Surface phonon-polaritons (SPhPs) are coupled states of optical phonons and electromagnetic waves that can significantly enhance the thermal energy transport along the interface of polar nanomaterials , , , . Theoretical , ,  and experimental ,  studies have shown that the propagation length of SPhPs can be as high as a few milimeters, which is much larger than the phonon mean free path, for a wide variety of materials at room temperature , . This sizeable difference on the mean free path yields a SPhP thermal conductivity of nanofilms , ,  and a SPhP thermal conductance of nanowires  that could be comparable to or even higher than their phonon counterparts , , , , , , , .
SPhP energy transport is determined by the permittivity and geometry of the involved polar nanomaterials and hence it can be modified by material discontinuities. Given that SPhPs are electromagnetic waves that propagate along the material interface, their energy transport is expected to increase as the material size is scaled down to nanoscales due to the strengthening of the surface effects. This is the case of the SPhP thermal conductivity of suspended nanofilms, which increases with the inverse third power of the film thickness mainly , , and takes the value of 4 W·m−1 K−1 for a 40-nm-thick thin film of SiO2 at 500 K , . This is more than twice the phonon thermal conductivity of SiO2 and increases for thinner films and higher temperatures. In a nanowire, on the other hand, SPhPs propagate ballistically and their contribution to the nanowire thermal conductance is quantised and given by , which is comparable to the phonon contribution at room temperature , . This comparability between the phonon and SPhP thermal conductivities and thermal conductances makes it difficult to experimentally distinguish and observe the SPhP heat flux. A nanostruture that is able to support the SPhP propagation and attenuate the phonon heat conduction is therefore desirable.
The purpose of this work is to theoretically demonstrate that the SPhP energy transport can efficiently be observed in a three-dimensional (3D) crystal made up of polar nanorods. This is done by determining the SPhP thermal conductance of the proposed crystal and showing that its values are as high as those of the quantum of thermal conductance of polar nanowires.
2 SPhP Energy Transport
Let us consider a 3D assembly of polar nanorods in contact, as shown in Figure 1. The nanorods of diameter 2a and length 2b 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) . 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 , which have been recently applied to build 3D phononic crystals , .
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 , , 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 T1 and T2 (T1>T2), 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 
where f is the Bose–Einstein distribution function  and ωmin and ωmax stand for the lowest and highest frequencies allowing the existence (βR >0) and propagation (Λ>0) of SPhPs. Equation 1 shows that G increases with the SPhP wave vector βR , 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 , , indicates that the 3D effect on G appears through the nondimensional factor which is associated with the radial confinement of SPhPs to the nanorod’s surface . Thus, Equation 1 physically establishes that the stronger the confinement, the higher the energy transport of SPhPs.
The wave vector βR =Re(β) is given by the dispersion relation of the SPhPs propagating along the z axis of the nanorods with relative permittivity ε1 and embedded in a dielectric medium with relative permittivity ε2. 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 , , the following dispersion relation for the complex wave vector β is obtained , :
where X=2ak2, K=β/k2, fj (K, X)=Lij (eiX(1+K))+Lij (eiX(1−K)), and is the polylogarithm function of order j=2, 3 . The wave vector and normalised polarizability ω being the excitation frequency, c the speed of light in vacuum, and αs the nanorod electrostatic polarizability given by 
The absorption of energy by the polar nanorods is accounted for their complex permittivity ε1, which turns αe and K=KR +iKI , into complex parameters as well. The real (KR >0) and imaginary (KI >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=(2k2KI )−1  is much larger than the center-to-center distance between the nanorods (Λ>>2a), the condition KIX<<1 is fulfilled and the linear approximation of the Taylor series expansion of Equation 2 in powers of KI X allows the decoupling of its real and imaginary parts to yield
where the functions are independent of KI and given by
Clj (θ) being the Clausen’s functions of order j=1, 2, 3 , . For low frequencies and/or small interparticle distances, such that X≪1, the contribution of vanishes and the SPhP propagation is driven by mainly. Equation 5 allows us to analytically determine the SPhP propagation length Λ=(2k2KI )−1, after solving numerically Equation 4 for KR =βR /k2.
The obtained dispersion relation and propagation length are now numerically quantified for a SPhP crystal with SiC nanorods surrounded by air (ε2=1). The permittivity ε1 of the crystalline SiC is well described by the damped harmonic oscillator model , 
where ωL =182 Trad/s and ωT =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 . The maximum of the imaginary part of ε1 occurs at ω=ωT , which indicates that SiC absorbs more energy from the SPhPs at this frequency. Furthermore, the real part Re(ε1) takes negative values within the frequency interval ωT <ω<ωL , which renders the main contribution to the propagation and energy transport of SPhPs , as shown below. These values of ε1 are expected to be valid for nanoparticle sizes larger than 5 nm  and at temperatures lower than 600 K .
3 Results and Discussion
The dispersion relation and propagation length of SPhPs travelling along the SPhP crystal are shown in Figure 2a and b, respectively. The frequency spectrum is determined by the conditions of existence (βR >0) and propagation (Λ>0) of SPhPs, which yield ωmin≈ωT and ωmax≈ωL . At low frequencies, the SPhP wave vector βR tends to be parallel to the light line (βR =k2) and therefore SPhPs propagate pretty much like photons. As the frequency increases, βR takes larger values and separates from the light line, which enhances the confinement of the SPhPs to the nanorod’s surface  and their energy transport, as established by Equation 2. The major contribution to the SPhP thermal conductance arises hence from the high-frequency regime. This confinement strengthens as the nanorod radius a reduces due to the stronger dipole interactions for higher aspect ratios b/a. However, the increase of βR comes along with the reduction of the SPhP propagation length Λ, as shown in Figure 2b. The trade-off between βR and Λ indicates that the excitation frequencies of SPhPs should be high enough to enhance their confinement and energy transport, but low enough not to significantly reduce their propagation length. Note that Λ is nearly independent of a, within all the frequency ranges of SPhP propagation. This is due to the large aspect ratio of the nanorods (b/a≥10), in which the propagation length (see Equation 5) increases linearly with b, becomes weakly dependent on a, and reaches its longest asymptotic values. Shorter propagation lengths are obtained for smaller aspect ratios, which indicate that nanorods with high aspect ratios (b/a>10) are able to maximise the propagation distance of SPhPs as well as to enhance their confinement (Fig. 2a). For ω=150 Trad/s and (a, b)=(50, 1000) nm, the propagation length Λ=90 μm (900 nanorods), which represents the maximum length of the SPhP crystal, allows the exchange of thermal energy between the thermal baths.
Figure 3a and b show the SPhP thermal conductance G of the SiC crystal as a function of the nanorod radius a and temperature T, respectively. Note that G increases as a reduces due to the increasing SPhP confinement shown by the dispersion relation in Figure 2a. This enhancement of G is strengthened by the temperature increase of the crystal, especially from 300 K to 500 K. When a is scaled down from 100 nm to 20 nm, G increases by more than one order of magnitude, which points out that long rods are the suitable candidates to observe a sizeable energy transport by SPhPs. This is confirmed in Figure 3b, which shows that the SPhP thermal conductance of a crystal generally increases with the aspect ratio b/a. For (b, a)=(10 μm, 100 nm), G is comparable to the quantum of thermal conductance G0, which shows that the proposed SPhP crystal can be a polariton conductor as good as a polar nanowire, within a wide range of temperatures. However, the advantage of this crystal is its ultralow phonon heat conduction, which can facilitate the observation of the SPhP heat transport through the measurement of G, in a similar way as was done with G0 .
The thermal conductance due to the propagation of SPhPs in a 3D crystal made up of SiC nanorods has been theoretically determined and shown to be comparable to the quantum of thermal conductance of polar nanowires. A polariton thermal conductance of 0.55 nW·K−1 has been found for a crystal with nanorods at 500 K and diameter (length) of 200 nm (20 μm). This relatively high thermal conductance along with the ultralow phonon counterpart demonstrates that the energy transport of these polaritons could be unambiguously observed in the proposed polariton crystal.
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Published Online: 2017-01-25
Published in Print: 2017-02-01