## Abstract

In this paper, we point out that the phonon polaritons in hexagonal boron nitride may not have to be volume-confined. Surface-confined phonon polaritons can occur at the edges of ultra-thin boron nitride films. Based on numerical calculations, we have investigated the effective indexes of such phonon polariton modes, and found that these modes normally have much lower losses than the volume-confined ones. Surface-confined phonon polaritons in boron nitride nanoribbons are also investigated that can be seen as the coupling of the phonon polariton modes at two edges of the nanoribbon, and the corresponding losses can be further reduced. Then, we give a deep discussion on the volume-confined and surface-confined phonon polaritons. In the end, the effects of defects and substrates on such modes are presented. The proposed low-loss surface-confined phonon polaritons may be useful in phonon polariton based photonic devices.

© 2016 Optical Society of America

## 1. Introduction

Plasmonic nanostructures can confine lights into subwavelength volumes with enhanced electric fields at the metal surfaces [1,2]. In today’s optical technologies, the application of plasmonic devices may be hindered by many reasons. One is that the free electrons in metal would cause serious ohmic losses, which cannot only reduce the electric field enhancements, but also the propagation lengths. One strategy is to use low-loss material instead of metal to build photonic devices. In the infrared region, dielectrics with phonon polaritons are promising candidates. Polar dielectrics normally have phonon modes, near the resonant frequencies of which the permittivities may change signs and become negative. This leads to so-called phonon polaritons [3]. Like surface plasmon polaritons in metal, the electric fields can be confined at the surface with enhanced intensities.

Phonon polaritons are often regarded as counterparts to plasmon polaritons for photonic applications in the infrared region. They have the ability to confine infrared lights into deep subwavelength volumes [4–6], create large field enhancements [7,8], and achieve strong light-matter interactions [9–12]. The lights excite the lattice vibrations in the crystal structure leading to charge oscillations at the material surface. Unlike plasmon polaritons relying on free electrons of metal, phonon polaritons often involve different classes of materials, like semiconductors and insulators. Thus, improved functionalities such as low losses and high quality factors can be expected in the photonic devices built from these materials. Phonon polaritons can be mainly classified into two kinds: one is surface-confined, and the other is volume-confined. Surface-confined phonon polaritons usually only have maximum electric field intensities at the material surface. Volume-confined phonon polaritons, however, have strong electric fields in the bulk material. Phonon polaritons in photonics usually lead to two effects: (1) in isotropic polar dielectrics, the permittivity of the material may become negative, supporting surface-confined phonon polaritons; (2) in anisotropic polar dielectrics, components in the permittivity tensor may have opposite signs, supporting volume-confined phonon polaritons. The anisotropic materials in the second case are so-called hyperbolic materials, whose iso-frequency surfaces of the electromagnetic modes are hyperboloids near the phonon resonances.

Recently, it is discovered that hexagonal boron nitride (h-BN) is a natural hyperbolic material in the Reststrahlen bands [13]. 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; and in-plane E_{1u} phonon mode, enabling type II hyperbolic response [11]. Such phonon modes lead to negative permittivity components in the dielectric tensor. Compared with artificial hyperbolic metamaterials which often suffer from huge ohmic losses and low-yield fabrication processes, these naturally formed hyperbolic materials may be more useful in real-world applications. In recent years, using near-field optical microscopic the propagation of phonon polaritons has been observed both in thin h-BN films and nanowires [14–18]. Even the hybrid modes in h-BN and graphene composites have been investigated, showing promising characteristics [19–25].

Despite the fact that h-BN is a hyperbolic material which with no doubt supports volume-confined phonon polaritons. However, surface-confined phonon polaritons can exist at the boundaries. Such modes only occur at the discontinuities of the material. In this paper, we numerically investigate the surface-confined phonon polaritons at the edges of thin h-BN films, which are referred to as edge phonon polaritons. The effective indexes of these modes indicate that they have much lower losses compared with the volume-confined modes. Edge phonon polaritons in h-BN nanoribbons are also investigated. Nanoribbon has two edges. Thus, two edge modes would couple with each other, further reducing the imaginary part of the effective index, i.e., reducing the loss. Then, a deep discussion about the surface-confined and volume-confined phonon polaritons is given, further clarifying the differences between these two kinds of modes. In reality, the edges of h-BN films may be imperfect, and the sidewalls may have rough surfaces. In the end, we also consider the effects of structural defects as well as substrates on the edge phonon polaritons. Our results may be useful in the application of h-BN based photonic devices.

## 2. Hyperbolicity in h-BN

h-BN is a layered material where hexagonal boron nitride planes are combined through Van der Waals forces. The schematic of h-BN is shown in Fig. 1(a), where the axis perpendicular to the boron nitride plane is called c-axis. h-BN has two phonon modes related to hyperbolicity in the infrared. One is out-of-plane A_{2u} phonon mode, which has transverse optic mode frequency ${\omega}_{TO}=780c{m}^{-1}$ and longitudinal optic mode frequency ${\omega}_{LO}=830c{m}^{-1}$ [11]. Between these two mode frequencies, the permittivity parallel to the c-axis becomes negative while the permittivity perpendicular to the c-axis remains positive (${\epsilon}_{\parallel}<0$,${\epsilon}_{\perp}>0$), as indicated by the blue and red curves in Fig. 1(b). The other one is in-plane E_{1u} phonon mode, which has transverse optic mode frequency ${\omega}_{TO}=1370c{m}^{-1}$ and longitudinal optic mode frequency${\omega}_{LO}=1610c{m}^{-1}$ [11]. Between these two mode frequencies, the permittivity parallel to the c-axis becomes positive while the permittivity perpendicular to the c-axis is negative (${\epsilon}_{\parallel}>0$,${\epsilon}_{\perp}<0$), as indicated in the same figure. In the calculation of the edge modes, we took the formulas in [11] to derive the permittivities of h-BN, which is

## 3. Edge phonon polaritons at the boundaries of thin h-BN films

We first present the edge phonon polaritons at the boundaries of thin h-BN films. The calculations were performed using a mode solver based on finite element method (FEM). h-BN was modeled as an anisotropic dielectric film with a thickness of several nanometers. In simulation, the whole area was surrounded by absorbing boundaries. Figure 2(a) shows the electric field intensity profile of the edge phonon polariton mode we have found at the boundary of a 5 nm-thick h-BN film at 1400 cm^{−1}. This mode propagates along z axis (not plotted in Fig. 1(a)) with fields equally concentrated at the top and bottom corners of the edge, as shown in the x-y plane. The thickness is roughly 1400 times smaller compared with the corresponding excitation wavelength which is about 7 μm. There hardly exist any fields leaking into the bulk material. The electric fields are tightly confined at the surface. In calculation, we have compared these modes with other volume-confined modes, and found that such surface-confined edge modes indeed had lower losses [14,15]. The effective indexes of the edge modes with different film thicknesses are plotted in Figs. 2(b) and 2(c). Figures 2(b) and 2(c) respectively indicate the real (denoted as *n _{eff,r}*) and the imaginary (denoted as

*n*) parts of the indexes. As we can see, the boundary of h-BN film can support edge phonon polaritons even if the thickness is only 1nm. From Fig. 2(b), we may have two conclusions: one is that the effective index increases its real part as the frequency becomes higher; the other is that the edge modes in thinner films have larger mode effective indexes. This implicates that one can control the wavelength of the edge phonon polariton mode by carefully choosing the thickness of the film. In all cases, the corresponding excitation wavelength is about 7 μm. The phonon polariton modes have wavelengths that are two orders of magnitude smaller. Thus, the mode areas are roughly more than 10

_{eff,i}^{4}times smaller than their diffraction limits. These edge modes can confine infrared lights into extreme subwavelength volumes, more importantly, with low propagation losses. Figure 2(c) shows the imaginary parts of the effective indexes corresponding to the modes in Fig. 2(b). In all cases, the imaginary part reaches the lowest value at some frequency near the center of this range, while it becomes larger at the low as well as the high frequency end. The increase of the imaginary part near 1370 cm

^{−1}is caused by the material loss. Approaching the resonant frequency, the imaginary part of the permittivity reaches its maximum, so does the loss. The increase of the imaginary parts near 1550 cm

^{−1}is, however, caused by the dispersion. Normally, the effective index increases its real part and inevitably also increases its imaginary part [2]. Physically speaking, the permittivity perpendicular to the c-axis approaches zero from a negative value, weakening the ability of containing electromagnetic fields at the edges. The index must increase and so does the loss. As we have seen in Figs. 2(b) and 2(c), the complex effective indexes depend on the geometric parameter of the edge, i.e., the thickness of the h-BN film. The thickness-dependence of the edge modes can be understood as follows. The electromagnetic waves at the top and bottom corners of the edge are strongly coupled. When the thickness varies, the coupling between these two waves is also strongly modified, changing both the real and imaginary parts of the effective index. In this way, the propagation wavelength and the loss of the edge mode both depend on the film thickness.

One should notice that these edge modes actually exist in the whole upper Reststrahlen band. We only plotted the modes in the spectrum from 1370 cm^{−1} to 1550 cm^{−1} because the losses are serious outside this range. Thus, we chose to neglect the high-loss range (from 1550 to 1610 cm^{−1}) and only plotted the mode effective indexes in such range where losses are relatively low. In calculation, we have searched the entire Reststrahlen band and found that thin h-BN films support edge phonon polaritons only in the upper Reststrahlen band. This can be understood as follows. In the upper Reststrahlen band, the in-plane permittivity of h-BN film is negative while the permittivity perpendicular to the film is positive. This makes h-BN thin films similar to graphene sheets. Graphene has a two-dimensional conductivity. The in-plane effective permittivity of graphene can be calculated from this conductivity and turns out to be negative. Thus, thin h-BN films and graphene sheets are very much alike in the upper Reststrahlen band. Since graphene supports edge plasmons [26], the edge phonon polaritons in thin-BN films may also be expected.

We used the parameters of bulk h-BN in the calculations. As the thickness of h-BN film decreases, this layer contains only a few atomic layers. Although large deviations of the permittivities as well as mode effective indexes may occur, our results may still be meaningful. These edge modes only depend on the in-plane phonon mode [18]. For few-layer h-BN, the out-of-plane phonon mode may be largely affected but the in-plane phonon mode may still be preserved. To strictly proof the existence of these modes in few-layer h-BN, one needs a new calculation method involving quantum mechanics. However, it is highly possible that these edge modes still exist even in monolayer h-BN sheets.

In order to quantify the transmission performance of these modes, we calculated the ratio of the real to the imaginary part of the effective index (denoted as *n _{eff,r}/n_{eff,i}*), see Fig. 3(a). This ratio describes how far measured by its wavelength can the phonon polariton mode propagates before it loses its energy. The edge modes in films with thicknesses in the range from 1 nm to 5 nm all possess such ratios over 20, which mean that edge phonon polaritons have excellent transmission performances even if the thickness of the film is only 1nm. Such deep subwavelength confinements together with such high transmission performances are rather rare in plasmonics [2]. Figure 3(b) shows the effective index at specific frequency as a function of the film thickness. Blue and red curves respectively denote the real and imaginary parts of the index. As the film becomes thinner, the real part increases, while the imaginary part decreases. Such abnormal behavior can be explained by Fig. 2(c). At 1400 cm

^{−1}, the edge mode in the film with a thickness of 1nm has the lowest loss. Compared to the volume-confined phonon polaritons in h-BN nanowires with round cross-sections, the losses of these edge phonon polaritons are significantly smaller. The complex effective indexes of the phonon polariton modes in solid suspended h-BN nanowires can be found in [14], where the imaginary and real parts are comparable. Thus, the losses of these phonon polariton modes are serious. Such nanowire configuration might hardly be useful in real-world photonic devices. The corresponding ratio

*n*is about 4 for a suspended solid h-BN nanowire with a diameter of 40nm, which means that the phonon polaritons in such nanowire can only propagate by few wavelengths. As the radius decreases, the loss would become more significant. However, edge phonon polaritons in this paper have ratios

_{eff,r}/n_{eff,i}*n*over 20, even if the thickness of the film is only 1 nm. Such distinguished characteristic of low loss is a key finding in our investigations.

_{eff,r}/n_{eff,i}## 4. Edge phonon polaritons in h-BN nanoribbons

In nanoribbons, two edge modes are coupled with each other. In this section, we present the edge phonon polaritons in h-BN nanoribbons. Figure 4(a) is the schematic of our h-BN nanoribbon; the enlarged view shows its molecular structure. There are two geometric parameters: width denoted as *d* and thickness denoted as *t*. The electric field intensity profile of the edge mode at 1400 cm^{−1} in a nanoribbon with $t=5nm,d=25nm$is shown in Fig. 4(b). The fields are concentrated at the four corners, and almost no fields exist in the bulk material. We have calculated the edge modes in two nanoribbons: one has $t=5nm,d=25nm$, the results are shown in Fig. 4(c) (labeled as 5nm, 25nm); the other has $t=1nm,d=5nm$, the results are shown in Fig. 4(d) (labeled as 1nm, 5nm). Blue and red curves respectively indicate the real and the imaginary parts of the effective indexes. From these figures, one can see that the mode effective indexes in both cases increase as the wavenumber approaches 1550 cm^{−1}, just as the cases in Fig. 2. To quantify the transmission performance of these edge modes, we plot the ratio of the real to the imaginary part of the effective index in Fig. 4(e). For both nanoribbons, despite of their huge differences in size, the ratio *n _{eff,r}/n_{eff,i}* can all reach over 30 in some frequency ranges. Compared with the data in Fig. 3(a), we conclude that the edge phonon polaritons in h-BN nanoribbons have better transmission performances than those at the boundaries of thin h-BN films, i.e., the losses of these modes are relatively lower. In nanoribbons, two edges are close enough for causing the coupling between these two edge modes. Such coupling leads to relatively more electromagnetic fields spreading outside the material. Thus, the material loss can be further reduced. In the end of this section, we show in Fig. 4(f) the effective index as a function of the width in a nanoribbon with $t=5nm$, and the wavenumber is set at 1400 cm

^{−1}. When the width increases, both the real and the imaginary parts of the index decrease, while the ratio

*n*roughly remains unchanged.

_{eff,r}/n_{eff,i}Again, the losses of edge phonon polaritons in h-BN nanoribbons are significantly lower than volume-confined phonon polaritons. The ratio *n _{eff,r}/n_{eff,i}* can reach over 30 even if the thickness of the nanoribbon is only 1 nm. In h-BN nanowires, such ratio is only about 4 [14]. As the diameter decreases, the loss would become more significant, and this ratio would be further reduced. Thus, surface-confined edge phonon polaritons in h-BN nanoribbons indeed possess much better transmission performances than the volume-confined phonon polaritons in bulk h-BN.

## 5. Discussion

In this section, we provide more details about the volume-confined and surface-confined phonon polaritons in h-BN. In h-BN nanowires, phonon polariton modes are found [14,15]. However, these modes are volume-confined and differ from the surface-confined edge phonon polariton modes in this paper. Actually, we have analytically calculated the phonon polariton modes with zero azimuthal indexes in suspended h-BN nanowires and found multiple modes. By drawing the complex effective index as a two-dimensional map (x axis represents the real part, while y axis represents the imaginary part of the index), we found that the locations of these modes on such map are approximately in a straight line. Figure 5(a) shows the complex effective indexes of the phonon polariton modes in a solid h-BN nanowire with a diameter of 200 nm. The wavenumber is 1400cm^{−1}. Only the first three modes are presented. The locations of the modes are marked and indicated by the black arrows. A black dash line is drawn through all three modes. The corresponding electric field intensity distributions of these three modes are respectively plotted in Figs. 5(b)-5(d). This reminds us that these modes are actually of the same kind. One might notice that the first mode (as shown in Fig. 5(b)) has electric fields confined at the nanowire surface. It is exactly the phonon polariton mode that has been widely studied by many researches [14,15]. Although most of the electric fields are confined at the surface, we still consider this mode as a volume-confined one. There are mainly two points that lead us to such conclusion: one is that it is merely the lowest order mode, there are no intrinsic differences compared with the high order ones which are clearly volume-confined modes; two is that such mode disappears if the nanowire contacts with a thick h-BN slab, the electromagnetic fields would leak into the bulk material. Moreover, the ratios *n _{eff,r}/n_{eff,i}* of these three modes are rather similar.

As for edge phonon polariton modes, situations are quite different. Edges are actually the boundaries of half-infinite h-BN films. Thus, the edge phonon polariton modes are certainly surface-confined; otherwise, they would leak into the bulk material. Here, we show the phonon polariton modes in a h-BN nanoribbon with $t=5nm,d=25nm$. The wavenumber is 1400cm^{−1}. Both the volume-confined and surface-confined phonon polariton modes exist. Figure 5(e) is the electric field intensity distribution of a volume-confined mode. Most of the fields are inside the material. Thus, the loss is serious. The complex effective index of this mode is 12.7 + 194.4*i*. Practically, such mode cannot propagate along the nanoribbon. For the surface-confined mode, the electric field intensity distribution is plotted in Fig. 5(f). It is exactly the mode investigated in this paper. One can see that most of the fields are at the surface, and almost no fields are inside the bulk material. Since material loss is the only mechanism that compromises the transmission performance, such surface-confined modes with no doubt have lower losses. The complex effective index of this mode is 38.8 + 2.0*i*. Clearly, the surface-confined mode has much larger ratio *n _{eff,r}/n_{eff,i}* than the volume-confined one. Based on these arguments, we consider the edge modes in this paper as surface-confined phonon polariton modes, which are different from the volume-confined ones.

## 6. The effects of defects and substrates on the edge phonon polaritons

#### 6.1 Defects

We created a 2 nm-wide step at the edge of a 4 nm-thick h-BN film as a defect. The electric field distributions of the edge modes are strongly modified. The upper panel in Fig. 6(a) is the electric field intensity profile of the edge mode in a perfect h-BN film. The fields are symmetrically confined at the top and bottom corners. For the edge with a 2 nm-wide step-like defect, the fields are mostly confined at the thinner part and partially be scattered into the bulk material (see the lower panel in Fig. 6(a)). The corresponding effective indexes are plotted in Fig. 6(b), where the solid and dash curves respectively denote the real and imaginary parts. As we can see, such defect can largely affect the edge modes. As the fields are mostly confined at the thinner part of the edge, the real part of the mode effective index should become larger. It is similar to the case where the thickness of the film is reduced (see Fig. 2(b)). This can also cause the increase of the imaginary part, i.e., the increase of the loss. The scattering of the fields into the bulk h-BN is another mechanism leading to more losses.

As in nanoribbon, such defect can occur at one or both edges. The edge modes are also strongly affected. Electric field intensity profiles of these edge modes are shown in Fig. 6(c). In this figure, the nanoribbon in (i) is perfect, while there are defects on (ii) one and (iii) two sides of the nanoribbon. With defects, the electric fields are mostly confined at the thinner parts. The corresponding effective indexes of these edge modes are shown in Fig. 6(d). As we can see, such defects can lead to the increases of both the real and imaginary parts of the indexes, especially at the high-frequency end of the upper Reststrahlen band.

We further point out that the roughness of sidewalls in h-BN films may have crucial effects on the propagation of the edge modes. To simulate a rough sidewall, we have designed a nanoribbon with serrated edges and compared the propagation of the edge modes in such system with that in a perfect nanoribbon. The results are shown in Fig. 6(e). The left panel shows the *E _{z}* components of the phonon polariton modes propagating along a perfect nanoribbon. Wave-like field distributions are rather obvious. Enlarged view shows the structural details where the edge is smooth. The right panel, however, is a nanoribbon with serrated edges. The structural details are shown in the enlarge view where triangles are etched away from the material creating rough sidewalls. The

*E*components of the phonon polariton modes propagating along such nanoribbon show no wave-like field distributions. The edge mode can only pass through few such defects, and the propagation loss is serious. The triangles that we etched away from the material were extremely small compared to the wavelength of the edge mode. Thus, our design is a good approximation for the rough sidewalls in h-BN. As we have seen, such roughness may cause serious damages to the propagation of the edge phonon polariton modes.

_{z}If the boron nitride sheets are not in registry, the phonon modes may be largely affected or even be destroyed. However, as long as the in-plane phonon mode exists and the in-plane relative permittivity is negative, thin h-BN films and nanoribbons still support edge phonon polariton modes.

#### 6.2 Dielectric and metallic substrates

In the presence of substrates, the edge modes are also largely modified. We investigated the edge modes in 4 nm-thick h-BN films on Au, Si and SiO_{2} substrates. The corresponding electric field intensity profiles are shown in Fig. 7(a). For Au substrate, the metal would reflect almost all of the electric fields and lead to field concentrations in the h-BN slab. For dielectric substrates, the effects depend on the material. High refractive index material such as Si can also lead to field concentrations in the h-BN slab, while low refractive index material such as SiO_{2} seems to have little effects on the field distributions. The corresponding effective indexes are shown in Fig. 7(b). Compared with the suspended h-BN film, SiO_{2} substrate indeed does not affect the edge modes very much. Only at the high frequency end can the SiO_{2} substrate lead to small increases of both the real and imaginary parts. For high refractive index dielectric and metallic substrates, the edge modes are significantly modified; the wavelengths of the edge modes become much shorter.

We further investigated the effects of substrate roughness on the edge modes in h-BN nanoribbons. The rough surfaces are created by etching the smooth material surface with nanopores. The electric field intensity profiles are shown in Fig. 7(c), and the corresponding effective indexes are plotted in Fig. 7(d). Solid and dash curves respective indicate the real and imaginary parts of the effective indexes. We only consider the cases of Si and SiO_{2} substrates with smooth as well as rough surfaces. For SiO_{2} substrate, the edge mode in h-BN nanoribbon seems to be unaffected, and the electric field distributions look like those in suspended h-BN nanoribbons. The roughness of the SiO_{2} surface also does not change the field distributions very much, see (i) and (ii) in Fig. 7(c). As shown in Fig. 7(d), the effective indexes of the edge modes on smooth and rough SiO_{2} substrates are quite close. However, as for Si substrate, the roughness could have significant impacts. The electric field intensity distributions of the edge modes on smooth and rough Si substrates are rather different, see (iii) and (iv) in Fig. 7(c). The roughness of the Si surface would cause serious scattering of the fields leading to more losses. As shown in Fig. 7(d), the effective indexes of the edge modes on smooth and rough Si substrates are quite different. One can see that the imaginary part of the effective index is much higher on rough Si substrate (the dash pink curve in Fig. 7(d)). In application, dielectric substrate with low refractive index may be a good choice since the roughness of the surface is not crucial to the propagation of the edge modes.

## 7. Conclusion

In this paper, we have investigated the edge phonon polaritons at the boundaries of thin h-BN films and nanoribbons. These modes are surface-confined and exhibit much lower losses than volume-confined phonon polaritons. The electric fields are concentrated around the corners with almost no intensities in the bulk. Thus, the field confinements as well as the transmission performances can both be largely improved. Structural defects can significantly affect the propagation of these edge modes, which should be removed as much as possible. The substrate can also modify the field distributions and the effective indexes. For a dielectric substrate with low refractive index, the roughness of the material surface is not crucial to the propagation of the edge modes. In practical systems, these edge modes can be excited in many ways, for example, by electromagnetic scattering from grating structures or tips which have already been investigated in plasmonics. Our proposed surface-confined edge phonon polaritons may become beneficial in the application of h-BN based nanophotonic devices such as infrared nanowaveguides and nanoantennas.

## Funding

Startup Fund of Hangzhou Dianzi University (No. KYS075616011).

## Acknowledgments

We appreciate the insightful discussions through emails with Prof. Behnood Ghamsari. Dr. Qing Hu (Department of Mechanical Engineering, Massachusetts Institute of Technology) also offered help during the calculation.

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