1. Introduction

Recently, there has been a growing interest in the study of polaritons in two-dimensional materials (graphene [1–3], hBN [4,5] and MoO₃ [6–10] etc.) due to the unique properties of polaritons, such as strong electromagnetic confinement, hyperbolic dispersion and enhanced light-matter interaction at nanoscale [11–13]. Materials with reduced dimensionality [14–16] such as nanowire and ribbon, presents novel opportunities for the control of electromagnetic fields at the nanoscale, and generally achieved tunable and highly confined propagating polaritons. The recent investigation of polaritons in various one-dimensional structures, such as InAs nanowires [17,18], hBN ribbons [19,20], hBN nanotubes [21–23], and carbon nanotubes [24–26], has shown potential in facilitating polaritons with exceedingly high confinement. This, in turn, allows for stronger interaction between light and matter at the nanoscale.

Hyperbolic phonon polaritons (HPhPs) supported in anisotropic materials such as MoO₃, have recently attracted significant attention [6,8,27–29]. The in-plane hyperbolicity, significant wavelength confinement, and ultra-long lifetime of HPhPs enable unique applications such as negative refraction [30,31], nanoscale directional energy transfer [8,32], and integrated flat optics [33–35] in the field of nanophotonics. However, there have been few reports [36,37] on the study of hyperbolic phonon polaritons (HPhPs) in one-dimensional structures, except for the recently documented guided polaritons along the forbidden direction in MoO₃ nanoribbons. This discovery has unveiled the novel properties of HPhPs in one-dimensional structures. Compared with one-dimensional nanoribbons structure, rolled nanotubes, synthesized by rolling up van der Waals (vdW) materials [38–40], exhibit a continuous surface that enables the propagation of phonon polaritons in different directions.

Here, we theoretically purpose and numerical study the HPhPs in rolled one-dimensional MoO₃ nanotubes. Compared to the HPhPs in an infinite MoO₃ thin film, the HPhPs in a rolled MoO₃ tube exhibit low propagation loss and maintain the same electromagnetic confinement in the one-dimensional structure. By tailoring the rolling direction of the two-dimension MoO₃ film, we were able to achieve a tunable electromagnetic confinement of HPhPs in the one-dimensional rolled tube. Through the implementation of full-wave numerical simulations, we observed the canalized phonon polaritons mode in the rolled twisted bilayer MoO₃ nanotube, which can propagate in a spiraling manner along the curved surface of the nanotube. The spiraling forward angle of canalized phonon polaritons is determined by the rolling direction of the twisted bilayer MoO₃ thin film, allowing for a tunable electromagnetic field energy distribution, spiral direction and cycle in the rolled nanotubes. Our findings highlight the significant potential of rolled MoO₃ nanotubes as versatile platforms for advanced light manipulation and the development of nanophotonic circuits.

2. Results and discussion

MoO₃, a prototypical van der Waals material [6,7,11], is weakly bound by van der Waals forces between each layer, and displays a significant in-plane difference in lattice spacing between the [100] and [001] directions. As a result, it yields three principal dielectric values ɛ_x(ω), ɛ_y(ω), and ɛ_z(ω), corresponding to the crystallographic directions [100], [001], and [010], respectively. Therefore, in the case of a one-dimensional MoO₃ nanotube wrapped with a two-dimensional thin film as shown in Fig. 1(a), the three principal dielectric values in the cylindrical coordinate system ɛ_r(ω), ɛ_φ(ω), and ɛ_a(ω) correspond to the crystallographic directions [010], [100], and [001], respectively. Generally, the dielectric function of MoO₃ can be written as a Lorentz model [7,41]:

 

Fig. 1. Permittivity of MoO₃ thin film and rolled tube. (a) Schematic of MoO₃ thin film and rolled nanotube along [001] direction. (b) The real part of permittivity of MoO₃ thin film and rolled tube.

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The real part of permittivity of MoO₃ thin film and rolled tube were plotted in Fig. 1(b), where the related parameters of Lorentz model are obtained from the literature [7]. As the color shaded area in Fig. 1(b) shown, MoO₃ featuring three Reststrahlen bands within the infrared spectral range. The real part of permittivity in RB1 and RB2 along the [100] and [001] crystallographic direction is opposite in sign, give rise to the hyperbolic phonon polaritons (HPhPs). To investigate the phonon polaritons in the MoO₃ rolled tube, we conducted the mode analysis calculations with the commercial finite-element software COMSOL in four distinct MoO₃ structures: the rolled tube, solid tube, nanoribbon, and thin film. The rolled tube was wrapped with a 100 nm MoO₃ film and has a radius of 200 nm; the solid tube has a diameter of 100 nm; the cross-section of the ribbon is a square with a width of 100 nm; the thin film has a thickness of 100 nm. Figure 2(a)-(d) depicts the absolute value of electric filed |E| at a frequency of 920 cm⁻¹ within each structure, where the electric filed |E| localized in MoO₃ and decays rapidly far away from the surface indicating the fundamental phonon polaritons mode.

 

Fig. 2. Dispersion of HPhPs in MoO₃ rolled nanotube. (a)-(d) The absolute value of electric filed |E| distribution for the fundamental phonon polaritons mode at the frequency of 920 cm-1 in MoO₃ rolled tube, solid tube, ribbon and thin film, respectively. The scale bar is 100 nm. (e) Dispersion of MoO₃ HPhPs corresponding to the structure of rolled tube (black hollow square), solid tube (black hollow circle), ribbon (black hollow triangle) and thin film (false color plot) along the [001] (left panel) and [100] (right panel) crystallographic direction.

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To further study the phonon polaritons in the rolled nanotube, we calculated the dispersion of HPhPs for each structure along different crystal directions in air ambient. The dispersion of HPhPs in rolled tube, solid tube and ribbon were calculated by the finite element method (FEM). In contrast, the imaginary part of the Fresnel reflectivity coefficients Im(r_p) of the 100 nm thickness of MoO₃ along different crystallographic directions were calculated [42] to represent the dispersion of phonon polaritons.

As shown in Fig. 2(e), the electromagnetic confinement k_p/k₀ of HPhPs in MoO₃ solid tube and ribbon increases with reduced dimensionality. More importantly, the dispersion of HPhPs in rolled tube maintains consistency with that of the thin film layer, which contrasts with the reported results in the rolled graphene tube [43]. This consistency can be attributed to the nature of phonon polaritons (HPhPs) in MoO₃. The HPhPs in MoO₃ behave as a volume mode [14,15], thus the rolled tube exhibits an equivalent structure in the cylindrical coordinate system compared to the thin film in the Cartesian coordinate system. On the contrary, the surface plasmons in graphene are directly influenced by the surrounding field.

Another remarkable observation is that the quality factor [44,45] Q = Re(k_p)/Im(k_p) of HPhPs in rolled tube is higher than that in the thin film. Figure 3 shows that within the hyperbolic dispersion regions RB1 and RB2, the maximum value of the quality factor for HPhPs in rolled tubes, solid tubes, and ribbons along the [001] (left panel) and [100] (right panel) crystallographic directions is approximately 1.25 (50/40) times higher than that in thin films. We infer that it can be attributed to the higher surface attenuation of the thin film, since energy loss at both two infinite surfaces. Besides, we found that the maximum quality factor value (corresponding to a lowest propagation loss of phonon polaritons) occurs in the middle of Reststrahlen bands range. Therefore, we suggest that the loss is primarily composed of interface loss and intrinsic loss two components. At the lower edge of Reststrahlen bands, phonon polaritons exhibits smaller confinement, thus most of the energy dissipated from the interface into the air. Otherwise, at the upper edge of Reststrahlen bands, the intrinsic loss caused by the absorption of longitudinal optical phonon dominates, resulting in a higher loss of phonon polaritons. This remarkable lower propagation loss of phonon polaritons in MoO₃ rolled tubes enhances their potential for applications in nanophotonic devices.

 

Fig. 3. Quality factor of HH in rolled tube, solid tube, ribbon and film along the [001] (left panel) and [100] (right panel) crystallographic direction.

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Remarkably, the strong in-plane anisotropy of MoO₃ allows for the generation of diverse rolled tube configurations by altering the rolled direction angle α between the rotation axis and the [100] crystallographic direction, as depicted in Fig. 4(a). To further investigate the phonon polaritons in the MoO₃ rolled tube, we calculated the dispersion of HPhPs in the MoO₃ rolled tube and thin film along different propagation direction, respectively. Figure 4(b) shows the dispersion of HPhPs in MoO₃ rolled tube with a 30° rolled angle α (black hollow square) and MoO₃ thin film with the same angle between propagation direction and [100] direction (false color plot). Apparently, the dispersion of MoO₃ rolled tube can be tune by tailoring the rolled angle. The confinement of HPhPs in MoO₃ rolled tube with different rolled angle at the frequency of 920 cm^¬1 was illustrated in Fig. 4(c). It displays that the confinement increases with a larger rolled angle α, where the threshold angle [46] α_th= arctan $({({ - {\varepsilon_{[{001} ]}}/{\varepsilon_{[{100} ]}}} )^{1/2}})$ (α_th∼54° at the frequency of 920 cm⁻¹). The phonon polaritons can propagate with the angle smaller than α_th, within the blue shaded area in Fig. 4(d). It is also notable that a new propagation phonon polaritons mode occurs in the RB1 range due to the propagation direction k_p cross with the IFC of MoO₃ in RB1 range as illustrated in Fig. 4(d).

 

Fig. 4. HPhPs in MoO₃ rolled tube along different rolled angle α. (a) Schematic of MoO₃ rolled tube along different angle α with [100] direction. (b) Dispersion of HPhPs in MoO₃ rolled tube with a 30° rolled angle (black hollow square) and 0° rolled angle (black dashed line) and MoO₃ thin film with 30° angle between propagation direction and [100] direction (false color plot). (c) Dispersion of MoO₃ rolled tube with different rolled angle (black hollow square) and MoO₃ thin film with varied angles between propagation direction and [100] direction (false color plot) at a frequency of 920 cm-1. (d) The iso-frequency curve (IFC) of HPhPs in a 100 nm thin film at different frequencies.

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As discussion above, the phonon polaritons in MoO₃ rolled tube exhibits almost the same behavior with that in two-dimensional MoO₃ thin film, and significantly shows a low propagation loss by contrast. Therefore, the MoO₃ rolled tube shows a great application potential with enabling in-plane superlensing [32,46], negative refraction [30] application occurs in the two-dimensional MoO₃ layer due to the low propagation loss. Hence, we suggest a MoO₃ rolled tube wrapped with twisted bilayer MoO₃ thin film in order to achieve a canalized phonon polaritons [8,47] propagating in the one-dimensional nanotube structure. Figure 5(a) displays the schematic of rolled twisted bilayer MoO₃ nanotube. A twisted bilayer MoO₃ thin film with different twisted angle θ was rolling into one-dimensional nanotube with a radius of 0.6 um, the thickness of each layer is 100 nm and the rotation axis was parallel to the [100] direction. We then utilize a vertical dipole to excite the phonon polaritons in the twisted bilayer MoO₃ slabs with different twisted angle at the frequency of 920 cm⁻¹, the electric field E_z 50 nm above the slab were depicted in Fig. 5(c). Obviously, a photonic topological transition appears with the varied twisted angle θ, and the canalized phonon polaritons mode occurs at the twisted angle of 70°. In contrast, the phonon polaritons in rolled twisted bilayer MoO₃ nanotube was excited by a vertical dipole as the schematic shows in Fig. 5(b). Consequently, the two-dimensional photonic topological transition can be realized in the surface of one-dimensional rolled twisted bilayer MoO₃ nanotube. This also make it a great possible to realize the negative refractive with a graphene/MoO₃ rolled nanotube, and suggest a promising platform for arbitrary polariton manipulation in one-dimensional system. Moreover, the apparently canalized phonon polaritons was observed with the twisted angle of 70° in the MoO₃ nanotube, which propagate spiraling along the nanotube.

 

Fig. 5. Canalized phonon polaritons in a rolled twisted bilayer MoO₃ nanotube. (a) Schematic of rolled twisted bilayer MoO₃ nanotube. (b) Schematic of numerical excite the canalized phonon polaritons in a rolled twisted bilayer MoO₃ nanotube. (c) The electric field Ez with different twisted angle θ of bilayer MoO₃ slabs. (d) The radial electric field Er of rolled twisted bilayer MoO₃ nanotube with different twisted angle θ. The MoO₃ tube is rolled along the [100] direction. The scale bar in (c) and (d) is 2 um.

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Furthermore, the spiraling phonon polaritons propagate along the rolled twisted (fixed twisted angle θ=70°) bilayer MoO₃ with different rotation direction simulations were conducted. Figure 6(a) illustrated the schematic of rolled twisted bilayer MoO₃ nanotube with different rotation direction. The twisted angle θ was fixed to 70° and the angle between the [100] direction and rotation axis was defined as the rolled angle β.

 

Fig. 6. Guided spiraling phonon polaritons in rolled twisted bilayer MoO₃ nanotube. (a) Schematic of rolled twisted bilayer MoO₃ nanotube with different rolled angle β between the [100] direction. (b) and (c) The radial electric field Er and absolute value of electric field |E| for rolled twisted bilayer MoO₃ nanotube with different rolled angle β. The scale bar in (b) and (c) is 2 um.

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Undoubtedly, the spiraling phonon polaritons can be guided by the rolled angle β. Figure 6(b) and (c) display the radial electric field E_r and absolute value of electric field |E| 50 nm above the surface of nanotube. By varying the rolled angle β, the spiraling phonon polaritons propagates along the nanotube at different spiral angles, thereby allowing the tailoring of the spiral motion direction and cycle. This approach also provides a method to adjust the electromagnetic field energy spread in a long distance (β=15° in Fig. 6(c)) or localized in a short range (β=90° in Fig. 6(c)).

3. Conclusions

In conclusion, our theoretical study on hyperbolic phonon polaritons (HPhPs) in rolled one-dimensional molybdenum trioxide (MoO₃) nanotubes has revealed several key insights. We reveal that the HPhPs in MoO₃ rolled nanotube exhibit smaller propagation losses than that in the MoO₃ thin film and offer tunable electromagnetic confinement along the rolled direction. Furthermore, by rolling the twisted bilayer MoO₃ into nanotube, we have successfully achieved a canalized phonon polaritons mode in the nanotube, and a guided spiraling phonon polaritons propagate along the tube. Indeed, the rolled MoO₃ nanotube opens a new avenue for transferring the novel properties of phonon polaritons from traditional two-dimensional into one-dimensional configuration. Our findings highlight the significant potential of rolled MoO₃ nanotubes as versatile platforms for various applications in light manipulation and nanophotonics circuits. Specifically, the similar rolled nanotube with graphene/MoO₃ or black phosphorus layer structure can be employed for achieving application such as superlensing, negative refraction in one-dimensional structure, thus opening up exciting possibilities for advanced nanophotonic devices.