005; pp. 108–111. [Google Scholar]
- Guo, Y.; Wang, M. Heat transport in two-dimensional materials by directly solving the phonon Boltzmann equation under Callaway’s dual relaxation model. Phys. Rev. B
**2017**, 96, 134312. [Google Scholar] [CrossRef][Green Version] - Guo, Y.; Tachikawa, S.; Volz, S.; Nomura, M.; Ordonez-Miranda, J. Quantum of Thermal Conductance of Nanofilms due to Surface-Phonon Polaritons. Under Rev.
**2021**. [Google Scholar] - Kourganoff, V. Basic Methods in Transfer Problems, 1st ed.; Dover Publications: New York, NY, USA, 1963. [Google Scholar]
- Ordonez-Miranda, J.; Volz, S.; Nomura, M. Surface Phonon-Polariton Heat Capacity of Polar Nanofilms. Phys. Rev. Appl.
**2021**, 15, 054068. [Google Scholar] [CrossRef] - Yeh, C.; Shimabukuro, F.I. The Essence of Dielectric Waveguides; Springer: New York, NY, USA, 2008. [Google Scholar]

**Figure 1.**Scheme of a polar nanowire supporting the propagation of both SPhPs and phonons due to the temperature difference ${T}_{h}>{T}_{c}$ imposed by two thermal baths.

**Figure 2.**Real and imaginary parts of the relative permittivity $\epsilon ={\epsilon}_{R}+i{\epsilon}_{I}$ of SiN, as a function of frequency [23]. The yellow zone stands for the band in which ${\epsilon}_{R}<0$.

**Figure 3.**Frequency spectrum of the wavevector and propagation length of SPhPs propagating along a SiN nanowire suspended in air (${\epsilon}_{0}=1$). The blue dashed line stands for the wavevector of light in vacuum.

**Figure 4.**(

**a**) Temperature and (

**b**) heat flux profiles in a SiN nanowire with representative lengths l. The solid and dashed-dot lines in (

**b**) represent the respective SPhP and phonon heat fluxes, whereas the dashed one stands for the total heat flux ${q}_{t}$. Calculations were carried out for a SiN nanowire with a radius $a=50$ nm and a typical phonon thermal conductivity of ${k}_{ph}$ = 1 W/m·K.

**Figure 5.**Frequency spectrum of the transmission probability of SPhPs propagating along a SiN nanowire of different lengths.

**Figure 6.**(

**a**) SPhP thermal conductance spectrum along with its integrated counterpart as a function of the (

**b**) length and (

**c**) temperature of a SiN nanowire suspended in air with a radius $a=50$ nm. The solid lines represent the predictions of Equation (17) and the dots the stand for the numerical results obtained with the DOM+FDM described in Section 2.1. Calculations in (

**a**) were carried out for $T=300$ K.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Guo, Y.; Nomura, M.; Volz, S.; Ordonez-Miranda, J. Heat Transport Driven by the Coupling of Polaritons and Phonons in a Polar Nanowire. *Energies* **2021**, *14*, 5110.
https://doi.org/10.3390/en14165110

**AMA Style**

Guo Y, Nomura M, Volz S, Ordonez-Miranda J. Heat Transport Driven by the Coupling of Polaritons and Phonons in a Polar Nanowire. *Energies*. 2021; 14(16):5110.
https://doi.org/10.3390/en14165110

**Chicago/Turabian Style**

Guo, Yangyu, Masahiro Nomura, Sebastian Volz, and Jose Ordonez-Miranda. 2021. "Heat Transport Driven by the Coupling of Polaritons and Phonons in a Polar Nanowire" *Energies* 14, no. 16: 5110.
https://doi.org/10.3390/en14165110