Let us consider a 3D assembly of polar nanorods in contact, as shown in Figure 1. The nanorods of diameter 2*a* and length 2*b* are modelled like spheroidal particles with large aspect ratio (*b*/*a*>>1) to strengthen the propagation of SPhPs along the +*z* axis. The thermal energy transport along this crystal of packed nanorods is expected to be dominated by SPhPs for two main reasons: First, the relatively small diameter of the nanorods, along with their multiple interfaces, act as thermal barriers that significantly reduce the phonon thermal conductivity (<0.05 Wm^{−1} K^{−1}) [22]. Second, the high surface area-to-volume ratio (2/*a*+1/*b*) of the nanorods favours the propagation of SPhPs along their surfaces. This crystal could be fabricated by the techniques such as 3D printing or high-resolution stereolithograpy [23], which have been recently applied to build 3D phononic crystals [24], [25].

Figure 1: SPhP crystal made up of polar nanorods excited by two thermal baths. The purple glow around the nanorods stands for the coupled electromagnetic field induced by their surface polarization.

In general, the polarizable molecules of polar nanorods undergo multipole (dipole, quadrupole, octupole, etc.) interactions, which strengthen as their center-to-center distances are reduced. However, for the sake of simplicity and to maintain our analytical approach, we are going to consider only the dominant dipole interactions in this work. Taking into account that the higher-order multipole interactions can enhance the polariton thermal conductance rendered by the dipole interactions [26], [27], our results for nanorods in contact will represent the minimum (lower bound) for the energy transport of SPhPs. To quantify the SPhP energy, we place the proposed crystal in thermal contact with two thermal baths set at the temperatures *T*_{1} and *T*_{2} (*T*_{1}>*T*_{2}), as shown in Figure 1. Under this thermal excitation, the oscillating electrical dipoles of the polar nanorods emit an electric field, which induces the excitation of neighbouring electrical dipoles that keep the propagation of the field (SPhP) along the +*z* axis. Assuming that the SPhP propagation length *Λ* is longer than the crystal length along the *z* axis, the thermal conductance *G* of the SPhP crystal along this axis is given by [28]

$$G=\frac{ab}{8\pi}{\displaystyle {\int}_{{\omega}_{\text{min}}}^{{\omega}_{\text{max}}}\hslash \omega {\beta}_{R}^{2}\frac{\partial f}{\partial T}d\omega}\mathrm{.}$$(1)

where *f* is the Bose–Einstein distribution function [5] and *ω*_{min} and *ω*_{max} stand for the lowest and highest frequencies allowing the existence (*β*_{R} >0) and propagation (*Λ*>0) of SPhPs. Equation 1 shows that *G* increases with the SPhP wave vector *β*_{R} , which depends on the permittivity and size of the nanorods. The comparison of the thermal conductance in Equation 1 with that for a linear chain of nanoparticles [26], [27], indicates that the 3D effect on *G* appears through the nondimensional factor $ab{\beta}_{R}^{2}\mathrm{/}\text{}4,$ which is associated with the radial confinement of SPhPs to the nanorod’s surface [27]. Thus, Equation 1 physically establishes that the stronger the confinement, the higher the energy transport of SPhPs.

The wave vector *β*_{R} =Re(*β*) is given by the dispersion relation of the SPhPs propagating along the *z* axis of the nanorods with relative permittivity *ε*_{1} and embedded in a dielectric medium with relative permittivity *ε*_{2}. Both media are assumed to be nonmagnetic and that the electrical dipoles inside the nanorods are aligned parallel to the *z* axis, as established by the driving thermal gradient imposed along this axis by the thermal baths. In view of this alignment, the dipole–dipole interactions occur along this direction mainly and therefore the SPhP propagation in the 3D crystal can be modelled as that along a single chain of nanorods with longitudinal polarization. Furthermore, the length of the nanorods is considered to be much larger than their radius, and hence the SPhP heat transport along the vertical direction is small in comparison with the one along the *z* axis. By solving the Maxwell equations under the proper boundary conditions required for the existence of SPhPs [1], [5], the following dispersion relation for the complex wave vector *β* is obtained [27], [28]:

$$-\text{\hspace{0.17em}}i+{\alpha}_{e}^{-\text{\hspace{0.17em}}1}=\frac{3}{{X}^{3}}[{f}_{3}\mathrm{(}K\mathrm{,}\text{\hspace{0.17em}}X\mathrm{)}-iX{f}_{2}\mathrm{(}K\mathrm{,}\text{\hspace{0.17em}}X\mathrm{)}]\mathrm{,}$$(2)

where *X*=2*ak*_{2}, *K*=*β*/*k*_{2}, *f*_{j} (*K*, *X*)=*Li*_{j} (*e*^{iX(1+K)})+*Li*_{j} (*e*^{iX(1−K)}), and $L{i}_{j}\mathrm{(}z\mathrm{)}={\displaystyle {\sum}_{n\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{\infty}{z}^{n}\mathrm{/}{n}^{j}}$ is the polylogarithm function of order *j*=2, 3 [29]. The wave vector ${k}_{2}=\sqrt{{\epsilon}_{2}}\omega \mathrm{/}c$ and normalised polarizability ${\alpha}_{e}={\alpha}_{s}{k}_{2}^{3}/\mathrm{(}6\pi {\epsilon}_{2}\mathrm{)},$ *ω* being the excitation frequency, *c* the speed of light in vacuum, and *α*_{s} the nanorod electrostatic polarizability given by [30]

$${\alpha}_{s}=\frac{8\pi {a}^{2}b}{3}\frac{{\epsilon}_{1}-{\epsilon}_{2}}{{\epsilon}_{1}+{\epsilon}_{2}}\mathrm{.}$$(3)

The absorption of energy by the polar nanorods is accounted for their complex permittivity *ε*_{1}, which turns *α*_{e} and *K*=*K*_{R} +*iK*_{I} , into complex parameters as well. The real (*K*_{R} >0) and imaginary (*K*_{I} >0) parts of the normalised wave vector *K* are associated with the propagation (along the +*z* direction) and attenuation factors, respectively. For the case of interest, in which the SPhP propagation length *Λ*=(2*k*_{2}*K*_{I} )^{−1} [12] is much larger than the center-to-center distance between the nanorods (*Λ*>>2*a*), the condition *K*_{I}*X*<<1 is fulfilled and the linear approximation of the Taylor series expansion of Equation 2 in powers of *K*_{I} *X* allows the decoupling of its real and imaginary parts to yield

$$\text{Re}\mathrm{(}{\alpha}_{e}^{-\text{\hspace{0.17em}}1}\mathrm{)}=\frac{3}{{X}^{3}}[{g}_{3}^{+}\mathrm{(}{K}_{R}\mathrm{,}\text{\hspace{0.17em}}X\mathrm{)}+X{g}_{2}^{+}\mathrm{(}{K}_{R}\mathrm{,}\text{\hspace{0.17em}}X\mathrm{)}]\mathrm{,}$$(4)

$${K}_{I}=-\text{\hspace{0.17em}}\frac{{X}^{2}}{3}\frac{\text{Im}\mathrm{(}{\alpha}_{e}^{-\text{\hspace{0.17em}}1}\mathrm{)}}{{g}_{2}^{-}\mathrm{(}{K}_{R}\mathrm{,}\text{\hspace{0.17em}}X\mathrm{)}-X{g}_{1}^{-}\mathrm{(}{K}_{R}\mathrm{,}\text{\hspace{0.17em}}X\mathrm{)}}\mathrm{,}$$(5)

where the functions ${g}_{j}^{\pm}$ are independent of *K*_{I} and given by

$${g}_{j}^{\pm}\mathrm{(}{K}_{R}\mathrm{,}\text{\hspace{0.17em}}X\mathrm{)}={\text{Cl}}_{j}[X\mathrm{(}1+{K}_{R}\mathrm{)}]\pm {\text{Cl}}_{j}[X\mathrm{(}1-{K}_{R}\mathrm{)}]\mathrm{,}$$(6)

$${\text{Cl}}_{1}\mathrm{(}\theta \mathrm{)}=-\text{\hspace{0.17em}}\mathrm{ln}|2\mathrm{sin}\mathrm{(}\theta /2\mathrm{)}|\mathrm{,}$$(7)

$${\text{Cl}}_{2}\mathrm{(}\theta \mathrm{)}={\displaystyle {\int}_{0}^{\theta}{\text{Cl}}_{1}\mathrm{(}{\theta}^{\prime}\mathrm{)}d{\theta}^{\prime}}\mathrm{,}$$(8)

$${\text{Cl}}_{3}\mathrm{(}\theta \mathrm{)}={\displaystyle \sum _{n\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{\infty}\frac{\mathrm{sin}\mathrm{(}n\theta \mathrm{)}}{{n}^{3}}}\mathrm{,}$$(9)

Cl_{j} (*θ*) being the Clausen’s functions of order *j*=1, 2, 3 [29], [31]. For low frequencies and/or small interparticle distances, such that *X*≪1, the contribution of ${g}_{2}^{+}$ vanishes and the SPhP propagation is driven by ${g}_{3}^{+}$ mainly. Equation 5 allows us to analytically determine the SPhP propagation length *Λ*=(2*k*_{2}*K*_{I} )^{−1}, after solving numerically Equation 4 for *K*_{R} =*β*_{R} /*k*_{2}.

The obtained dispersion relation and propagation length are now numerically quantified for a SPhP crystal with SiC nanorods surrounded by air (*ε*_{2}=1). The permittivity *ε*_{1} of the crystalline SiC is well described by the damped harmonic oscillator model [5], [32]

$${\epsilon}_{1}\mathrm{(}\omega \mathrm{)}={\epsilon}_{\infty}\mathrm{(}1+\frac{{\omega}_{L}^{2}-{\omega}_{T}^{2}}{{\omega}_{T}^{2}-{\omega}^{2}-i\Gamma \omega}\mathrm{)}\mathrm{,}$$(10)

where *ω*_{L} =182 Trad/s and *ω*_{T} =149 Trad/s are the longitudinal and transversal optical frequencies, respectively; *Γ*=0.892 Trad/s is a damping constant and *ε*_{∞}=6.7 is the high frequency permittivity [33]. The maximum of the imaginary part of *ε*_{1} occurs at *ω*=*ω*_{T} , which indicates that SiC absorbs more energy from the SPhPs at this frequency. Furthermore, the real part Re(*ε*_{1}) takes negative values within the frequency interval *ω*_{T} <*ω*<*ω*_{L} , which renders the main contribution to the propagation and energy transport of SPhPs [5], as shown below. These values of *ε*_{1} are expected to be valid for nanoparticle sizes larger than 5 nm [34] and at temperatures lower than 600 K [35].

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