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 AZD6738 band offset [30]. The bandgap energy of Al x Ga1 – 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 AZD4547 separation of variables. Thus, the wave function can be written as (4) 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): (5) 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. (6) where Case 2: E > V 01. (7) 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. (8) (9) (10) After determining the eigenvalues and wave functions, the third-order susceptibility for two energy levels, ground and first excited states, the model should be described [32, 33]. Thus, the density matrix 4SC-202 method [34, 35] is used, and the nonlinear third-order susceptibility corresponding to optical mixing between two incident light fields with frequencies Baf-A1 chemical structure ω 1 and ω 2 appears in Equation 11: (11) where q is electron charge,

N is carrier density, α fg = 〈ψ f|r|ψ 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 third-order susceptibility of QEOEs, we take ω 1 = 0, ω 2 = -ω in Equation 11. The third-order nonlinear optical susceptibility χ (3)(-ω, 0, 0, ω) is a complex function. The nonlinear quadratic electro-optic effect (DC-Kerr effect) and EA frequency dependence susceptibilities are related to the real and imaginary part of χ (3)(-ω, 0, 0, ω) [20–22]. (12) 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 electro-optic effect and electro-absorption 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 × 1024 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 1.5-ps relaxation times, respectively.