1. Introduction

Phonon polaritons are coupled states of lattice vibrations and electromagnetic waves [1–3]. These modes usually exist near the optic phonon mode of polar materials [2,3]. Due to the resonances, the permittivities of the materials may become negative, showing high reflections in specific frequency ranges. These phonon polaritons have the ability to confine the electromagnetic energy in deep-subwavelength volumes. Compared with surface plasmons in metal, phonon polaritons do not involve free charge oscillations, which may avoid ohmic losses and be used to build nanophotonic devices with enhanced functionalities [2,3].

In the frequency range of phonon polaritons, the permittivity of the material may become negative. According their electromagnetic responses, there are mainly two kinds: one is that the optic phonon modes lead to an isotropic negative permittivity, typical example is silicon carbide (SiC), which has a negative isotropic permittivity near 920cm⁻¹ [4]; the other is that the optic phonon modes lead to an anisotropic permittivity tensor, each component of the permittivity tensor exhibits negative values in different ranges of frequencies, typical example is hexagonal boron nitride (h-BN) [5]. The second kind is often regarded as hyperbolic materials, of which the iso-frequency surfaces of the electromagnetic modes are hyperboloids. H-BN is a nature hyperbolic material. It has two phonon modes in the infrared that are related to hyperbolicity: out-of-plane A_2u phonon mode, enabling type I hyperbolic response; in-plane E_1u phonon mode, enabling type II hyperbolic response [6,7]. The phonon polariton modes in h-BN are propagative inside the bulk material. Using scattering-type near field microscopes, phonon polaritons in h-BN nanowires and thin films have been discovered; exotic optical phenomena have been revealed, such as subwavelength wave guiding and focusing [6–19], tunable plasmon-phonon interaction [20–27], super-resolution imaging [28], negative refraction [29], enhanced thermal radiation as well as absorption [30–36], and slow-light effect [37]. Thin h-BN films support multiple phonon polariton modes denoted by an index (denoted as n) indicating the number of nodes of the tangential electric field across the slab: the fundamental mode has no node inside the material (n = 0), while high-order modes possess several nodes (n > 0) [22]. Due to its hyperbolicity, phonon polaritons in h-BN thin films always exist, even in the monolayer; as the film becomes thinner, the fundamental mode becomes more separated from high-order ones in the spectra. These phonon polariton modes have no cutoff frequencies which should normally occur in small dielectric waveguides.

However, we have found that the phonon polaritons in cylindrically curved h-BN thin films show quite different phenomena: they possess an extra index, azimuthal index, denoted as m which does not exist in planar structures; high-order modes show cutoff frequencies, below which the modes stop propagating; compared with h-BN nanowires, the length in the lateral dimension (thickness) is small, which causes the mode with n = 0 well separated from the ones with n > 0, independent from the azimuthal index m. In this paper, we investigate such cylindrically curved h-BN thin films; only the modes with n = 0 are concerned. The phonon polariton modes possess an azimuthal index m; high-order ones with m > 0 may have cutoff frequencies in the upper Reststrahlen band depending on their sizes. In dimer structures, gap phonon polaritons as well as other special modes occur due to the coupling between the neighboring structures; for large dimers, multiple gap phonon polariton modes appear in-between. Then, the coupling between the cylindrically curved h-BN thin film with a substrate is considered. In the end, we vary the geometric parameters of the structures and give some insightful discussions about the phonon polariton modes. Our investigations show that the curvature can indeed strongly affect the phonon polariton modes in h-BN thin films

2. Phonon polaritons in cylindrically curved h-BN

Thin h-BN film is treated as a curved anisotropic material. Without loss of generality, we focus on the case where the thickness of h-BN is 10 nm. Such thickness makes sure that the bulk parameters of h-BN still hold and the optical response, i.e., the hyperbolicity, remains unchanged compared with bulk h-BN. The core is assumed to be silicon, which is a lossless dielectric in the frequency range considered in this paper. The permittivity of the dielectric core merely shifts the dispersion relations, or equivalently the mode effective indexes, of the phonon polariton modes; the main conclusions in this paper still hold if the core is changed to other materials. In the framework of electrodynamics, the phonon polariton modes in such structure can be found by using the waveguide theory.

Figure 1(a) is the schematic of such structure, where the h-BN thin film is curved into a cylinder with a dielectric core. Cylindrical coordinates are used, and the z-axis is along the axial direction. Given its molecular structure, the c-axis originally perpendicular to the boron nitride plane in planar h-BN thin films is now pointing in the radial direction. Thus, in the cylindrical coordinate system, the permittivity in the radial direction is ε_// (the permittivity parallel to the c-axis), i.e., ε_r = ε_//; while the permittivities in the azimuthal and axial directions are ${\epsilon }_{\perp }$(the permittivity perpendicular to the c-axis), i.e., ε_θ = ε_z = ε_⊥, as shown in Fig. 1(b). Due to the anisotropy of h-BN, a transformation of the permittivity tensor is required before the calculation, which maps the diagonal permittivity tensor in the cylindrical coordinate system into the one in the Cartesian coordinate system. In calculations, thin h-BN films are treated as anisotropic materials with bulk permittivities [12,22]:

 

Fig. 1 (a) Schematic of the cylindrically curved h-BN thin film, where the c-axis originally perpendicular to the boron nitride plane in planar h-BN thin films is now in the radial direction. A dielectric core has been incorporated. (b) Model for the cylindrically curved h-BN thin films. For h-BN layer, the permittivities in the axial and azimuthal directions are ε_⊥, while the permittivity in the radial direction is ε_//. (c) The permittivities ε_// and ε_⊥ of h-BN near the Reststrahlen bands. Only the real parts are plotted.

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In the upper Reststrahlen band, h-BN possesses negative in-plane permittivities, which makes thin h-BN layers similar to metallic films. This leads to the fact that the phonon polariton modes in cylindrically curved thin h-BN films only exist in the upper Reststrahlen band. In calculations, we have actually searched for the modes in both the lower and upper Reststrahlen bands, and confirmed such judgement. Planar h-BN films support multiple propagative phonon polariton modes due to its hyperbolicity; however, in cylindrically curved thin h-BN films, the phonon polariton modes would possess an azimuthal index denoting the periodicity in the azimuthal direction, which is missing in the planar structures. This index indicates the orbital angular momentum carried by the electromagnetic waves. As one would see, it also causes the cutoff of the m > 0 modes.

We have calculated the phonon polariton modes in a small-size structure with a core of 40 nm in diameter covered with 10 nm-thick h-BN; the results are shown in Figs. 2(a) and 2(b). Only the first three modes have been plotted, indicated by their azimuthal indexes m = 0, m = 1 and m = 2; the corresponding normalized electric field intensities are shown as (i), (ii) and (iii) in Fig. 2(a), respectively. The m = 0 mode has a constant field intensity profile around the structure, while the m = 1 and m = 2 modes respectively have dipole- and quadrupole- like field profiles, which are common in circular waveguide structures. As one can see, the electric fields in all cases are confined near the h-BN layer with peak intensities at the material surface, indicating the fact that these are indeed phonon polariton modes. The real parts of the effective indexes n_eff,r as well as the ratios n_eff,r/n_eff,i of the above modes are shown in Fig. 2(b). n_eff,r/n_eff,i, or equivalently Re[q]/Im[q], is a quantity describing how long measured by the phonon polariton wavelength the mode can propagate [12]. With the chosen parameters, the m = 0 phonon polariton mode has a dispersion relation across the entire upper Reststrahlen band; however, the m = 1 and m = 2 modes stop propagation near 1400 cm⁻¹ and 1450 cm⁻¹, respectively. Since the loss of the material is relatively low in this frequency range, such stops can only be caused by the dispersion, i.e., cutoff behaviour of the modes. Cutoff frequencies of the phonon polariton modes in h-BN have not been reported by other researches yet. The ratios n_eff,r/n_eff,i can reach over 20 for the m = 2 mode and can become much larger for the m = 0 and m = 1 modes, which are enough for many applications. Near the cutoff frequency, the ratio n_eff,r/n_eff,i drops rapidly, corresponding to the stop of the propagation of the phonon polaritons.

 

Fig. 2 (a) The normalized |E| of the phonon polariton modes with m = 0, m = 1 and m = 2 at 1500 cm⁻¹. The dielectric core has a diameter of 40 nm, and the thickness of h-BN is 10 nm. (b) The effective indexes n_eff,r and the ratios n_eff,r/n_eff,i of the modes in (a). (c) The normalized |E| of the phonon polariton modes with m = 0, m = 1 and m = 2 at 1500 cm⁻¹. The dielectric core has a diameter of 200 nm, and the thickness of h-BN is still 10 nm. (d) The effective indexes n_eff,r and the ratios n_eff,r/n_eff,i of the modes in (c).

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As for large-size structures, the modes possess similar electric field intensity distributions with the small-size ones. We have calculated the phonon polariton modes in a structure with a core of 200 nm in diameter covered with 10 nm-thick h-BN. Again, only the first three modes have been plotted, as shown in Fig. 2(c). The corresponding n_eff,r and n_eff,r/n_eff,i are plotted in Fig. 2(d). For such large-size structure, the modes with different azimuthal indexes are nearly degenerate, having quite similar n_eff,r across the upper Reststrahlen band. Only small deviations exist for the m = 2 mode at the low-frequency end, see the solid blue curve in Fig. 2(d). The ratios n_eff,r/n_eff,i for the m = 0 and m = 1 modes are also quite similar; while, for the m = 2 mode, the curve is a little lower. Highly degenerate phonon polariton modes are expected if the size of the cylindrically curved h-BN thin film becomes even larger.

The cutoff behaviour of high-order modes can be understood as follows. For planar h-BN films on substrate, the complex wavevector $q+ik$of the phonon polariton modes can be derived in the quasi-static limit [5]: $q+ik=-\psi [arctan({\epsilon }_a/{\epsilon }_{\perp }\psi )+arctan({\epsilon }_s/{\epsilon }_{\perp }\psi )+n\pi ]/d$, where $\psi =\sqrt{{\epsilon }_{\Vert }}/i\sqrt{{\epsilon }_{\perp }}$, ${\epsilon }_a$ and ${\epsilon }_s$ are respectively the permittivities of the environment (air) and the substrate. d is the thickness of the h-BN film, and n is an integer (n = 0,1,2...). For cylindrically curved h-BN, the wavevector has two components: k_z and k_θ, where k_θ = m/R. m is the azimuthal index, and R is the effective radius of the structure. Assuming that the total wavevectors of the planar and the cylindrically curved h-BN films are the same, i.e., the curvature does not change the wavevector, one has $k_z^2+k_{\theta }^2=k_z^2+{(m/R)}^2={(q+ik)}^2$. As the frequency decreases, ${(q+ik)}^2$ approaches zero ($k_z^2+{(m/R)}^2={(q+ik)}^2\to 0$, as shown in Ref [5].). Thus, for m = 0, k_z can always be real; however, for m > 0, k_z must become imaginary at some frequency since${(m/R)}^2$ is a positive constant, which results in the cutoff behaviour of the high-order modes.

In the upper Reststrahlen band, h-BN thin film possesses negative in-plane permittivity, which makes the structure in Fig. 1(a) similar to graphene-wrapped dielectric nanowires, but only in the metallic regime. The imaginary part of graphene conductivity can be positive or negative depending on its parameters [38–40]. In the metallic regime, graphene’s positive conductivity may also lead to negative in-plane effective permittivity, resulting in the plasmonic modes in graphene-wrapped nanowires [38,40], similar to the phonon polariton modes in this paper. However, different from graphene, h-BN is a hyperbolic material which supports phonon polariton modes with extremely short wavelengths. As one would see in the following sections, multiple gap phonon polariton modes exit in dimers, which is not common in plasmonic structures. Also, as the thickness of h-BN increases, the high-order modes are no longer propagative, showing cutoff frequencies; this is interesting because usually the increasement of the size does not lead to the cutoff of the high-order modes.

3. Phonon polaritons in dimers

Dimers are formed by closing two structures together, leaving a gap in-between. In this section, we investigate the phonon polaritons in dimer structures, where the gap is only 5 nm.

For small dimers, we used two structures as shown in Fig. 2(a) and put them close to each other forming a 5 nm gap. The thickness of h-BN is 10 nm, and the diameter of the core is 40 nm. Three representative modes are plotted in Fig. 3(a) which shows their corresponding E_z (left column) and |E| distributions (right column): the gap phonon polariton mode denoted as “1st mode”, the anti-symmetric mode denoted as “2nd mode”, and the symmetric mode denoted as “3rd mode”. The gap phonon polariton mode is confined within an ultra-small volume with high electric field intensities in the gap (see the color bar of the |E| distributions of 1st mode in Fig. 3(a)). The corresponding effective indexes n_eff,r as well as the ratios n_eff,r/n_eff,i of the modes in Fig. 3(a) are shown in Fig. 3(b). The anti-symmetric and symmetric modes are nearly degenerate, i.e., both n_eff,r and n_eff,r/n_eff,i are quite similar, except for those near 1400 cm⁻¹. The gap phonon polariton mode has the largest index n_eff,r (solid black curve), and also the largest ratio n_eff,r/n_eff,i (dashed black curve). According to the |E| distributions, the gap mode has most of the fields outside the lossy material; thus, ultra-strong field confinement and low propagation loss can be simultaneously achieved.

 

Fig. 3 (a) E_z and |E| of the three representative modes in the small dimer structure, where the diameter of the core is 40 nm and the thickness of h-BN is 10 nm. (b) The effective indexes n_eff,r and the ratios n_eff,r/n_eff,i of the modes in (a). (c) E_z and |E| of the four representative modes in the large dimer structure, where the diameter of the core is 200 nm and the thickness of h-BN is still 10 nm. (d) The effective indexes n_eff,r and the ratios n_eff,r/n_eff,i of the modes in (c).

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For large dimers, we placed two structures as shown in Fig. 2(c) close to each other forming a 5 nm gap. The thickness of h-BN is 10 nm, and the diameter of the core is 200 nm. Four representative modes are plotted. Figure 3(c) shows their E_z (left column) and |E| distributions (right column): the first gap phonon polariton mode denoted as “1st mode”, the anti-symmetric mode denoted as “2nd mode”, the symmetric mode denoted as “3rd mode”, and the second gap phonon polariton mode denoted as “4th mode”. For such large structure, multiple gap phonon polariton modes can exist. This may be caused by the fact that the coupling area becomes larger as the diameter of the core increases, which leads to several gap phonon polariton modes. The first gap mode has anti-symmetric E_z profile along the dimer; the second gap mode has anti-symmetric E_z profiles both in the directions along and vertical to the dimer. From their |E| distributions, one can see that the gap phonon polaritons may lead to large field enhancements. For example, it can reach over 70 for the first gap mode and over 30 for the second gap mode. The corresponding effective indexes n_eff,r as well as the ratios n_eff,r/n_eff,i of the modes in Fig. 3(c) are shown in Fig. 3(d). As one can see, the 2nd, 3rd, and 4th modes have quite similar indexes n_eff,r as well as ratios n_eff,r/n_eff,i. However, the 1st mode, the first gap phonon polariton mode, possesses the largest index n_eff,r, and also the largest ratio n_eff,r/n_eff,i in the whole spectrum, which means that the electric fields are tightly confined in the gap with relatively low propagation loss.

4. Phonon polaritons coupled with substrate

On substrate, the phonon polariton modes can be strongly modified. In this section, a silicon substrate has been used. The cylindrically curved h-BN is 10 nm-thick with a dielectric core of 200 nm in diameter. The h-BN layer touches the substrate, i.e., the gap between the cylindrically curved h-BN and the substrate is zero.

Three representative phonon polariton modes are shown in Fig. 4(a), respectively denoted as 1st, 2nd and 3rd modes, which can be distinguished from their E_z (left column) and |E| distributions (right column): the 1st mode is the first gap phonon polariton mode; the 2nd mode possesses electric fields mostly on the upper region of the structure; the 3rd mode is the second gap phonon polariton mode. The corresponding indexes n_eff,r and ratios n_eff,r/n_eff,i are plotted in Fig. 4(b). When coupled with the substrate, the first gap phonon polariton mode no longer has anti-symmetric E_z distributions, which is different compared with the first gap modes in dimers (compare the E_z distribution of the 1st mode in Fig. 4(a) with those of the 1st modes in Figs. 3(a) and 3(c)); the second gap phonon polariton mode has anti-symmetric E_z distribution (see the E_z distribution of 3rd mode in Fig. 4(a)). The electric fields are highly enhanced near the touching point. The existence of the substrate can strongly affect the mode profiles as well as the effective indexes; again, multiple gap phonon polariton modes can be observed due to the strong coupling between the cylindrically curved h-BN thin film and the substrate.

 

Fig. 4 (a) E_z and |E| distributions of the three representative modes, where the whole structure is placed on Si substrate. (b) The effective indexes n_eff,r and the ratios n_eff,r/n_eff,i of the modes in (a). The diameter of the dielectric core is 200 nm; the thickness of the h-BN is 10 nm.

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5. The effects of geometric parameters

In this section, we investigate the effects of geometric parameters on the phonon polariton modes. The wavenumber is fixed at 1500 cm⁻¹.

For isolated structures (see Fig. 2), we firstly increase the core radius from 20 nm to 100 nm while keeping the h-BN thickness at 10 nm. As shown in Fig. 5(a), both n_eff,r and n_eff,r/n_eff,i of the m = 0, m = 1 and m = 2 modes become larger; such phenomenon is similar to graphene-wrapped nanowires [38–40]. Then, we increase the thickness of the h-BN while keeping the core radius fixed at 20 nm. As shown in Fig. 5(b), the effective indexes n_eff,r of all modes decrease. The ratios n_eff,r/n_eff,i for the m = 1 and m = 2 modes drop rapidly, indicating the cutoff of these two modes; but, for the m = 0 mode, this ratio remains large. These phenomena can be understood as follows: when the thickness of the h-BN layer increases, the wavevector $q+ik$ decreases according to the fact that such wavevector is proportional to the inverse of the h-BN thickness [5]; for modes with m > 0, k_z may no longer be real and become imaginary, leading to the cutoff of the high-order modes.

 

Fig. 5 (a) n_eff,r and n_eff,r/n_eff,i of the m = 0, m = 1, and m = 2 modes as functions of the core radius, while the thickness of the h-BN layer is 10 nm. (b) n_eff,r and n_eff,r/n_eff,i of the m = 0, m = 1, and m = 2 modes as functions of the h-BN thickness, while the radius of the core is fixed at 20 nm. (c) n_eff,r and n_eff,r/n_eff,i of the 1st, 2nd, 3rd and 4th modes as functions of the dimer gap. (d) n_eff,r and n_eff,r/n_eff,i of the 1st, 2nd and 3rd modes as functions of the gap with the substrate. All the data in this figure are plotted at 1500 cm⁻¹.

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For dimer structures, we change the gap from 0 nm to 10 nm, as shown in Fig. 5(c). The core radius is 100 nm and the thickness of h-BN is 10 nm. Only the gap modes are strongly affected, which is normal since their electric fields are mostly confined within the gap. The 2nd and 3rd modes have field distributions at the two edges of the dimer; thus, changing the gap would not modify their effective indexes. As one can see from Fig. 5(c), the slops of all curves become small if the gap is larger than 10 nm, which means that the coupling between the neighboring structures is weak at such distance. Strong couplings of phonon polariton modes only exist within a range of a few nanometers. For the structure on the silicon substrate, we vary the gap from 0 nm to 8 nm. As shown in Fig. 5(d), the indexes n_eff,r of the gap modes decrease rapidly as the gap becomes larger. Again, only the gap phonon polariton modes are strongly modified; the 2nd mode which has most of its fields confined in the upper region of the structure is not affected. Strong coupling of the gap modes with substrates can only happen within a range of a few nanometers.

6. Conclusion

Hexagonal boron nitride supports phonon polariton modes which may be largely affected after being cylindrically curved. Another index, the azimuthal index m, arises which describes the periodicity in the azimuthal direction. High-order modes with m > 0 may have cutoff frequencies depending on the size of the structure. In dimers, gap phonon polaritons may occur; for large-size dimers, multiple gap phonon polariton modes can exist. After putting such cylindrical h-BN on a substrate, multiple gap phonon polariton modes may also appear. By varying the geometric parameters, it is proved that strong couplings can happen only if two structures are extremely close to each other, for example, separated with a gap of a few nanometers. Our investigations may open the way to search for new phonon polariton modes in h-BN thin films with complex structures.