Rashba effect on linear and nonlinear optical properties of a cylindrical core/shell heterojunction quantum dot

Rashba effect may play an important role in the nonlinear optical properties of heterojunction quantum dots. In this work, we have theoretically examined the effects of Rashba spin-orbit interaction on an electron in a cylindrical core/shell quantum dot (CCSQD). The modifications of various properties of cylindrical core/shell quantum dot such as transition energies, dipole transition matrix elements and linear and nonlinear optical properties due to change in Rashba coupling parameter, magnetic field and effective Rydberg energy were studied. We solved the Schrödinger equation using numerical methods and obtained energy eigenvalues as functions of the aforementioned parameters. It was observed that, the magnetic field has a considerable effect on absorption coefficients and refractive index. It was also observed that increasing the magnetic field shifts the resonances towards higher energies. Additionally, increasing in the Rashba coupling coefficient (α_R) was found out to result an increase in absorption coefficients and refractive index. Our results demonstrated that, we can manipulate optical properties of cylindrical core/shell quantum dot using an external magnetic field.

1 Introduction

In the last decade, heterojunctions formed by heterogeneous semiconductors with different bandgaps have been the focus of attention of researchers in the field of optoelectronics and solid-state devices. Low dimension systems of heterojunction semiconductors provide an extra degree of freedom which opens the door to novel phenomena that are not possible with homogeneous structures [1]. Nanoscale heterostructure-based optoelectronic devices enable precise control over the fundamental properties such as the width of the bandgap, refractive index, effective mass and mobility of charge carriers, and the energy spectrum of electrons. These nanostructures have a diverse range of applications. GaMnAs-based ferromagnetic semiconductors are the most popular systems for investigating spin-dependent transport phenomenon [2]. Double heterostructure (DH) based lasers were developed because, unlike homojunction lasers, they could provide uninterrupted lasing action at high temperatures due to the fact that DH structures have the capacity to attain a high density of carrier injection and population inversion by “double injection” [3]. Semi-insulating heterogeneous semiconductor nanostructures are used for insulated device isolation in integrated circuits and as the current impeding layer in heterostructure lasers due to their high resistivity, short carrier lifetime, and enhanced defect densities [4]. Heterojunction based semiconductor nanostructures are commonly used in various devices such as solar cells [5], gas sensors [6], bipolar transistors [7], wide-dynamic-range image sensors [8] and photocatalysts [9].

Low dimensional (LD) systems of heterojunction structures made with semiconductors are often utilized to confine the motion of charge carriers in one dimension (1D), two dimensions (2D), or in all three dimensions (3D), according to the dimensions of low dimensional systems. Depending on the dimensions of quantum confinement of charge carriers, we can create quantum wells (1D confinement), quantum wires (2D confinement), or quantum dots (3D confinement) [10]. LD systems grant a wide range of functionality to nanoelectronic devices. For instance, recently, researchers working in the field of artificial intelligence have utilized structural, compositional, and chemical tunability of such systems to create advanced neuromorphic devices [11]. These exceptional properties of LD systems help in the fabrication of nanostructures with precisely designed density-of-states which results in increased optical efficiency [12]. One such heterojunction LD system that has played a crucial role in the advancement of semiconductor-based devices is a quantum dot (QD). Quantum dots are quasi-zero dimensional (0D) structures that have discreet atom-like energy levels. The atom-like energy structure of QDs produces a vivid optical spectrum [13] and a few carrier arrangements help us to obtain ultimate control over carrier densities [14]. There have been recent studies that have reported manipulation of spin states [15] and optical properties [16] of quantum dots by using different parameters such as laser field and wetting layer elements, respectively. Spin states of coupled single-electron quantum dots have been proposed to implement one, and two quantum-bit gates for quantum computing [17]. Gao et al. proposed a groundbreaking study where they describe the development of nanoparticle probes based on semiconductor QD for cancer targeting and imaging [18]. Y.Z. Hu et al. used theoretical and experimental results to provide the first indication of the existence of biexciton states in QDs [19]. Nonlinear optical properties enable us to utilize quantum heterostructures in a wide range of applications incorporating high-order harmonic generation, multi-photon imaging, and optical parametric oscillations [20]. Due to the carrier confinement in low dimension systems, the nonlinear optical response of quantum dots presents an exciting field of study with promising applications. Miller and Chemla considered phonon broadening in QDs and proved that strong local field effects could provide optical bistability without any extrinsic feedback [21]. In colloidal quantum dots, nonlinear optical properties have also been found to strongly influence spectral luminescence [22]. Second-harmonic generation (SHG) has also been observed when placing an impurity on the repulsive part of the confining potential in 2-D quantum rings [23].

Core-shell quantum dots (CSQDs) have been researched extensively due to the fact that their linear and nonlinear properties can be manipulated by controlling the thickness, impurity, and dielectric environment of the shell. For instance, in ZnO/ZnS CSQD, increasing the shell width results in a redshift of threshold energy and a significant rise in nonlinear absorption coefficients as well [24]. Shell thickness can also be used to optimize charge separation yield, lifetime, and recombination rates in CSQDs [25]. CSQD can also be used to create unique quantum structures. For example, by introducing an inorganic semiconducting layer over the CSQD having a bandgap higher than that of the shell, a novel quantum structure called quantum dot quantum well (QDQW) can be fabricated [26]. In GaN/AlN CSQD, it has been observed that in addition to shell thickness, hydrostatic pressure can also be used to modify the nonlinear optical properties [27].

In recent years the studies that involve manipulation of spin in QDs are being focused with plenty of attention due to possible applications of spintronics in futuristic devices [28–30]. The peculiar features of spin-based devices, such as low power consumption, high degree of functionality, etc., provide superiority over the properties based on the electronic charge devices. In spintronic devices, the Spin-Orbit Interactions (SOIs) play an essential role in designing the devices. The two distinct SOIs, namely Rashba and the Dresselhaus effects, are either present in the crystals or are induced by external fields. Depending on the asymmetries involved, either or both play a dominant role in the spin manipulation [31]. Indeed, the Rashba term is considered as the momentum-depend splitting of the spin band in both systems: bulk and low dimensional systems (heterostructures, surface states, and QD). This spin-orbit coupling is independent of structures of the shape or geometry. For more details, we refer the readers to Refs. [32–36].

Let us comment that Pan Sun et al. [37] have reported optical and magnetic properties of doped wide-band-gap GaN. They have shown that the photoluminescence (PL) intensity increased under an external magnetic field. This issue may open up a suite of potential applications in producing magneto-optical devices. More recently, Mkrtchyan et al. [38] have studied the interband and intraband optical transitions in an asymmetric bi convex lens-shaped QD made in InAs in the presence of an external magnetic field. The authors have demonstrated a broadening of the absorption peaks with increasing temperature and a decrease in the peaks amplitudes with magnetic field strength increases. They have shown that the SHG and THG nonlinear parameters of the system decreased with an increase in the magnetic field.

A significant study concerning the magneto-refractive effects of PbS quantum dots (QDs) core fiber with different particle sizes was investigated using the DFT method [39]. The defect structures of PbS nanoclusters were constructed, and the spin magnetic moments of sulfur (S) and plumbum (Pb) were analyzed. In this study, the authors show that the magnetic moments are mainly induced by the spin interaction between Pb 6s, 6p, and S 3p states. In addition, they have demonstrated that spin magnetism shows a weakening trend as the particle size increases. Their experimental investigations show that the magneto-refractive effect is strong as the size of PbS QDs decreases which indicates that this material can be used for magnetic field sensing applications.

On the other hand, we understand that advances in fabrication of nanoscale structures such as molecular beam epitaxy and metal oxide chemical vapor deposition has made it possible to design the quantum heterostructures of required shapes and sizes. So fabrication of cylindrical core/shell Quantum dot experimentally a possibility [40–42]. Recently Iorio et al [43] reported experimental results on suspended Quantum wire in external magnetic field where they have manipulated the orientation of Spin-Orbit interactions in external magnetic field. Of course the magnetic field strength in their case is quite small. Also there are experimental realisations where spin-orbit couplings are studied experimentally on nanowires [44, 45]. In view of the above, we can conclude that the study undertaken here can be studied experimentally.

In the present work, we study a GaN/AlN cylindrical core-shell quantum dot (CCSQD) while including the effects of Rashba SOI and using the finite element method to obtain the eigenvalues of the electron in the CCSQD. After finding the energy eigenvalues, we calculated the changes in absorption coefficients (AC) and the refractive index (RI). We used the compact density matrix and perturbation theory approach for this. Finally, we find second harmonic generation (SHG) and third harmonic generation (THG) susceptibilities, respectively. Our motivation was to study the variations in the aforementioned physical properties with changes in the Rashba parameter, Rydberg free energy, and the magnetic field.

2 Theory

2.1 Electronic properties

We will consider an electron confined in a single GaN/AlN CCSQD, which is structured such that a large band offset exists between core and shell materials, as in GaN/AlN $(E_g^{\mathit{GaN}}=3.4eV)$ [46]. The single QDs are surrounded by the dielectric material shell with large band gap. Therefore, infinite confining potentials can be considered as a good approximation for our model. The structure is submitted to a uniform magnetic field directed along to the z-axis.

In the frame work of Effective Masse Approximation (EMA), the Hamiltonian of the system, which considers the effects of the Zeeman and Rashba effects can be written as:

$\mathcal{H}={\mathcal{H}}_0+\frac{1}{2}{g}^{\ast }{\mu }_{B}B{\sigma }_z+{\mathcal{H}}_R(1)$

where ${\mathcal{H}}_0$ is the unperturbed Hamiltonian of a single-particle electron in a CCSQD:

${\mathcal{H}}_0=\frac{1}{2m_e^{\ast }}{\left(\vec{p}-\frac{e}{c}\vec{A}\right)}^2+V_w\left(r_e\right)(2)$

with $\vec{A}=B/2(-y,x,0)$ the vector potential corresponding to the applied magnetic field expressed in the Coulomb gauge, $m_e^{*}$ is the effective mass of the electron, e is the absolute value of the electron charge. In the structures where the band offsets between the core and shell materials is very important, the electron confining potentials in the xy-plane V (ρ_e) and in the z-direction V (z_e) were assumeed the following forms [48]:

$\begin{array}{c}{V}_w\left({\rho }_e\right)=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill \text{\hspace{0.17em}for\hspace{0.17em}}{R}_c\le {\rho }_e\le R_s\hfill \\ \hfill \infty \hfill & \hfill ;\text{\hspace{0.17em}otherwise\hspace{0.17em}}\hfill \end{array}\right.and\\ V_w\left(z_e\right)=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill \text{\hspace{0.17em}for\hspace{0.17em}}\left|z\right|\le \frac{H}{2}\hfill \\ \hfill \infty \hfill & \hfill ;\text{\hspace{0.17em}otherwise\hspace{0.17em}}\hfill \end{array}\right.\end{array}(3)$

The second term in the Hamiltonian [1] describes the Zeeman contribution where g* is the Landé g factor for the bound electron, μ_B is the Bohr magneton, σ_z is the z component of the Pauli matrices vector σ. ${\mathcal{H}}_R$ is the 2D Rashba spin-orbit interaction Hamiltonian.

${\mathcal{H}}_R=\frac{{\alpha }_R}{\hslash }\left[\widehat{\sigma }\times \left(\widehat{p}-\frac{e}{c}\widehat{A}\right)\right](4)$

Where α_R is Rashba SOI parameter.

With the introduction of the RSOI, the Hamiltonian of the system is expressed as a 2 × 2 matrix as follows:

$\left(\begin{array}{cc}\hfill {\mathcal{H}}_{11}\hfill & \hfill {\mathcal{H}}_{12}\hfill \\ \hfill {\mathcal{H}}_{21}\hfill & \hfill {\mathcal{H}}_{22}\hfill \end{array}\right){\mathrm{\Psi }}_e=E_{\pm }^e{\mathrm{\Psi }}_e(5)$

Where ${\mathrm{\Psi }}_e={\mathrm{\Psi }}_{\uparrow }\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}\right)+{\mathrm{\Psi }}_{\downarrow }\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right)$ are the eigenstates of the system. $\left(\genfrac{}{}{0.0pt}{}{1}{0}\right)$ and $\left(\genfrac{}{}{0.0pt}{}{0}{1}\right)$ are the eigenfunctions of σ_z. Ψ_↑ and Ψ_↓ are the wave functions with spin-up and spin-down, respectively. The matrix elements of the equation [5] are:

${\mathcal{H}}_{11}=\frac{-{\hslash }^2}{2m_e^{\ast }}\left(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}\right)+\frac{e^2{B}^2}{8m_e^{\ast }{c}^2}\left(x^2+y^2\right)+\frac{ieB\hslash }{2m_e^{\ast }c}\left(x\frac{\partial }{\partial y}-y\frac{\partial }{\partial x}\right)+\frac{1}{2}{g}^{\ast }{\mu }_{B}B+V_w^e\left(x,y\right)(6)$

${\mathcal{H}}_{22}=\frac{-{\hslash }^2}{2m_e^{\ast }}\left(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}\right)+\frac{e^2{B}^2}{8m_e^{\ast }{c}^2}\left(x^2+y^2\right)+\frac{ieB\hslash }{2m_e^{\ast }c}\left(x\frac{\partial }{\partial y}-y\frac{\partial }{\partial x}\right)-\frac{1}{2}{g}^{\ast }{\mu }_{B}B+V_w^e\left(x,y\right)(7)$

${\mathcal{H}}_{12}={\alpha }_R\left(\frac{\partial }{\partial x}-i\frac{\partial }{\partial y}-\frac{eB}{2\hslash c}\left(x+iy\right)\right)(8)$

${\mathcal{H}}_{21}=-{\alpha }_R\left(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y}+\frac{eB}{2\hslash c}\left(-x+iy\right)\right)(9)$

In cylindrical coordinate system, the energy and the wave function are given by the solutions of Schrödinger equation $\mathcal{H}{\mathrm{\Psi }}_e(\rho ,\phi ,z)=E{\mathrm{\Psi }}_e(\rho ,\phi ,z)$, we remark that the Hamiltonian of the system can be written as a sum of the two independent parts:

$\left[\mathcal{H}\left(\rho ,\phi \right)+\mathcal{H}\left(z\right)\right]{\mathrm{\Psi }}_e\left(\rho ,\phi ,z\right)=E{\mathrm{\Psi }}_e\left(\rho ,\phi ,z\right)(10)$

It is worth montioning that, the application of the EMA constitute an important dabate in strong confinement characterized by the sizes less than a*. This approximation is only questionable when it is coupled with the variational method. In our case, even if we use the EMA, the resolution of the Schrödinger equation is done by using the finite element method which gives a very close results to other method as tight binding or kp theory mainly when we use an Extra-fine meshing. As a result, the wave function can be expressed as a product of the two independent parts:

${\mathrm{\Psi }}_e\left(\rho ,\phi ,z\right)=F\left(\rho ,\phi \right)g\left(z\right)(11)$

Substituting the wave function (Eq. 11) in Eq. 10, we obtained the following equations

$\left[\frac{-{\hslash }^2}{2m^{\ast }}\left(\frac{{\partial }^2}{\partial {\rho }^2}+\frac{1}{\rho }\frac{\partial }{\partial \rho }+\frac{1}{{\rho }^2}\frac{{\partial }^2}{\partial {\phi }^2}\right)-\frac{eB\hslash }{2m^{\ast }{c}^2}{L}_z+\frac{e^2{B}^2}{8m^{\ast }c}{\rho }^2\right]F\left(\rho ,\phi \right)=E_{\rho }F\left(\rho ,\phi \right)(12)$

Using variable separation, we obtained:

$\left[\frac{-{\hslash }^2}{2m^{\ast }}\left(\frac{{\partial }^2}{\partial {\rho }^2}+\frac{1}{\rho }\frac{\partial }{\partial \rho }-\frac{m^2}{{\rho }^2}\right)-\frac{eB\hslash }{2m^{\ast }{c}^2}m+\frac{e^2{B}^2}{8m^{\ast }c}{\rho }^2\right]R\left(\rho \right)=E_{\rho }R\left(\rho \right)(13)$

Using the effective atomic unit, we can rewrite the Bohr radius as $a^{*}={\hslash }^2\epsilon /m_e^{*}{e}^2$ and effective Rydberg energy as $R^{*}={\hslash }^2/2m_e^{*}{(a^{*})}^2$. In these units, the Hamiltonian of our system can be written as:

$-\left(\frac{{\partial }^2}{\partial {\rho }^2}+\frac{1}{\rho }\frac{\partial }{\partial \rho }\right)F\left(\rho \right)+\left(\frac{m^2}{{\rho }^2}+\frac{{\gamma }^2{\rho }^2}{4}\right)F\left(\rho \right)-\left(E_{\rho }+m\gamma \right)F\left(\rho \right)=0(14)$

and

$-\frac{{\partial }^2}{\partial z^2}g\left(z\right)=E_{z}g\left(z\right)(15)$

Where $\gamma =\frac{\hslash {\omega }_c}{2R^{*}}$ and ${\omega }_c=\frac{eB}{m_e^{*}c}$ is the effective cyclotron frequency. The solutions of Eqs 14, 15 are as follows:

$F\left(\rho \right)=\frac{1}{\rho }\left[A{W}_{t,m/2}\left(\frac{\gamma {\rho }^2}{2}\right)+B{M}_{t,m/2}\left(\frac{\gamma {\rho }^2}{2}\right)\right](16)$

Where $t=m+\frac{E_{mn}}{2\gamma }$and

$g\left(z\right)=\mathit{sin}\left(\frac{l\pi z}{H}+\frac{l\pi }{2}\right)(17)$

Here, A and B are constants that can be obtained by boundary conditions and through normalization of the wave functions. Also, $W_{t,m/2}(\frac{\gamma {\rho }^2}{2})$ and $M_{t,m/2}(\frac{\gamma {\rho }^2}{2})$ are the Whittaker function of first and second class, whereas, W_m/2 and M_m/2 are the mth-order Whittaker function. We recall that the radial wave function must vanish at the inner (ρ = a) and outer (ρ = b) radial boundary. These conditions are introduced by the following transcendental equation

$\left[W_{t,m/2}\left(\frac{\gamma a^2}{2}\right)M_{t,m/2}\left(\frac{\gamma b^2}{2}\right)-M_{t,m/2}\left(\frac{\gamma a^2}{2}\right)W_{t,m/2}\left(\frac{\gamma b^2}{2}\right)\right]=0(18)$

The electron eigenenergies are related to the parameters E_mn by the following relation:

$E_{\mathit{lmn}}=\left[E_{mn}^2+{\left(\frac{l\pi }{H}\right)}^2\right];l=\mathrm{1,2},\cdot \cdot \cdot ,m=0,\pm 1,\pm 2,\cdot \cdot \cdot .(19)$

Now, we consider the Rashba Hamiltonian and write the Schrodinger equation of the system as

$\left(\begin{array}{cc}\hfill H_0+\frac{1}{2}{g}^{\ast }{\mu }_{B}B\hfill & \hfill H_{12}^R\hfill \\ \hfill H_{21}^R\hfill & \hfill H_0-\frac{1}{2}{g}^{\ast }{\mu }_{B}B\hfill \end{array}\right)\left(\begin{array}{c}\hfill C_{+}{\mathrm{\Psi }}_{+}\hfill \\ \hfill C_{-}{\mathrm{\Psi }}_{-}\hfill \end{array}\right)=E_{\pm }\left(\begin{array}{c}\hfill C_{+}{\mathrm{\Psi }}_{+}\hfill \\ \hfill C_{-}{\mathrm{\Psi }}_{-}\hfill \end{array}\right)(20)$

where

$\begin{array}{ccc}\hfill H_{21}^R& =\hfill & {\alpha }_R{e}^{-i\phi }\left[\frac{\partial }{\partial \rho }-\frac{i}{\rho }\frac{\partial }{\partial \phi }+\frac{eB}{2\hslash }\rho \right]\hfill \\ \hfill H_{12}^R& =\hfill & {\alpha }_R{e}^{i\phi }\left[-\frac{\partial }{\partial \rho }-\frac{i}{\rho }\frac{\partial }{\partial \phi }+\frac{eB}{2\hslash }\rho \right]\hfill \end{array}(21)$

and C₊, C₋ are the expansion coefficients.

One can obtain the following system of algebraic equations if we let Φ(ρ, z) = F(ρ)g(z)

$\begin{array}{ccc}\hfill & \hfill & \left[E_{\mathit{lmn}}-E_{\pm }+\frac{1}{2}{g}^{\ast }{\mu }_{B}B\right]C_{+}\mathrm{\Phi }\left(\rho ,z\right)+{\alpha }_R\left[\frac{\partial }{\partial \rho }+\frac{m+1}{\rho }+\frac{eB}{2\hslash }\rho \right]C_{-}\mathrm{\Phi }\left(\rho ,z\right)=0\hfill \\ \hfill & \hfill & \left[E_{\mathit{lmn}}-E_{\pm }-\frac{1}{2}{g}^{\ast }{\mu }_{B}B\right]C_{-}\mathrm{\Phi }\left(\rho ,z\right)+{\alpha }_R\left[-\frac{\partial }{\partial \rho }+\frac{m}{\rho }+\frac{eB}{2\hslash }\rho \right]C_{+}\mathrm{\Phi }\left(\rho ,z\right)=0\hfill \end{array}(22)$

We have multiplied the system equations by Φ^∗(ρ, z) integrating over the whole space and used the above relations to obtain the energy eigenvalues E_±.

2.2 Nonlinear optical properties

In this section, we recall the main results of the electric-dipole approximation that describes a transition from the ground state to the first excited state. Since this theory is a natural extension of a well-known theory of the optical absorption [49, 50], we will only present its main results. By neglecting the influence of the temperature and restricting the calculation only to one-photon transitions at 0K, we determined the optical absorption coefficients (ACs) and the refractive index changes (RICs). Generally speaking, this designation one photon transition at 0 K is usually used when the phonons effect is neglected. In a monochromatic electromagnetic field with frequency ω, the transition probability between two states i and f is generally known as the oscillator strength and can be given by the Fermi golden rule. The mathematical formulations for linear photoabsorption coefficient (α⁽¹⁾) and the third-order nonlinear photoabsorption coefficient (α⁽³⁾) have been derived from the density matrix approach and the perturbation theory. Thus, the expression for α, the total absorption coefficient (AC) is as follows [48, 51, 52]:

$\alpha \left(\omega ,I\right)={\alpha }^{\left(1\right)}\left(\omega \right)+{\alpha }^{\left(3\right)}\left(\omega ,I\right),(23)$

Where,

${\alpha }^{\left(1\right)}\left(\omega \right)=\omega \sqrt{\frac{\mu }{\epsilon }}\frac{\hslash {\mathrm{\Gamma }}_{\text{fi}}|M_{\text{fi}}{|}^2\sigma }{{\left(E_{\text{fi}}-\hslash \omega \right)}^2+{\left(\hslash {\mathrm{\Gamma }}_{\text{fi}}\right)}^2}(24)$

and

$\begin{array}{c}{\alpha }^{\left(3\right)}\left(\omega ,I\right)=-\omega \sqrt{\frac{\mu }{\epsilon }}\left(\frac{I}{2{\epsilon }_0{n}_{\text{r}}c}\right)\frac{4\sigma \hslash {\mathrm{\Gamma }}_{\text{fi}}{\left|M_{\text{fi}}\right|}^4}{{\left[{\left(E_{\text{fi}}-\hslash \omega \right)}^2+{\left(\hslash {\mathrm{\Gamma }}_{\text{fi}}\right)}^2\right]}^2}\\ \left[1-\frac{{\left(M_{\text{ff}}-M_{ii}\right)}^2}{4{\left|M_{\text{fi}}\right|}^2}\left\{\frac{3E_{\text{fi}}^2-4\hslash \omega E_{\text{fi}}+{\hslash }^2\left({\omega }^2-{\mathrm{\Gamma }}_{\text{fi}}^2\right)}{E_{\text{fi}}^2+{\left(\hslash {\mathrm{\Gamma }}_{\text{fi}}\right)}^2}\right\}\right]\end{array}(25)$

where E_fi = E_f − E_i denotes the energy transition between an initial state (i) and a final state (f). M_fi = e < Ψ_f|ρ|Ψ_i > is the transition matrix element between the state i and state f. Note that the parameters used above are: c is the speed of light in a vacuum, σ_s is the electron density related to the occupied volume by the relation, I is the intensity of the incident electromagnetic radiation, ω is the angular frequency of the laser radiation, μ is the permeability of the system, n_r is the relative refractive index of semiconductor and ɛ₀ is the permittivity of the free space Γ_fi is the line width which is also defined as the inverse of the relaxation time τ_fi and known as the relaxation rate of initial and final states. It was found that the propagation of electromagnetic radiation through a material also produces changes in the refractive index of the material. The sum of linear and nonlinear third-order refractive index change is denoted as the total refractive index change (RIC).

$\frac{\mathrm{\Delta }n\left(\omega \right)}{n_{\text{r}}}=\frac{\mathrm{\Delta }{n}^{\left(1\right)}\left(\omega \right)}{n_{\text{r}}}+\frac{\mathrm{\Delta }{n}^{\left(3\right)}\left(\omega \right)}{n_{\text{r}}}(26)$

Where,

$\frac{\mathrm{\Delta }{n}^{\left(1\right)}\left(\omega \right)}{n_{\text{r}}}=\frac{1}{2{\epsilon }_0{n}_{\text{r}}^2}\frac{\sigma {\left|M_{\text{fi}}\right|}^2\left(E_{\text{fi}}-\hslash \omega \right)}{{\left(E_{\text{fi}}-\hslash \omega \right)}^2+{\left(\hslash {\mathrm{\Gamma }}_{\text{fi}}\right)}^2}(27)$

and

$\begin{array}{ll}\hfill \frac{\mathrm{\Delta }{n}^{\left(3\right)}\left(\omega ,I\right)}{n_{\text{r}}}& =-\frac{\mu cI\sigma {\left|M_{\text{fi}}\right|}^4}{{\epsilon }_0{n}_{\text{r}}^3}\frac{\left(E_{\text{fi}}-\hslash \omega \right)}{{\left[{\left(E_{\text{fi}}-\hslash \omega \right)}^2+{\left(\hslash {\mathrm{\Gamma }}_{\text{fi}}\right)}^2\right]}^2}\times \left[1-\frac{{\left(M_{\text{ff}}-M_{ii}\right)}^2}{4{\left|M_{\text{fi}}\right|}^2\left(E_{\text{fi}}^2+{\left(\hslash {\mathrm{\Gamma }}_{\text{fi}}\right)}^2\right)}\right.\hfill \\ \hfill & \left.\times \left\{E_{\text{fi}}\left(E_{\text{fi}}-\hslash \omega \right)-{\left(\hslash {\mathrm{\Gamma }}_{\text{fi}}\right)}^2-{\left(\hslash {\mathrm{\Gamma }}_{\text{fi}}\right)}^2\times \frac{\left(2E_{\text{fi}}-\hslash \omega \right)}{\left(E_{\text{fi}}-\hslash \omega \right)}\right\}\right]\hfill \end{array}(28)$

To have a complete description of the nonlinear optical responses of a system excited by an electromagnetic radiation E(t) = E₀e^j(ωt) + E₀e^−j(ωt) polarized along the z-axis, we used the development of the polarization as power series of the field strength.

$P\left(t\right)=\equiv {\epsilon }_0\left[{\chi }_{\omega }^{\left(1\right)}E\left(t\right)+{\chi }_{\omega }^{\left(2\right)}{E}^2\left(t\right)+{\chi }_{\omega }^{\left(3\right)}{E}^3\left(t\right)+\dots \right](29)$

where ${\chi }_{\omega }^{(1)}$ is the linear susceptibility while ${\chi }_{\omega }^{(2)}$ and ${\chi }_{\omega }^{(3)}$ are defined as the second and third order nonlinear optical susceptibilities.

The partial development of Eq. 29 leads to the usual expansion:

$P\left(t\right)={\epsilon }_0{\chi }_{\omega }^{\left(1\right)}{E}_0{e}^{i\omega t}+{\epsilon }_0{\chi }_0^{\left(2\right)}|E_0{|}^2+{\epsilon }_0{\chi }_{2\omega }^{\left(2\right)}{e}^{2i\omega t}+{\epsilon }_0{\chi }_{\omega }^{\left(3\right)}{E}^3{e}^{i\omega t}+{\epsilon }_0{\chi }_{3\omega }^{\left(3\right)}{E}^3\left(t\right)e^{3i\omega t}+c.c(30)$

${\chi }_0^{(2)}{\chi }_{2\omega }^{(2)}{\chi }_{\omega }^{(3)}$ and ${\chi }_{3\omega }^{(3)}$ are the optical rectification, the second harmonic generation (SHG), third order optical susceptibility and the third order harmonic generation (THG), respectively. The mathematical formulation of different contributions in the polarization can be derived by using the compact density matrix method. In this study, we have ignored the non-resonant terms as they have trivial contributions. We have focused on the three following effects [53, 54]:

${\chi }_{\omega }^{\left(1\right)}=\frac{\hslash \sigma }{{\epsilon }_0}\frac{|M_{21}{|}^2}{\hslash \omega -E_{21}-j\hslash {\mathrm{\Gamma }}_{21}}(31)$

${\chi }_{2\omega }^{\left(2\right)}=\frac{\sigma }{{\epsilon }_0}\frac{M_{12}{M}_{23}{M}_{31}}{\left(\hslash \omega -E_{21}-j\hslash {\mathrm{\Gamma }}_{21}\right)\left(2\hslash \omega -E_{31}-j\hslash {\mathrm{\Gamma }}_{31}\right)}(32)$

and

${\chi }_{3\omega }^{\left(3\right)}=\frac{\sigma }{{\epsilon }_0}\frac{M_{12}{M}_{23}{M}_{34}{M}_{41}}{\left(\hslash \omega -E_{21}-j\hslash {\mathrm{\Gamma }}_{21}\right)\left(2\hslash \omega -E_{31}-j\hslash {\mathrm{\Gamma }}_{31}\right)\left(3\hslash \omega -E_{41}-j\hslash {\mathrm{\Gamma }}_{41}\right)},(33)$

where interstate damping terms are as follow—Γ₂₁ = Γ_fi, Γ₃₁ = Γ_fi/2, and Γ₄₁ = Γ_fi/3. We note that the different dipolar matrix elements M_xy is also called oscillator strengths between two allowed transitions i and f.

3 Results and discussion

The parameters used in the numerical calculations for GaN core/shell QD are as follows: $m_e^{\ast }=0.19m_0$, g^∗ = 2.06, ɛ = 8.9, a^∗ = 2.48 nm, R^∗ = 32.62 meV, Γ_fi = 0.2 ps⁻¹, n_r = 3.08, I = 2 × 10⁵W/m² and σ = 4.6 × 10²²m⁻³ [46, 55]. We have calculated the spectrum of CCSQD using the method described in the previous section. The energy levels and the transition matrix elements, and found that the energy levels as well as the transition matrix elements stongly depend on the material parameters and on the α_R and also on the applied magnetic field B. Figure 1 shows a schematic representation of the system’s setup.

FIGURE 1

FIGURE 1. (A) Schematic presentation of cylindrical core/shell QD (B) The band diagram for radial confinement.

The problem of threshold magnetic field strength in semiconductor materials is related to the dimensionless parameter γ, (0 ≺ γ ≺ 1). Generally speaking γ define the strength of the applied magnetic field: (γ ≪ 1 corresponds to low magnetic fields) and (γ tend to 1 corresponds to high magnetic fields). We have already discussed this important point in our previous papers see for example [47]. The relation between B(Tesla) and γ writes as: $B(T)=[(\frac{m_e^{\ast }}{m_e}){\mu }_B^{-1}{R}^{\ast }]\gamma$, (${\mu }_B=\frac{e\hslash }{2m_{e}c}=5.796\times {10}^{-2}meV{T}^{-1}$ is the Bohr magneton). By using the physical parameters of GaN, B(T) = 106.93γ. In our case, we have choose three low values of (γ = 0.09, 0.18 and 0.28) Which correspond to B = 10, 20 and 30 T, respectively.

Figure 2 shows the variation in transition energies as a function of Rashba parameter (α_R) for R_c = 1a^∗ and R_s = 2a^∗ and height H = 10^∗. Different colors represent different strength of magnetic field, i.e., black, red and blue for fields of 10T, 20T and 30T, respectively. It was observed that on increasing Rashba parameter, there is shift towards higher energy irrespective of the value of the applied magnetic field. This shifting towards higher energies is further enhanced by increasing the magnetic field, as clearly seen in the graphs.

FIGURE 2

FIGURE 2. Transition energies as functions of the Rashba parameter and R_c = 1a*, R_s = 2a*, H = 10a*.

Figure 3 represents the effect of α_R on transition matrix elements of few transitions from initial (i) to final state (f) which are denoted by M_fi. The variations of matrix elements M₂₁, M₂₃, M₃₄, M₃₁ and M₄₁ with α_R for different magnetic fields are shown in Figures 3A–E, respectively. It is seen that when δ_if = ±1, ±2, there is smooth variation in electric dipole moment with α_R. These changes in M_fi can be either increasing (Figure 3A) or decreasing (Figures 3B–D). The change in α_R produces a change in energy levels and wavefunctions. These changes are expected to be smooth in the aforementioned cases as SOI coupling remains quite small compared to other interactions. However, for M₄₁, first we see a dip with increasing α_R for all magnetic fields, and then the value of the transition matrix element increases with α_R for all magnetic fields. This increase is almost the same for all strengths of magnetic field.In Figure 4, first four low-lying states of the CCSQD in the presence of different SOI strengths are shown. As shown in figure, first row corresponds to α_R = 3 × 10⁻¹¹ eV m and the next two rows correspond to α_R = 4 × 10⁻¹¹ eV m and α_R = 5 × 10⁻¹¹ eV m, respectively. We have kept the magnetic field fixed at 20T for these results. It can be seen from the figure that the wavefunctions Ψ₂ and Ψ₄ show more splitting compared to Ψ₁ and Ψ₃. This splitting may be attributed to increased SOI effect as we increase the Rashba parameter.Figure 5 shows similar results but for different values of magnetic fields, i.e., B = 10T, B = 20T and B = 30T, respectively. Here, we have kept α_R = 5 × 10⁻¹¹ eV m. Here, again, we can see that more splitting is observed in wave functions Ψ₂ and Ψ₄ as compared to Ψ₁ and Ψ₃. This may be due to increased SOI effects due to the magnetic field.

FIGURE 3

FIGURE 3. Different matrix elements (A) M₂₁, (B) M₂₃, (C) M₃₄, (D) M₃₁, and (E) M₄₁ for CCSQD with α_R for different magnetic fields and R_c = 1a*, R_s = 2a*, and H = 10a*..

FIGURE 4

FIGURE 4. Lowest four wave functions for CCSDQ with R_c = 1a*, R_s = 2a*, H = 10a*, for different values of α_R, for B = 20T.

FIGURE 5

FIGURE 5. Lowest four wave functions for CCSDQ with R_c = 1a*, R_s = 2a*, H = 10a*, for different values of B, for α_R = 5 × 10⁻¹¹ eV m.

The calculated transition energies and off-diagonal electric dipole matrix elements constitute the source of understanding the properties of the interstate optical coefficients investigated in this work. In the subsequent plots, we have investigated the behaviour of the ACs, RI changes, SHG and THG when the rashba parameter is varied for a constant magnetic field (B = 20T). The role of the effect of α_R in the changes of AC and RI shown in Figures 6A,B. We see that the increase in α_R leads to a shift towards higher energies. The reason is that the transition energy increases with the spin orbit coupling. This phenomenon is explained previously in the figure of E_fi, concerning the variation of the amplitude increase, due to the variation in the matrix element M₂₁. In Figure 7 we have shown the changes in SHG and THG coefficients with photon energy, again keeping the magnetic field constant at 20T for α_R = 3, 4 and 5 × 10⁻¹¹ eV m. A blue shift is visible in peaks for the SHG and the THG coefficients on increasing the Rashba parameter.

FIGURE 6

FIGURE 6. The linear, the third order nonlinear and the total absorption coefficients (A) and the refractive index changes (B) as functions of incident photon energy with different RSOI strengths. Here the parameters of CCSQD are as: R_c = 1a*, R_s = 2a*, H = 10a*.

FIGURE 7

FIGURE 7. The second (A) and the third (B) harmonic generation as functions of incident photon energy with different SOI strengths α_R = 3, 4 and 5 × 10⁻¹¹ eV m and R_c = 1a*, R_s = 2a*, H = 10a*.

In Figure 8, the variation of AC and RIC with photon energy are presented. Figures 8A,B shows the variation of AC and RIC of a CCSQD as a function of photon energy for three values of height when α_R = 5 × 10⁻¹¹ eV m, B = 20 T, R_c = 1a*, R_s = 2a*. It is well observed that a shift towards high energies when the height H decreases, this displacement is caused by the increase in transition energies. Moreover, it can be observed that the amplitude increases as the height increases. It is observed due to the variation of the dipole matrix element with the height. It is noticed that the magnetic field has a significant effect on AC and RIC. In Figures 6A,B, we can see that the curves show resonances when E_fi becomes comparable with ℏω. At such energies, an increase in the magnetic field leads to a displacement of the resonances towards higher energies. This shift is due to the increased transition energy. Because as B increases, sub level energies also increase and consequently, their transition energy E₂₁ also increases. By increasing energy sub-level and reducing overlap between the wave functions, M₂₁ decreases and the peaks are reduced by increasing B.

FIGURE 8

FIGURE 8. The linear, third order nonlinear and the total absorption coefficients (A) and refractive index changes (B) as functions of incident photon energy with different values of magnetic field B = 10, 20, and 30T and SOI strength α_R = 5 × 10⁻¹¹ eV m, R_c = 1a*, R_s = 2a*, H = 10a*.

In Figure 9, the variation of SHG and THG coefficients for different values of magnetic field, i.e., B = 10T, 20T and 30T for fixed value of α_R = 5 × 10⁻¹¹ eV m with the photon energy is shown. Again, as expected, a shift towards higher energies is observed in the peaks. This is due to increase in SOI on increasing the magnetic field. In Figure 10, we studied the linear, the third order nonlinear and the total ACs and RICs as a function of photon energy for different heights (H). Here we keep the magnetic field and Rashba parameter fixed. It is well observed that there is a shift towards higher energies when the height H decreases, This change is caused by the increase in transition energies. Moreover, it can be observed that the amplitude decreases as the height increases. This can be explained to be happening due to the variation of the dipole matrix element with the height.

FIGURE 9

FIGURE 9. Second (A) and third (B) harmonic generation as functions of incident photon energy with different values of magnetic field B = 10, 20, and 30T and SOI strengths α_R = 5 × 10⁻¹¹ eV m, R_c = 1a*, R_s = 2a*, H = 10a*.

FIGURE 10

FIGURE 10. The linear, the third order nonlinear and the total absorption coefficients (A) and refractive index changes (B) as functions of incident photon energy for different heights H = 8, 9 and 10a* of the CCSQD with R_c = 1a*, R_s = 2a*. The SOI strength is α_R = 5 × 10⁻¹¹ eV m, and B = 20T

In Figure 11, we studied the variation of SHG and THG for different values of H, again keeping magnetic field and Rashba parameter fixed at constant values. Here, we observed variations similar to Figure 10. It is noticed again that there is a shift towards higher energies when the height H decreases. But here, unlike ACs and RICs, amplitude is found to increase with the increase in height H.

FIGURE 11

FIGURE 11. The Second ahrmonic generation (A) and third harmonic generation (B) as function of incident photon energy with different heights H = 8, 9 and 10a* of the CCSQSD for α_R = 5 × 10⁻¹¹ eV m, R_c = 1a*, R_s = 2a*, B = 20T.

4 Conclusion

In this study, we have investigated the influence of Rashba spin orbit interaction on an electron in cylindrical core/shell quantum dot. The transition energy changes significantly when we vary the Rashba parameter and magnetic field. When we increase the magnetic field or Rashba parameter, a shift towards higher energies is observed in transition energies. Variations in the Rashba coupling parameter, magnetic field, and effective Rydberg energy also cause measurable changes in the systems linear and nonlinear optical properties. As we increase the magnetic field, peaks of the absorption coefficients and refractive index changes show a blue shift. Second and third harmonic generation peaks also show shifts towards higher values of energy on increasing the magnetic field and Rashba parameter. Through our results, we have shown that it is possible to manipulate the above-mentioned properties of cylindrical core/shell quantum dots by utilizing the Rashba spin orbit coupling effects that become dominant as we increase the magnetic field. For requirements for a specific article type please refer to the Article Types on any Frontiers journal page. Please also refer to Author Guidelines for further information on how to organize your manuscript in the required sections or their equivalents for your field.

Data availability statement

The data presented in this study are available on reasonable request to the corresponding author.

Author contributions

Conceptualization, LMP, VP, and EF; data curation, MK; formal analysis, VN and MK; methodology, KL, V, and JE; project administration, EF, VP, and DL; resources, DL; software, MK, VN, and JE; supervision, EF, VP, and DL; validation, LMP, VP, DL, and EF; visualization, LMP, KL, DL, and EF; writing-original draft, MK, GL, and VN; writing—review and editing, VN, VP, DL, and EF. All authors have read and agreed to the published version of the manuscript.

Funding

LMP acknowledges financial support from ANID through Convocatoria Nacional Subvención a Instalación en la Academia Convocatoria Año 2021, Grant SA77210040. DL acknowledges partial financial support from Centers of Excellence with BASAL/ANID financing Grant AFB180001, CEDENNA.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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