Abstract
Quadratic electrooptic effects (QEOEs) and electroabsorption (EA) process in a GaN/AlGaN spherical quantum dot are theoretically investigated. It is found that the magnitude and resonant position of thirdorder nonlinear optical susceptibility depend on the nanostructure size and aluminum mole fraction. With increase of the well width and barrier potential, quadratic electrooptic effect and electroabsorption process nonlinear susceptibilities are decreased and blueshifted. The results show that the DC Kerr effect in this case is much larger than that in the bulk case. Finally, it is observed that QEOEs and EA susceptibilities decrease and broaden with the decrease of relaxation time.
Background
Semiconductor quantum dots with their excellent optoelectronic properties are now mostly used for various technologies such as biological science [1–4], quantum dot lasers [5, 6], lightemitting diodes (LEDs) [7], solar cells [8], infrared and THZIR photodetectors [9–14], photovoltaic devices [15], and quantum computing [16, 17]. GaN and AlN are members of IIIV nitride family. These wide bandgap semiconductors are mostly appropriate for optoelectronic instrument fabrication.
Thirdorder nonlinear optical processes in ZnS/CdSe coreshell quantum dots are investigated in [18–20]. It is shown that the symmetry of the confinement potential breaks due to large applied external electric fields and leads to an important blueshift of the peak positions in the nonlinear optical spectrum. The effect of quantum dot size is also studied, and it is verified that large nonlinear thirdorder susceptibilities can be achieved by increasing the thickness of the nanocrystal shell.
The authors of [21, 22] studied the quadratic electrooptic effects (QEOEs) and electroabsorption (EA) process in InGaN/GaN cylinder quantum dots and CdSeZnSCdSe nanoshell structures. They have found that the position of nonlinear susceptibility peak and its amplitude may be tuned by changing the nanostructure configuration. The obtained susceptibilities in these works are around ${10}^{17}\frac{{\mathit{m}}^{2}}{{\mathit{v}}^{2}}$ and 10^{15} esu, respectively.
In reference [23], selffocusing effects in wurtzite InGaN/GaN quantum dots are studied. The results of this paper show that the quantum dot size has an immense effect on the nonlinear optical properties of wurtzite InGaN/GaN quantum dots. Also, with decrease of the quantum dot size, the selffocusing effect increases.
In a recent paper [24], we have shown that with the control of GaN/AlGaN spherical quantum dot parameters, different behaviors are obtained. For example, with the increase of well width, thirdorder susceptibility decreases. The aim of this study is to investigate our proposed GaN/AlGaN quantum dot nanostructure from quadratic electrooptic effect and electroabsorption process points of view. In this paper, we study thirdorder nonlinear susceptibility of GaN/AlGaN semiconductor quantum dot based on the effective mass approximation. The numerical results have shown that in the proposed structure, the thirdorder nonlinear susceptibilities near 2 to 5 orders of magnitudes are increased.
The organization of this paper is as follows. In the 'Methods’ section, the theoretical model and background are described. The 'Results and discussion’ section is devoted to the numerical results and discussion. Summarization of numerical results is given in the last section.
Methods
In this section, theoretical model and mathematical background of the thirdorder nonlinear properties of a new GaN/AlGaN quantum dot nanostructure are presented. The geometry of a spherical centered defect quantum dot and potential distribution of this nanostructure are shown in Figure 1. We consider three regions consisting of a spherical well (with radius a), an inner defect shell (with thickness b  a), and an outer barrier (with radius b). The proposed spherical centered defect quantum dot can be performed by adjusting the aluminum mole fraction.
In this paper, the potential in the core region is supposed to be zero, and the potential difference between two materials is constant [25]. There are various methods for investigating electronic structures of quantum dot systems [26–28]. The effective mass approximation is employed in this study. The timeindependent Schrödinger equation of the electron in spherical coordinate can be written as [29].
where m_{ i }^{∗} and V_{ i }(r) are effective mass and potential distribution in different regions. They are obtained as follows [30]:
and
where xd and xb are defect and barrier regions of aluminum molar fraction, respectively. The rest mass of electron is denoted by m_{e}, and ΔE_{c}(x) = 0.7 × [E_{g}(x)  E_{g}(0)] is the conduction band offset [30]. The bandgap energy of Al_{ x }Ga_{1  x}N is E_{g}(x) = 6.13x + (1  x)(3.42  x) (expressed in electron volts) [30, 31]. In a spherical coordinate, Schrödinger Equation 1 can be readily solved with the separation of variables. Thus, the wave function can be written as
where n is the principal quantum number, and ℓ and m are the angular momentum numbers. Y_{ ℓm }(θ, ϕ) is the spherical harmonic function and is the solution of the angular part of the Schrödinger equation. By substituting Equation 4 into Equation 1, the following differential equation is obtained for R_{ nℓ }(r):
In order to calculate R_{ nℓ }(r), the two E < V_{01} and E > V_{01} cases must be considered. With change of variables and some mathematical rearranging, the following spherical Bessel functions in both cases are obtained:
Case 1: E < V_{01}.
where
Case 2: E > V_{01}.
where
For the whole determination of eigenenergies and constants that appeared in the wave function, R_{ nℓ }(r) should satisfy the following boundary, convergence, and normalization conditions.
After determining the eigenvalues and wave functions, the thirdorder susceptibility for two energy levels, ground and first excited states, the model should be described [32, 33]. Thus, the density matrix method [34, 35] is used, and the nonlinear thirdorder susceptibility corresponding to optical mixing between two incident light fields with frequencies ω_{1} and ω_{2} appears in Equation 11:
where q is electron charge, N is carrier density, α_{fg} = 〈ψ_{f}rψ_{g}〉 indicates the dipole transition matrix element, ω_{o} = (E_{f}  E_{g})/ħ is the resonance frequency between the first excited and ground states (transition frequency), and Γ is the relaxation rate. For the calculation of thirdorder susceptibility of QEOEs, we take ω_{1} = 0, ω_{2} = ω in Equation 11. The thirdorder nonlinear optical susceptibility χ^{(3)}(ω, 0, 0, ω) is a complex function. The nonlinear quadratic electrooptic effect (DCKerr effect) and EA frequency dependence susceptibilities are related to the real and imaginary part of χ^{(3)}(ω, 0, 0, ω) [20–22].
These nonlinear susceptibilities are important characteristics for photoemission or detection applications of quantum dots.
Results and discussion
In this section, numerical results including the quadratic electrooptic effect and electroabsorption process nonlinear susceptibilities of the proposed spherical quantum dot are explained. In our calculations, some of the material parameters are taken as follows. The number density of carriers is N = 1 × 10^{24} m^{3}, electrostatic constant is ϵ = (0.3x + 10.4)ϵ_{o}[30, 31], and typical relaxation constants are ℏΓ = 0.27556 and 2.7556 meV which correspond to 15 and 1.5ps relaxation times, respectively.
The quadratic electrooptic effect and electroabsorption process susceptibilities as functions of pump photon wavelength at 15ps relaxation time are illustrated in Figure 2. In these figure, the solid and dashed lines show 15 and 30Å well widths, respectively. It is clear that with the increase of the well width, both QEOEs and EA susceptibilities decreased and blueshifted. These behaviors can be related to quantum confinement effect. Because of the increase of well width, the centered defect acts as small perturbation.
The thirdorder susceptibility of GaN/AlGaN quantum dot versus pump photon wavelength with different barrier potentials as parameter is shown in Figure 3. The thirdorder susceptibility is decreased and blueshifted by the increasing barrier potential. These are related to energy levels and dipole transition matrix element behaviors by dot potential. See Figures four and twelve of [24]. So, the resonance wavelength and magnitude of the thirdorder susceptibility can be managed by the control of well width and confining quantum dot potential.
Same as Figure 2, we illustrate the quadratic electrooptic effect and electroabsorption process susceptibilities as functions of pump photon wavelength at 1.5ps relaxation time in Figure 4. By comparing Figures 2 and 4, it is observed that the QEOEs and EA susceptibilities decrease and broaden with decreasing relaxation time.
In Figure 5, we show the effect of confining quantum dot potential on thirdorder susceptibility. As can be seen with increasing barrier potential, the thirdorder susceptibility is decreased and blueshifted. Fullwidth at half maximum (FWHM) of thirdorder susceptibility in Figure 5 is approximately ten times broader than the FWHM in Figure 3.
The effect of relaxation constant (ħ Γ) is demonstrated for two well sizes in Figure 6. It can be seen that the peak of the thirdorder susceptibility is decreased by the increase of the relaxation rate. It is clear from Equation 11 that the thirdorder susceptibility has an inverse relationship with relaxation constant. Also, the difference between the peak of susceptibilities in a = 15 Å and a = 30 Å is decreased with the increase of relaxation rate.
Conclusions
In this paper, we have introduced spherical centered defect quantum dot (SCDQD) based on GaN composite nanoparticle to manage electrooptical properties. We have presented that the variation of system parameters can be tuned by the magnitude and wavelength of quadratic electrooptic effects and electroabsorption susceptibilities. For instance, the results show an increase of well width from 15 to 30 Å; the peaks of the both QEOEs and EA susceptibilities are decreased $\left(7.218\times {10}^{12}{\scriptscriptstyle \frac{{\mathit{m}}^{2}}{{\mathit{V}}^{2}}}\phantom{\rule{0.37em}{0ex}}\mathrm{to}1.062\times {10}^{12}{\scriptscriptstyle \frac{{\mathit{m}}^{2}}{{\mathit{V}}^{2}}}\right)$ and blueshifted (59.76 to 37.29 μm). With decreasing dot potential, the thirdorder susceptibility is increased $\left(2.444\times {10}^{12}\frac{{\mathit{m}}^{2}}{{\mathit{v}}^{2}}\phantom{\rule{0.5em}{0ex}}\mathrm{to}\phantom{\rule{0.5em}{0ex}}7.218\times {10}^{12}\frac{{\mathit{m}}^{2}}{{\mathit{v}}^{2}}\right)$ and red shifted (45.25 to 59.76 μm). The effect of relaxation constant (ħ Γ) which is verified by the peak of the thirdorder susceptibility is decreased by the increasing relaxation rate. These behaviors can be related to the quantum confinement effect and inverse impact of relaxation constant.
Abbreviations
 EA:

electroabsorption
 FWHM:

fullwidth at half maximum
 LEDs:

lightemitting diodes
 QEOEs:

quadratic electrooptic effects
 SCDQD:

spherical centered defect quantum dot
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Acknowledgements
The authors thank the Department of Physics, Tabriz Branch, Islamic Azad University, and the Department of Medical Nanotechnology, Faculty of Advanced Medical Science of Tabriz University for all the supports provided. This work is funded by the Grant 20110014246 of the National Research Foundation of Korea.
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MK conceived of the study and participated in its design and coordination. AV assisted in the numerical calculations. AA and YH participated in the sequence alignment and drafted the manuscript. SWJ supervised the whole study. All authors read and approved the final manuscript.
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Kouhi, M., Vahedi, A., Akbarzadeh, A. et al. Investigation of quadratic electrooptic effects and electroabsorption process in GaN/AlGaN spherical quantum dot. Nanoscale Res Lett 9, 131 (2014). https://doi.org/10.1186/1556276X9131
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Keywords
 Quadratic electrooptic effects
 Thirdorder susceptibility
 Spherical quantum dot
 Relaxation time