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.

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 properties9,10 and even interface superconductivity.11 

Since the pioneer work by Fuch and Kleivers12 on the Fröhlich SO phonon modes in ionic crystal slabs under the condition of long wavelength (phonon wavelength λph is far larger than constant of crystal lattice ac, λphac13), 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., ℏωSO ∈ [ℏωTO, ℏωLO]. 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 [ℏωTO, ℏωLim] (ωLim is a characteristic cutoff frequency and ωTO < ωLim < ωLO). Whereas SO modes within the high-energy sub-interval of [ℏωLim, ℏωLO] 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 structures42 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, [ωTO, ωLO], into two distinct sub-intervals: [ωTO, ωLim] and [ωLim, ℏωLO]. These sub-intervals correspond to the positive curvature of NW structures and the negative curvature of NH structures, respectively. Notably, the cutoff frequency point ωLim 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 ωLim decreases as the dielectric constant of the surrounding medium increases. This observation suggests the possibility of regulating ωLim 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.

Let us first derive and display the phonon states and dispersive relationships of SO modes in NW and NH. Here, we consider two Q1D nanostructure models, i.e., a NW and a NH [see Figs. 1(a) and 1(b)]. Both the models with the same radius R are located in a dielectric medium with dielectric constant of ϵd. Within the framework of dielectric continuum model (DCM), electrostatic potential of all the phonon modes should satisfy the Laplace equation, ϵ(ω)∇2Φ(r) = 0, based on the Maxwell equations for free oscillations [free charge density ρ0(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
ε0ϵt(ω)1ρρρρ+1ρ22φ2+ϵz(ω)2z2Φ1NW (r)=0,ρR,ϵd1ρρρρ+1ρ22φ2+2z2Φ2NW(r)=0,ρ>R.
(1)
FIG. 1.

Sketch map of the wurtzite nitride NW (a) and NH models (b) embedded in a nonpolar dielectric matrix. Two dimensional cross-section and charge distribution diagrams of NW (c), NH (d), and plane heterostructure (e).

FIG. 1.

Sketch map of the wurtzite nitride NW (a) and NH models (b) embedded in a nonpolar dielectric matrix. Two dimensional cross-section and charge distribution diagrams of NW (c), NH (d), and plane heterostructure (e).

Close modal
For the anisotropic nitride NH, the phonon potentials should satisfy the Laplace equation,
ϵd1ρρρρ+1ρ22φ2+2z2Φ1NH(r)=0,ρR,ε0ϵt(ω)1ρρρρ+1ρ22φ2+ϵz(ω)2z2Φ2NH (r)=0,ρ>R.
(2)
In Eqs. (1) and (2), ϵd is the dielectric constant of the nonpolar dielectric matrix, and ϵz(ω) [ϵt(ω)] is the dielectric function in the direction along (perpendicular to) the c axis of the wurtzite material, which is given by
ϵv(ω)=ϵvω2ωv,L2ω2ωv,T2,v=z,t.
(3)
In Eq. (3), ωz,L, ωz,T, ωt,L, and ωt,T are the zone center characteristic frequencies of A1(LO), A1(TO), E1(LO), and E1(TO) modes, respectively. The Fröhlich SO phonon states in wurtzite positive curvature NW and negative curvature NH systems are given by
Φm,kzNW(r)=Aexp(ikzz)exp(imφ)ϕm,kzNW(ρ)=Aexp(ikzz)exp(imφ)×Km(kzR)Im(γkzρ)ρ<RIm(γkzR)Km(kzρ)ρR
(4)
and
Φm,kzNH(r)=Bexp(ikzz)exp(imφ)ϕm,kzNH(ρ)=Bexp(ikzz)exp(imφ)×Km(γkzR)Im(kzρ)ρ<RIm(kzR)Km(γkzρ)ρR,
(5)
respectively, where m is the azimuthal quantum number, kz is the free wave-number in z direction, and Im(x) and Km(x) are the first and second kind modified Bessel functions, respectively. The γ function in Eqs. (4) and (5) is defined as17,
γ(ω)=signϵz(ω)ϵt(ω)ϵt(ω)ϵz(ω),
(6)
where ϵt(ω) and ϵz(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 [ℏωt,TO, ℏωz,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 [ωt,TO, ωz,LO].17 
FIG. 2.

γ(ω) as a function of ω (a). Dispersive energies ℏωSO of the SO modes in AlN (b), GaN (c), and InN (d) NW (red curve) and NH (blue curve) as a function of kzR.

FIG. 2.

γ(ω) as a function of ω (a). Dispersive energies ℏωSO of the SO modes in AlN (b), GaN (c), and InN (d) NW (red curve) and NH (blue curve) as a function of kzR.

Close modal
The continuum condition of the normal component of dielectric displacement vector at the surface ρ = R will lead to the following equations:
ϵdIm(γkzR)Km1(kzR)+Km+1(kzR)+ϵtγKm(kzR)×Im1(γkzR)+Im+1(γkzR)=0
(7)
and
ϵdKm(γkzR)Im1(kzR)+Im+1(kzR)+ϵtγIm(kzR)×Km1(γkzR)+Km+1(γkzR)=0.
(8)

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

By following the standard quantized steps as those given in Refs. 16 and 17, one can obtain the Fröhlich electron–SO phonon interaction Hamiltonians He–SO in positive curvature NW and negative curvature NH structures, and they are given by
He-SONW=m,kz[Γm,kzNW(ρ)exp(imφ)exp(ikzz)bm(kz)+H.c.],
(9)
with the electron–SO phonon coupling function Γm,kzNW(ρ) in NW, i.e.,
Γm,kzNW(ρ)=CSO,mNW(kz)ϕm,kzNW(ρ)
(10)
and
He-SONH=m,kz[Γm,kzNH(ρ)exp(imφ)exp(ikzz)bm(kz)+H.c.],
(11)
with the electron–SO phonon coupling function Γm,kzNH(ρ) in NH, i.e.,
Γm,kzNH(ρ)=CSO,mNH(kz)ϕm,kzNH(ρ).
(12)
The electron–SO phonon coupling constant CSO,m NW(kz) in NW is given by
CSO,mNW(kz)=2πωSOekzKm(kzR)1ϵtϵ1ϵtϵ01InttNW+1ϵzϵ1ϵzϵ01IntzNW1/2
(13)
with
InttNW=0Rρdργ2[Im12(γkzρ)+Im12(γkzρ)],IntzNW=0R2ρdρIm2(γkzρ).
(14)
The electron–SO phonon coupling constant CSO,mNH (kz) in NH is given by
CSO,mNH(kz)=2πωSOekzIm(kzR)1ϵtϵ1ϵtϵ01InttNH+1ϵzϵ1ϵzϵ01IntzNH1/2,
(15)
with
InttNH=R+ρdργ2[Km12(γkzρ)+Km12(γkzρ)],IntzNH=R+2ρdρKm2(γkzρ).
(16)
In Eqs. (13) and (15), e denotes an electron charge.

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

TABLE I.

Zone-center energies (in meV) of polar optical phonons and dielectric constants for wurtzite GaN, AlN, and InN materials.43,44

MaterialωtTωzTωtLωzLϵϵ0
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 
MaterialωtTωzTωtLωzLϵϵ0
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 waves46), 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.

The dispersive energies ℏωSO of the SO modes in AlN, GaN, and InN NW (red curve) and NH (blue curve) as a function of kzR are shown in Figs. 2(b)2(d), respectively. The nitride NWs and NHs are placed in vacuum, i.e., ϵd = 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 kzR, while that in NW is a monotonic and incremental function of kzR. As kzR is quite large, both the blue and red curves approach a certain limited characteristic energy ℏωLim, 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 ℏωLim for quite large kzR can be obtained via Eqs. (7) and (8). Based on the limited properties of the Bessel functions, Im(x) and Km(x), i.e., Im(x) and Km(x), respectively, will approach ex/2πx and πex/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.,
ϵz(ω)ϵt(ω)ϵd=0.
(17)
Equation (17) is just the dispersive equation of SO modes in nitride plane HS.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 kz is quite large, this means that the phonon wave-length λph becomes quite small, i.e., λphR. 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 ϵi(ω)=ϵ(ω2ωi,LO2)/(ω2ωi,TO2) (i = t, z) into Eq. (17), the cutoff frequency ωLim can be worked out. For wurtzite AlN HS located in vacuum, the value of ωLimAlN is equal to ∼106.60 meV. As stated above, the energies of SO modes in the Raman and PL experiments18,25,26 of AlN NW and nanorod positive structures are within the sub-interval of [ωt,TOAlN,ωLimAlN], i.e., [83.13, 106.60] meV. For example, Sahoo et al.18 observed an SO phonon mode at 850 cm−1 (105.4 meV) in AlN nanotips by using the Raman spectroscopy technique. One scholar of our group25 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 103 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 [ωt,TOAlN,ωLimAlN], i.e., [83.13,106.60] meV. Similarly, it is observed from Figs. 2(c) and 2(d) that, the cutoff point frequencies ωLim 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 [ωt,TOGaN,ωLimGaN] for GaN NWs27–31 and [ωt,TOInN,ωLimInN] for InN NWs32 were found. However, the SO phonon modes whose energies are within another sub-interval of [ωLim, ωz,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 ℏωSO of SO modes as functions of the curvature K of AlN nanostructures [Figs. 3(a)3(c)] and dielectric constant ϵd of dielectric medium [Fig. 3(d)] are shown in Fig. 3. Three different parameters of dielectric constants, ϵd = 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., kz = 1 × 105 (blue), 1.5 × 105 (black), and 2 × 105 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 ℏωSO of SO modes monotonously decrease from the characteristic energy ℏωt,TO to another one ℏωz,LO. The three curves intersect at K = 0. In addition, the intersection just divides the interval of [ωt,TO, ωz,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 ωLim 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 kz. Comparing Fig. 3(a) to Figs. 3(b) and 3(c), it is found that with the increase in dielectric constant ϵd, the cutoff point frequency ωLim decreases obviously. For example, as ϵd increases from 1 to 4, ωLim 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 ωLim of positive and negative curvature nanostructures can be regulated by the dielectric constant of a dielectric environment.

FIG. 3.

Dispersive energies ℏωSO of SO modes as functions of the curvature K of AlN nanostructures (a)–(c) and dielectric constant ϵd of the dielectric medium (d).

FIG. 3.

Dispersive energies ℏωSO of SO modes as functions of the curvature K of AlN nanostructures (a)–(c) and dielectric constant ϵd of the dielectric medium (d).

Close modal

From Fig. 3(d), we observe that the dispersive energies ℏω SO of SO modes and the cutoff point frequency ωLim in AlN NW and NH structures are a monotonic and degressive function of ϵd. The cutoff frequency point ωLim is the boundary of NW with positive curvature and NH with negative curvature. Moreover, the smaller the value of kzR, the more obvious is the dispersion of the SO mode. On the contrary, the bigger the value of kzR, the more remarkably the dispersive curves of SO modes in NW and NH approach the curve of ωLim. This further reveals that ωLim 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 ΓkzSO(ρ) (also the electron–phonon coupling function16,17) in AlN NW [Fig. 4(a)] and NH [Fig. 4(b)] as a function of ρ are shown in Fig. 4 when ϵd = 1 and kz = 4 × 105 cm−1. Six different curvature parameters, K = ±0.5 × 105 cm−1 (blue curves), ±0.67 × 105 cm−1 (red curves), and ±1 × 105 cm−1 (black curves) are chosen when drawing the figures. In each embedded graph, the peak width at half height of electrostatic potential ΓkzSO(ρ) in the AlN material layer dAlN and dielectric medium dMed 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 dMed decreases obviously, while that in AlN dAlN 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.

FIG. 4.

Distributions of electrostatic potential ΓkzSO(ρ) in AlN NW (a) and NH (b) as a function of ρ. Six different curvature parameters, K = ±0.5 × 105 cm−1 (blue curves), ±0.67 × 105 cm−1 (red curves), and ±1 × 105 cm−1 (black curves) are chosen. dAlN and dMed denote the peak width at half height of ΓkzSO(ρ) in the AlN layer and dielectric medium, respectively.

FIG. 4.

Distributions of electrostatic potential ΓkzSO(ρ) in AlN NW (a) and NH (b) as a function of ρ. Six different curvature parameters, K = ±0.5 × 105 cm−1 (blue curves), ±0.67 × 105 cm−1 (red curves), and ±1 × 105 cm−1 (black curves) are chosen. dAlN and dMed denote the peak width at half height of ΓkzSO(ρ) in the AlN layer and dielectric medium, respectively.

Close modal

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 σNW in NWs with positive curvature is the highest, while σNH in NHs with negative curvature is the lowest. The surface charge density σPL in planar HS falls somewhere in the middle, resulting in the following order: σNW > σPL > σNH. 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, ωSONW<ω SOPL<ωSONH). 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., ωSOPL=ωLim), 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 ΓkzSO(ρ) in AlN NW [Fig. 5(a)] and NH [Fig. 5(b)] as a function of ρ are shown in Fig. 5 when K = ±1 × 105 cm−1 and kz = 4 × 105 cm−1. The black, red, and blue curves correspond to the dielectric constants ϵd = 1, 2, and 4, respectively. Similarly, the peak width at half height of ΓkzSO(ρ) in the AlN semiconductor dAlN and that in dielectric medium dMed as a function of ϵd 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 ϵd than that in a dielectric medium. This can be seen quite clearly in the embedded graphs. With the increase in ϵd, the values of dMed (red curves) nearly keep unchanged, but the values of dAlN (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 ϵd 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 ϵd. 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.

FIG. 5.

Electrostatic potentials ΓkzSO(ρ) in AlN NW (a) and NH (b) as a function of ρ when K = ±1 × 105 cm−1 and kz = 4 × 105 cm−1. The black, red, and blue curves correspond to the dielectric constants ϵd = 1, 2, and 4, respectively.

FIG. 5.

Electrostatic potentials ΓkzSO(ρ) in AlN NW (a) and NH (b) as a function of ρ when K = ±1 × 105 cm−1 and kz = 4 × 105 cm−1. The black, red, and blue curves correspond to the dielectric constants ϵd = 1, 2, and 4, respectively.

Close modal

It should be noted that although the fixed free wave-number kz = 4 × 105 cm−1 is selected in the discussion of ΓkzSO(ρ) 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 kz. In general, the wave-number kz 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 kz (5 × 105 and 6 × 105 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 kz.

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., λphac. 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.

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 [ℏωTO, ℏωLO] of SO modes, which is commonly observed in low-dimensional semiconductor systems,13–23 can be further refined into two distinct sub-ranges: [ℏωTO, ℏωLim] and [ℏωLim, ℏωLO]. The low-energy (high-energy) sub-range corresponds to SO modes in NWs (NHs) exhibiting positive (negative) curvature. Notably, the cutoff frequency ωLim 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 ωLim 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 InN32 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 NWs18,25–32 but also can be used in the design and development of optical and optoelectronic devices based on SO modes of Q1D nanostructures.

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).

The authors have no conflicts to disclose.

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).

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

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