Effects of hydrostatic pressure and temperature on the second-harmonic generation of spherical quantum dots with inversely quadratic Hellmann potential

Abstract

We have theoretically investigated second-harmonic generation (SHG) of spherical quantum dots with inversely quadratic Hellmann potential under the influence of hydrostatic pressure and temperature in the reported work. The Schrödinger equation which from the definition is solved using Nikiforov–Uvarov (N–U) method, and then energy level and wave function are derived. The compact density matrix method and iterative method are used to calculate the nonlinear optical SHG coefficient. The numerical results show that under different constraint parameters, the resonance peak of SHG coefficient moves to high energy or low energy, that is, redshift or blueshift. In addition, the peak of SHG coefficient will increase or decrease with the change of parameters.

Appendix: Overview of Nikiforov–Uvarov (N–U) method

The N–U method is mainly used to solve particular forms of second-order differential equations. This Schrödinger equation conforms to the following format

$$ \psi^{\prime\prime}\left( z \right) + \left[ {E - V(z)} \right]\psi \left( z \right) = 0 $$

(33)

the above formula can be transformed into the following hypergeometric equation through coordinate transformation

$$ \psi^{\prime\prime}\left( s \right) + \frac{{\overline{\tau }\left( s \right)}}{\sigma \left( s \right)}\psi^{\prime}\left( s \right) + \frac{{\overline{\sigma }\left( s \right)}}{{\sigma^{2} \left( s \right)}}\psi \left( s \right) = 0 $$

(34)

where $\sigma \left( s \right)$and $\overline{\sigma }\left( s \right)$are second-degree polynomials, $\overline{\tau }\left( s \right)$is a first-degree polynomial, and $\psi \left( s \right)$can be written as

$$ \psi \left( s \right) = \varphi \left( s \right)\chi \left( s \right) $$

(35)

where $\varphi \left( s \right)$denotes the derivative of logarithm [22], whose form is

$$ \frac{{\varphi^{\prime}\left( s \right)}}{\varphi \left( s \right)} = \frac{\xi \left( s \right)}{{\sigma \left( s \right)}} $$

(36)

where $\xi \left( s \right)$ represents first-order polynomials.

Simultaneous Eqs. (2) and (3), we can get the hypergeometric equation

$$ \sigma \left( s \right)\chi^{\prime\prime}\left( s \right) + \overline{\tau }\left( s \right)\chi^{\prime}\left( s \right) + \lambda \chi \left( s \right) = 0 $$

(37)

In Eq. (5), when the condition that n is a constant value is satisfied, we can gain hypergeometric-type functions according to the Rodrigue relation

$$ \chi_{n} \left( s \right) = \frac{{C_{n} }}{{\rho_{n} }}\frac{{{\text{d}}^{n} }}{{{\text{d}}s^{n} }}\left[ {\sigma^{n} \left( s \right)\rho_{n} \left( s \right)} \right] $$

(38)

among $C_{n}$ is normalized constant, $\rho_{n} \left( s \right)$ expresses weight function and must meet the following conditions

$$ \frac{d}{ds}\left[ {\sigma^{n} \left( s \right)\rho_{n} \left( s \right)} \right] = \tau \left( s \right)\rho \left( s \right) $$

(39)

$$ \tau \left( s \right) = \overline{\tau }\left( s \right) + 2\xi \left( s \right) $$

(40)

The parameter b and function $\xi \left( s \right)$ have the following relationship

$$ \xi \left( s \right) = \frac{{\sigma^{\prime} - \overline{\tau }}}{2} \pm \sqrt {\left( {\frac{{\sigma^{\prime} - \overline{\tau }}}{2}} \right)^{2} - \overline{\sigma } + k\sigma } $$

(41)

$$ \lambda = k + \xi^{\prime}\left( s \right) $$

(42)

when the discriminant is zero, we can obtain a new eigenequation

$$ \lambda = \lambda_{n} = - n\frac{{{\text{d}}\tau }}{{{\text{d}}s}} - \frac{{n\left( {n - 1} \right)}}{2}\frac{{{\text{d}}^{2} \sigma }}{{{\text{d}}s^{2} }},\;n = 1,2, \ldots $$

(43)