1. Introduction

In recent years, with the progress in semiconductor nanotechnology, such as molecular-beam epitaxy and metal-organic chemical-vapor deposition, various kinds of low-dimensional complex heterostructures can be fabricated. For example, in 1993, utilizing the method of wet chemical synthesis, Eychmüller et al.^[1] synthesized spherical quantum dot quantum wells (QDQW): an inhomogeneous quantum dot (QD) with a central core barrier and several shell wells. Experimental and theoretical studies^[2-7] revealed that the inner configurations for electrons and holes make many physical properties in QDQWs obviously different from those in quantum wires (QW) and QDs. In 1997 and 1998, under reactive laser ablation, inhomogeneous coaxial cylindrical quantum-well wires (QWW) were successfully synthesized by Suenaga and Zhang's group^{[8, 9]}, which were also called quantum cables (QC). Due to their peculiar properties, QC systems have attracted great attention by many authors. Zeng et al.^[10] predicted the single-electron subband properties in a five-layer GaAlAs/GaAs QC system, and the systems exhibit some interesting and unique behavior unexpected in other nanostructures. Under a dielectric continuum approximation, Zhang et al.^[11] study the interface optical (IO) and surface optical (SO) phonon modes, as well as the Fröhlich electron-IO (or -SO) phonon interaction Hamiltonian, in a multilayer coaxial cylindrical QC. Results revealed that the dispersion frequencies of IO or SO modes sensitively depend on small $k_{z}$ (the wavevector in the z direction) and m (the azimuthal quantum number), while for large $k_{z}$ and m, the frequency for each mode converges to the limit frequency value of the IO (or SO) mode in a single heterostructure. The phonons with higher frequencies are found to contribute more significantly to the electron-phonon interaction.

Because of its convenience and the absence of damage to the materials being measured, electron Raman scattering (ERS) has become a powerful tool to investigate different physical properties of low dimensional semiconductor systems^[12-14]. Based on resonant Raman scattering, Raman injection lasers and some other opto-electronic devices can be produced^[15]. Experimental research on Raman scattering in nanocrystals has been extensively reported^[16-20]. Tenne et al.^[21] investigate the phonon spectra of self-assembled InAs QDs in an AlAs matrix with different growth temperatures by Raman spectroscopy. The Raman spectroscopy data show that QDs grown at low temperatures appear to have smaller sizes and increasing the temperature first leads to increasing QDs size. Milekin et al.^[22] studied the interface modes in InAs/Al(Ga)As QDs structures with symmetrical and asymmetrical barriers localized in the vicinity of corrugated QD/matrix and wetting layer/matrix interfaces. Results reveal that the observed interface phonon frequency positions are in good agreement with those calculated using the dielectric continuum model. To interpret experimental results, many theoretical physicists usually focus on the calculation of the differential cross section (DCS) for Raman scattering^[23-26]. Thus, the calculation of the DCS for ERS remains a rather interesting and fundamental issue to achieve a better understanding of the man-made semiconductor nanostructures. Zhong et al.^[27] calculate the ERS cross-section in CdS/HgS cylindrical QDQW. The rich spectra obtained in Ref. [27] provide direct information about the electron band structure. Liu et al.^{[28, 29]} calculate the first-order Raman scattering process in quantum discs and quantum wires within the framework of dielectric continuum model. The Raman spectra they obtained give the direct information about the longitudinal optical (LO) and IO phonon respectively and have a good agreement with the experiment results.

In the present paper, we consider the Fröhlich electron-phonon interaction in the framework of the dielectric continuum model. We calculate the Raman intensity in a three-layer coaxial cylindrical Al$_{x}$Ga$_{1-x}$As/GaAs/vacuum quantum cable. Electron states are assumed to be confined within the well. Three types of optical phonons, named $\text{IO}_{1}$, $\text{IO}_{2}$ and SO modes have been investigated. The contributions of phonons to the first-order ERS are analyzed. The singularities are found to be sensitively size-dependent for small QC and by varying the size of the well it is possible to control the frequency shift in the Raman spectrum. The selection rules for the Raman scattering processes and the phonon behavior for different well size are also studied. The calculated results are compared with those of experiments.

2. Model and theory

We consider a three-layer coaxial cylindrical Al$_{x}$Ga$_{1-x}$-As/GaAs/vacuum QC system. As shown in Fig. 1, $R_{1}$, $R_{2}$ are the inner and outer radius. The electron states are considered to be confined within the well. In the envelope function approximation neglecting electron-hole Coulomb interaction the wave function for electrons can be written as

$ $$\begin{equation} \begin{split} \psi_{\rm nl}(\boldsymbol{r})={}&C^{-1}_{\rm nl}\rm{e}^{{\rm i}l\theta}\rm{e}^{{\rm i}k_{z}z} \\ &{}\times \begin{cases} T_{\rm l}(\beta_{\rm nl}R_{1})J_{\rm l}(\alpha_{\rm nl}r), \quad r\leqslant R_{1}, \\ J_{\rm l}(\alpha_{\rm nl}R_{1})T_{\rm l}(\beta_{\rm nl}r), \quad R_{1}<r\leqslant R_{2}, \\ 0, \quad r> R_{2}, \end{cases} \end{split} \end{equation}$$ $

(1)

$ $$\begin{equation} T_{\rm l}(\beta_{\rm nl}r)=J_{\rm l}(\beta_{\rm nl}r)-\frac{J_{\rm l}(\beta_{\rm nl}R_{2})}% {Y_{\rm l}(\beta_{\rm nl}R_{2})}Y_{\rm l}(\beta_{\rm nl}r), \end{equation}$$ $

(2)

$ $$\begin{equation} \begin{split} C^{2}_{\rm nl}=&\frac{1}{2\pi L}\Big[T_{\rm l}(\beta_{\rm nl}R_{1})^{2} \int_{0}^{R_{1}}J_{\rm l}(\alpha_{\rm nl}r)^{2}r{\rm d}r \\ &+J_{\rm l}(\beta_{\rm nl}R_{1})^{2} \int_{R_{1}}^{R_{2}}T_{\rm l}(\beta_{\rm nl}r)^{2}r{\rm d}r\Big], \end{split} \end{equation}$$ $

(3)

where $\alpha_{\rm nl}=\sqrt{2\mu_{1}(V_{\rm c}-E_{\rm nl})/\hbar^{2}}$, $\beta_{\rm nl}=\sqrt{2\mu_{2}E_{\rm nl}/\hbar^{2}}$; $J_{\rm l}$ and $Y_{l}$ are the lth-order Bessel functions of the first and second kind; $\mu_{1}$ ($\mu_{2})$ is the effective mass in the dot (well); L is the length of the QC ($L\gg R_{2}$); $C_{\rm nl}$ is the normalization factor, and $V_{\rm c}$ represents the band offset. Taking as matching boundary conditions the continuity of the function $\psi$ and the current $\partial\psi/\mu_{\rm i}\partial r$ at the interface, the energy eigenvalues $E_{\rm nl}$ are determined by the secular equation,

$ $$\begin{equation} \frac{1}{\mu_{1}}T(\beta_{\rm nl}R_{1})J_{l}(\alpha_{\rm nl}r)'\Big|_{r\, =\, R_{1}}= \frac{1}{\mu_{2}}J(\alpha_{\rm nl}R_{1})T_{l}(\beta_{\rm nl}r)'\Big|_{r\, =\, R_{1}}. \end{equation}$$ $

(4)

Within the framework of the dielectric continuum approach, the Fröhlich electron-phonon interaction can be written as^[11]

$ $$\begin{equation} H_{\text{e}-\text{IO, SO}}=-\sum\limits_{m_{p}, q_{z}} \Gamma _{m_{p}, \;q_{z}}^{\text{IO, SO}}(r)[\rm{e}^{{\rm i}m_{p}\varphi}\rm{e}^{{\rm i}q_{z}z}C^{+}_{m_{p}}(q_{z})+H.C.], \end{equation}$$ $

(5)

where

$ $$\begin{equation} \begin{split} \Gamma_{m_{p}, \;q_{z}}^{\text{IO, SO}}(r)={}&N_{m_{p}, q_{z}} \\ &{}\times\begin{cases} g_{1}I_{m_{p}}(q_{z}r), \quad r\leqslant R_{1}, \\ g_{2}I_{m_{p}}(q_{z}r)+h_{2}K_{m_{p}}(q_{z}r), \quad R_{1}<r\leqslant R_{2}, \\ K_{m_{p}}(q_{z}r), \quad r>R_{2}, \\ \end{cases} \end{split} \end{equation}$$ $

(6)

and

$ $$\begin{equation} \begin{split} |N_{m_{p}, \;q_{z}}|^{2}=\, &\frac{2\hbar\omega e^{2}}{k_{z}L}\Bigg[\left(\frac{1}{\epsilon_{1}-\epsilon_{10}}-\frac{1}{\epsilon_{1}-\epsilon_{1\infty}}\right)^{-1} \\& \times \text{Int}(1, m_{p}, q_{z})\\& +\left(\frac{1}{\epsilon_{2}-\epsilon_{20}}-\frac{1}{\epsilon_{2}-\epsilon_{2\infty}}\right)^{-1}\text{Int}(2, m_{p}, q_{z})\Bigg]^{-1} , \end{split} \end{equation}$$ $

(7)

with

$ $$\begin{equation} \begin{split} \text{Int}(1, m_{p}, q_{z})={}& \int_{0}^{R_{1}}r{\rm d}rg_{1}^{2}\big[I_{m_{p}-1}(q_{z}r)^{2}+I_{m_{p}+1}(q_{z}r)^{2}\\ &{}+2I_{m_{p}}(q_{z}r)^{2}\big], \end{split} \end{equation}$$ $

(8)

$ $$\begin{equation} \begin{split} \text{Int}(2, m_{p}, q_{z})={}& \int_{0}^{R_{1}}r{\rm d}r\Big[g_{2}^{2}(I_{m_{p}-1}(q_{z}r)^{2}+I_{m_{p}+1}(q_{z}r)^{2} \\ &{}+2I_{m_{p}}(q_{z}r)^{2})+h_{2}^{2}(K_{m_{p}-1}(q_{z}r)^{2} \\ &{}+K_{m_{p}+1}(q_{z}r)^{2}+2K_{m_{p}}(q_{z}r)^{2})\\ &{}+2g_{2}h_{2}(2I_{m_{p}}(q_{z}r)K_{m_{p}}(q_{z}r)\\ &{}+I_{m_{p}-1}(q_{z}r)K_{m_{p}-1}(q_{z}r)\\ &{}-I_{m_{p}+1}(q_{z}r)K_{m_{p}+1}(q_{z}r))\Big], \end{split} \end{equation}$$ $

(9)

$ $$\begin{equation} g_{2}=\frac{(\epsilon_{2}-\epsilon_{\rm d})K_{m_{p}}(q_{z}R_{2})K_{m_{p}}(q_{z}R_{2})'} {[I_{m_{p}}(q_{z}R_{2})K_{m_{p}}(q_{z}R_{2})'-K_{m_{p}}(q_{z}R_{2})I_{m_{p}}(q_{z}R_{2})']\epsilon_{2}}, \end{equation}$$ $

(10)

$ $$\begin{equation} h_{2}=\frac{\epsilon_{2}K_{m_{p}}(q_{z}R_{2})I_{m_{p}}(q_{z}R_{2})'-\epsilon_{\rm d}I_{m_{p}}(q_{z}R_{2})K_{m_{p}}(q_{z}R_{2})'} {[K_{m_{p}}(q_{z}R_{2})I_{m_{p}}(q_{z}R_{2})'-I_{m_{p}}(q_{z}R_{2})K_{m_{p}}(q_{z}R_{2})']\epsilon_{2}} , \end{equation}$$ $

(11)

$ $$\begin{eqnarray} g_{1}=g_{2}+\frac{K_{m_{p}}(q_{z}R_{1})}{I_{m_{p}}(q_{z}R_{1})}h_{2}, \end{eqnarray}$$ $

(12)

where $\epsilon_{\rm d}$ is the dielectric function of the vacuum.

3. Raman scattering intensity

There are different types of Raman scattering process. In this paper, we put emphasis on the three-step process (Stokes emission): in the first step, the system absorbs one $\omega_{\rm i}$ photon from the incident radiation and creates an EHP state with the electron transition from the initial state $|i\rangle$ to a virtual state $|\nu\rangle$; in the second step, the electron undergoes another transition from state $|\nu\rangle$ to another virtual state $|\mu\rangle$ and emits an optical phonon with frequency $\vartheta_{\rm p}$; in the final step, the EHP recombines with emission of a secondary-radiation photon with frequency $\omega_{\rm s}$.

The Raman cross section can be given by the third-order perturbation theory^[26]

$ $$\begin{equation} \frac{{\rm d}^{2}\sigma}{{\rm d}\Omega {\rm d}\omega_{\rm s}} =\frac{\eta(\omega_{\rm i})\eta^{3}(\omega_{\rm s})}{4\pi^{2}\hbar^{3}c^{4}}\sum\limits_{f}|K_{\rm if}|^{2} (\hbar\omega_{\rm i}-\hbar\vartheta_{\rm p})^{2}. \end{equation}$$ $

(13)

Since the time order of quantum creation (annihilation) can be changed optionally, there are six permutations open to us, and the transition matrix elements $K_{\rm if}$ consist of six terms

$ $$\begin{equation} K_{\rm if}\equiv\sum\limits_{\mu, \nu}W_{\rm if}(\mu, \nu), \end{equation}$$ $

(14)

with

$ $$\begin{equation} \begin{split} W_{\rm if}(\mu, \nu)= \, &\frac{\langle f|H_{\rm eF}(\omega_{\rm s})|\mu\rangle\langle\mu|H_{\rm e-ph}|\nu\rangle\langle\nu|H_{\rm eF}(\omega_{\rm i})|i\rangle} {(\hbar\omega_{\rm i}+E_{\rm i}-E_{\mu}-\hbar\vartheta_{\rm p}+{\rm j} \Gamma_{\mu})(\hbar\omega_{\rm i}+E_{\rm i}-E_{\nu}+{\rm j}\Gamma_{\nu})} \\ &+\frac{\langle f|H_{\rm eF}(\omega_{\rm s})|\mu\rangle\langle\mu|H_{\rm eF}(\omega_{\rm i})|\nu\rangle\langle\nu|H_{\rm e-ph}|i\rangle} {(E_{\rm i}-E_{\nu}-\hbar\vartheta_{\rm p}+{\rm j}\Gamma_{\nu})(\hbar\omega_{\rm i}+E_{\rm i}-E_{\mu}-\hbar\vartheta_{\rm p}+{\rm j}\Gamma_{\mu})} \\ &+\frac{\langle f|H_{\rm e-ph}|\mu\rangle\langle\mu|H_{\rm eF}(\omega_{\rm s})|\nu\rangle\langle\nu|H_{\rm eF}(\omega_{\rm i})|i\rangle} {(\hbar\omega_{\rm i}+E_{\rm i}-E_{\nu}+{\rm j}\Gamma_{\nu})(E_{\rm i}-E_{\mu}-\hbar\vartheta_{\rm p}+{\rm j}\Gamma_{\mu})} \\ &+\frac{\langle f|H_{\rm eF}(\omega_{\rm i})|\mu\rangle\langle\mu|H_{\rm e-ph}|\nu\rangle\langle\nu|H_{\rm eF}(\omega_{\rm s})|i\rangle} {(E_{\rm i}-E_{\nu}-\hbar\omega_{\rm i}+\hbar\vartheta_{\rm p}+{\rm j}\Gamma_{\nu})(E_{\rm i}-E_{\mu}-\hbar\omega_{\rm i}+{\rm j}\Gamma_{\mu})} \\ &+\frac{\langle f|H_{\rm eF}(\omega_{i})|\mu\rangle\langle\mu|H_{\rm eF}(\omega_{\rm s})|\nu\rangle\langle\nu|H_{\rm e-ph}|i\rangle} {(E_{\rm i}-E_{\mu}-\hbar\omega_{\rm i}+{\rm j}\Gamma_{\mu})(E_{\rm i}-E_{\nu}-\hbar\vartheta_{\rm p}+{\rm j}\Gamma_{\nu})} \\ &+\frac{\langle f|H_{\rm e-ph}|\mu\rangle\langle\mu|H_{\rm eF}(\omega_{i})|\nu\rangle\langle\nu|H_{\rm eF}(\omega_{\rm s})|i\rangle} {(E_{\rm i}-E_{\nu}-\hbar\omega_{\rm i}+\hbar\vartheta_{\rm p}+{\rm j}\Gamma_{\nu})(E_{\rm i}-E_{\mu}-\hbar\vartheta_{\rm p}+{\rm j}\Gamma_{\mu})}, \end{split} \end{equation}$$ $

(15)

where $j=\sqrt{-1}$; $\mathit{\Gamma}_{\mu}$ and $\mathit{\Gamma}_{\nu}$ are the inhomogeneous line widths of the electron transitions; $\eta(\omega)$ is the refraction index as a function of radiation frequency $\omega$; $|\nu\rangle$ and $|\mu\rangle$ denote the intermediate states with energy $\varepsilon_{\nu}$ and $\varepsilon_{\mu}$, respectively.

In the initial state $|i\rangle$ we assume a completely occupied valence band and an unoccupied conduction band, and one incident photon, so the energy $\varepsilon_{\rm i}$ for the initial state is

$ $$\begin{equation} \varepsilon_{\rm i}=\hbar\omega_{\rm i}. \end{equation}$$ $

(16)

In final state $|f\rangle$, we have a secondary-radiation emitted photon with energy $\hbar\omega_{\rm s}$, and a phonon with energy $\hbar\vartheta_{\rm p}$ and electron back to the initial state. Thus, we can get the energy $\varepsilon_{\rm f}$ for the final state,

$ $$\begin{equation} \varepsilon_{\rm f}=\hbar\omega_{\rm s}+\hbar\vartheta_{\rm p}. \end{equation}$$ $

(17)

According to the energy conservation law, we have $\varepsilon_{\rm i}=\varepsilon_{\rm f}$, hence

$ $$\begin{equation} \hbar\omega_{\rm i}=\hbar\omega_{\rm s}+\hbar\vartheta_{\rm p}. \end{equation}$$ $

(18)

For the intermediate states, the energy $\varepsilon_{\nu(\mu)}$ can be written as

$ $$\begin{equation} \varepsilon_{\nu}=E_{n_{\nu}l_{\nu}t_{\nu}}+E_{\rm g}, \end{equation}$$ $

(19)

$ $$\begin{equation} \varepsilon_{\mu}= E_{n_{\mu}l_{\mu}t_{\mu}}+\hbar\vartheta_{\rm p}+E_{\rm g}. \end{equation}$$ $

(20)

In the following, we calculate the matrix elements of the DCS in scattering configurations $Z(X, X)\overline{Z}$.

$ $$\begin{equation} \langle\nu|\hat{H}_{\rm eF}(\omega_{\rm i})|i\rangle=\frac{{\rm j}e}{\eta(\omega_{\rm i})}\sqrt{\frac{2\pi}{\hbar\omega_{\rm i}}}\Pi_{\nu {\rm i}}, \end{equation}$$ $

(21)

$ $$\begin{equation} \langle f|\hat{H}_{\rm eF}(\omega_{\rm s})|\mu\rangle=\frac{{\rm j}e}{\eta(\omega_{\rm s})}\sqrt{\frac{2\pi}{\hbar\omega_{\rm s}}}\Pi_{\mu {\rm f}}, \end{equation}$$ $

(22)

with

$ $$\begin{equation} \begin{split} \Pi_{\nu i}=\, &\frac{2\pi LE_{\nu {\rm i}}}{C_{n_{\nu}l_{\nu}}C_{n_{i}l_{i}}} \Big[T_{l_{\nu}}(\beta_{n_{\nu}l_{\nu}}R_{1})T_{l_{i}}(\beta_{n_{i}l_{i}}R_{1}) \\ &\times\int_{0}^{R_{1}}J_{l_{\nu}}(\alpha_{n_{\nu}l_{\nu}}r)J_{l_{i}}(\alpha_{n_{i}l_{i}}r)r^{2}{\rm d}r \\ &+J_{l_{\nu}}(\alpha_{n_{\nu}l_{\nu}}R_{1})J_{l_{i}}(\alpha_{n_{i}l_{i}}R_{1})\\ &\times\int_{R_{1}}^{R_{2}}T_{l_{\nu}}(\beta_{n_{\nu}l_{\nu}}r)T_{l_{i}}(\beta_{n_{i}}l_{i}r)r^{2}{\rm d}r \Big]\delta_{l_{\nu}, l_{i}}\delta_{k_{z_{\nu}}, k_{z_{i}}} , \end{split} \end{equation}$$ $

(23)

where

$ $$\begin{equation} E_{\nu {\rm i}}=E_{\nu}-E_{\rm i}, \end{equation}$$ $

(24)

$E_{\mu {\rm f}}$ and $\Pi_{\mu {\rm f}}$ have similar expressions with $E_{\nu{\rm i}}$ and $\Pi_{\nu {\rm i}}$.

$ $$\begin{equation} \langle\mu|\hat{H}_{\rm e-ph}^{\text{IO, SO}}(\omega_{\rm s})|\nu\rangle=\, \Lambda_{\text{IO, SO}}^{\mu\nu}, \end{equation}$$ $

(25)

and

$ $$\begin{equation} \begin{split} \Lambda_{\text{IO, SO}}^{\mu\nu}=\, &\frac{2\pi L} {C_{n_{\nu}l_{\nu}}C_{n_{\mu}l_{\mu}}}\delta_{l_{\mu}, l_{\nu}-l_{p}}\delta_{k_{z_{\mu}}, k_{z_{\nu}}-q_{z}} \\ &\times\sum\limits_{l_{p}, \, q_{z}}N_{l_{p}, q_{z}} \Big\{T_{l_{\nu}}(\beta_{n_{\nu}l_{\nu}}R_{1})T_{l_{\mu}}(\beta_{n_{\mu}l_{\mu}}R_{1}) \\ &\times\int_{0}^{R_{1}}J_{l_{\mu}}(\alpha_{n_{\mu}l_{\mu}}r) g_{1}I_{m_{p}}(q_{z}r)J_{l_{\nu}}(\alpha_{n_{\nu}l_{\nu}}r)r{\rm d}r \\ &+J_{l_{\mu}}(\alpha_{n_{\mu}l_{\mu}}R_{1})J_{l_{\nu}}(\alpha_{n_{\nu}l_{\nu}}R_{1}) \\ &\times\int_{R_{1}}^{R_{2}}T_{l_{\mu}}(\beta_{n_{\mu}l_{\mu}}r) \left[g_{2}I_{m_{p}}(q_{z}r)+h_{2}K_{m_{p}}(q_{z}r)\right]\\ & \times T_{l_{\nu}}(\beta_{n_{\nu}l_{\nu}}r)r{\rm d}r\Big\} . \end{split} \end{equation}$$ $

(26)

The selection rules for the scattering processes are shown from Eqs. (23) to (26).

Finally, after cumbersome calculations we obtain expressions of the DCS for the three-step Raman scattering process

$ $$\begin{equation} \frac{{\rm d}^{2}\sigma}{{\rm d}\Omega {\rm d}\omega _{\rm s}} =\left[\frac{{\rm d}^{2}\sigma }{{\rm d}\Omega {\rm d}\omega _{\rm s}}\right]_{\text{IO}}+\left[\frac{{\rm d}^{2}\sigma }{{\rm d}\Omega {\rm d}\omega _{\rm s}}\right]_{\text{SO}}, \end{equation}$$ $

(27)

$ $$\begin{equation} \left[\frac{{\rm d}^{2}\sigma}{{\rm d}\Omega {\rm d}\omega _{\rm s}}\right]_{\text{IO, SO}} =\frac{e^{4}\eta(\omega_{\rm s})(\omega_{\rm i}-\vartheta_{\rm p})^{2}}{\hbar^{3}c^{4}\eta(\omega_{\rm i})\omega_{\rm i}\omega _{\rm s}} \left|\sum\limits_{\mu, \nu}\Upsilon^{\mu\nu}_{\text{IO, SO}}\right|^{2}, \end{equation}$$ $

(28)

where

$ $$\begin{equation} \begin{split} \Upsilon^{\mu\nu}_{\text{IO, SO}}= \, &\frac{\Pi_{\mu f}\Lambda_{\text{IO, SO}}^{\mu\nu}\Pi_{\nu {\rm i}}} {(\hbar\omega_{\rm i}-\varepsilon_{\nu {\rm i}}+{\rm j}\Gamma_{\nu})(\hbar\omega_{\rm i}-\hbar\vartheta_{\rm p}-\varepsilon_{\mu {\rm i}}+{\rm j}\Gamma_{\mu})} \\ &-\frac{\Pi_{\mu {\rm f}}\Pi_{\mu\nu}\Lambda_{\text{IO, SO}}^{\nu {\rm i}}}{( \varepsilon_{\nu {\rm i}}+\hbar\vartheta_{\rm p} -{\rm j}\Gamma_{\nu})(\hbar\omega_{\rm i}-\hbar\vartheta_{\rm p}-\varepsilon_{\mu {\rm i}}+{\rm j}\Gamma_{\mu})} \\ &-\frac{\Lambda_{\text{IO, SO}}^{\mu {\rm f}}\Pi_{\mu\nu}\Pi_{\mu {\rm i}}} {(\hbar\omega_{\rm i}-\varepsilon_{\nu {\rm i}}+{\rm j}\Gamma_{\nu})( \varepsilon_{\mu {\rm i}}+\hbar\vartheta_{\rm p}-{\rm j}\Gamma_{\mu})} \\ &+\text{three similar terms}. \end{split} \end{equation}$$ $

(29)

Compared to the first three terms, the contribution of the last three terms in Eq. (28) is small enough that it can be neglected. Therefore, in the present work, we consider just the contribution of the first three terms of Eq. (28) during the calculation of $\Upsilon^{\mu\nu}_{\tau}$ for the determination of the DCS.

4. Result and discussion

In this section, the resonant DCS given by Eqs. (27) and (28) is calculated numerically for a three-layer coaxial cylindrical Al$_{0.4}$Ga$_{0.6}$As/GaAs/vacuum QC. The parameters used in the computation are^[30]: $\epsilon_{0}$ = 13.18, $\epsilon_{\infty}$ = 10.89, $\mu_{1}$ = 0.067$m_{0}$, $\hbar\omega_{\rm TO}$ = 33.29 meV for GaAs material and $\epsilon_{0}=12.74, \epsilon_{\infty}=10.51, \mu_{2}=0.10m_{0}, \hbar\omega_{\text{TO}}=37.63$ meV for Al$_{0.4}$Ga$_{0.6}$As material; $V_{\rm c}$ = 0.21 eV; $\Gamma_{\mu}=\Gamma_{\nu}$ = 1 meV.

In Fig. 2, under the scattering configuration $X(Z, Z)\overline{X}$, we show the DCS for the IO and SO phonons as a function of the Raman shift in a three-layer coaxial cylindrical Al$_{0.4}$Ga$_{0.6}$As/GaAs/vacuum QC. The inner radius of the QC we select here is kept at 4.0 nm and the outer radius $R_{2}=$ 8.0 nm, 10.0 nm, 12.0 nm shown by an arrow along the dash curve in the figure. Two types of IO phonons signed $\text{IO}_{1}$ and $\text{IO}_{2}$ have been investigated. From this figure, one can easily find that when increasing radius $R_{2}$, the contribution of the IO and SO phonons to the DCS decreases. However, more calculations reveal that as $R_{2}\rightarrow\infty$, for the coupling between the IO and SO phonons quickly becomes weaker and can then be considered negligible, the intensities for the IO phonons will decline to some constant while for the SO phonons, the DCS will decrease to zero. The electron-IO (SO) phonon interaction is related to the distribution of the electron wavefunction around the interface (surface). When the radius increases, the wider spacial spreading of electron wave function in the well makes its coupling with the interface phonon weaker. When $R_{2}\rightarrow\infty$, the coupling between the electron and the IO phonons will approach a constant and not depend on radius $R_{2}$, while for the SO phonons, the electron-SO phonon interaction can be negligible when radius $R_{2}$ is large enough. Moreover, the figure also shows that when increasing the radius $R_{2}$, shown in Fig. 1 by arrows, the peaks for $\text{IO}_{2}$ and $\text{SO}$ phonons are red shifted, while those for $\text{IO}_{1}$ shift to a higher frequency. However, when the radius $R_{2}$ is large enough, the peaks for both of the $\text{IO}$ and $\text{SO}$ phonons will localize at certain values, i.e., IO$_{1}$ at 283.87 cm^-1, IO₂ at 316.32 cm^-1 and SO at 292.48 cm^-1, which agrees well with the results studied by the dielectric continuum model shown in Fig. 3. This shift can be explained by the effect of size-selective Raman scattering. The effect of localization becomes significant in the small-size low dimensional systems, which results in a notable low-frequency shift of optical phonons in the QC because of their frequencies dispersion. This phenomenon tells us that by varying the size of the QC it is possible to control the frequency shift in the Raman spectrum. The same effect was also observed for various low dimensional structures by Raman scattering experiments^[15-17]. Compared Fig. 2(a) with Fig. 2(b), the DCS related the SO phonons is about 2 orders smaller than that related to the IO phonons and decrease quickly as $R_{2}$ increase, thus in our models, the IO phonons make the main contribution to the Raman intensity.

Figure 3 shows the dispersion frequencies of the IO and SO phonons as functions of the radius $R_{2}$ with $m_{p}=0$ and $R_{1}=$ 4.0 nm, which are calculated using the dielectric continuum model. This figure shows that the variation of the dispersion frequencies is obvious in small radius, while for large $R_{2}$, the value of IO and SO phonon frequency approaches certain values, respectively. From Fig. 3, we can see that the frequencies related to $\text{IO}_{2}$ and $\text{SO}$ phonons slightly decrease as $R_{2}$ rises, while for $\text{IO}_{2}$ the frequencies increase slightly. When $R_{2}\rightarrow \infty$, the Al$_{0.4}$Ga$_{0.6}$As/GaAs interface becomes a QW heterostructure^[29] and the coupling between IO and SO phonons can be negligible, thus the frequencies for $\text{IO}_{1}$ and $\text{IO}_{2}$ phonons converge respectively to 35.17 meV and 39.19 meV, i.e., the frequencies of the IO phonons in QW heterostructure^[29]. For the SO phonons, as $R_{2}\rightarrow \infty$, the SO phonons do not feel the curvature of the spherical surfaces^[31]. Therefore, for about $R>12$ nm, the frequencies of SO phonons will converge to the limit value 36.24 meV, i.e., the frequencies of the surface optical phonons in single planar heterostructure. These results studied by the dielectric continuum model are in good accordance with the Raman spectra we have analyzed above.

5. Conclusions

In this work, we have investigated theoretically the DCS for electron Raman scattering process involving IO (SO) phonon-assisted transitions in a three-layer coaxial cylindrical Al$_{0.4}$Ga$_{0.6}$As/GaAs/vacuum QC system. Three types of optical phonons signed $\text{IO}_{1}$, $\text{IO}_{2}$ and SO have been observed. It is shown that the IO phonons make the main contribution to the DCS and as the size of the QC increase to infinity, for the coupling between the IO and SO phonons can be negligible, the contributions of the IO phonons to the DCS will become size-independent while those for the SO phonons, the DCS will decrease to zero quickly. The picks we get in the Raman spectra are sensitively radius-dependent for QC with small size. The effect of size-selective Raman scattering has been analyzed and the phonon behavior has been discussed. The frequencies for both IO and SO phonons are dispersed at small radius $R_{2}$ and converge as $R_{2}\rightarrow\infty$.