Energy of Fröhlich surface optical (SO) phonon in quasi-one-dimensional (Q1D) nanostructures remains doubtful in terms of Raman and photoluminescence experimental data. Based on a notion of the curvature proposed, the confusion is clearly clarified. It is found that the energy interval of SO modes previously accepted in the quantum system could be further divided into two sub-intervals based on the positive and negative curvature of nanowire (NW) and nanohole (NH). Furthermore, the cutoff energy and width of energy sub-intervals in NW and NH can be modulated by altering the dielectric constant of the surrounding medium. Moreover, the physical mechanism of curvature and dielectric effects on the energies of SO phonon in NW and NH are comprehended reasonably from a perspective of electrostatic potential distribution. The calculated energies of SO modes in low-energy sub-interval are fully consistent with the Raman and PL experimental results for AlN, GaN, and InN NWs. It is predicted that SO modes of high-energy sub-interval could be observed in the NH structure. The current theoretical scheme and numerical results not only extend and deepen the knowledge of the energy of the SO phonon but also can be used in the design and development of optical and optoelectronic devices based on SO modes of Q1D nanostructures.

## I. INTRODUCTION

The significance of comprehending the vibrating properties of crystal material lies in the involvement of phonons in numerous physical processes, ranging from thermal and electrical transport to optoelectronic features, superconductivity mechanisms, and phase transitions.^{1–4} In addition, recent theoretical predictions and experiments have led to the discovery and verification of phonon angular momentum and chiral phonons, which can be utilized for precise measurements of the gyromagnetic ratio, investigations of spin–phonon coupling, and the study of the thermal Hall effect.^{5,6} In low-dimensional nanostructures, apart from the ordinary longitudinal optical (LO) and transverse optical (TO) phonon modes, the surface optical (SO) phonon modes and related other confined phonon modes usually emerge due to breakdown of symmetry.^{7} The SO phonon modes play a crucial role in understanding interface thermal conductance,^{8} electrical transport and optoelectronic properties^{9,10} and even interface superconductivity.^{11}

Since the pioneer work by Fuch and Kleivers^{12} on the Fröhlich SO phonon modes in ionic crystal slabs under the condition of long wavelength (phonon wavelength *λ*_{ph} is far larger than constant of crystal lattice *a*_{c}, *λ*_{ph} ≫ *a*_{c}^{13}), more insight into the properties of SO mode in nanostructures has been gained.^{14–24} For example, the energies of the SO mode are between those of bulk TO and LO phonon modes, i.e., *ℏω*_{SO} ∈ [*ℏω*_{TO}, *ℏω*_{LO}]. SO modes in nanostructures with a perfect surface are not easily observed in experiments due to the constraints of momentum conservation. Moreover, the behaviors of the SO mode are quite sensitive to the surface situations, such as the dielectric medium, surface morphology, and size of nanostructure.

However, the comprehension of SO phonon characteristics, influencing factors, and physical nature in low-dimensional quantum structures remains incomplete. For example, Raman and PL spectra experiments conducted on ordinary nanowires (NWs)^{25–32} have consistently shown that the energies of SO modes fall within the low-energy sub-interval of [*ℏω*_{TO}, *ℏω*_{Lim}] (*ω*_{Lim} is a characteristic cutoff frequency and *ω*_{TO} < *ω*_{Lim} < *ω*_{LO}). Whereas SO modes within the high-energy sub-interval of [*ℏω*_{Lim}, *ℏω*_{LO}] have been rarely observed, and the underlying reasons for this observation remain unclear. In addition, the physical mechanism by which dielectric media reduce the energy of SO phonon modes is not fully understood and discussed.^{15,21,22} These issues are crucial for comprehending phonon properties and for the design and application of optoelectronic devices utilizing SO phonons, thus necessitating further physical clarification.

In general, nanoscaled quasi-one-dimensional (Q1D) nanostructures fall into two categories, i.e., positive curvature nanowire (NW) and negative curvature nanohole (NH).^{33–35} While the SO phonon modes in positive curvature NWs have been extensively studied,^{22–32} those in negative curvature NH structures remain unexplored. Moreover, NHs with negative curvatures hold immense potential in solar cell, lasing, and other optoelectronic applications.^{36–38} This work introduces the concepts of positive and negative curvatures to analyze and explain the energy interval of SO phonon modes in Q1D nanostructures. Furthermore, a perspective on the spatial distribution of the electrostatic potential is offered to elucidate the influence of dielectric media on the energy of SO modes.

In fact, curvature effect on surface phonon and plasmon frequencies,^{39} phonon thermal conductivity of dielectric NWs,^{40} dispersive properties of nanotubes,^{41} and phonon polaritons in van der Waals layered structures^{42} were studied. However, curvature effect was not addressed in relation to the energy intervals of SO phonon modes in Q1D nanostructures. The experimental results indicate a remarkable decrease in the frequencies of SO modes in quantum systems as the dielectric constant of the medium increases.^{15,21} Despite these findings, the microscopic mechanism and physical nature of these features remain incompletely understood. This knowledge gap highlights the significance of the topic chosen for discussion in this article.

The significance of this work lies in three primary aspects. First, it divides the traditionally accepted frequency intervals of SO modes, [*ω*_{TO}, *ω*_{LO}], into two distinct sub-intervals: [*ω*_{TO}, *ω*_{Lim}] and [*ω*_{Lim}, *ℏω*_{LO}]. These sub-intervals correspond to the positive curvature of NW structures and the negative curvature of NH structures, respectively. Notably, the cutoff frequency point *ω*_{Lim} marks the limit frequency of SO modes in planar heterostructures (HS), thereby enriching and expanding our understanding of SO phonon frequency intervals in Q1D nanostructures. Second, it is observed that the cutoff frequency *ω*_{Lim} decreases as the dielectric constant of the surrounding medium increases. This observation suggests the possibility of regulating *ω*_{Lim} through manipulation of the dielectric constant of the medium. Finally, the frequency of the SO modes is significantly influenced by the curvature and dielectric constant of the medium. To gain a deeper understanding of this phenomenon, we offer a comprehensive analysis and discussion rooted in the spatial distribution of the phonon potential. This fresh perspective offers valuable insights into the physics behind the modulation of SO mode frequency by nanostructure curvature and medium dielectric properties.

## II. THEORY

*R*are located in a dielectric medium with dielectric constant of

*ϵ*

_{d}. Within the framework of dielectric continuum model (DCM), electrostatic potential of all the phonon modes should satisfy the Laplace equation,

*ϵ*(

*ω*)∇

^{2}Φ(

**r**) = 0, based on the Maxwell equations for free oscillations [free charge density

*ρ*

_{0}(

**r**) = 0].

^{7,16,17}Under cylindrical coordinates (

*ρ*,

*φ*,

*z*), the Laplace equation of phonon potentials for the anisotropic nitride NW is given by

*ϵ*

_{d}is the dielectric constant of the nonpolar dielectric matrix, and

*ϵ*

_{z}(

*ω*) [

*ϵ*

_{t}(

*ω*)] is the dielectric function in the direction along (perpendicular to) the

*c*axis of the wurtzite material, which is given by

*ω*

_{z,L},

*ω*

_{z,T},

*ω*

_{t,L}, and

*ω*

_{t,T}are the zone center characteristic frequencies of A

_{1}(LO), A

_{1}(TO), E

_{1}(LO), and E

_{1}(TO) modes, respectively. The Fröhlich SO phonon states in wurtzite positive curvature NW and negative curvature NH systems are given by

*m*is the azimuthal quantum number,

*k*

_{z}is the free wave-number in

*z*direction, and

*I*

_{m}(

*x*) and

*K*

_{m}(

*x*) are the first and second kind modified Bessel functions, respectively. The

*γ*function in Eqs. (4) and (5) is defined as

^{17}

^{,}

*ϵ*

_{t}(

*ω*) and

*ϵ*

_{z}(

*ω*) are the dielectric function in the radial and axial directions,

^{16,17}respectively. The oscillating and decaying features of phonon states are directly related to the sign of

*γ*(

*ω*). Taking into account the decaying properties of the SO phonon state on both sides of the heterostructure, the sign of

*γ*(

*ω*) should be positive. Hence, the SO modes can appear in the energy interval of [

*ℏω*

_{t,TO},

*ℏω*

_{z,LO}] via the figure of

*γ*(

*ω*) ∼

*ω*shown in Fig. 2(a). This is just the mathematical origin that the SO mode frequencies fall into the frequency interval [

*ω*

_{t,TO},

*ω*

_{z,LO}].

^{17}

*ρ*=

*R*will lead to the following equations:

This Eq. (7) [(8)] is just the dispersive equation of SO modes in positive curvature NW (negative curvature NH) structures.

*H*

_{e–SO}in positive curvature NW and negative curvature NH structures, and they are given by

*e*denotes an electron charge.

## III. NUMERICAL RESULTS AND DISCUSSIONS

Now, let us discuss the curvature and dielectric effects on the dispersive spectra and electrostatic potential distributions as well as the physical mechanism and essence behind these observations for the SO modes in Q1D nanostructures. The typical wurtzite nitride semiconductors are chosen to perform numerical calculation. The material parameters of AlN, GaN, and InN are presented in Table I.^{43,44}

Material . | ω_{tT}
. | ω_{zT}
. | ω_{tL}
. | ω_{zL}
. | ϵ_{∞}
. | ϵ_{0}
. |
---|---|---|---|---|---|---|

GaN | 69.25 | 65.91 | 91.83 | 90.97 | 5.35 | 9.20 |

AlN | 83.13 | 75.72 | 113.02 | 110.3 | 4.77 | 8.50 |

InN | 59.02 | 55.92 | 73.40 | 72.90 | 8.40 | 15.30 |

Material . | ω_{tT}
. | ω_{zT}
. | ω_{tL}
. | ω_{zL}
. | ϵ_{∞}
. | ϵ_{0}
. |
---|---|---|---|---|---|---|

GaN | 69.25 | 65.91 | 91.83 | 90.97 | 5.35 | 9.20 |

AlN | 83.13 | 75.72 | 113.02 | 110.3 | 4.77 | 8.50 |

InN | 59.02 | 55.92 | 73.40 | 72.90 | 8.40 | 15.30 |

Due to the fact that SO modes with smaller azimuthal quantum numbers, such as *m* = 0, in contrast to those with larger *m* values (*m* ≥ 1), are easily observed in the experiments,^{14,15,19,45} the lowest azimuthal quantum number *m* = 0 is considered in the following discussion. Physically, the smaller the azimuthal quantum number, the higher the symmetry of the phonon potential, and thus the easier the phonon modes are likely to be detected experimentally. Based on the similar characteristics of the electrostatic potential of phonon modes and the electronic wave function (both can be considered as probability density waves^{46}), this can be understood from the electronic wave function distribution of hydrogen atoms. When the azimuthal quantum number *m* is 0, the wave function of hydrogen atom becomes almost spherical symmetry with the highest symmetry; while as *m* increases, the symmetry decreases significantly.^{47} This is the physical reason that we choose the SO modes with *m* being 0 in the following discussion.

*ℏω*

_{SO}of the SO modes in AlN, GaN, and InN NW (red curve) and NH (blue curve) as a function of

*k*

_{z}

*R*are shown in Figs. 2(b)–2(d), respectively. The nitride NWs and NHs are placed in vacuum, i.e.,

*ϵ*

_{d}= 1. From Fig. 2(b), it is found that the blue curve is always above the red one. This means that dispersive energy of the SO mode in NH is usually higher than that in NW. Moreover, the dispersive energy of SO modes in NH is a monotonic and degressive function of

*k*

_{z}

*R*, while that in NW is a monotonic and incremental function of

*k*

_{z}

*R*. As

*k*

_{z}

*R*is quite large, both the blue and red curves approach a certain limited characteristic energy

*ℏω*

_{Lim}, 106.6 meV. This clearly reveals that in terms of the positive and negative curvatures, the ordinary energy interval [83.13,110.3] meV of SO modes in AlN nanostructure are divided into two sub-intervals, [83.13,106.6] and [106.6, 110.3] meV. The former sub-interval corresponds to the positive curvature NW, and the latter one corresponds to the negative curvature NH. In fact, the limited characteristic energy

*ℏω*

_{Lim}for quite large

*k*

_{z}

*R*can be obtained via Eqs. (7) and (8). Based on the limited properties of the Bessel functions,

*I*

_{m}(

*x*) and

*K*

_{m}(

*x*), i.e.,

*I*

_{m}(

*x*) and

*K*

_{m}(

*x*), respectively, will approach $ex/2\pi x$ and $\pi e\u2212x/2x$ as

*x*→ ∞.

^{48}Using these relations, both the dispersive Eqs. (7) and (8) of SO modes in NW and NH will be reduced to the following concise form, i.e.,

^{43}Hence, the cutoff point frequency of the positive and negative curvature is exactly the frequency of the SO modes in a planar HS. This clearly explains the mathematical reason why the dispersive energies of SO modes in NW and NH approach the energy of SO modes in a planar HS. From a viewpoint of physics, it is also a natural result. In fact, for a certain radius

*R*of NW and NH, as the phonon wave-number

*k*

_{z}is quite large, this means that the phonon wave-length

*λ*

_{ph}becomes quite small, i.e.,

*λ*

_{ph}≪

*R*. In this situation, the phonons cannot feel and distinguish the curving degree and sign (positive and negative) of NW and NH; they only feel the plane heterostructure. This is just the physical reason that both the dispersive energies of SO modes in positive curvature NW and negative curvature NH approach the result in planar HS.

^{43}

Substituting the dielectric function $\u03f5i(\omega )=\u03f5\u221e(\omega 2\u2212\omega i,LO2)/(\omega 2\u2212\omega i,TO2)$ (*i* = *t*, *z*) into Eq. (17), the cutoff frequency *ω*_{Lim} can be worked out. For wurtzite AlN HS located in vacuum, the value of $\omega LimAlN$ is equal to ∼106.60 meV. As stated above, the energies of SO modes in the Raman and PL experiments^{18,25,26} of AlN NW and nanorod positive structures are within the sub-interval of [$\omega t,TOAlN,\omega LimAlN$], i.e., [83.13, 106.60] meV. For example, Sahoo *et al.*^{18} observed an SO phonon mode at 850 cm^{−1} ($\u223c105.4$ meV) in AlN nanotips by using the Raman spectroscopy technique. One scholar of our group^{25} grew and measured the AlN NW grown on Si substrates, and a SO phonon mode with energy of about 100 meV was found in the experiment of PL spectra. Recently, Gacević *et al.*^{26} fabricated and measured the AlN NWs, and several phonon replicas with energies $\u223c103$ meV were found in the PL spectra. These phonon modes are ascribed to be the SO modes in the AlN NW systems.^{23} It is obvious that all these SO phonon in AlN structures are within the energy sub-interval of [$\omega t,TOAlN,\omega LimAlN$], i.e., [83.13,106.60] meV. Similarly, it is observed from Figs. 2(c) and 2(d) that, the cutoff point frequencies *ω*_{Lim} of the positive and negative curvature are 88.06 and 71.63 meV for Q1D GaN and InN nanostructures, respectively. In addition, a series of SO modes with energies in the sub-intervals [$\omega t,TOGaN,\omega LimGaN$] for GaN NWs^{27–31} and [$\omega t,TOInN,\omega LimInN$] for InN NWs^{32} were found. However, the SO phonon modes whose energies are within another sub-interval of [*ω*_{Lim}, *ω*_{z,LO}], have not been observed in AlN, GaN, and InN nanostructures by now to the best of our knowledge. Hence, we anticipate that they could be observed in the Q1D negative curvature NH structures in terms of our theories.

The dispersive energies *ℏω*_{SO} of SO modes as functions of the curvature *K* of AlN nanostructures [Figs. 3(a)–3(c)] and dielectric constant *ϵ*_{d} of dielectric medium [Fig. 3(d)] are shown in Fig. 3. Three different parameters of dielectric constants, *ϵ*_{d} = 1, 2, and 4, are chosen when drawing Figs. 3(a)–3(c), respectively. Moreover, the blue, black, and red curves respectively correspond to three different wave-numbers of SO modes, i.e., *k*_{z} = 1 × 10^{5} (blue), 1.5 × 10^{5} (black), and 2 × 10^{5} cm^{−1} (red). It is noted that the curvature *K* of NW and NH is defined as *K* = ±1/*R*, and the symbol “+” (“−”) represents the positive (negative) curvature of NW (NH). From Figs. 3(a)–3(c), it is seen that with the increase in *K*, the dispersive energies *ℏω*_{SO} of SO modes monotonously decrease from the characteristic energy *ℏω*_{t,TO} to another one *ℏω*_{z,LO}. The three curves intersect at *K* = 0. In addition, the intersection just divides the interval of [*ω*_{t,TO}, *ω*_{z,LO}] into two sub-intervals, which correspond to the sub-intervals for the SO mode in NH and NW, respectively. It is obvious that the cutoff point frequencies *ω*_{Lim} is determined by Eq. (17). It is interesting to note that for NW (NH) with a positive (negative) curvature, the red (blue) curves are always above on the blue (red) ones, and the black ones are in the middle. This shows that for a certain *K*, the energies of the SO phonon mode in NW (NH) increase (decrease) with the increase in wave-number *k*_{z}. Comparing Fig. 3(a) to Figs. 3(b) and 3(c), it is found that with the increase in dielectric constant *ϵ*_{d}, the cutoff point frequency *ω*_{Lim} decreases obviously. For example, as *ϵ*_{d} increases from 1 to 4, *ω*_{Lim} decreases from 106.60 to 98.40 meV. This is completely consistent with the experimental results in regard to the energy dependence of the SO mode on dielectric medium.^{15,21} Moreover, this also reveals that the cutoff point frequency *ω*_{Lim} of positive and negative curvature nanostructures can be regulated by the dielectric constant of a dielectric environment.

From Fig. 3(d), we observe that the dispersive energies *ℏω* _{SO} of SO modes and the cutoff point frequency *ω*_{Lim} in AlN NW and NH structures are a monotonic and degressive function of *ϵ*_{d}. The cutoff frequency point *ω*_{Lim} is the boundary of NW with positive curvature and NH with negative curvature. Moreover, the smaller the value of *k*_{z}*R*, the more obvious is the dispersion of the SO mode. On the contrary, the bigger the value of *k*_{z}*R*, the more remarkably the dispersive curves of SO modes in NW and NH approach the curve of *ω*_{Lim}. This further reveals that *ω*_{Lim} is the cutoff frequency of positive and negative curvature nanostructures.

In order to deeply understand the physical mechanism and nature of curvature influence on the dispersive energy of SO modes, the distributions of electrostatic potential $\Gamma kzSO(\rho )$ (also the electron–phonon coupling function^{16,17}) in AlN NW [Fig. 4(a)] and NH [Fig. 4(b)] as a function of *ρ* are shown in Fig. 4 when *ϵ*_{d} = 1 and *k*_{z} = 4 × 10^{5} cm^{−1}. Six different curvature parameters, *K* = ±0.5 × 10^{5} cm^{−1} (blue curves), ±0.67 × 10^{5} cm^{−1} (red curves), and ±1 × 10^{5} cm^{−1} (black curves) are chosen when drawing the figures. In each embedded graph, the peak width at half height of electrostatic potential $\Gamma kzSO(\rho )$ in the AlN material layer *d*_{AlN} and dielectric medium *d*_{Med} as a function of *K* are plotted. In Figs. 4(a) and 4(b), it is shown clearly that the electrostatic potentials of SO modes mainly localized at the surfaces of NW and NH, and they decay on both sides of the surfaces. This is just the reason that these phonon modes are named SO modes. In the embedded graphs, it is observed that, with the increase in *K*, the peak width at half height in dielectric medium *d*_{Med} decreases obviously, while that in AlN *d*_{AlN} increases remarkably. This illustrates that as *K* increases, the electrostatic potential distributions of SO modes are gradually compressed from the dielectric medium to AlN material layer, which results in the dispersive energies of SO mode decrease correspondingly.

In essence, the electrostatic potential of the SO phonon mode within a crystal can be envisioned as a surface charge density wave. This wave encompasses the entire lattice vibration of nanostructures, drawing from the principles of polaronic effect and charge–phonon interaction.^{10,49} For a heterostructure surface composed of a semiconductor and a nonpolar dielectric medium, the energy of the SO phonon modes is influenced by the charge density wave arising from lattice vibrations. Classically, electrostatic charge tends to accumulate on the outer surface of conductive materials.^{50} Consequently, the surface charge density *σ*_{NW} in NWs with positive curvature is the highest, while *σ*_{NH} in NHs with negative curvature is the lowest. The surface charge density *σ*_{PL} in planar HS falls somewhere in the middle, resulting in the following order: *σ*_{NW} > *σ*_{PL} > *σ*_{NH}. This ordering is graphically represented in Figs. 1(c)–1(e). Here, we postulate that increasing charge density wave in phonon mode leads to a decrease in dispersive energy for SO phonon modes. This suggests that the lowest energies of SO modes occur in NW, followed by plane HS, and the highest energies occur in NH (namely, $\omega SONW<\omega \u2009SOPL<\omega SONH$). In a half-infinite planar HS, the dispersive energy of the SO mode coincides with the cutoff energy for both positive and negative curvature nanostructures (i.e., $\omega SOPL=\omega Lim$), providing further evidence of how curvature affects the dispersive energy of SO phonon modes in nanostructures.^{43} This understanding sheds light on the underlying mechanism of curvature’s influence on phonon modes in these structures.

The electrostatic potentials $\Gamma kzSO(\rho )$ in AlN NW [Fig. 5(a)] and NH [Fig. 5(b)] as a function of *ρ* are shown in Fig. 5 when *K* = ±1 × 10^{5} cm^{−1} and *k*_{z} = 4 × 10^{5} cm^{−1}. The black, red, and blue curves correspond to the dielectric constants *ϵ*_{d} = 1, 2, and 4, respectively. Similarly, the peak width at half height of $\Gamma kzSO(\rho )$ in the AlN semiconductor *d*_{AlN} and that in dielectric medium *d*_{Med} as a function of *ϵ*_{d} are displayed in each embedded graph. From Figs. 5(a) and 5(b), we observe that distributions of the SO phonon potential in AlN material layer are affected more remarkably by *ϵ*_{d} than that in a dielectric medium. This can be seen quite clearly in the embedded graphs. With the increase in *ϵ*_{d}, the values of *d*_{Med} (red curves) nearly keep unchanged, but the values of *d*_{AlN} (black curves) decrease obviously. This reveals that the electrostatic potentials of SO modes, both in NW and NH, converge more and more to the surface of the nanostructures as *ϵ*_{d} increases. This directly causes the increase in surface charge density near the surface. From a purely physical perspective, this is also a natural result, i.e., the introduction of the dielectric medium increases the density of the surface polarized charge. In terms of the idea proposed above, the increase in surface charge density results in the decrease in energies of SO modes in NW and NH with the increase in *ϵ*_{d}. This conclusion also agrees with the previous theoretical results and experimental observation.^{15,21,22} This clearly explains the origin and essence of the influence of dielectric medium on the energy of SO modes in nanostructures.

It should be noted that although the fixed free wave-number *k*_{z} = 4 × 10^{5} cm^{−1} is selected in the discussion of $\Gamma kzSO(\rho )$ in Figs. 4 and 5, this does not affect the obtained conclusions in regard to the influence of the curvature and the dielectric medium on the SO phonon energy. Based on Eqs. (10) and (12), the electron–SO phonon coupling function is related the wave-number *k*_{z}. In general, the wave-number *k*_{z} of Q1D nanostructures strongly affects the peak value of the electron–phonon coupling function, but does not change the spatial distribution of the electrostatic potential.^{51} Indeed, we calculated the electrostatic potential distribution of SO modes in AlN NW and NH in the situation of some different *k*_{z} (5 × 10^{5} and 6 × 10^{5} cm^{−1}), obtaining the same conclusions as shown in Figs. 4 and 5. Therefore, we believe that the conclusions are reliable and independent on free wave-number *k*_{z}.

Finally, it is necessary to briefly discuss the concept of Fröhlich SO modes. The Fröhlich electron–phonon coupling is defined to be the interaction between electron and long-wavelength LO phonon in bulk materials.^{13} It emphasizes that the wavelength of the phonon is much greater than the lattice constant of the crystal, i.e., *λ*_{ph} ≫ *a*_{c}. Recently, the concept of Fröhlich electron–phonon coupling was extended to the Fröhlich optical phonon modes, including the LO and SO as well as other phonon modes in nanostructures.^{52–54} Here, the concept of Fröhlich SO modes also implies that the wavelength of the SO phonon is much larger than the lattice constant, so that the SO phonon modes of the system can be treated in a DCM.

## IV. CONCLUSIONS

In conclusion, this study extends and deepens the prior knowledge of Fröhlich SO phonon vibration energy in Q1D nanostructures. By considering the positive and negative curvatures of NW and NH structures, the energy range [*ℏω*_{TO}, *ℏω*_{LO}] of SO modes, which is commonly observed in low-dimensional semiconductor systems,^{13–23} can be further refined into two distinct sub-ranges: [*ℏω*_{TO}, *ℏω*_{Lim}] and [*ℏω*_{Lim}, *ℏω*_{LO}]. The low-energy (high-energy) sub-range corresponds to SO modes in NWs (NHs) exhibiting positive (negative) curvature. Notably, the cutoff frequency *ω*_{Lim} precisely aligns with the frequency of SO mode in a half-infinite planar HS, where the curvature *K* equals zero.^{43} This cutoff frequency is comprehensively analyzed from both mathematical and physical perspectives. Furthermore, the cutoff frequency *ω*_{Lim} and the widths of the SO mode sub-ranges in NW and NH can be modulated by adjusting the dielectric constant of the surrounding medium.

Furthermore, to comprehend the physical mechanism and essence by which curvature and dielectric effects influence the dispersive energies of SO modes in NW and NH, we propose that the high (low) charge density wave of the phonon mode leads to the low (high) dispersive energy of SO phonon modes. This idea is supported by the numerical results of potential distribution on AlN, GaN, and InN NW and NH. As the dielectric constant of the medium rises, the density of surface polarized charge at NW and NH surfaces also increases, leading to a decrease in the dispersive energy of SO modes. This observation aligns with previous theoretical and experimental findings.^{15,21,22} Our work offers a plausible explanation for the SO phonon energies observed in Q1D AlN,^{18,25,26} GaN,^{27–31} and InN^{32} NWs with positive curvatures in PL and Raman experiments. We hope that this work will spur further more experimental explorations on SO phonon modes and especially anticipate that the SO phonon modes of high-energy sub-interval could be observed in NHs with negative curvature. The current theoretical scheme and numerical results not only clarify the confusion of the SO phonon energy located in low-energy sub-interval for nitride NWs^{18,25–32} but also can be used in the design and development of optical and optoelectronic devices based on SO modes of Q1D nanostructures.

## ACKNOWLEDGMENTS

This work was jointly supported by the National Key R&D Program of China (Grant No. 2021YFB3600200), the Guangdong Major Project of Basic and Applied Basic Research (Grant No. 2023B0303000012), the National Key Research and Development Program (Grant No. 2023YFB04400), the Special Item of Key Areas of Colleges and Universities of Guangdong Province (Grant No. 2022ZDZX1066), the Innovation Team Project of colleges and universities of Guangdong Province (Grant No. 2023KCXTD070), and the Basic and Applied Basic Research Project of Guangzhou City (Grant No. 202102080359).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Li Zhang**: Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). **Z. W. Liang**: Data curation (equal); Investigation (equal); Methodology (equal). **Qi Wang**: Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). **J. J. Shi**: Conceptualization (equal); Supervision (equal); Validation (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

## REFERENCES

*Modern Condensed Matter Physics*

*Phonon in Nanostructures*

*Theory of Solid State*

*Handbook of Properties and Data of Advanced Semiconductor Materials*

*Quantum Mechanics*

*Bessel Functions*

*Physics*