1. Introduction

Study of phonons in any reduced dimensional semiconductor system has been a subject of interest, in theoretical and experimental aspects, for the past few decades^{[1, 2, 3, 4]}. The studies reveal that the electronic properties, oscillator strengths and the spontaneous lifetimes are significantly altered by the phonon interaction with the charge carriers. Interaction between the charge carriers and the phonon plays a vital role in any low dimensional semiconductor system and it shows the manifestation in the behaviour of electronic and optical properties. In addition, this effect is found to be more pronounced when the external perturbations are included in the Hamiltonian of the system. It is known that the formation of polaron occurs when the charged particle is combined with the self-induced polarization in any polar semiconductor. The polaron phenomenon strongly influences the electron spectrum and the polaronic effects are more pronounced in low dimensional semiconductor systems than their bulk counterparts. The effects of LO phonon modes and two types of surface optical phonon modes in a cylindrical quantum dot have been established previously^[5]. Some investigations of the polaron studies found that the contribution of surface optical phonons is quite small^[6]. The polaronic energy can be found by the Stokes-shift line in the resonantly excited spectrum of a quantum dot^{[7, 8]}. The formation of the polaron has been observed experimentally in a self-assembled InAs quantum dot using magneto-spectroscopy with the s-p polaron transition^[9].

Studies of the impurity states in any low dimensional semiconductor system clarified the important aspects on theoretical and experimental research works and their opto-electronic properties are modified significantly. Considerable interest has evolved in studying the optical and electrical properties of wide band gap polar semiconductors for the potential applications in opto electronic devices^{[10, 11, 12, 13]}. Polaron states in cylindrical and spherical quantum dots with the parabolic confinement potentials have been investigated employing Feynman variational principle^[14]. The polar optical vibrational modes, in a finite quantum dot, have been discussed earlier^[15]. The electron-phonon interaction leading to anti-crossing level of the energy levels was first observed in the presence of magnetic field strength^[16]. Polaron induced anti-crossing energy levels, in a quantum dot, have been observed on the base of two level system^[17].

In the present work, the effect of LO phonon on the electronic and the optical properties is studied in the Cd$_{0.8}$Zn$_{0.2}$Se/ZnSe quantum dot taking into consideration the geometrical confinement effects and the magnetic field strength. The magneto-polaron induced hydrogenic binding energy as a function of dot radius is calculated. The oscillator strength and the spontaneous lifetime with the geometrical confinement are observed in the presence of magnetic field strength and the phonon contribution. The dependence of magnetic field on the optical gain as a function of photon energy is found with and without the inclusion of phonon contribution. The model of our calculation to obtain the energy eigen values and the optical properties are brought out in Section 2. The results and discussion are explained in Section 3. The summary and results are in the last section.

2. Model and calculation

Within the single band effective mass approximation, the Hamiltonian describing the electron and the LO-phonon with their lattice interaction, in a Cd$_{0.8}$Zn$_{0.2}$Se/ZnSe quantum dot, is given by

$\hat {H}=\hat {H}_{\rm e} +\hat {H}_{\rm ph} +\hat {H}_{\rm e-ph} ,$

(1)

where $H_{\rm e}$ is the Hamiltonian of a donor impurity. In the presence of a magnetic field, it can be written as

$H_{\rm e} =\frac{1}{2m_{\rm e}^\ast (E)}\left( {\boldsymbol{p}-\frac{e\boldsymbol{A}}{c}} \right)^2-\frac{e^2}{\varepsilon r}+V(r),$

(2)

where $\boldsymbol{p}$ is the momentum operator, the vector potential is defined as $\boldsymbol{A}=\frac{1}{2}\boldsymbol{B} \times \boldsymbol{r}$, $q$ is the electronic charge, $r$ is the distance between the electron and the impurity, $m_{\rm e}^\ast (E)$ is the energy-dependent effective mass of the electron, and $\varepsilon$ is the dielectric constant of the inner dot. The parabolic quantum dot is considered and the confined potential is taken as $V=\frac{V_{\rm 0B} r^2}{R^2}$ for $\vert r \vert$ $\leqslant$ $R$ and $V=V_{\rm 0B}$ for $\left| r \right|>R$ where $R$ is the radius of the quantum dot and $V_{\rm 0B}$ is the barrier height of the parabolic dot. The ratio of conduction-band to the band gap difference between CdSe and ZnSe semiconducting materials is considered as 70 %^[18]. The expression for the energy-dependent effective mass of electron is given by^[19]

$\frac{1}{m_{\rm e}^\ast (E)}=\frac{1}{m_0 }\frac{E_{\rm g} (E_{\rm g} +\Delta )}{3E_{\rm g} +2\Delta}\left({\frac{2}{E+E_{\rm g} }+\frac{1}{E+E_{\rm g} +\Delta }} \right),$

(3)

where $E$ is the confined energy in the conduction band, $m_{0}$ is the conduction band edge effective mass, $\Delta$ is the spin-orbit splitting energy, and $E_{\rm g}$ is the band gap of the inner material.

The strained dependent energy gap of the Cd$_{0.8}$Zn$_{0.2}$Se semiconducting material is given by^{[20, 21]}

$$\begin{split} \label{eq4} E_{\rm g}^{\rm s} = {}& E_{\rm g}^0 +\left[2a_{\rm c} \left( {C_{11} -C_{12} } \right)/C_{11} -b\left( {C_{11} +2C_{12} } \right)/C_{11} \right] \\& \Xi +\delta E_{\rm c}, \end{split}$$

(4)

where $\Xi$ is the lattice mismatch between the inner dot and the outer barrier material, $a_{\rm c}$ is the hydrostatic deformation potential at $\Gamma$ point, $b$ is the shear stress deformation potential, $C_{ij}$ are the elastic constants, and $E_{\rm g}^0$ is the band gap energy of the unstrained inner quantum dot material. $\delta E_{\rm c}$ is the shift of the conduction band due to the strain. Taking into account the shifting of the average conduction band energy, the corresponding expression is given by

$\delta E_{\rm c} =a_{\rm c} (2\varepsilon _{\vert \vert } +\varepsilon _\bot ),$

(5)

where $a_{\rm c}$ is the hydrostatic deformation potential for the conduction band, $\varepsilon _{\vert \vert }$ is the in-plane strain component and $\varepsilon _\bot$ is the perpendicular strain component to the plane interface. The strain components are expressed as

$\varepsilon _{\vert \vert } =\frac{a_{\vert \vert } -a}{a},$

(6)

and

$\varepsilon _\bot =-2\frac{C_{12} }{C_{11} }\varepsilon _{\vert \vert } .$

(7)

The Frhlich interaction has been considered throughout the calculations. The Hamiltonian of the non-interacting LO-phonon can be expressed as

$H_{\rm ph} =\sum\limits_{\boldsymbol{q}} {\hbar \omega _{\rm LO} a_{\boldsymbol{q}}^+ a_{\boldsymbol{q}} },$

(8)

where $a_{\boldsymbol{q}}^+ (a_{\boldsymbol{q}})$ is the creation (annihilation) operator for an LO phonon with the wave vector $\boldsymbol{q}$ and the frequency $\omega _{\rm LO}$. It describes a non-interacting optical phonon system. These operators satisfy the commutation relations $\left[{a_{\boldsymbol{q}}^+, a_{\boldsymbol{q}} } \right]=\delta _{qq'}$. This gives Bose statistics to the phonons. The energy value for the LO phonon is taken as $\hbar \omega _{\rm LO}$ $=$ 36.75 meV at 4.2 K.

The electron-phonon interaction, $H_{\rm e-ph}$ is given by^[22]

$H_{\rm e-ph} =\sum\limits_{mlk_z } {\left\{ {\left[{V_{\rm LO}^\ast (r){\rm e}^{{\rm i}m\phi }{\rm e}^{{\rm i}k_z z}-V_{\rm LO}^\ast (r_i )} \right]a_{ml}^+ (k_z )+h.c} \right\}},$

(9)

where $V_{\rm LO}^\ast$ is the Fourier coefficient of the electron-phonon interaction due to the LO phonons. The matrix elements, $V_{\rm LO}^\ast$ refer the strength of the interaction and, in fact, they depend on the wave vector. The coupling parameter, $V_{\rm LO}^\ast$, is given by

$V_{\rm LO}^\ast = \Gamma _{\rm LO}^{\ast ml} (k_z )J_m \left( {\dfrac{\chi _m^l r}{R}} \right), \quad r\ne 0, \\[2mm] \Gamma _{\rm LO}^{\ast ml} (k_z ), \quad r=0,$

(10)

where $\Gamma _{\rm LO}^{\ast ml}$ is the coupling coefficient of the electron-phonon interaction given by

$\Gamma _{\rm LO}^{\ast ml} (k_z )=\frac{{\rm i}\sqrt 2 e(\hbar \omega _{\rm LO} )^{1/2}} {\sqrt R (\chi _m^{l^2} +k_z^2 R^2)^{1/2}J_{m+1} (\chi _m^l )} \left( {\frac{1}{\varepsilon _0 }-\frac{1}{\varepsilon _\infty }} \right)^{1/2},$

(11)

where $m=$ 0, $\pm$1, $\pm$2, $\cdots$, $l=$ 1, 2, 3, $\cdots$, $R$ is the radius of the quantum dot, and $J_{\rm m}$ is the $m$th order Bessel function of the first order. $\chi _{\rm m}^{l}$ is the $l$th root of $J_{\rm m}$.

We follow the variational method developed by Pines^{[23, 24]} using the expression for unitary transformation as

$U_0 =\exp \left\{ { {\sum\limits_{m, l, q_z } {(a_{\boldsymbol{q}}^+ +a_{\boldsymbol{q}} )[V_{\rm LO} (\rho )/\hbar \omega _{\rm LO} }]}} \right\},$

(12)

$U_1 =\exp \left[{\rm i}\left( K_z -\sum\limits_{m, l, q_z } q_z Za_{\boldsymbol{q}}^+ a_{\boldsymbol{q}} - \sum\limits_{n, k_{z _z}} {q_z Zb_{\boldsymbol{q}}^+ b_{\boldsymbol{q}} } \right) \right],$

(13)

and

$U_2 =\exp \left[{\sum\limits_{m, l, q_z } {f_{ml}^{\rm LO} (q_z , z)a_{\boldsymbol{q}}^+ -h.c} } \right],$

(14)

where $f_{ml}^{\rm LO}$ is the parameter to be determined by minimizing the expectation of the bound polaron Hamiltonian. Thus, the effective Hamiltonian for the bound polaron in the phonon vacuum state can be derived as

$H_{\rm eff} =\left\langle 0 \right|U_2^{-1} U_1^{-1} \hat {H}U_1 U_2 \left| 0 \right\rangle .$

(15)

The trial wave function for the taken system is the product of the wave function chosen as $\psi (r)$ and the phonon vacuum state $\left| 0 \right\rangle$. The trail wave function is chosen as

$\psi _{nlk} (\rho , \phi , z)= \begin{cases} N_1 \exp (ikz)\exp (\pm il\phi )J_l (r_{\rm nl} r), \quad r<R, \\[5mm] N_1 \dfrac{J_l (r_{\rm nl} R)}{k_l (b_{\rm nl} R)}\exp (\pm il\phi )K_l (b_{\rm nl} r), \quad r\geqslant R, \\ \end{cases}$

(16)

where $N_1$ is the normalization constant and $r_{nl}$ and $b_{nl}$ are given by

$r_{nl} =\sqrt {\frac{2m_{\rm e}^\ast E_{nlk} }{\hbar ^2}},$

(17)

and

$b_{nl} =\sqrt {\frac{2m_{\rm e}^\ast [V_0 -E_{nlk}]}{\hbar ^2}},$

(18)

where $E_{nlk}$ is the ground state lowest binding energy which is obtained as the first root of the transcendental equation.

The trial wave function, considering the Coulomb interaction, is given by

$\psi _{nlk} (\rho , \phi , z)= \begin{cases} N_2 \exp (ikz)\exp (\pm il\phi )J_l (r_{nl} r)\exp (-\delta r), \\ \quad \quad \quad \quad \quad \quad r<R , \\[2mm] N_2 \dfrac{J_l (r_{nl} R)}{k_l (b_{nl} R)}\exp (\pm il\phi )K_l (b_{nl} r)\exp (-\delta r), \\ \quad \quad \quad \quad \quad \quad r>R, \\ \end{cases}$

(19)

where $N_2$ is the normalization constant and $\delta$ is the variation parameter.

The Schrdinger wave equation, time independent, is given by

$H\psi _{nlm} =E\psi _{nlm} ,$

(20)

where $H$ is the Hamiltonian of the system as given in Equation~(1), and $E$ and $\psi$ are the eigen energies and the wave functions of the problem respectively. The energies are calculated using the variation approach taking the proper choice of the wave functions. In the presence of magnetic field strength, by introducing the effective Rydberg $R_y^\ast =m^\ast e^4/2\hbar ^2\varepsilon ^2$ as the unit of energy and the effective Bohr radius $R^\ast =\hbar ^2\varepsilon /m^\ast e^2$ as the length unit, the Hamiltonian given in Equation (2) becomes,

$H=-\nabla ^2+\frac{\gamma ^2}{4}r^2\sin ^2\theta +\gamma L_z +\frac{V(r)}{R_y^\ast }-\frac{l(l+1)}{r^2}-\frac{2}{r},$

(21)

where $\gamma$ is the measure of magnetic field defined as $\gamma =\frac{\hbar \omega _{\rm c}}{2R_y^\ast }$, here $\omega _{\rm c}$ is the cyclotron frequency.

The ground (1s) and first excited (2p) one electron quantum dot energy states are considered in order to have infrared transitions assuming $\Delta l=\pm 1$. By analogy to a hydrogenic-like atom, the ground state energy $E_{10}$ ($n=$ 1, $l =$ 0) can be assigned as 1s and the first excited state, energy $E_{11}$ ($n =$ 1, $l=$ 1) which corresponds to 2p. The wave functions of the first state and the related transition energies are obtained in order to have infrared transitions assuming $\Delta l=\pm 1$ with $n, l, m$ are the associated quantum numbers.

$\int {\psi _i^\ast \psi _j {\rm d}r=0}, \quad i\ne j.$

(22)

Thus, the binding energy of a hydrogenic donor impurity is given by

$E_{\rm b} (B)=E_{\rm s} +\gamma -\left\langle \psi \right|H\left| \psi \right\rangle _{\rm min } ,$

(23)

where $E_{\rm s}$ is the lowest binding energy which is obtained without adding the impurity.

The optical absorption, in the calculations, is based on the Fermi's golden rule. The transition rate for the photon for an impurity, from the initial state $\psi _i$ to the final state $\psi _j$ for absorption, in the CdSe/ZnSe parabolic quantum dot is given by the expression^{[25, 26]}

$T_{ij} =\frac{4\eta ^3}{3\varepsilon \hbar ^3c^2}\alpha E_{ij}^3 \left| {r_{ij} } \right|^2,$

(24)

where $\eta$ is the refractive index, $\alpha$ is the fine structure constant, $\varepsilon$ is the dielectric constant of the material, $E_{ij}$ is the transition energies from the initial, $i$th state to final, $j$th state and $\left| {r_{ij} } \right|^2=\left| {\left\langle {\psi _j \left| r \right|\psi _i } \right\rangle } \right|^2$.

The investigation of the oscillator strength gives additional information on the fine structure and the selection rules of the optical absorption coefficients in any optical transitions. It is expressed as^[27]

$f=\frac{2m_0 (E_{\rm f} -E_{\rm i} )}{\hbar ^2}\left| {\left\langle {\psi _j \left| r \right|\psi _i } \right\rangle } \right|^2,$

(25)

where $m_0$ is the electron mass in vacuum and $E_{\rm f} -E_{\rm i}$ is the transition energy from i state to f state. The charge carriers in a doped semiconductor are considered to be important for observation of intersubband optical absorption. The optical transitions can be achieved between the energy levels in the formation of discrete energy levels with the subsequent quantum confinement in any reduced semiconductor system. Eventually, the effects of geometrical confinement and the external perturbations will alter the intersubband optical absorption spectra drastically. The calculation of transition probability per unit time of an electron from one state to another state is carried out by Fermi Golden rule. The strength of the intersubband optical transition energy is associated with the Fermi Golden rule. It shows the transition probability of the impurity taking place from an initial state to the final state. The expression for the linear optical absorption coefficient for the intraband transitions in the conduction band is given by^{[28, 29]}

$$\begin{split} \label{eq23} \alpha (\omega )= {}& \frac{\omega \mu ce^2}{\eta }\left| {M_{\rm fi} } \right|^2\frac{m^\ast k_{\rm B} T}{R\pi \hbar ^2} \\[2mm]& \times \ln {\frac{1+\exp \left[{(E_{\rm F} -E_{\rm i} )/k_{\rm B} T} \right]}{1+\exp \left[{(E_{\rm F} -E_{\rm f} )/k_{\rm B} T} \right]}} \\[2mm]& \times \frac{\hbar /\Gamma _{\rm in} }{(E_{\rm f} -E_{\rm i} -\hbar \omega )^2+(\hbar /\Gamma _{\rm in} )^2}, \end{split}$$

(26)

where $e$ is the absolute value of the electron charge, $\mu$ is the permeability of the dot material, $c$ is the speed of light in free space, $\Gamma _{\rm in}$ is the relaxation time, $E_{\rm f(i)}$ is the final and initial state energy, $\omega$ is the angular frequency of optical radiation, $\eta$ is the refractive index, $k_{\rm B}$ is the Boltzmann constant, $T$ is the temperature, and $E_{\rm F}$ represents the Fermi energy. The $M_{\rm fi}$ is the matrix element given by

$M_{\rm fi} =\int {\psi _{\rm f}^\ast (r)r\psi _{\rm i}^\ast {\rm d}r} ,$

(27)

3. Results and discussion

The magnetic-field-induced donor binding energy is obtained solving the Hamiltonian using variational technique taking into account the effect of geometrical confinement and the phonon contribution. Some optical properties with the incoming photon energies, in the Cd$_{0.8}$Zn$_{0.2}$Se/ZnSe quantum dot, are computed with and without the phonon effect. The barrier height of the conduction band is found to be 1.35 eV and the relaxation time $\Gamma _{\rm in}$ is taken as 0.1 ps. The measure of magnetic field, one gamma is found to be 45.05 Tesla. The effective Redberg energy of the electron is found to be 18.9 meV and the effective Bohr radius is obtained as 38.2 {\AA}. The band offset ratio between the inner and the outer dot materials is taken as 70 : 30. The material parameters of Cd$_{0.8}$Zn$_{0.2}$Se and ZnSe are listed in Table 1.

In Figure 1, we present the variation of binding energy of a hydrogenic donor as a function of dot radius in a Cd$_{0.8}$Zn$_{0.2}$Se/ZnSe quantum dot for various magnetic field strengths with and without the inclusion of phonon effects. In the figure, the solid line represents the effects without the phonon contribution and the dashed line denotes the effects with the inclusion of phonon contribution. It is noticed that the donor binding energy is enhanced when the dot radius of the quantum dot decreases, it happens up to a critical value below which the binding energy is found to decrease with the reduction of dot radius. The binding energy is found to be more with the inclusion of effects due to the magnetic field strength and the phonon contribution for all the dot radii^[30]. The penetration of wave function into the barrier is the cause for the reduction of binding energy in the strong confinement region. The enhancement of donor binding with the magnetic field is observed in all the dot radii. The squeezing of the electronic wave function is the cause for this enhancement of binding energy due to the magnetic field. In addition, the effect of magnetic field is more pronounced for larger dots. It is also observed that the donor binding energy is enhanced with the inclusion of electron-phonon contribution^[31].

Figure 2 shows the variation of polaron-induced oscillator strength as a function of dot radius in a Cd$_{0.8}$Zn$_{0.2}$Se/ZnSe quantum dot in the presence of magnetic field strength. The dashed curve represents the effects with the inclusion of phonon contribution and the solid line represents without the effect of LO phonon interaction. The dependence of oscillator strength on the geometrical confinement, magnetic field strength and the phonon effects are brought out here. It is well known that the oscillator strength is contributed by the transition matrix elements. In fact, the discrete energy levels become broader and broader with the reduction of geometrical confinement in the dot and it causes the values of dipole matrix element to decrease sharply. The oscillator strength is found to increase with the magnetic field strength and the phonon effect. It is because the enhancement of binding energy occurs with the magnetic field and the phonon contribution.

The variation of spontaneous lifetime of 2p-1s like transition of a donor impurity as a function of dot radius in a Cd$_{0.8}$Zn$_{0.2}$Se/ZnSe quantum dot in the presence of magnetic field strength with and without the phonon contribution is shown in Figure 3. The values of spontaneous lifetime of the transition rate are found to be in picoseconds and it is observed that these values become higher when the geometrical confinement becomes smaller. Similarly the spontaneous lifetime is more pronounced when the effects of LO phonon contribution are incorporated. Longer lifetime is found when the phonon contribution is included in the Hamiltonian^[32]. The fast relaxation process of intersubband optical transition, in reduced quantum structures, is considered to be one of the technological advantages for the potential applications for fabricating novel optoelectronic devices.

Figure 4 displays the variation of polaron-induced gain as a function of photon energy in the presence of magnetic field strength in a Cd$_{0.8}$Zn$_{0.2}$Se/ZnSe quantum dot. The many body effects are included for calculating the optical gain spectra, the transition matrix elements and thereby the optical gain is found. The electron density is taken as 2 $\times$ 10$^{18}$ cm$^{-3}$. It is observed that 1.55 $\mu$m wavelength is achieved for 22 {\AA} quantum dot radius in the Cd$_{0.8}$Zn$_{0.2}$Se/ZnSe quantum dot. Moreover, this telecommunication wavelength can be altered by changing the magnetic field strength and the phonon contribution for a constant dot radius and vice-versa^[33]. This wavelength is suitable for fiber optical communication networks. It is observed that the peak value shows the higher amplitude when the magnetic field strength is increased and it moves towards the higher photon energy when the phonon contribution is included. It is because the larger matrix element is achieved in the influence of magnetic field strength and the phonon contribution and thereby the Fermi level increases with the electron density. The optical transition energies between the levels increase. Hence, the magneto-polaron induces the intersubband optical absorption resonant peak and suffers a blue shift^[34]. The dependence of magneto-LO phonon on the optical properties is brought out here.

In Figure 5, we present the variation of absorption wavelengths of the absorption spectrum with the dot radius in the presence of magnetic field strength in a Cd$_{0.8}$Zn$_{0.2}$Se/ZnSe quantum dot with and without the phonon corrections. It is noted that the absorption coefficient moves towards the higher photon energy as the dot radius is decreased. In fact, the confinement effect induces the blue shift of the absorption peak resonant and it is well known. It is also observed that the telecommunication wavelength, 1.55 $\mu$m, is achieved with the dot radius of 22 {\AA} in the Cd$_{0.8}$Zn$_{0.2}$Se/ZnSe quantum dot and this wavelength can be reduced when the magnetic field and the phonon effects are included. Thus, the difference between the energy levels of the donor impurity can be altered with the application of magnetic field strength. The resonant interaction will take place when this difference of energy equals the energy of the LO-phonon. Moreover, the strong coupling regime is achieved when the energy between levels matches with the LO phonon energy^[35]. Hence, we can conclude that the suitable telecommunication wavelength can be achieved with the application of external perturbations.

4. Conclusion

In conclusion, the dependence of magnetic field strength and the LO phonon, on the electronic and the optical properties, has been discussed in a Cd$_{0.8}$Zn$_{0.2}$Se/ZnSe quantum dot taking into consideration the geometrical confinement effect. The magneto-polaron-induced hydrogenic binding energy as a function of dot radius has been computed. The phonon-dependent oscillator strength and the spontaneous lifetime have been found taking into account the effects of spatial confinement and the magnetic field strength. The optical gain as a function of photon energy has been computed in the presence of magnetic field and the phonon contribution. It is shown that the telecommunication wavelength can be achieved and tuned when the external perturbations are included in the calculations of II-VI wide band gap semiconductor quantum dot. It is believed that the results would stimulate further experimental research works on the light emitters and the modulators in the fiber optical communication networks.